structural phase transitions and landau's theory

FOREWORD

Investigations of crystalline materials and phase transitions of them have been in

the focus of scientists for quite a long time. Those materials fascinate because of

their shape, transparency, color, hardness, and many other physical properties.

Especially in the vicinity of phase transitions the dependences of certain material

coefficients on temperature and/or pressure are often remarkable, technically

useable, and can also be well described by theoretical means.

Such materials - in particular single crystals - are in many cases fairly easy to

obtain and can be used to prove theoretical predictions.

The classical tool of choice is the theory named after the famous Russian

scientist Landau. This phenomenological approach is stringently based on group-

theoretical and symmetry-based considerations, and can be vastly applied to

model phase transitions.

I worked on the attached collection of papers since about year 2000, exclusively

during my leisure time. Some of the papers have been repeatedly refined or

slightly modified over the years, because after a while I always needed “a certain

distance” to the topics in order to be able to see the bigger picture again and not

to get lost in too many side aspects. Also some calculations were cumbersome.

That’s why I lost sometimes the passion to continue, but it luckily re-flourished

after a while.

One of my focus points was on ferroelastic / ferroelastoelectric transitions,

because they are just my favorites. On the other side I wanted to describe step-

by-step and very practically how to compile the energy expressions and how to

derive from them the temperature dependences of the material coefficients and

alike.

Hempel

December 2021

PS: Interested parties shall feel free to make use of the attached considerations,

but referencing to this source is highly appreciated.

Contact: [email protected]

Table of Contents

1. The Landau Theory of Symmetry Changes at Phase Transitions

(10 pages)

2. The Landau Theory of Phase Transitions - Case Discussions

(49 pages)

3. Calculation of Material Coefficients at Phase Transitions

(25 pages)

4. The Landau Theory of Ferroelastic Phase Transition of Rb4LiH3(SO4)4

(43 pages), with Annex for Improper Case (22 pages)

5. The Landau Theory of Ferroelastoelectric Phase Transition of

(NH4)2CuCl4 . 2H2O, (42 pages)

6. The Landau Theory of successive Phase Transitions of LiKSO4

(57 pages)

Hempel, 2017 Page 1 of 10

The Landau Theory of Symmetry Changes Phase Transitions

1. Introduction

It is assumed that a crystal undergoes a symmetry change at a critical point of temperature

TC and/or pressure pC.

The change of symmetry from a high temperature phase to a low temperature phase and

vice versa is called Phase Transition (PT).

Typically such PT can be described by the appearance of an Order Parameter (OP) at TC

when entering the low temperature phase.

In particular cases the OP can be spontaneous polarization, magnetization, elastic strain,…

Practically PTs are observed, where either the OP changes continuously in the vicinity of TC

(PT of 2nd kind) or discontinuously (PT of 1st kind).

2. Basic Assumptions

• The basic idea of Landau’s theory is to describe PTs of 2nd kind on the level of a

phenomenological theory

• Thereby the crystal exhibits the symmetry group Go in the high temperature phase and

transforms at TC to symmetry group G1 by losing certain symmetry elements (g) of Go.

• G1 is to be a subgroup of Go and the crystal transforms from one to another thermo-



• dynamic equilibrium state.

• Even at PTs of 2nd kind the symmetry of the crystal itself changes

discontinuously at TC

• Generally the symmetry elements (g) of a group G of a crystal leaves the

function ρ(x) (represents the probability density that an atom is located at

position x) invariant:

(R|t) ρ(x) = ρ ((R|t)-1 x) = ρ(x) with (R|t)=g G (1)

• Whereby: R represents the rotational part of space group element g

t represents the translation part of space group element g

• For the high temperature phase we can write:

(R|t) ρo(x) = ρo ((R|t)-1 x) = ρo(x) with (R|t)=g Go T≥TC (2)

• And for the low temperature phase:

(R|t) ρ1(x) = ρ1 ((R|t)-1 (x) = ρ1(x) with (R|t)=g G1 T<TC (3)

• Thus the density function in the low temperature phase can be written:

ρ1(x) = ρo(x)+δρ(x) (4)



• If we assume that δρ(x) is small, we can develop it according to terms of

basis functions of the irreducible representations (IRs) of group Go:

(5)

• The transform according to the respective IR D

• Next step is the introduction of a potential function compatible with symmetry

Go above, below and at the transition point.

• If we assume that is small - especially in the vicinity of the PT-point -

a Taylor expansion looks like:

F = F (T, p, ρo, ρ1) = F(o) + F(1) + F(2) + F(3) + F(4) + … (6)

• Depending on F has to be at minimum both, in the high and low

temperature phases

• F(o) stands for the identity representation of Go and depends on

F(o) cannot describe any PT.

( ) ( )xcx lki

k l i

lki

,,*

*

,,*

=

Basis

function i

of one

IR D

Multiplicity

of DAll IRs D

related to

one star *k

All stars *k

ooo c =

lki

,,*

( )x

( )x



• The other terms of eq. (6) have to be invariant under the symmetry operations

of Go too

• The transformations among the of within an IR D can be considered

to happen among the coefficients instead (i.e. acc. to the same rules) (7)

• Since must hold for T≥TC the are related to the OP (8)

• The 1st order term F(1) depends on (for fixed ,

and d being the dimension of D). The respective invariant must be ruled out

because:

a) such term cannot be related to an ID, but always to a

reducible representation (for simple explanation see e.g. /3/)

b) if a 1st order term is present, the potential F cannot assume

any minimum value, depending on the

• The term for fixed relates to the

2nd order invariant. There can be only one! (9)

• Together with eq. (9) a general expression of the invariant F(2) of eq. (6) is:

0),(,,* =pT c lki

( )( ) ( )=

=

d

i

lki

lki ccf

1

2,,*,,*2 lk ,,*

lkic ,,*

)(,,* x lki

lk

ic ,,*

( )( ) =

=

d

i

lki

lki ccf

1

,,*,,*1 lk ,,*

lkic ,,*

0

,,* lkic

F i.e.



(10)

• whereby the coefficients are not known. Generally their number

equals the number of the IRs.

• Typically only one ID and therefore one set is active / responsible for

a particular phase transition. Thus we put the respective

• Considering also higher order terms, finally F (ref. to eq. (6)) can be written for

the specific PT

(11)

• Looking now at eq. (11) it becomes obvious that all can only be zero (in

high temperature phase!) if the are all positive definite

• On the other side (below critical point), the can become unequal zero only

if the related is negative definite → has to change its sign

at the critical temperature TC (or pressure pC)

• Taking into account these requirements, the easiest function can be

(12)

Note: Generally the energy function F (eq. (11)) must be invariant under all symmetry operations (g) of the high

temperature phase Go. The Subscripts v and s ensure that the sums stretch over all possible invariants

),(,,* pT A lk

( )Co TTp AA −= )(

( )lki

lk

k llk

cf pT AF ,,*)2(,,*

*,,*

)2( ),(

=

lk

ic ,,*

ApT A lk ),(,,*

),(,,* pT A lk

),(,,* pT A lk

lkic ,,*

ilk

i cc ,,*

( ) ( ) ( ) ( ) ( ) ( ) ( ) ...,,,,,, )4()3(

1

2)( ++++== =

i

ss

si

vv

v

d

ii

o i cf pTCcf pTBcpTApTFcpTFF

A lk ,,*



• 3rd order terms*) in eq. (11) can be on principle allowed provided the

symmetrized cube [D]3 of ID D contains the identity representation of

group Go. The so called Landau Condition stipulates that for PTs of 2nd

order, 3rd order invariants must not be allowed in F

• This can be quickly checked by calculating the corresponding reduction

coefficient (see e.g. /02/):

(13)

• Existence of the 4th order terms*) can be formally checked through inspection

of the related reduction coefficient

• Important is to note: if the energy expression shall describe the PT under

equilibrium/minimum conditions, then F(4) (see eqs. (6) & (11)) has to be (14)

positive definite in total

*) Notes:

• The explicit form of the 3rd and 4th order invariants can be derived manually, which might be cumbersome sometimes

• Existence of a 3rd order invariant always means that the PT must be of 1st order. However, the Landau Theory and its

principles are usually applicable, as shown elsewhere (e.g. /03/).

• If a PT is of 1st kind, but no 3rd order invariant is allowed, then the energy expression has to be developed up the 6th

degree F(6), and 4th-term coefficient C has to be set negative, 6th order coefficient positive.

( ) ( ) possibleinvariant order 3rd

possibleinvariant order 3rd no gD

GoGgo 0

01 3

=



• Another important precondition of the Landau Theory is the requirement for

spatial homogeneity of the crystal. This is called Lifshitz Condition.

• If the related reduction coefficient calculates to be zero then spatial

homogeneity is expected.

(15)

• Also then any invariant containing terms like cannot

appear in eq. (11) (16)

• Generally, Landau stated that the real IDs (or physically IDs, being the sum

of a complex ID and its conjugated complex ID: ) which fulfil the

Landau- and Lifshitz-Conditions are named active or acceptable represen-

tations are able to trigger continuous phase transitions (2nd kind)**) (17)

*) Notes:

• If such gradient term(s) should be allowed by symmetry, the Landau approach can be applied as well. Then the

energy expression depends also on spatial changes of the OP, measurement dimensions, and not just on the unit cell.

• The OP(s) can be modulated and described by wave lengths which are not commensurable with the dimensions of the

crystal lattice (unit cell). Such phases are called incommensurate phases.

**) Note:

• Since ρo and δρ are real values (ref. to eqs. (1)&(4)), the basis functions i and the coefficients ci are also real.

( ) ( ) ( ) 0ggDG

vP

Ggo o

=

)(21

x

cc-

x

cc

P

ij

P

ji

)

DD



3. Special Case: Proper Phase Transitions

• PTs, described by the star (-point of the Brillouin zone of the high temperature

phase) are called ferrodistortive because the number of atoms of the primitive unit

cell doesn’t change at the transition.

• Also the translational symmetry doesn’t change.

• Proper PTs mean that specific physical properties like electric polarization,

magnetization, elastic strain can play the role of the order parameter itself.

• In case of polarization the related PT is to be ferroelectric and in case of strain

ferroelastic, respectively.

• The representations of the physical properties (e.g. polarization: polar vector ,

strain: polar tensor of rank 2: ) exhibit specific values for the characters

for individual symmetry elements g (see e.g. /2/).

• To check whether a certain active ID D can lead to a ferroelectric, ferroelastic or

other low temperature phase, the related reduction coefficients need to be looked at

(whereby represent the character of the ID of the concerned symmetry element

g):

(18)

0k =

)( v PD

2)( v PD

( ) ( )

=oGg

v P

o

E ggG

m )(1

( )( )gv P

)(

( )g



• The next topic is how to define in detail the components of polarization, strain, etc. that

they can be utilized to describe the PT under consideration

• As shown elsewhere (e.g. /2/) certain projection operators are capable to generate the

searched basis functions of an ID

(19)

• Application onto an arbitrary function f(x) (e.g. polarization, strain, …) yield to

a decomposition of the function according to its irreducible parts:

(20)

( ) ( )

=oGg

ij

o

ij gTgDG

d

Conjugated complex

element ij of D

Dimension of D

Operator related

to symmetry

element g

( ) ( )xfxf ijij =



4. Literature

/01/ Landau, L.D, Lifshitz, E.M., Lehrbuch der Theoretischen Physik, Statistische Physik,

Akademie-Verlag Berlin, 1984

/02/ Kocinsky, J., “Theory of Symmetry Changes at Continuous Phase Transitions”,

Elsevier, Amsterdam-Oxford-New York, 1983

/03/ Toledano P., Toledano J.-C., “The Landau Theory of Phase Transitions“, World

Scientific Publishing, 1987

/04/ Izyumov Yu. A., Syromyatnikov V. N., “Phase Transitions and Crystal Symmetry“,

Kluwer Academic Publishers, Dordrecht / Boston / London, 1990

/05/ Gufan, Yu. M., “Thermodynamic Theory of Phase Transitions”, Publisher: University of

Rostov on Don, 1982

Hempel, 2015 - 2016 Page 1 of 49

Landau Theory of Phase Transitions

Landau Theory – Case Discussions & Phase Diagrams

Table of Contents

1. Case: Q2 – Q4

2. Case: Q2 – Q3 – Q4

3. Case: Q2 – Q4 – Q6

4. Case: 2-component Order Parameter w/o cubic term

5. Case: 2-component Order Parameter with cubic term

6. Case: 2 coupled 1-component Order Parameters

7. Literature

Hempel, 2015 - 2016 Page 2 of 49


1. Case: Q2 – Q4

This case deals with the Free Energy Expansion (FEE) in dependence of the Order

Parameter (OP) Q.

F = AQ2 + BQ4 *) (1)

with A = Ao(T-To), To- Transition Temperature, Ao and B being positive constants.**)

(2)

The equation of state requires: (3)

This yields two solutions for Q

Solution I: Q = 0 (describes Paraphase) (4)

Solution II: Q2 = -A/(2B) (describes Ferrophase) (5)

The stability condition of Solutions I & II requires:

(6)

For Solution I (ref. to eq. 4) follows: (7)

For Solution II (ref. to eq. 5) follows: (8)

*) Note: In FEE a linear term of Q is not allowed. Otherwise the equation of state (eq. 3) can never be fulfilled.

**) Note: B must be positive to guarantee that FEE stays positive for large values of OP Q.

042 3 =+=

BQAQ

Q

F

02

2

+

212BQ2A 0Q

F

oTT for stable 0A

oTT for stable 0A

Hempel, 2015 - 2016 Page 3 of 49


Next task is to investigate the validity of Solutions I & II.

Solution I: Q = 0 → no limitations (9)

Solution II: Q2 = -A/(2B) → since B>0 must be A<0 (10)

To construct the Phase Diagram, the Phase Transition Line (PTL) has to be calculated

acc. to condition that the FEE of Para- and Ferrophase has to be equal.

F (Q = 0) = F (Q2 = -A/(2B)) (11)

0 = A (12)

The Phase Transition Line is characterized by A = 0, the two phases don’t overlap but

coincide and acc. to eq. (5) the OP has no jump at PTL, which mean that the investigated

Phase Transition (PT) is of 2nd kind. The Phase Diagram looks like:

PTL: A = 0

A

B

Q2 = -A/(2B)

Q = 0

Hempel, 2015 - 2016 Page 4 of 49


2. Case: Q2 – Q3 – Q4

This case deals with the Free Energy Expansion (FEE) in dependence of the

Order Parameter (OP) Q.

F = AQ2 + CQ3 + BQ4 *) (1)

with A = Ao(T-To), To- Transition Temperature, Ao and B being positive constants.

The constant C can be positive or negative. (2)


*) (4)

This yields two principal solutions for Q:

Solution I: (describes Paraphase) (5)

Solution II: (6)

(describes Ferrophase)

*) Note: In FEE and in the equation of state (term in brackets of eq. (4)) it becomes visible, that the term (7) CQ3 respectively CQ is invariant if simultaneously the signs of Q and C change.

(+C)(+Q) = (-C)(-Q) or respectively (-C)(+Q) = (+C)(-Q).

0432 32 =++=

BQCQAQ

Q

F

( ) 0432 2 =++ BQCQAQ

0=Q

22

2

9

321

8

3

8

3

264

9

8

3

C

AB

B

C

B

C

B

A

B

C

B

CQQ −−=−−==

Hempel, 2015 - 2016 Page 5 of 49



(8)

For Solution I (ref. to eqs. 3,5) follows: (9)

For the Solutions II follows with eqs. (8) and (4):

(10)

which together results in (11)

Eq. (11) can be re-written as: (12)

Stability is guaranteed in 2 cases:

1st case: and (13)

and

2nd case: and (14)

062

2

++

212BQCQ2A 0Q

F

oTT for stable 0A

06 ++ 212BQCQ2A

0432 2 =++ BQCQA

083 2 + BQCQ

( ) 083 + BQCQ

0Q 083 + BQC

0Q 083 + BQC

Hempel, 2015 - 2016 Page 6 of 49


Summary of stability constraints after discussion of solution II (see eq. (6)):

(15)

Summary:

a) Q+ is stable and positive if (16)

b) Q- is stable and negative if (17)

Another feature of the solutions Q+ and Q- as it can be easily shown using

eq. (6):

(18)

Q+

Q-

B

CA

32

9 2

possiblenot

0A

possiblenot

possiblenot

( ) ( )00

)0()0(

=−

=−

−+

−+

C QC Q

and

C QC Q

B

CA

32

9 2

B

CA

32

9 2

B

CA

32

9 2

B

CA

32

9 2

B

CA

32

9 2

00

00

−

+

Q C if

and

Q C if

Hempel, 2015 - 2016 Page 7 of 49


Next task is to investigate the validity of Solutions.

Solution I: Q = 0 (19)

Solution II: and (20)

Both solutions are real and valid if the root radicand stays positive.

This requires:

(21)

Simultaneous consideration of eqs. (16), (17), (18), (21) leads finally to:

→ For : (22)

→ For : (23)

To construct the Phase Diagram, the Phase Transition Line (PTL) has to be

calculated acc. to condition that the FEEs of Para- and Ferrophase have

to be equal.

F (Q = 0) = F (Q ≠ 0) (24)

B

CA

32

9 2

29

321

8

3

8

3

C

AB

B

C

B

CQ −+−=+ 29

321

8

3

8

3

C

AB

B

C

B

CQ −−−=−

+Q

−QB

CA C

32

9,0

2

B

CA C

32

9,0

2

Hempel, 2015 - 2016 Page 8 of 49


From eqs. (1) and (4) we obtain:

(25)

The solution Q2=0 can only occur provided A≡0 in Q+ (ref. to eq. (6)). This is not

possible because Q+ is stable already for bigger values of A:

The solution (A+CQ+BQ2)=0 leads together with eqs. (4), (23) to A = C2/(4B) (26)

which is the equation of PTL.

Since the stability limits of Para- and Ferrophase do overlap (ref. to eqs. (9),

(22), (23)) the Phase Transition is of 1st kind.

Also from eqs. (6), (26) we calculate for the jump of OP at the transition line:

or with eq. (25) (27)

The actual phase transition temperature calculates from eq. (2) together

with eq. (26) as:

(28)

( )220 BQCQAQ ++=

4320 BQCQAQ ++=

C

AQPTL

2−=

B

CQPTL

2−=

o

oCPTLBA

CTTT

4

2

+==

B

CA

32

9 2

Hempel, 2015 - 2016 Page 9 of 49


A

B

PTL: A = C2/(4B)

Stability limit

Paraphase A = 0

Stability limit

Ferrophase A = 9C2/(32B)Q = 0

Q ≠ 0

Since the stability regions of both phases do overlap, the OP exhibits a jump at the PTL and

the investigated Phase Transition (PT) is of 1st kind.

This is always the case if a 3rd order OP-Term in FEE appears.

A

C

PTL: A = C2/(4B)

Stability limit

Paraphase A = 0

Q = 0

Q-(C>0) ≠ 0Q+(C<0) ≠ 0

Stability limit

Ferrophase A = 9C2/(32B)

Hempel, 2015 - 2016 Page 10 of 49


3. Case: Q2 – Q4 – Q6

This case deals with the Free Energy Expansion (FEE) in dependence of the Order

Parameter (OP) Q.

F = AQ2 + BQ4 + DQ6 (1)

with A = Ao(T-To), To- Transition Temperature, Ao and D being positive constants.

The constant B can be positive or negative. (2)


(4)

This yields two solutions for Q:

Solution I: (describes Paraphase) (5)

Solution II:

(describes Ferrophase) (6)

0642 53 =++=

DQBQAQ

Q

F

( ) 0642 42 =++ DQBQAQ

0=Q

D

A

D

B

D

BQQ

393 2

222 −−==

2

22 31

33 B

AD

D

B

D

BQQ −−==

Hempel, 2015 - 2016 Page 11 of 49



(7)

For Solution I (ref. to eq. 5) follows: (8)

For the Solutions II eqs. (7) and (4) are valid:

(9)

(10)

Subtraction: eq. (9) – eq. (10) (11)

Therefore stability is guaranteed if is valid. (12)

Since the expression has to be positive definite too.

For the Paraphase (Q=0) follows from eq. (9) that A≥0. (13)

With the solution of eq. (6) certain cases can be discussed and are summarized

for the Ferrophase in the following table:

030 4

2

2

++

DQ12BQ2A 0

Q

F 2

CTT for stable 0A

( ) 022 + QQ3DB

0642 42 =++ DQBQA

030 4 ++ DQ12BQ2A 2

( ) 03 22 + QDQB

( )23DQB +02 Q

Hempel, 2015 - 2016 Page 12 of 49


(14)

Summary:

is stable if (15)

is stable if (16)


Solution I: Q = 0 → no limitations (17)

Solution II: and (ref. to eq. (6)) are valid if the root radicand remains

positive. This calculates:

02 Q 03 2 + DQB

02 +Q

02 −Q

0,0

0,0

AB

AB

ok always B

possiblenot B

,0

,0

possiblenot B

ok always ,B

,0

0

ok always B

possiblenot ,B

,0

0

2+Q 0,0 AB

0B2−Q

2+Q 2

−Q

Hempel, 2015 - 2016 Page 13 of 49


(18)

Simultaneous consideration of eqs. (15), (16), (18) leads finally to:

→ For : B>0 and A≤0 (19)

→ For : B<0 and (20)


calculated acc. to condition that the FEE of Para- and Ferrophase has to be equal.

F (Q = 0) = F (Q ≠ 0) (21)

(22)

Solution Q2 = 0: this requires A = 0 and materializes only for (23)

(ref. to eqs. (6), (19))

→ describes a Phase Transition of 2nd kind.

D

BA

B

AD

30

31

2

2−

2

−Q

2

+Q

D

BA

3

2

( )22 20 BQAQ +=

2

+Q

Hempel, 2015 - 2016 Page 14 of 49


Solution (2A+BQ2) = 0: this requires A = B2/(4D) (equation of PTL) and

can be only materialized for (ref. to limitation

for A as stipulated in eqs. (18, 20)).

→ describes a Phase Transition of 1st kind. (24)

Inserting eq. (24) into Q- (ref. to eq. (6)) yield for the jump of the OP at the PTL:

(25)

With eq. (24) this can be re-written to be:

(26)


with eq. (24) as:

(27)

2

−Q

D

BQ PTL

22 −=

B

AQ PTL

22 −=

o

oCPTLDA

BTTT

4

2

+==

Hempel, 2015 - 2016 Page 15 of 49


The Phase Diagram for solution looks like:

It represents a Phase Transition of 2nd kind.

The Phase Diagram for solution looks like:

It represents a Phase Transition of 1st kind.

2

+Q

PTL: A = 0

A

B

Q+2 ≠ 0

Q = 0

2

−QA

PTL: A = B2/(4D) B

Q-2 ≠ 0

Stability limit

Ferrophase A = B2/(3D)

Stability limit

Paraphase A = 0

Q = 0

Hempel, 2015 - 2016 Page 16 of 49


4. Case: 2-component Order Parameter w/o cubic term

This case deals with the Free Energy Expansion (FEE) in dependence of two Order

Parameter (OP) Components Q1, Q2.

F = A(Q12 +Q2

2 ) + B1(Q14 +Q2

4 ) + B2Q12Q2

2 (1)

with A = Ao(T-To), To- Transition Temperature, Ao being positive.

The equations of state write:

(2)

(3)

This yields to totally 4 possible phases, depending on combinations of (Q1, Q2) :

Phase 0: (0, 0) → describes Paraphase (4)

Phase 1: ( , 0) → describes Phase 1 *) (5)

Phase 2: ( , ) → describes Phase 2 (6)

Phase 3: (Q1,Q2) → describes Phase 3 (7)

*) Note: Another possible Phase (0, Q‘) is identical to Phase 1. This originates from the symmetry of FEE

regarding Q1, Q2.

( ) 022 2

12

2

212

2

=++=

QBQBAQ

Q

F

( ) 022 2

22

2

111

1

=++=

QBQBAQ

Q

F

Q

Q Q

Hempel, 2015 - 2016 Page 17 of 49


Phase 1: ( , 0)

From eqs. (2) & (3) follows

(8)

Phase 2: ( , )


(9)

Phase 3: (Q1,Q2)

From the difference of eqs. (2) & (3) follows

(10)

Eq. (10) can only be fulfilled if , which results either in

and describing Phase 2 (as above) or in describing

Phase 3. Other values are not permitted.

The calculation of the stability condition requires exploitation of 2nd derivatives

of FEE:

02 21 =+ QBA

1

2

2B

AQ −=

0)2( 2

21 =++ QBBA

21

2

2 BB

AQ

+−=

22

2

1 QQ =

Q

Q Q

( ) ( ) 02 2

2

2

121 =−− QQBB

''21 QQQ ==

121 , QQ QQ 2 −=−=

Hempel, 2015 - 2016 Page 18 of 49


(11)

(12)

(13)

For OPs with 2 components the stability requires:

and (14)

Phase 0: (0, 0) Inserting Q1=Q2=0 into eqs. (11)-(14) leads to

the condition A>0 for phase 0 (15)

Phase 1: ( , 0) Inserting from eq. (8) into eq. (11) leads to

A<0 (16)

( )

( )

212

21

2

2

12

2

212

2

2

2

22

2

112

1

2

4

62

62

QQBQQ

F

QBQBAQ

F

QBQBAQ

F

=

++=

++=

021

2

Q

F

0

22

2

12

221

2

21

2

Q

F

QQ

FQQ

F

Q

F

Q Q

Hempel, 2015 - 2016 Page 19 of 49


Inserting from eq. (8) into eq. (14) leads to

(17)

Phase 2: ( , ) Inserting from eq. (9) into eq. (11) leads to

& also AB1≤0 (18)

Phase 3: Inserting from eq. (9) into eq. (14) leads to

(19)


Phase 0: (0, 0) no restrictions (20)

Phase 1: ( , 0) From eq. (8) follows that either

A<0 and B1>0 or A>0 and B1<0 must hold.

But eq. (16) requires A<0 which results together

with B1>0 and eq. (17) that (21)

Phase 2: From eq. (9) follows

& also A>0 and 2B1+B2<0 or (22)

Phase 3: A<0 and 2B1+B2>0 (23)

( )0

2

2

1

21 −

B

BB

( ) ( ) 022 2121 +− BBBB

0)2( 21 −BB

Q Q

Q

Q

( )QQ ,

Q

Q

−=

−=

12

21

QQ

QQ

−=

−=

12

21

QQ

QQ

Hempel, 2015 - 2016 Page 20 of 49


Eqs. (22, 23) together with eq. (19) yield

A>0 and 2B1+B2<0 and 2B1-B2<0 or (24)

A<0 and 2B1+B2>0 and 2B1-B2>0 (25)

This can be re-written to be

A>0 and 2B1<-B2 and 2B1<B2 or (26)

A<0 and 2B1>-B2 and 2B1>B2 (27)

Eqs. (26, 27) can be rewritten as:

A>0, 2B1<B2<-2B1 → not possible since B1>0 or (28)

A<0, -2B1<B2<2B1 → possible since B1>0 (29)

Summary:

Phase 0: (0, 0) A>0, no further restrictions (30)

Phase 1: ( , 0) A<0, B1>0, and 2B1<B2 (31)

Phase 2: ( , ) A<0, B1>0, and -2B1<B2<2B1 (32)

Phase 3: A<0, B1>0, and -2B1<B2<2B1 (33)

Q

Q Q

−=

−=

12

21

QQ

QQ

Hempel, 2015 - 2016 Page 21 of 49




F (0, 0) = F ( , 0) and

F (0, 0) = F ( , ) (34)

Both expressions of eq. (34) are fulfilled if A=0, where the stability limits of the

adjacent phases coincide. Therefore those Phase Transitions are of 2nd Kind.

The transition between Ferrophases 1 and 2 cannot be described within this

model since there is no solution of the form A = A(B1, B2). To do this, higher

order terms of (Q1, Q2) would have to be considered in FEE (ref. to eq. (1)).

PTL: A = 0A

B2

+2B1-2B1

Phase 0

Phase 2 / 3 Phase 1

Q Q

Q

Hempel, 2015 - 2016 Page 22 of 49


5. Case: 2-component Order Parameter with cubic term

This case deals with the Free Energy Expansion (FEE) in dependence of two Order

Parameter (OP) Components Q1, Q2.

F = A(Q12 +Q2

2 ) + C(Q13 - 3Q2

2Q1) + B(Q14 +Q2

4) (1)

with A = Ao(T-To), To- Transition Temperature, Ao being positive, C can be positive

or negative, and B must be positive.


(2)

(3)

This yields to totally 6 possible phases, depending on combinations of (Q1, Q2) :


Phase 1: ( , 0) → describes Ferrophase 1 (5)

Phase 2: (0, ) → describes Ferrophase 2 (6)

Phase 3: ( , ) → describes Ferrophase 3 (7)

Phase 4: ( , ) → describes Ferrophase 4 (8)

Phase 5: (Q1, Q2) → describes Ferrophase 5 (9)

( ) ( ) 0432 2

2

2

11

2

2

2

11

1

=++−+=

QQBQQQCAQ

Q

F

( ) 0462 2

2

2

12212

2

=++−=

QQBQQCQAQ

Q

F

Q

Q

Q Q

Q Q −

Hempel, 2015 - 2016 Page 23 of 49


Phase 1: ( , 0)


(10)

Phase 2: (0, )


solvable only if C = 0 (3rd order term ruled out!)

and yields (11)

Phase 3: ( , )


solvable only if C = 0 (3rd order term ruled out!)

and yields (12)

Phase 4: ( , ) Same situation as for phase 3. (13)

0432 2 =++ QBQCA

B

A

B

C

B

CQQ

264

9

8

32

2

−−==

03 2 =− QC

B

AQ

22 −=

042 3 =+ QBQA

04 2 =+ QBA

043 2 =+− QBQCA

B

AQ

42 −=

Q

Q

Q

Q Q −

Q

Hempel, 2015 - 2016 Page 24 of 49


Phase 5: (Q1, Q2)

From eqs. (2) & (3) follows → (14)

Inserting eq. (14) in eq. (2) yields:

(15)

This expression looks similar like the state equation

for phase 1 (ref. to eq. (10)). Indeed eq. (15) can be

transformed exactly into the state equation by putting

→ (16)

Result: the solutions (Q’, 0), and from eq. (14)

yield the same results, except some changes in the

coefficients.

The related Ferrophases differ only in their arrangements

with regard to the Paraphase.

These sets of combinations of the OP-components are

summarized as type which describes phase 1. (17)

Therefore in the following chapters only phase 1 needs

to be investigated.

22

213 QQ =03 2

221 =−QQ

083 211 =+− BQCQA

QQ −=2

11 0432 2 =++ QBQCA

),3( QQ

0,Q

Hempel, 2015 - 2016 Page 25 of 49



of FEE:

(18)

Phase 0: (0, 0) A>0 (19)

Phase 1: ( , 0) Mind that only phase 1 needs to be investigated

Inserting (Q1=Q‘, Q2=0) in eq. (18) and making use of

the state equation eqs. (10, 16) yield:

(20)

212

21

2

2

2

2

112

2

2

2

2

2

112

1

2

86

)3(462

)3(462

QBQCQQQ

F

QQBCQAQ

F

QQBCQAQ

F

+−=

++−=

+++=

0

9

34

21

2

22

2

21

2

=

−=

−−=

QQ

F

QCQ

F

QCAQ

F

Q

Hempel, 2015 - 2016 Page 26 of 49


Stability requires:

and (21)

and results in: and (22a,b)

Eq. (22a) reformed to and multiplied by itself yields:

(23)

If the equation of state (eq. (16)) is rearranged to be

and if this expression is introduced into eq. (23) the 1st stability condition

comes out to be:

(24)

If eq. (22a) holds then eq. (22b) requires: → (25)

Eq. (25) therefore demands that Q’<0 if C>0 and Q’>0 if C<0. (26)

Furthermore, following relations must hold:

021

2

Q

F0

22

2

12

221

2

21

2

Q

F

QQ

FQQ

F

Q

F

034 −− QCA ( ) ( ) 0934 −−− QCQCA

09 − QC 0QC

B

CA

32

9 2

034 + QCA

024916 222 ++ QACQCA

22 322416 QABQACA −=+

Hempel, 2015 - 2016 Page 27 of 49


(27)

which mean that changes the sign if C changes its sign. This can be

easily proven, using eq. (10).



Phase 1: ( , 0) From eq. (10) follows that either the root radicand needs

to be positive. The requires:

(29)

and coincides with one stability limit (ref. to eq. (24)).



F ( , 0) = F (0, 0) (30)

B

CA

32

9 2

Q

Q

Q

00

00

'

'

−

+

Q C if

and

Q C if ( )

)0()0(

0)0(

''

''

=−

=−

−+

−+

C QC Q

and

C QC Q

Hempel, 2015 - 2016 Page 28 of 49


(31)

This can be reformed to be

Inserted into the state equation (eq. (16)) yields

(32)

Now the jump of the OP at the PTL calculates:

(33)

Inserting eq. (33) into eq. (31) yields the expression of the PTL:

(34)

If this is reinserted into eq. (32) another expression for the OP-jump

is obtained:

(35)


with eq. (34) as: (36)

0432 =++ QBQCQA

QCQAQB −−= 22

04432 =−−+ QCAQCA

C

AQPTL

2−=

B

CA

4

2

=

B

CQPTL

2−=

o

oCPTLBA

CTTT

4

2

+==

Hempel, 2015 - 2016 Page 29 of 49


Since the stability regions of the Para- and Ferrophases do overlap, the OP exhibits a jump

at the PTL and the investigated Phase Transitions (PTs) are of 1st kind. This is always

the case if a 3rd order OP-Term in FEE appears.

A phase transition between the two Ferrophases [ ] cannot be described with this

model and would require the addition of higher OP-terms in FEE.

Stability limit

Ferrophase A = 9C2/(32B)

A

C

PTL: A = C2/(4B)

Q‘ = 0

Stability limit

Paraphase A = 0 Q‘-(C>0) ≠ 0Q‘+(C<0) ≠ 0

−+ QQ

Hempel, 2015 - 2016 Page 30 of 49


6. Case: 2 coupled 1-component Order Parameters

This case deals with the Free Energy Expansion (FEE) in dependence of two coupled

Order Parameters (OPs) – Q and P.

F = AQ2 + BQ4 + CQ6 + DP2 + EP4 + FQ2P2 (1)

with A = Ao(T-ToQ), ToQ- Transition Temperature regarding Q, Ao, C, E being positive, and

B, F can be positive or negative, D=Do(T-ToP), ToP- Transition Temperature regarding P.


(2)

(3)

This yields to totally 3 possible phases, depending on combinations of (Q, P) :


Phase 1: ( , 0) → describes Ferrophase 1 (5)

Phase 2: (0, P) → describes Ferrophase 2 (6)

Phase 3: ( , P) → describes Ferrophase 3 (7)

02642 253 =+++=

FQPCQBQAQ

Q

F

0242 23 =++=

PFQEPDP

P

F

Q

Q

Hempel, 2015 - 2016 Page 31 of 49


Phase 1: ( , 0)


(8)

Phase 2: (0, P)


(9)

Phase 3: ( , P)


(10)

with (11,12)

032 42 =++ QCQBA

2

22 31

33 B

AC

C

B

C

BQQ −−==

02 2 =+ EPD

E

DP

22 −=

032 242 =+++ FPQCQBA

02 22 =++ QFEPD

2

222 2448

11212

−−

−==

FDECCAE

ECECQQ

22

22Q

E

F

E

DP −−= 24 FBE −=

Q

Q

Hempel, 2015 - 2016 Page 32 of 49



of FEE:

(13)

(14)

(15)

In particular *):

and (16a,b)

Phase 0: (0, 0) From eqs. (13-15) follows A>0and D>0 for phase 0 (17a,b)

Phase 1: ( , 0) From eq. (13, 16a) together with eq. (2) follows

(ref. also to case 3 of this paper) and (18)

yield that stability is given for if A<0 and for

if (19)

*) Note: As shown in the literature (ref. to Gufan et al.) the inequations are usually evaluated, but in

several cases it is appropriate to investigate the expressions as equations, to find the stability limits.

FQPPQ

FF

FQEPDP

FF

FPCQBQAQ

FF

QP

PP

QQ

4

2122

230122

2

22

2

2

242

2

2

=

=

++=

=

+++=

=

02

2

=

QQF

Q

F0

2

22

2

2

2

P

F

QP

FPQ

F

Q

F

02 + QBA2

−Q2

+Q

C

BA

3

2

Q

Hempel, 2015 - 2016 Page 33 of 49


Eq. (16b) as calculates

(20)

Since the 1st factor is negative (ref. to eq. (18))

the expression also has to be negative or (21)

zero, too.

Using eq. (8) this yields the requirement:

(22)

From eqs. (18, 21) follows that:

→ if A>0 then B<0 and therefore ∆<0 (ref. to eq. (12)) (23)

→ if B>0 then A<0 (24)

→ if D>0 then F<0 (25)

→ if F>0 then D<0 (26)

Phase 2: (0, P) From eq. (13, 16a) together with eq. (3) follows

and requires (27)

Eq. (16b) as calculates

(28)

02 − QPPPQQ FFF

( ) ( ) 0022 22 −++ QFDQBA

)22( 2QFD +

DF

BD

F

CA

23 2

2+−=

022 2 + FPA DE

FA

2

02 − QPPPQQ FFF

( ) ( ) 0012222 22 −++ EPDFPA

Hempel, 2015 - 2016 Page 34 of 49


Since the 1st factor is positive (ref. to eq. (27))

the expression also needs to be positive (29)

or zero. Using eq. (9) this yields the requirement:

(30)

From eqs. (27, 29) follows that:

→ if A<0 then F>0 (31)

→ if F<0 then A>0 (32)

→ if D<0 then E>0 (E is always positive ref. to eq. (1)) (33)

Phase 3: ( , P) First step is to investigate FQQ (ref. to eq. (16a)).

From eq. (13), but easier from eq. (2) via application

of the chain rule the 2nd derivative FQQ is obtained

(34)

The 1st factor of eq. (34) requires to be:

(ref. to validity of solution below – eqs. (48,49)) (35)

The 2nd factor of eq. (34) together with eq. (10) yields for:

with ∆<0 being always stable if root radicand

of eq. (10) is positive (36a)

with ∆>0 being stable if following relation holds

(36b)

0D

( )2122 EPD +

( ) 03 22 + QCBQ

02 Q

2

−Q

2

−Q

EC

B

C

BD

E

FA

632

2 +−

Q

Hempel, 2015 - 2016 Page 35 of 49


with ∆>0 being always stable if root radicand

of eq. (10) is positive (37a)

with ∆<0 being stable if following relation holds

(37b)

Moreover from the 2nd factor of eq. (36) follows:

→ if B<0 then C>0 (but C is always positive ref. to eq. (1)) (38)

Next step is the investigation of .

This leads to the expression

(39)

The 1st factor of eq. (39) to be positive or zero leads to the

stability equation (which equals eq. (22)).

(40)

whereby the 2nd factor of eq. (39) to be also positive or

zero yields:

(41)

( ) 01216 222 + QCEPQ

2

2

32D

F

CD

F

BA −=

02 − QPPPQQ FFF

2

2

482 CED

E

FA

+=

2

+Q

2

+Q

EC

B

C

BD

E

FA

632

2 +−

Hempel, 2015 - 2016 Page 36 of 49




Phase 1: ( , 0) From eq. (8) follows that the root radicand needs

to be positive for . The requires: (43)

Moreover the expression for has to be positive definite.

→ For : either B>0 and A<0 or B<0 and A>0 (44)

→ For : B<0 is required (45)

Phase 2: (0, P) From eq. (9) follows that D<0 (46)

Phase 3: ( , P) From eq. (10) follows that the root radicand of has

to be positive:

(47)

Moreover the expression for has to be positive definite.

→ For : either ∆>0 and or ∆<0 and (48)

→ For : ∆<0 is required (49)

C

BA

3

2

2

2

482 CED

E

FA

+

2

Q

2

+Q

2

−Q

2

Q

DE

FA

2 D

E

FA

2

2

QQ

Q

2

+Q

2

−Q

2

Q

Hempel, 2015 - 2016 Page 37 of 49


Additionally – to ensure that P2 is also positive definite -

from eq. (11) follows that:

→ has to be positive definite (ref. to eqs. (48,49)) (50)

→ from eq. (11) follows D+F <0, which yields: (51)

→ D<0 and F<0 or (52)

→ D<0 and F>0 or (53)

→ D>0 and F<0 (54)

To construct the Phase Diagram, the Phase Transition Lines (PTLs) have to be

calculated acc. to condition that the FEEs of the adjacent phases have to be equal.

PTL 0-1: - Phase 0 exists if A≥0 (ref. to eqs. (17a,b,42) (55)

- if B>0 than Phase 1 is stable for A<0 (ref. to eq. (24)

and the OP is (ref. to eq. (44)). (56)

→ the stability limits of Phases 0 and 1 coincide at A=0

and the PT is therefore of 2nd kind. (57)

- if B<0 than Phase 1 is stable if (ref. to eq. (19)),

→ the stability limits overlap and the PT is of 1st kind,

described by OP (ref. to eq. (45)). (58)

- the respective PTL calculates according to

F ( , P=0) = F ( , P=0) (59)

2

+Q

C

BA

3

2

2

−Q

2Q

Q

2Q

0=Q

Hempel, 2015 - 2016 Page 38 of 49


and yield A = B2/(4C) (ref. also to case 3of this paper) (60)

PTL 1-2: From F ( , P=0) = F ( , P) follows after cumbersome

calculations for the PTL:

(61)

This PTL shall go continuously over to the PTL as derived

for the PT between Phases 0 and 1 (ref. to eqs. (55-58))

To check this out, D will be set to zero. From eq. (61)

follows: (62)

Rewriting eq. (62) results in:

(63)

and has two different solutions for A:

1. A=0 → describes the case where B>0 (ref. to eqs. (56,57))

The OP in Phase 1 is (64)

2. A = B2/(4C) → describes case where B<0 (ref. to eq. (58))

The OP in Phase 1 is (65)

032

23222432334

22524243246

=++

+−−

CDCDEB

ADCBECAEBACE

022 24243246 =− CAEBACE

( ) 04 22 =− BCAA

2

+Q

2

−Q

Q 0=Q

Hempel, 2015 - 2016 Page 39 of 49


PTL 2-3: From F ( , P) = F ( , P) follows after cumbersome

calculations for the PTL:

(66)

PTL 1-3: Since the stability limits of Phases 1 and 3 coincide

at (ref. to eqs. (22,41))

the PT is therefore of 2nd kind. (67)

PTL 0-3: From F (Q=0, P=0) = F ( , P) follows after intense

calculations for the PTL (ref. to Gufan et al.):

(68)

with E

FDAA

2

~−=

2

2

32D

F

CD

F

BA −=

03

~32

~~2

222232

222223226

=+

−−+

DEBD

ADCACACE

2

2

642 CED

E

FA

+=

0=Q Q

Q

Hempel, 2015 - 2016 Page 40 of 49


Summary:

Looking at the stability condition for Phase 3 (ref. to eq. (39)):

It turns out that in case the phase transition lines can only be obtained from

. The accompanied PTs are all of 2nd kind. Remembering that

(ref. to eq. (12)) it requires B>0 and (“weak coupling”).

The case (“strong coupling” between OPs and P) materializes if B>0 and

or respectively if and B>0 or B<0. In this case new PTLs appear.

One word on the symmetry effects on PTs described by two 1-component OPs with a

lowest order interaction term like .

This term is of course compatible with all symmetries. If we assume that and belong

to different irreducible representations of the high symmetry group of the Paraphase,

depending on the magnitude of interaction of the two Order Parameters they can either

appear successively or simultaneously (“triggered PT”), when coming from the Paraphase.

In the Phase where the two OPs are both present the symmetry is characterized by the

intersection of the symmetry elements accompanied by the onsets of and .

This means that for triggered PTs the lowering of symmetry is greater than the lowering of

symmetry connected with one OP only.

( ) 01216 222 + QCEPQ

0

022 == PQ24 FBE −=

BEFBE 22 +−

BEF 2+ EBF 2−

0 Q

22 PQ Q P

Q P

Hempel, 2015 - 2016 Page 41 of 49


On the following summary slide an overview is provided about all stability limits and validity

constraints for the individual Phases, followed by graphical representations of different

cases – with particular focus on the cases with .

Let’s look at interesting examples:

1. In all discussed cases the PTL 1-3 describes completely (sometimes partly) a PT of 2nd

kind. This is because the stability limits of Phases 1 and 3 coincide there( ).

PT of 2nd kind means, that there is no “jump-like” appearance of the OP at the PTL

1-3 when coming from Phase 1 (decreasing D). Indeed both OPs ( for Phase 1 and

for Phase 3) exactly coincide at PTL 1-3 (same numerical values), and is going

to grow like a typical 2nd order PT OP on top of - as D further decreases. When

starts to exist in Phase 3 simultaneously OP P becomes present (acc. to eq. (11)).

2. If looking at sequential PTs – e.g. 0→1→2 (ref. to slide 41) the 1st transition is of 2nd kind

( appears smoothly). If A and D further decrease the subsequent PT 1→2 is of 1st

kind (provided this transition happens above the Three-Phase Point Q). Exactly at the

PTL 1-2 drops to zero (because of lost stability) and appears with a jump.

Remarkable is the variety of possible PTs in dependence on the coefficients B and F. Even

direct PTs between Phases 0 and 3 can be described (“triggered PT”). Assuming that

coefficients A and D depend (or not) only on temperature (T) then all relevant PTs lie on the

thermodynamic path in the A-D-Plane, described either by A(T) & B=const. or by B(T) &

A=const. or by A(T) & B(T).

0

2

2

32D

F

CD

F

BA −=

Q 2

Q

2Q

2Q

2Q

2Q

2+Q

2P2

+Q

Hempel, 2015 - 2016 Page 42 of 49


Phase

Sta

bili

ty

1 Not stable Not stable Not stable

2 Not stable Not stable Not stable Not stable

3 Not stable Not stable

---

Stable if: and

Va

lidity

1

--- --- ---

--- --- ---

2 --- --- --- ---

3

--- --- ---

--- --- ---

2

+Q

2

2

32D

F

CD

F

BA −=

2

−Q 2

+Q 2

−Q2P

C

BA

3

2

0D ,2

DE

FA

0A

limits no & 0

EC

B

C

BD

E

FA &

6320

2 +−

limits no & 0

EC

B

C

BD

E

FA &

6320

2 +−

2

2

482 CED

E

FA

+=

0) (means

AB

or AB

0&0

0&0

0) (means

B

0

0D

C

BA

3

2

2

2

482 CED

E

FA

+=

DE

FA &

or DE

FA &

20

20

0

2

2

32D

F

CD

F

BA −=

Hempel, 2015 - 2016 Page 43 of 49


The PTL 2-3 (1st kind) goes over at Point Q to PTL 1-2 (also 1st kind) and then approaches smoothly the PTL 0-1 at A=0.

At The Three Phase Point Q PTL 2-3 (ref. to eq. (66)) and PTL 1-3 (2nd kind, ref. to eq. (67)) meet. Also the PTL 1-2 (eq.

(61)) goes through Point Q.

CE

FD

24

)12( =

2

22

192

3)12(4

CE

FA

+=

Note: ‘+‘ is valid

2+Q

22 &PQ−

2P

Hempel, 2015 - 2016 Page 44 of 49


The PTL 2-3 (1st kind) goes over at Point Q to PTL 1-2 (also 1st kind) and then approaches smoothly the PTL 0-1 at

A=B2/(4C). At The Three Phase Point Q PTL 2-3 (ref. to eq. (66)) and PTL 1-3 (2nd kind, ref. to eq. (67)) meet. Also the

PTL 1-2 (eq. (61)) goes through Point Q.

CE

FD

24

)12( =

2

22

192

3)12(4

CE

FA

+=

Note: ‘+‘ is valid

2−Q

22 &PQ−

2P

Hempel, 2015 - 2016 Page 45 of 49


The PTL 1-3 (2nd kind) goes over at the Tricritical Point T to PTL 1-3 (1st kind) and then becomes at Point L the PTL 0-3

(1st kind, “triggered” PT, ref. also to Holakovsky). Further at Point N this PTL becomes the PTL 2-3 (1st kind).

Since B>0 the PT between Phases 0 and 1 is always of 2nd kind (PTL 0-1).

CE

FD

12

=

2

22

48

2

CE

FA

+=

2+Q

22 &PQ−

2P

Hempel, 2015 - 2016 Page 46 of 49


The PTL 1-3 (2nd kind) ends at the Tricritical Point T and goes over into the 1st kind stability line of Phase 3 (blue dots). The

Point T lies above PTL 0-1 (1st kind, A = B2/(4C)) if F>-(2|B|E)1/2 and also above the Three-Phase Point L. This means that

in this case the transition between Phases 1 and 3 will be always of 2nd kind (PTL 1-3). The PTL 0-3 (1st kind) is a

“triggered” PT (ref. also to Holakovsky). Further at Point N the PTL 2-3 (1st kind) begins.

CE

FD

12

=

2

22

48

2

CE

FA

+=

2−Q

22 &PQ−

2P

Hempel, 2015 - 2016 Page 47 of 49


With the settings B>0, F>0 and ∆>0 all PTs are of 2nd kind. In case C=0 the PTL 1-3 become straight a line.

2+Q

22 &PQ+

2P

Hempel, 2015 - 2016 Page 48 of 49


With the settings B>0, F<0 and ∆>0 all PTs are of 2nd kind. In case C=0 the PTL 1-3 become straight a line. This behavior

includes the PT-sequence 0→2→3 (path: A=constant and D decreasing), where all PTs are of 2nd kind (ref. to Holakovsky).

2+Q

22 &PQ+

2P

Hempel, 2015 - 2016 Page 49 of 49


7. Literature



/02/ Gufan, Yu. M., “Thermodynamic Theory of Phase Transitions”, Publisher: University of

Rostov on Don, 1982



/04/ Gufan Yu. M, Larin E. S., Sov. Solid State Phys. 22(2), 270(1980)

/05/ Gufan Yu. M., Torgashev V. I., Sov. Solid State Phys. 22(6), 951(1980)

/06/ Holakovsky, J., phys. stat. sol. (b) 56, 615(1973)

Temperature Dependences of Material Coefficients

at Structural Phase Transitions

Hempel, 2007 & 2017 Page 1 of 25

Temperature Dependences of Material Coefficientsat Structural Phase Transitions

Table of Contents

1. Theory

2. Elastic Coefficents

3. Dielectric Impermeabilities

4. Piezoelectric Coefficients

5. Clarifying Example Calculation

6. Calculation of Temperature Dependences

7. Summary



Hempel, 2007 & 2017 Page 2 of 25

Issue

Provided the Thermodynamic Potential function, suitable to describe

the Phase Transition under consideration, has been properly derived –

How can the temperature dependences of certain material coefficients

be predicted ?



Hempel, 2007 & 2017 Page 3 of 25

1. Theory

• Basing on standard group theoretical principles the Free Energy Power Expansion

has been derived as

F = F (Qi, Sj, Pk) with (1)

Qi – Order Parameter (OP) components

Sj – Deformation Tensor components (j=1...6, Voigt notation)

Pk – Polarization Vector components (k=1...3)

• For the sake of generality let‘s assume that external „forces“ are applied to the

crystal – according to:

Tj = F/ Sj (External Stress) (2)

Ek= F/ Pk (External Electrical Field) (3)

Note: For a mechanically and electrically „free crystal“ Tj and Ek has to be set = 0!



Hempel, 2007 & 2017 Page 4 of 25

• Solving the equation system (2), (3) we obtain

Sj = Sj (Qi, Ek, Tj) (4)

Pk = Pk (Qi, Ek, Tj) (5)

• Inserting (4) and (5) into (1) leads to the Free Energy Expression in the form:

F = F (Qi, Ek, Tj) with Ek, Tj being the external „forces“ (6)

Note: Qi are the only internal parameters the Free Energy is depending on.

• That Energy Expression (eq. 6):

• Is only suitable to determine the equilibrium values of the Order Parameter

Components (Qi), and therefore the possible changes of the crystal symmetry

• Has no „dynamic“ relevance because the Sj and Pk are not able to follow the

fast changes („vibrations“) of the Order Parameter („soft mode“)

(7) 0,0

Q

P

Q

S ki



Hempel, 2007 & 2017 Page 5 of 25

2. Elastic Coefficients

• To calculate the elastic coefficents, formulae (1) has to be re-considered:

F = F (Qi, Sj, Pk) (8)

taking into account the following conditions:

1. Via certain couplings the Qi can be dependent on Sj [Qi = Qi (Sj)]

2. Via electromechanical couplings Pk can be dependent on Sj

[Pk = Pk (Sj)]

• The elastic coefficients shall be calculated under the condition that the Order Parameter

(as the „driver“ of the phase transition) can freely move, i.e.

Ki = F/ Qi = 0 (9)

with Ki being the conjugated force with regard to Qi



Hempel, 2007 & 2017 Page 6 of 25

• Equation (8) has dynamic relevance because the Order Parameter can quasistatically

follow the „slow“ changes of the strain Sj

• When putting together the Free Energy Expression according to (1) or (8), the

corresponding material coefficients have to be entered under following conditions:

1. Bare elastic stiffnesses coij – at Qi = 0 and Pk = 0

2. Bare dielectric impermeabilities oij - at Qi = 0 and Sj = 0

3. Bare piezoelectric coefficients homn - at Qi = 0

• Finally the elastic stiffnesses calculate according to:

(10)

with cij being the stiffnesses under the influence of the order parameter‘s action (ref. to

(9) above) when the Order Parameter can freely move.

ji

ij SS

Fc

=

2

0

j

i

S

Q



Hempel, 2007 & 2017 Page 7 of 25

• The mechanical stresses calculate:

(11)

= 0 (ref. to (9))

• Finally the stiffnesses calculate:

(12)

influence of OP‘s influence of piezoelectric

action coupling

m

i

i im

m S

Q

Q

F

S

FT

+

=

+

+

=

j n

j

jmi n

i

imnm

mn S

P

PS

F

S

Q

QS

F

SS

Fc

222



Hempel, 2007 & 2017 Page 8 of 25

Goal is now to express:

From eq. (9) follows that the Order Parameter can freely move:

(13.1)

Suggested ‘Ansatz‘: (13.2)

Interchange of subscripts j k leads to (13.3)

Intersting into eq. (13.1) yield: (13.4)

?=

m

i

S

Q

0==

j

j

KQ

F

0=

jn Q

F

dS

d0

22

=

+

n

i

i ijjn S

Q

QQ

F

QS

F

ijj jnn

i RQS

F

S

Q

−=

2

−=

ik

k knn

i RQS

F

S

Q

2

0222

=

−

+

k

ik

kni ijjn

RQS

F

QQ

F

QS

F



Hempel, 2007 & 2017 Page 9 of 25

(13.5)

Comparison of coefficients results in:

(13.6)

and can be compactly re-written as:

(13.7)

Note: The term describing the piezoelectric coupling in eq. (12) can be calculated likewise.

jnkn

ikk iji QS

F

QS

FR

QQ

F

=

222

=

=

jk

jkR

QQ

Fik

k iji 1

02

kjiki ij

RQQ

F=

2



Hempel, 2007 & 2017 Page 10 of 25

• Compact, final expression (see e.g. /1, 2/) :

(14)

Note: The stiffnesses are those to be measured at constant electric Field at isothermal conditions

with

(15)

being the equations to determine the R-matrix components in (14), xy = 0 if x y,

xy = 1 if x = y, represents the Kronecker symbol

vn

2

kvvk, km

2

kn

2

ikki, im

2

nm

2

mn PS

FR

PS

F

QS

FR

QS

F

SS

Fc

−

−

=

kvk jk

vjiki ji

kj RPP

FR

QQ

F

=

=

22



Hempel, 2007 & 2017 Page 11 of 25

3. Dielectric Impermeabilities

• The calculation of the impermeabilities is analogue to the calculation of the elastic

coefficients as shown in Chapter 2.

• The tensor of the dielectric impermeabilities is related to the tensor of the dielectric

susceptibility like () = ()-1

(16)

• The electricl fields calculate:

(17)

= 0 (rf. to (9))

m

i

i im

m P

Q

Q

F

P

FE

+

=

ji

ij PP

F

=

2



Hempel, 2007 & 2017 Page 12 of 25

• Finally the impermeabilites calculate:

(18)

influence of OP‘s influence of piezoelectric

action coupling

• Compact expression after some treatment (see /1, 2/):

(19)

Note: The impermeabilities are those to be measured at const. mech. Stress at isothermal conditions

with

(20)

+

+

=

j n

j

jmi n

i

imnm

mn P

S

SP

F

P

Q

QP

F

PP

F 222

vn

kvvk kmkn

ikki imnm

mn SP

FR

SP

F

QP

FR

QP

F

PP

F

−

−

=

2

,

22

,

22

kvk jk

vjiki ji

kj RSS

FR

QQ

F

=

=

22



Hempel, 2007 & 2017 Page 13 of 25

4. Piezoelectric Modules

• The calculation of the piezomodules is analogue to the calculation of the elastic

coefficients and the impermeabilities as shown in chapters 2 and 3.

(21)

• The 1st derivatives of the Free Energy Expansion with regard to the polarization

components gives the electrical fields (see eq. (17))

(22)

= 0 (rf. to (9))

m

i

i im

m P

Q

Q

F

P

FE

+

=

ji

ij SP

Fh

=

2



Hempel, 2007 & 2017 Page 14 of 25

• The 2nd derivatives with regard to the strains give the piezoelectric modules:

(23)

influence of OP‘s

action

(24)

with

(25)

+

=

i n

i

imnm

mn S

Q

QP

F

SP

Fh

22

kn

ikki imnm

mn QS

FR

QP

F

SP

Fh

−

=

2

,

22

iki ji

kj RQQ

F

=

2



Hempel, 2007 & 2017 Page 15 of 25

5. Clarifying Example Calculation

• Let‘s assume the phase transition under consideration is driven by a 2-component

order parameter Qi (i = 1,2)

• Exemplarily the matrix components Rik (ref. to (13.7)) are calculated as follows:

1st Case kj

using the abbreviations: follows

iki ji

kj RQQ

F

=

2

k

j

k

j

RQQ

FR

QQ

F2

2

2

1

1

2

0

= +

...11

2

11 osa QQ

FF

=



Hempel, 2007 & 2017 Page 16 of 25

(a)

(b)

2nd Case k=j

(c)

(d)

Solution of this system of 4 equations (a)...(d) yields:

21221112

22211211

0

0

RFRF

RFRF

+=

+=

j

j

j

j

RQQ

FR

QQ

F2

2

2

1

1

2

1

= +

22221212

21211111

1

1

RFRF

RFRF

+=

+=

212122211

1212

2122211

1122

2122211

2211

RFFF

FR

FFF

FR

FFF

FR

=−

=

−=

−=



Hempel, 2007 & 2017 Page 17 of 25

6. Calculation of Temperature Dependences

• Depending on the coupling term between the OP-components (Qi) and the state

parameters (Si, Pj) in the Free Energy Expansion (1), following typical cases shall

be considered (e.g. ref. to /3, 4/):

1. True proper (ferroelastic) phase transition (2nd kind, OP identical to strain component S)

(30)

(31)

(32)

The stiffness calculate now acc. to eqs. (10) and (14) as:

(33)

o2

o

oo

TT for B

A-S

and TT for 0S to leads 0S

F

TTAA with BSASF

=

==

−=+= )(4

1

2

1 42

oooS

oooS

TT for )T-(T -2A-2Ac

and TT for )T-(T AAc

==

==



Hempel, 2007 & 2017 Page 18 of 25

2. Pseudo-proper (ferroelastic) phase transition (2nd kind)

(34)

(35)

(36)

(37)

Next step is the calculation of the free energy F’ of the stress free crystal by

inserting of eq. (35) into eq. (34). After some calculation yields:

(38)

BQAQ

F Stability

DSBQAQQ

F F of Mininum

Qc

D-S DQSc

S

F crystal free

TT AA with DQSSc BQAQF

oS

oS

oo2o

S

2

2

2

3

42

3:

:

0:""

)(2

1

4

1

2

1

+=

++=

=+==

−=+++=

)(4

1)(

2

1'

242

2

oS

'oS c

DA A mit BQQ

c

DA F −=+−=



Hempel, 2007 & 2017 Page 19 of 25

Reforming the expression for A’ results in a change of the transition temperature:

(39)

Searching for the minimum of F’ (eq. (38)) with respect to Q yield:

(40)

(41)

Now the elastic coefficients are calculated acc. to eq. (14).

(42)

For the Paraphase (T>TC) follows from eq. (42), using Q=0 (in eq. (37)) and

with reformed eq. (39) with regard to D:

(43)

)('2

ooS

oCCo Ac

DT T with )T-(TAA +==

C

C

TT for B

AQ

TT for Q

−=

=

'

0

2

2

QQ

oSQQ

oSS F

Dc

QS

FF

QS

Fcc

221

2

−=

−= −

)()( CoC

CoSS TT TT

TTcc

−+−

−=



Hempel, 2007 & 2017 Page 20 of 25

For the Ferrophase (T<TC) follows from eq. (42), using Q=-A’/B (in eq. (37)) and

with reformed eq. (39) with regard to D:

(44)

3. Improper (ferroelastic) phase transition (2nd kind)

It is well known that an improper transition is accompanied by at least two order

parameter components (ref. to /5/). For simplicity we confine our calculations

here to one order parameter component Q.

(45)

(46)

(47)

(48)

)(2)(

)(2

TT TT

TT cc

CoC

CoSS

−+−

−=

2DSBQAQ

F Stability

DSQBQAQQ

F F of Mininum

Qc

D-S DQSc

S

F crystal free

TT AA with SDQSc BQAQF

oS

oS

oo22o

S

++=

++=

=+==

−=+++=

2

2

2

3

22

42

3:

2:

0:""

)(2

1

4

1

2

1



Hempel, 2007 & 2017 Page 21 of 25

Next step is the is the calculation of the free energy F’ of the stress free crystal

by inserting of eq. (46) into eq. (45). After some calculation yields:

(49)

As it can be seen the transition temperature is not changed: To=TC

Searching for the minimum of F’ (eq. (49)) with respect to Q yield:

(50)

(51)

For the strain in the phases (ref. to eq. (46)) is valid:

(52)

)2

()2

(4

1

2

1'

24

22

oS

oS c

DB B' with Q

c

DB AQF −=−+=

C

oS

2

C

TT for

c

2D-B

A-

B

AQ

TT for Q

=−=

=

'

0

2

2

2Qc

DS

oS

−=



Hempel, 2007 & 2017 Page 22 of 25

Now the elastic coefficients are calculated acc. to eq. (14).

(53)

For the Paraphase (T>TC) follows from eq. (53), applying Q=0:

(54)

For the Ferrophase (T<TC) follows from eq. (53), using Q2 from eq. (51) and

with FQQ (see eq. (48)) where the expressions for Q2 and S (see eqs. (51), (52))

have been inserted:

(55)

QQ

oSQQ

oSS F

QDc

QS

FF

QS

Fcc

2221

2 4−=

−= −

oSS cc =

B

Dcc o

SS

22−=



Hempel, 2007 & 2017 Page 23 of 25

Proper Transition

Pseudo-proper Transition

Improper Transition



Hempel, 2007 & 2017 Page 24 of 25

7. Summary

• The formalism presented firstly by Slonczewski et al. to predict the temperature

dependences of isothermal elastic coefficients has been extended with respect to:

- Dielectric Impermeabilities

- Piezoelectric Modules

as well as with respect to:

- Influences of measuring conditions (constant Pj or Ej respectively

constant Tk (“free” crystal) or Sk (“clamped” crystal))

• The outlined formalism is fully applicable at phase transitions that are characterized

by multi-component order parameters.



Hempel, 2007 & 2017 Page 25 of 25

Literature

1. Rehwald, W., Adv. In Physics, Vol. 22, 1973, No.6, p. 721 ff.

„The study of structural phase transitions by means of ultrasonic experiments“

2. Slonczewski, J. C. and Thomas, H., Phys. Rev. B1, 3599 (1970),

„Interaction of elastic strain with the structural transition of SrTiO3“

3. Bulou, A., Rousseau, M., Nouet, J., Key Engineering Materials, Vol. 68, 133-186 (1992)

„Ferroelastic Phase Transitions and Related Phenomena“

4. Carpenter, M. A., Salje, E. K. H., et al. American Mineralogist, Vol 83, 2-22 (1998)

5. Wadhawan, V. K., Phase Transitions, Vol. 3, 3-103 (1982)

„Ferroelasticity and Related Properties of Crystals“

Phase Transition & Landau Theory of Rb4LiH3(SO4)4

Page 1 of 43Hempel, 2002 & 2016

Table of Contents

1. Observations made and models discussed so far

2. Crystal physical Basics

3. Landau Modelling

a. Proper Ferroelastic Transition

b. Pseudo-Proper Ferroelastic Transition

4. Comparison with experimental results and Summary

5. Literature

Annex: Investigation of an Improper Ferroelastic

Transition Model (4mmFmm2)


Page 2 of 43Hempel, 2002 & 2016

1. Observations made and models discussed

Rb4LiH3(SO4)4, abbreviated RLHS, was firstly reported to be ferroelastic by Wolejko

et al. /01,02/ and by Mroz et al. /03/.

They found that RLHS:

• Is of tetragonal symmetry by habitus (as grown), confirmed by X-ray investigation

• Exhibits a 2nd Order Phase Transition (PT) at 137 K

• Shows twinning in planes perpendicular to the c-axis. Two types of perpendicular

domain walls were observed along [010]- and [100]-directions

• Can be transferred to a mono-domain state by application of external stresses

along [010]- and [100]-directions, respectively

• Has a polar axis parallel to c-axis above and below the PT-point

• Shows a -type specific heat anomaly and almost no anomaly of the dielectric

permittivities at the PT-point

• It was concluded that RLHS should undergo a ferroelastic PT according to Aizu’s

species 4mmFmm2 /04/.


Page 3 of 43Hempel, 2002 & 2016

Other authors as of Hempel et al. /05/, Zúñiga et al. /06/ and later also Mroz et al.

/07/ discarded the possibility of Species 4mmFmm2 and proposed that the

symmetry change is correctly described by 4F2 (Space Groups (SGs): P41= C42,

P21=C22). The PT is of ferrodistortive (zellengleich, equitranslational) type which

means that the number of molecules per unit cell Z (Z=4) does not change through

the PT /08,09/.

This finding bases on following observations:

• X-ray and neutron diffraction investigations of RLHS

• The crystal exhibits a rotary power (optical activity) of -0,28°/mm along its c-axis

in the paraelastic phase (at T=293 K), which is generally not allowed for a

paraphase belonging to Point Group (PG) 4mm.

• Ferroelastic domain walls (“W’-walls” according to Sapriel’s classification /10/)

are present in planes perpendicular to c-axis. They are mutually perpendicular

and are rotated around c-axis by 35° (which contradicts to the observations

made by Wolejko et al. /01/ because in case 4mmFmm2 the domain walls would

be fix “W-walls”, mutually perpendicular and along either [100]- and [010]- or

[1-10]- and [110]- directions).


Page 4 of 43Hempel, 2002 & 2016

Symmetry elements of PG 4:

1, 413, 4

23, 4

33

Order of PG 4mm = 4

(symmetry generators are marked red)

Coordinate System (x(1), y(2), z(3)):

(a, c are the lattice parameters, tetragonal a=b)x

y

z

As result of the PT 4 → 2 the fourfold symmetry along z is lost, but c-axis remains

the polar axis. Both phases are piezoelectric.

Since the PT is of ferrodistortive type, the Landau modelling can be done just within

the framework of the Point Groups – here PG 4 and PG 2.

All physical quantities and coefficients are related to the coordinate system x, y, z.

2. Crystal physical Basics

z

y

x

ecc

eab

eaa

=

=

=


Page 5 of 43Hempel, 2002 & 2016

Since the ferroleastic phase exhibits symmetry of PG 2 (with two symmetry

elements:1, 423) the number of possible ferroelastic domains („states“) calculates as:

Species 4F2 leads to the spontaneous deformation tensors for the two states /04/:

According to Sapriel /10/ two mutually perpendicular domain walls are possible:

Those walls are denominated W‘-walls, because their orientation depends on the

spontaneous strain components and they again on the temperature.

22

4

2===

PG of order

4 PG of ordern

122 Sb and SS 2

1a with ab

ba

S ab

ba

S =−=

−

−−

=

−= )(

000

0

0

000

0

0

11221

( )a

bab pwith -y/p xypx

2 2/12++===


Page 6 of 43Hempel, 2002 & 2016

3.a Landau Modelling - Proper Ferroelastic Transition

First step to compile the Free Energy Expansion (FEE) for RLHS is to look up the

Irreducible Representations (IDs) of PG 4. According to Kovalev /11/:

ID 1 413 42

3 433

T1 1 1 1 1

T2 1 i -1 -i

T3 1 -1 1 -1

T4 1 -i -1 i

IDs T2 and T4 are complex. To get

physically meaningful IDs they

have to be transformed into real

form /08/. Combination (here

addition) of complex T2 with its

conjugate complex T4 leads to:

ID 1 413 42

3 433

A=Γ1 1 1 1 1

B=Γ3 1 -1 1 -1

E=E1+E2

(=Γ2+Γ4)

2 0 -2 0

ID A is the identical representation

and not of interest here.

Next step is to check which ID can

induce PG 2 symmetry.


Page 7 of 43Hempel, 2002 & 2016

As it can be easily shown (e.g. Kocinski /12/) that in case of equitranslational phase

transitions only those IDs will lead to a certain low symmetry phase (subgroup of PG

4) where the characters ( ) of the symmetry elements (g) of the subgroup (here

PG 2) are either:

=1 for one-dimensional IDs

or

= 0, 2 for two-dimensional IDs

→ Only ID B is able to induce the PG 2 Symmetry, consisting of the symmetry

elements 1, 423.

Next step is the inspection of the reduction coefficients to derive information

regarding permitted appearance of ferroelasticity and ferroelectricity. Acc. to /12/,

the calculation of

yields mE=2 regarding ferroelasticity (i.e. allowed!) and mE=0 regarding ferroelec-

tricity (i.e. not allowed!). Of course this is just a formal group-theoretical justification

of the obvious behaviour. Meaning, that ferroelasticity is always allowed if the crystal

system of the paraphase changes as a result of a PT.

)( g

)( g

)()(4

1g g m B

gPE =


Page 8 of 43Hempel, 2002 & 2016

Whether ID B is an active representation can be decided after consideration of the

Landau- and Lifshitz-Conditions.

It turns out that the Lifshitz-Condition is always fulfilled if the PT is of ferrodistortive

type (characterized by a PT-mode condensing at the Γ–Point of the Brillouin zone

and accompanied by a wave-vector ).

The Landau-Condition is always fulfilled for ferrodistortive PTs if the ID under

consideration is one-dimensional and real /12/.

Alternatively the following calculation should be done:

In the present case the result is zero, which confirms that the Landau-Condition isn’t

violated.

→ The ID B is an active representation and can induce the PT 4→2.

0=k

with g G

oGgB

o

0)(1 ?3

=

( )3233 )(6

1)()(

2

1)(

3

1)( g g g g g BBBBB ++=


Page 9 of 43Hempel, 2002 & 2016

Next step to get Free Energy Expansion (FEE) is to decompose the (ordinary)

strain tensor and (ordinary) polarisation vector into their irreducible parts regarding

PG 4.

Utilisation of the projection operator (see e.g. /12/)

with = P resp. S and

= operator related to

symmetry element gk

yield the following results:

Strain S: ½(S11+S22), S33, ½(S11-S22), S23, S13, S12 respectively

(in Voigt’s Notation /13/ and

normalized*)

*) Note: ½(S11+S22)=S1/2+S2/2 → |S1/2+S2/2|=(0,5²+0,5²)1/2=1/(2)1/2→ ½(S1+S2)/|S1/2+S2/2|=(S1+S2)/(2)1/2

=

3

2

1

P

P

P

P

=

332313

232212

131211

SSS

SSS

SSS

S

)()()()( xfgTgG

dxf

Gg

kijkij

k

=)(xf

)( kgT

65421

321 ,,,

2

)(,,

2

)(SSS

SSS

SS −+


Page 10 of 43Hempel, 2002 & 2016

Polarisation P: P1, P2, P3

When decomposing the strain tensor into its symmetry adjusted irreducible parts it

turns out that ID B induces two parts of the irreducible strain tensor, namely

In case of a proper ferroelastic transition one part plays the role to the primary OP

and couples linearly to the other secondary OP /14/. It is assumed that in both

possible cases the coupling between the two OP-parts is weak (i.e. the coefficient I

in eq. (2) below is small). Up to now it’s not clear which one represents the primary,

respectively secondary OP.

Each term of the FEE has to be invariant under the action of all the symmetry

elements of PG 4, but it is sufficient to prove this invariance under the action of the

“generators” of PG 4 only. These are here just the symmetry operation 413.

To this end the transformation properties of Si, Pk, and Q (Q to be Order Parameter

within a pseudo-proper model which will be discussed later) have been compiled:

621 ,

2

)(S

SS −


Page 11 of 43Hempel, 2002 & 2016

(1)

IDIrreducible

Part

After action

of

413

A

B

E

6

21 )(2

1

S

SS −

IDIrreducible

Part

After action

of

413

A

B , Q , -Q

E

3

0

0

P

5

4

S

S

3

21 )(2

1

S

SS +

3

21 )(2

1

S

SS +

6

21 )(2

1

S-

SS −−

4

5

S

S−

3

0

0

P

02

1

P

P

0

0

0

0

0

0

−

01

2

P

P

ID A induces the identical representation of PG 4 whereby ID B (one component, two

parts) induces PG 2 (incl. OP Q), and ID E induces PG 1 (two components).


Page 12 of 43Hempel, 2002 & 2016

The Free Energy Expansion (FEE) – reflecting the dynamics of the crystal - for a

proper ferroelastic transition can now be written:

F = FS + FP + FS-P

FS = A(S1-S2)2 + B(S1-S2)

4 + C(S1+S2)2 + DS3

2 + E(S42 +S5

2) +

+ GS62 + HS6

4 + I(S1-S2)S6 + J(S1+S2)S3 + K(S1-S2)2(S1+S2) +

+ L(S1-S2)2S3 + MS6

2(S1+S2) + NS62S3 + n(S1-S2)

2S32 (2)

FP = O(P12+P2

2) + PP32

FS-P = RS3P3 + T(S1+S2)P3 + U(P1S4+P2S5) + V(P1S5-P2S4) +

+ W(S1-S2)2(P1

2+P22) + XS6

2(P12+P2

2) + Y(S1-S2)2P3 + ZS6

2P3

We can distinguish between 2 cases where an acoustic mode is softening:

1.The driving/primary OP is the strain difference (S1-S2) and therefore the only

temperature–dependent parameter of eq. (1) is A = Ao(T-To). (3)

2.The driving/primary OP is the shear strain S6 and therefore the only

temperature–dependent parameter of eq. (1) is G = Go(T-To). (4)

Note: (S1-S2) and S6 cannot appear as primary OPs simultaneously, since they do not exhibit a common

coefficient in FEE eq. (2) (A ≠ G, see overleaf).


Page 13 of 43Hempel, 2002 & 2016

The following material coefficients are used in FEE:

(5)

A straightforward method to elaborate the respective irreducible parts of the material

coefficients can be referred to in /15/. Third order elastic constants are listed at

Brugger /16/ and 4th order constants at Chung /17/.

It has to be noted that the coefficients in FEE have to satisfy following conditions:

1. coij - bare elastic stiffnesses measured at constant polarization (and at

constant OP Q as discussed later in the pseudo-proper model)

( )

hV hU hT hR P

cc 24

1n O c

6

1N cM

oooo

o

o

oo

o

ooo

15143133

33

33

12331133

11

11366166

2

1

2

1

2

1

2

1

6

1

======

−=====

)()(

24

1

2

1

2

1

2

1

)(4

1)34(

192

1)(

4

1

1231131121111316

6666664433

012111122111211111211

oooooo

oooo

oooooo

cc 12

1L cc

24

1K cJ cI

cH cG cE cD

cc C ccc B cc A

−=−===

====

+=+−=−=


Page 14 of 43Hempel, 2002 & 2016

1.ßoij - bare dielectric impermeabilities measured at constant strain (and at

constant OP Q as discussed later in the pseudo-proper model)

2.hokl – bare piezoelectric modules are those measured at constant OP Q (as

discussed later in the pseudo-proper model)

Considering now that no external electrical (Ek) or mechanical (Ti) fields are applied,

the solution of the equation system:

(6)

yields the State Parameters in dependence of the OP for the “free crystal”:

1st case: OP = (S1-S2)

(7)

Note: Ki are constants, not explicitly shown here.

0

0

==

==

kk

ii

EP

F

TS

F

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) 02

22

1

22

1

0

5216

2

212

212

2

212

211

2

213

2

21221

2

21121

==−−=

−+−−=−+−=

−=−=+

−===

SS SSG

IS

SSK

SSS SSK

SSS

SSKS SSKSS

SSKP PP

4

3

3


Page 15 of 43Hempel, 2002 & 2016

Inserting the state parameters into FEE (eq. (2)) the following expression is obtained,

which is suited to find the equilibrium states of the crystal:

F = A‘(S1-S2)2 + B’(S1-S2)

4 (8)

with renormalized coefficients and B’. (9)

As result the actual PT-temperature is changed from To to (10)

It is assumed that the coupling coefficient I= is small and

therefore yields: ToTc.

The equilibrium values of the OP are easily derived from eq. (8):

(11)

Solution 1: Paraphase (12)

Solution 2: Ferrophase (13)

G

IAA

4'

2

−=

GA

ITT

ooc

4

2

+=

( )( ) ( )32121

21

'4'20 SSBSSASS

F−+−==

−

( ) 021 =−SS

( )'2

'2

21 B

ASS −=−

oc16


Page 16 of 43Hempel, 2002 & 2016

The material coefficients are now calculated acc. to /18,19/:

(14)

Note: - Stiffnesses are those at constant electric field (E), Impermeabilities at constant mech. stress (T).

- circled expressions exhibit throughout this paper “piezoelectric coupling terms” that describe the

differences between cEmn and cP

mn respectively between ßTmn and ßS

mn.

vn

2

kvvk, km

2

nm

2Emn PS

FR

PS

F

SS

Fc

−

=

vn

kvvk kmnm

Tmn SP

FR

SP

F

PP

F

−

=

2

,

22

nm

mn SP

Fh

=

2

kvk jk

vj RPP

F with

=

2

kvk jk

vj RSS

F with

=

2


Page 17 of 43Hempel, 2002 & 2016

Throughout the complete paper following abbreviations are used to describe the

piezoelectric coupling terms (acc. to eq. (14)):

(15)

It needs to be remembered that the expressions:

(case 1), and

(case 2) are temperature dependent!

( ) ( )

( ) ( ) 2131211331211

2131211121133

22112

1

oooooo

oooooo

cccccc

ccccccRR

−+−

−−++

==

22112112 RRRR −=−==

( ) 213121133

121133

2 oooo

oo

cccc

ccR

−+

+=

( ) 213121133

1332233113

2 oooo

o

cccc

cRRRR

−+

−====

0366326621661 ====== RRRRRR

)(441211 oooo TT AAcc −==−

oo cR

cRR

66

66

44

5544

11===

)(2266 ooo TT GGc −==


Page 18 of 43Hempel, 2002 & 2016

( )( ) ( )

( ) ( )

( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )

( )

( )

( )

−−−=

−−=

−==−

−

+−+−++−−=

−

+−++−−−++−=

−

+−++−−+++−=

−+−=−+=

−=−=−=

−

+−++−=

−

=

−

−

−

21

66

1661626

21

66

1661616

1211

2

2111221112111141211121112

2

211122111211114211121111211121122

2

211122111211114211121111211121111

122211

1

12111211

1

2

21112211121111412112

21

21

31

31

)(222

1 with

)34(16

1

2

1

2

1

)34(16

1

6

1

2

1

2

1

)34(16

1

6

1

2

1

2

1

)(2)(424

1

)(22

1

2

1

)34(16

1

2

1

SSc

ccc

SSc

ccc

TTAAcc

SScccKccccc

SScccKSSccccccc

SScccKSSccccccc

TTATTAccc

TTAcccc

SScccKccSS

F

o

ooE

o

ooE

oo

oo

oooooooE

oooooooooE

oooooooooE

ococo

EEE

OP

oo

ooEE

OP

ooooo

OP

(16)

Paraphase (note: TcTo)

Ferrophase (note: TcTo)

Inverse OP-Susceptibility


Page 19 of 43Hempel, 2002 & 2016

( ) ( )

( )

( )

( ) ( )

SSc

cZSSKcc

hhcc

hhcc

SSc

chZ SS

c

ccc

oo

ooE

o

oooE

o

oooE

oo

oo

o

ooE

2

2126633

21622

2156666

11

215

214

4455

11

215

214

4444

21

6633

163321

66

1636636

4

2

3

−−−+=

+−=

+−=

−+−−=

( )( )

( )( ) ( )

( )( ) ( )21

33

33211231131332

21

33

33211231131331

33

2332

21123311333333

2

6

1

2

6

1

SSh

YSScccc

SSh

YSScccc

hSScc

12

1cc

o

ooooE

o

ooooE

o

ooooE

−+−−−=

−−−−+=

−−−+=

morphic coefficient!

(17)


Page 20 of 43Hempel, 2002 & 2016

( )

( )

( ) ( ) 23333333113

2

21366

216

22

212

112

31113333

44

215

2142

21266

216

1122

44

215

2142

21266

216

1111

24

12

2

2

ooo

o

oooT

o

oo

o

ooT

o

oo

o

ooT

hRhhRSSc

cZSSYRhR

c

hhSS

c

cXW

c

hhSS

c

cXW

−−−−−−−=

+−−

++=

+−−

++=

( )

( )

( )21

66

1636

1425

1524

1515

1414

3333

213132

213131

2

2

2

SSc

cZh

hh

hh

hh

hh

hh

SSYhh

SSYhh

o

o

o

o

o

o

o

o

o

−−=

=

−=

=

=

=

−−=

−+=


(18)


Page 21 of 43Hempel, 2002 & 2016

2nd case: OP = S6

(19)

Note: Ki are constants, not explicitly shown here.

Inserting the state parameters into FEE (eq. (2)) the following expression is obtained,

which is suited to find the equilibrium state of the crystal:

F = G’S62 + H’S6

4 (20)

with renormalized coefficients and H’. (21)


It is assumed that the coupling coefficient I= is small and

therefore results: ToTc.

The equilibrium values of the OP is easily derived from eq. (20):

(23)

A

IGG

4'

2

−=

AG

ITT

ooc

4

2

+=

( )

( ) 02

4242

0

521

6

2

67

26

2

67

1

2

68

2

6721

2

6621

===−

+=−=

==+

===

SS S A

I- SS

SA

IS

KS S

A

IS

KS

SKS SKSS

SKP PP

46

3

3

3

66

6

'4'20 SHSGS

F+==

oc16


Page 22 of 43Hempel, 2002 & 2016



(26)

06 =S

'2

'26 H

GS −=

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) 26911221112111141211122211

26911221112111141211121112

26911221112111146

1211

161121111211121122

26911221112111146

1211

161121111211121111

'33

2

66

1

6666

1

26

33

22610662

6

21

)34(16

1

2

12

4

1

)34(16

1

2

1

2

1

)34(16

1

3

1

2

1

2

1

)34(16

1

3

1

2

1

2

1

)(22

1)(4

2

4

SK ccc Kccccc

SK ccc Kccccc

SK ccc KScc

cccccccc

SK ccc KScc

cccccccc

TTGH

ZTTGc

TTGcc

SZ

SKcS

F

oooooEEE

oooooooE

ooo

oo

oooooooE

ooo

oo

oooooooE

ocoocoE

OP

oooE

OP

o

oOP

+−++−=−+

+−+−++−−=

+−++

−

−+++−=

+−++

−

−−++−=

−+

−−==

−===

−+=

=

−

−

−

Paraphase (note: TcTo)

Ferrophase (note: TcTo)

Inverse OP-Susceptibility


Page 23 of 43Hempel, 2002 & 2016

( )( )

( )( )

( )( )

( )

( )o

oooE

o

oooE

o

ooE

o

o

oo

ooooE

ooo

oo

oo

ooooE

ooo

oo

oo

ooooE

ooE

ooE

hhcc

hhcc

Sh

ZScc

hS

cc

ccccc

Scc

chYS

cc

ccccc

Scc

chYS

cc

ccccc

Sc

cc

Sc

cc

11

2

15

2

144455

11

2

15

2

144444

6

33

33636636

33

2

332

62

1211

2

16123311333333

6

121133

16336

1211

161231131332

6

121133

16336

1211

161231131331

6166

1626

6166

1616

2

3

1

12

1

4

3

1

4

3

1

3

3

+−=

+−=

−=

−−

−+=

−−

−

−+=

−+

−

−−=

+−=

+=


(27)


Page 24 of 43Hempel, 2002 & 2016

( )

( )

( )2

333333311326

66

2262

1211

2162

112

31113333

44

215

2142

62

1211

216

1122

44

215

2142

62

1211

216

1111

24

48

42

42

ooo

ooo

oooT

o

oo

oo

ooT

o

oo

oo

ooT

hRhhRSc

ZS

cc

cYRhR

c

hhS

cc

cWX

c

hhS

cc

cWX

−−−−

−−=

+−

−++=

+−

−++=

morphic coefficient!6363333

142515246

1211

163132

141415156

1211

163131

2

4

4

ZSh hh

hh hh Scc

cYhh

hh hh Scc

cYhh

o

oo

oo

oo

oo

oo

oo

==

=−=−

+=

==−

−=

(28)


Page 25 of 43Hempel, 2002 & 2016

Now the Free Energy Expansion (FEE) for a pseudo-proper ferroelastic transition

(case 3) can be compiled (similar to Quirion /22/ and David /23/):

F = FOP + FS + FP + FS-P + FS-Q + FP-Q

FOP = A1Q2 + B1Q

4

FS = A(S1-S2)2 + B(S1-S2)

4 + C(S1+S2)2 + DS3

2 + E(S42+S5

2) +

+ GS62 + HS6

4 + I(S1-S2)S6 + J(S1+S2)S3 + K(S1-S2)2(S1+S2) +

+ L(S1-S2)2S3 + MS6

2(S1+S2) + NS62S3 + n(S1-S2)

2S32

FP = O(P12+P2

2) + PP32 (29)

FS-P = RS3P3 + T(S1+S2)P3 + U(P1S4+P2S5) + V(P1S5-P2S4) +

+ W(S1-S2)2(P1

2+P22) + XS6

2(P12+P2

2) + Y(S1-S2)2P3 + ZS6

2P3

FS-Q = r(S1-S2)Q + sS6Q + tS3Q2 + u(S1+S2)Q

2 + vS4S5Q + w(S42+S5

2)Q2

FP-Q = x(P12+P2

2)Q2 + yP32Q2

The driving OP is now Q (e.g. related to a soft optical mode) and couples linearly

with both (S1-S2) and S6 (acoustic modes). The only temperature–dependent

parameter in eq. (29) is A1 = A1o(T-To). (30)

.


Page 26 of 43Hempel, 2002 & 2016

Solution of the equation system (ref. to eq. (6)) yield the state parameters of the

“free crystal” in dependence of the OP Q.

(31)

Inserting the state parameters into FEE (eq. (29)) the following expression is for the

free crystal is obtained:

F = A1‘Q2 + B1‘Q

4 (32)

with renormalized coefficients

and B1’ (not explicitly shown here). (33)


( )

( )

QK

QK S Q

KQ

K S

QKQ IAG

sArI S QKQ

IAG

rGsI SS

S S QKS QKSS

QKP PP

19618

43

3

2222

4

2

4

2

00

0

182162

182161

2221

52

172

1621

21521

−=+=

=−

−==

−

−=−

====+

===

( )2

22

114

'IAG

AsGrrsI A with AAA

−

+−=+=

ooc A

ATT

1

−=


Page 27 of 43Hempel, 2002 & 2016

• The equilibrium values of the OP is easily derived from eq. (32):

(35)



• To calculate the temperature dependences of the material coefficients (ref.

to eq. (40, overleaf) also the 2nd derivatives FQQ (= inverse order parameter

susceptibilities) have to be used. They can be derived from FEE (eq. (29)),

whereby the equilibrium OP and state parameters (ref. to egs. (31, 36, 37))

have to be used:

FQQ = 2A1 + 12B1Q2 + 2tS3 + 2u(S1+S2) + 2yP3

2

FQQ = 2A1 + (12B1 + 2tK17 + 2uK16 + 2yK15)Q2 (38)

• This yields finally:

FQQ = 2A1o(T-To) for the Paraphase, and

FQQ = 4A1o(Tc-T) + 2A1o(Tc-To) for the Ferrophase (39)

(see also /20,21/)

311 '4'20 QBQA

Q

F+==

0=Q

'2

'

1

12

B

AQ −=


Page 28 of 43Hempel, 2002 & 2016

• The material coefficients which will be derived are those valid for isothermal

conditions.

• The stiffnesses are those at no applied electric field (Ei) and freely moving OP.

• The dielectric impermeabilities are at no applied mechanical stress (Tij) and

freely moving OP.

• The piezoelectric coefficients are those at freely moving OP.

vn

2

kvvk, km

2

n

2

m

2

nm

2

mn PS

FR

PS

F

QS

F

QQ

F

QS

F

SS

Fc

−

−

=

−12

kvk jk

vj RPP

F with

=

2

vn

kvvk kmnmnm

mn SP

FR

SP

F

QP

F

QQ

F

QP

F

PP

F

−

−

=

− 2

,

221222

kvk jk

vj RSS

Fwith

=

2

(40)

QS

F

QQ

F

QP

F

SP

Fh

nmnm

mn

−

=

− 21222


Page 29 of 43Hempel, 2002 & 2016

• Since (S1-S2) and S6 couples linearly with OP Q, as well as inter alia, it doesn’t

become easily obvious which combination of elastic stiffnesses has to go to

zero when approaching Tc.

• According to Boccara /24/ and others /20/ the eigenvalues of the stiffness matrix

of the PG 4, related to the eigenvector, which describes the transition to the

ferrophase with PG-symmetry 2, must become zero at Tc.

• First step is to look at the transformation matrix Dkl (k,l =1…6) which generates

the symmetry adapted strain vector (decomposition into irreducible parts) from

the ordinary strain, expressed as a 6 component vector. As shown on page 9

this is:

−

+

=

−

6

5

4

21

3

21

6

5

4

3

2

1

2

2

100000

010000

001000

00002

1

2

1000100

00002

1

2

1

S

S

S

SSS

SS

S

S

S

S

S

S


Page 30 of 43Hempel, 2002 & 2016

• Application of transformation matric Dkl onto the matrix of stiffnesses of PG 4

leads to the stiffness matrix in form of symmetry adapted (irreducible) parts, like:

• As shown previously the specific eigenvector being in charge of the PT 4F2

reads like:

cc

c

c

ccc

cc

ccc

c

oo

o

o

ooo

oo

ooo

okl

−

+

=

6616

44

44

161211

3313

131211

00200

00000

00000

20000

00002

00002

~

T

k SSS

S

−= 6

21 002

00~


Page 31 of 43Hempel, 2002 & 2016

• The eigenvalue equation:

leads to the specific eigenvalue (see also /24/):

whereby the solution with ‘+’ has to be discarded because the eigenvalue needs

to become zero at Tc.

It has to be noted that the stiffnesses and (as discussed in cases

1 and 2 above) will individually not become zero at the phase transition point.

* Note: The expression derived by Quirion et al. /22/ appears to be incorrect.

02

00200

00000

00000

20000

00002

00002

6

21

6616

44

44

161211

3313

131211

S

0

0

SS0

0

cc

c

c

ccc

cc

ccc

oo

o

o

ooo

oo

ooo

=

−

−

−

−

−−

−

−+

( )*

16

2

661211661211

2

82

1

+−−+−=

ooooooo ccccccc

oo cc 1211 −oc 66


Page 32 of 43Hempel, 2002 & 2016

( ) ( )

( ) ( )

( )( )

( ) ( )

( ) ( )

( )

( )

( )o

o

QQ

oooE

QQo

ooE

QQo

ooE

QQ

ooEEE

QQ

ooEEE

QQ

oE

QQ

oooE

QQ

oooE

h

F

QtQKcc

12

1cc

F

suQrQK

c

ccc

F

suQrQK

c

ccc

F

rQKcc

2

1ccc

4

1

F

r cc

2

1ccc

4

1

F

uQruQrQKcc

F

uQrQKQKcccc

F

uQrQKQKcccc

33

233

2222

18123311333333

18

66

1661626

18

66

1661616

22

201211122211

2

1211122211

2201212

22

20181121111122

22

20181121111111

4

2

31

2

31

2

2

22

2

6

1

2

6

1

−−−+=

+−−

−−=

+−

−=

−+−=−+

−−=−+

+−+−−=

+−−+−−=

+−+−+=

(41)

Paraphase

Ferrophase


Page 33 of 43Hempel, 2002 & 2016

( ) ( )

( ) ( )

( )

( )

QQo

oE

E

o

oooE

o

oooE

QQoo

oo

o

ooE

QQo

ooooE

QQo

ooooE

F

sQ

KZKcc

vQc

hhwQcc

hhwQcc

F

2tsQ- QK

c

chZ

c

ccc

F

tQuQrQK

hYcccc

F

tQuQrQK

hYcccc

22

33

2192

216666

45

11

215

2142

4455

11

215

2142

4444

18

6633

1633

66

1636636

18

33

331231131332

18

33

331231131331

4

2

2

2

3

222

6

1

222

6

1

−

−+=

=

+−+=

+−+=

+−=

+−−

−−−=

+−

−−+=



(42)


Page 34 of 43Hempel, 2002 & 2016

( )

( )

QQ

oooo

QQ

o

QQ

o

QQ

o

oooo

o

oT

o

oo

o

ooT

o

oo

o

ooT

F

QysKQZKh

hh hh hh hh

F

QytKhh

F

uQrQyKQYKhh

F

uQrQyKQYKhh

hRhhRhRQKc

ZQKYRyQ

c

hhQK

c

cXWxQ

c

hhQK

c

cXWxQ

315

1936

1425152415151414

415

3333

4315

183132

4315

183131

23333333113

23111

2219

66

222

182

112

3333

44

215

21422

18266

2162

1122

44

215

21422

18266

2162

1111

42

8

242

242

24

122

22

22

−=

=−===

−=

+−−−=

+−+=

−−−−−+=

+−

+++=

+−

+++=


(43)


Page 35 of 43Hempel, 2002 & 2016

S3, S1+S2, P3

Case 1: OP = S1-S2 Case 2: OP = S6

Case 3: OP = Q (pseudo-ferroelastic)

(Coefficients r, s, t, u, y of eq. (29 ) are

assumed to be positive. Arrows mean jumps.)

Qualitative summary of results:

S1, S2

1(2)

2(1)

11(22)

22(11)

c12

(S1-S2), S6

1-2(6)

6(1-2)

S3, S1+S2, P3

S1, S2

1(2)

2(1)

c12

(S1-S2), S6

1-2(6)

6(1-2)

S3, S1+S2, P3

c11, c22

S1, S2

1(2)

2(1)

c12

(S1-S2), S6

1-2(6)

6(1-2)

¼(c11+ c22 -2c12)

c33

c11, c22

TcTcTc

11(22)

22(11)1−

OP½(c11-c12)

¼(c11+ c22 -2c12)

¼(c11+ c22 -2c12)

¼(c11+ c22 -2c12)

½(c11-c12)

½(c11-c12)

¼(c11+

c22 -2c12)

c11

c22


Page 36 of 43Hempel, 2002 & 2016

c33, 11=22, 33

c16

c26

c44, c55

h33, h14, h15,

h24, h25,

Case 1: OP = S1-S2 Case 2: OP = S6 Case 3: OP = Q (pseudo-ferroelastic)

c31, c32

h31, h32

31(32)

c66

32(31)

c36, h36

c33, 11=22, 33

c16

c26

c44, c55

h33, h14, h15,

h24, h25,

c31, c32

h31, h32

31(32)

c66

32(31)

c36, h36

c16

c26

c44=c55,

11=22,

33, h33

c36, h36

c66

c45

c31, c32

h31, h32

h14, h15,

h24, h25

Tc

TcTc

( )1−

OP

31(32)

32(31)


Page 37 of 43Hempel, 2002 & 2016

4. Comparison with experimental results and summary

To compare the predictions with experimental results it has to be kept in mind that:

• In absence of external forces RLHS exhibits always a multidomain state in

the Ferrophase /01,02,03/, whereby the current predictions base on the

fact that the crystal is in a monodomain state of a well defined orientation

(ref. to slide 4). This means that the experimental results for the

Ferrophase need to be individually considered case by case, whether

applicable.

• In the Paraphase such particularities don’t appear, and therefore data

measured there, are assumed to be more useful to check theoretical

predictions.

• The predictions have been derived taking not into account the ordinary

pyroelectric- as well as the ordinary thermal expansion- effects.

• The measuring conditions can play an important role. E.g. the elastic

coefficients are usually measured at adiabatic conditions, whereby the

predictions are done for isothermal coefficients.


Page 38 of 43Hempel, 2002 & 2016

The measured state parameters (S1, S2, S3, P3) (ref. to /03,05,07/) fit nicely with the

predictions, but do not differ for the 3 cases discussed.

The elastic coefficient c66 (ciiρvii2) is in fair agreement with case 3, as well as

c44=c55. c33 shows a small jump downward at Tc which can also only be explained

within case 3 (see also overleaf).

ref. to Quirion /22/ref. to Hempel /05/


Page 39 of 43Hempel, 2002 & 2016

ref. to Hempel /05/

The elastic coefficient c11=ρv112 is nicely in line with the predictions of case 3 (see

also overleaf the data from Wolejko /02/ measured with pendulum devise).


Page 40 of 43Hempel, 2002 & 2016

ref. to Wolejko /02/ ref. to Hempel /05/

The coefficient-combination (c11-c12) follows qualitatively only the predictions derived

for case 3.

The dielectric permittivity 33 is in good agreement with all 3 cases.

33

( ) 11

1

11 cs −

( ) 33

1

33 cs −

( )1211 cc −

33


Page 41 of 43Hempel, 2002 & 2016

Summarizing it can be stated that – as proposed by Quirion et al. /22/ - the pseudo-

proper ferroelastic Landau model indeed describes best the so far obtained

experimental results.

Major arguments are:

• The (small) jump of c33 at Tc which can only be explained with this model.

• No linear but hyperbolic temperature-dependences have been experimentally

observed so far for the elastic coefficients c11, c12, c66 in the Paraphase which

contradicts the proper ferroelastic models

• The measured temperature dependence of c11 matches well with the prediction

of the pseudo-proper model.

To prove the model and to get more clarity on the correct Free Energy Expansion,

especially the temperature dependences of - incl. c11, c12, (c11-c12), c31, c66, c16, c36

as well as of selected piezoelectric coefficients should be investigated in more

detail.


Page 42 of 43Hempel, 2002 & 2016

/01/ Wolejko, T., et al., Ferroelectrics, 81, 175(1988)


/03/ Mroz, B., et al., J. Phys. Cond. Matter, 1, 4425(1989)

/04/ Aizu, K., J. Phys. Soc. of Japan, 28, 706(1970)

/05/ Hempel, H., et al., phys. stat. sol. a(110), 459(1988)

/06/ Zúñiga, F. J., et al., Acta Cryst., C46, 1199(1990)

/07/ Mroz, B., et al., Phys. Rev. B55, 11174(1997)

/08/ Janovec, V., et al., Czech J. Phys., B25, 1362(1975)

/09/ Boyle, L. L., et al., Acta Cryst., A28, 485(1972)

/10/ Sapriel, J., Phys. Rev. B12, 5128 (1975

/11/ Kovalev, O. V., “Representation of Crystallographic Space Groups”, Taylor &

Francis Ltd., 1993




Page 43 of 43Hempel, 2002 & 2016

/13/ Sirotin, Yu. I., Shaskolskaya, M. P., “Fundamentals of Crystal Physics”,

Mir Publishers, Moscow, 1982

/14/ Toledano, P., et al., Phys. Rev. B27, 5717(1983)

/15/ Hempel, H., PhD Dissertation, 1886

/16/ Brugger, K., J. of Appl. Phys., 36, 759(1965)

/17/ Chung, David, Y., Acta Cryst. A30, 1(1974)

/18/ Slonczewski, J. C., Thomas, H., Phys. Rev. B1, 3599(1970)

/19/ Hempel, H., internal paper

/20/ Bulou, A., et al., Key Engineering Materials 68, 133(1992)

/21/ Carpenter, M., A., American Mineralogist 83, 2(1998)

/22/ Quirion, G., J. Phys. Cond. Matter, 21, 455901(2009)

/23/ David, W. I. F., J. Phys. C: Solid State Phys. 16, 5093(1983)

/24/ Boccara, N., Annals of Physics, 47, 40(1968)


Hempel, 2002 & 2017 ANNEX Page 1 of 22

Annex

Investigation of an Improper Ferroelastic Transition Model

(Aizu Species: 4mmFmm2)

Preface:

• Initially a symmetry change according to Aizu Species 4mmFmm2 (/01/, /02/,

/03/) was assumed

• This was later discarded (/05/, /06/, /07/), but in current paper a symmetry

change 4mm→mm2 is exemplarily investigated, based on an improper model

to demonstrate the differences between the proper / pseudo-proper and an

improper transition behaviour.



Primitive 4mm Symmetry of the Paraelastic Phase

• 8 corresponding space groups do exist: Ci4v with i=1...8 (e.g. ref. to /08, /09/)

• Reciprocal lattice is calculated according to :aj x ak

bi = 2 i, j, k = 1, 2, 3ai (aj x ak)

• At tetragonal lattices there is bi || ai (i=1...3) reciprocal lattice is

primitive, too

• Now the BRILLOUIN Zone can be constructed

• Change of translational symmetry is described by certain lattive vectors (k-

vectors), that need to be in ‚symmetric position‘, i.e.:

kj = -kj + K with K being an entire reciprocal lattice vector (Kocinsky /09/)

• The complete set of those vectors can be identified either by trying out or,

more conveniently, by inspecting of Kovalev‘s tables /08/:

½ b1, ½ b2, ½ b3, ½ (b1+b2), ½ (b2+b3), ½ (b1+b3), ½ (b1+b2+b3), 0



• In order to describe the transition from 4mm to mm2, the k-vector with

inherent mm2-symmetry has to be selected via following procedure:

All 8 symmetry elements gk of 4mm (each represented as matrix

R(gk)) will be applied to all k-vectors in symmetric position. The set of

those symmetry elements that transform the selected vector into itself

or into an equivalent vector (acc. to the relation kj = kj +K) constitute

a point symmetry group. We are searching for mm2!

Example: ½ b1 = (/a, 0, 0)

-1 0 0 /a -/a

Symmetry 42 0 -1 0 0 = 0

element: 0 0 1 0 0

• Finally four k-vectors exhibit mm2-symmerty, namely:

½ b1, ½ b2, ½ (b2+b3), ½ (b1+b3)

• The fist two vectors belong to Brillouin-Zone point “X”, and the others to “R”

(/08/)



• In order to identify the interrelation between these vectors, the

corresponding ‘stars‘ have to be determined (e.g. ref. to Kocinski /09/)

• the star of an arbitrarily selected k-vector is obtained when all symmetry

elements (of the point group 4mm) act on it like k’ = Rk + K, e.g.:

0 -1 0 /a 0

1 0 0 0 = /a a.s.o.

0 0 1 0 0

= 41 = ½ b1 = ½ b2

• The corresponding star consists now of all nonequivalent k’-vectors resulting

from this operation; particularly we get:

Star (K1): ½ b1, ½ b2 (has got two arms)

Star (K4): ½ (b2+b3), ½ (b1+b3) (has got two arms, too)

Note: All other k-vectors in symmetrical position as mentioned above possess symmetry

4mm, and therefore cannot describe a transition to mm2. Consequently we have toexclude them from further considerations.



• Star K1 describes a doubling of the elementary cell in either x- or y-direction,

or in x- and y- direction simultaneously (causing a quadruplication of the

elementary cell in the ferroelastic phase)

• Star K4 behaves like K1 but with an additional doubling along z-direction

• Since nothing specific is known regarding the definitive symmetries of the para-

and ferrophase, we will confine our subsequent considerations to star K1

• Next step is the provision of the Full Irreducible Representations (FIRs) of the

space groups Ci4v with regard to star K1 (all FIRs of the Space Groups are

listed at Kovalev /08/). Their dimensions is two because the projective (small)

representations are all one-dimensional, but star K1 has two arms

• The structure of the related matrixes are identical for Ci4v with i=1,3,5,7 and

i=2,4,6,8 respectively

• Again for the reasons, that we don‘t know the exact space groups, we confine

the investigations to Ci4v with i=1,3,5,7

• In the following we will look at the FIRs (here provided for the generators, which

is sufficient)



Generators 41 m[100]

0 1 1 0

D1 1 0 0 1

0 1 -1 0

D2 1 0 0 -1 (1)

0 -1 1 0

D3 1 0 0 -1

0 -1 -1 0

D4 1 0 0 1

Notes: - The matrix structure is identical for all 4 FIRs for the space groups Ci4v with i=1,3,5,7 and

therefore the related Free Energy Expressions (FEE) are identical.

- D1-D4 are active representations, because they are real and the star K1 consists of two

arms and consequently the Landau condition is fulfilled. The Lifshitz is also obeyed since

the projective (small) representations are all one-dimensional (see e.g. /09/).

FIRs



Expression of the Free Energy Expression (FEE) with regard to Order

Parameter:

All Terms of the FEE) have to be invariant under the action of all

symmetry operations (incl. integral translations) of the high

temperature phase (para-phase)! We will consider here FIR D1 only.

• Since the above mentioned 4 FIRs are all 2-dimensional, we have to

consider an Order Parameter (OP) made up of 2 OP-Components

• Therefore we interpret the OP as a vector with 2 components p, q

p

OP =

q

1st Step - invariance under the symmetry generators (sufficient):

p p‘ q

(i) e.g D1(41) = =

q q‘ p



p p‘ p

(ii) D1(m[100]) = =

q q‘ q

2nd step - invariance under complete translations (translation operator T(tN)):

Vectors describing complete translations: tn= n1a1 + n2a2 + n3a3 with n1 n2 n3 –

integer and - basis vectors of the high

temperature unit cell (4mm)

(iii) T(tN) p = p‘ = e-ik1tn p = ± p (+ if n1=2, 4,…| - if n1=1, 3,…)

(iv) T(tN) q = q‘ = e-ik2tn q = ± q (+ if n2=2, 4,…| - if n2=1, 3,…)

k1, k2 are the arms of K1, whereby k1 relates to OP-component p and k2 to q. Since

the OP-terms have to be invariant under all possible complete translations tn –

this means for all values (and combinations) of n1 and n2.

This demand can only be satisfied with quadratic terms: p2, q2, p2q2, …

zyx eca eaa eaa === 321 ,,



3rd step – construction power terms of p and q (n-exponent, δi–constants):

!

(i) δ1 pn + δ2 qn = δ1 p‘n + δ2 q‘n → invariant if n even (see (iii), (iv)

above) and δ1=δ2 (see (i), (ii))

!

(ii) δpnqm = δp‘nq‘m → invariant if n,m even (see (iii) and (iv))

The Order Parameter Part (FOP) of the Free Energy Expression reads:

FOP = A(p2+q2) + B(p4+q4) + Cp2q2 + ... (2)

Remarks:

1. Expansion up to 4th power of p, q shall be sufficient because of

2nd order phase transition (experimentally evidenced)

2. According to Landau‘s theory is A=Ao(T-To) while B, C ... are

usually temperature independent coefficients

3. „Conditions of Landau and Lifshitz“ are obeyed (D1-D4 are

“active representations”) which allows a 2nd order and

commensurate phase transition



4th Step – Couplings of order parameter (OP) and macroscopic parameters:

• The order OP itself represents a microscopic parameter which is related to a

certain point of the BRILLOUIN Zone (at k0 in the present case – “X”-point)

• All macroscopic parameters are directly related to the -point‘s symmetry (which

reflects the symmetry of the high temperature point group under consideration –

here 4mm)

• The -Point of the BRILLOUIN Zone relates to k=0

• All Parameters related to k=0 are ‘per se’ and always invariant under complete

lattice translations of the high temperature symmetry

• All coupling terms of OP and macroscopic parameters must be simultaneously

invariant regarding the action of the symmetry generators as well as of complete

translations



Macroscopic Strain and Polarization (tensor of 2nd rank and ploar vector)

Application of projection operator ρTij (see /09/) on a function f(x):

with:

dT - Dimension of irreducible representation at -point

|G| - Order of group 4mm (here =8)

gk - Symmetry elements of 4mm

T*(gk)ij - complex conjugated matrix elements of irreducible representation

R(gk) - Symmetry operator connected with element gk

with f(x) = P (polarization vector) and Sij (strain tensor):

)()()()( x fg Rg G

dx f

Ggkijkij

k

=

=

3

2

1

P

P

P

P

=

332313

232212

131211

SSS

SSS

SSS

S



Result: 1. Irreducible parts of the polarization: P1, P2, P3

2. Irreducible parts of the strain tensor (in Voigt notation (/10/) and

normalized (see also /11/):

To finally construct the FEE in which all terms have to be invariant under the

action of the symmetry generators of the point Group 4mm (in particular the

elements: 41 and m[100]) and under complete translations related to star K1.

Therefore a transformation table for all parameters (OP, polarization, strain)

has been compiled.

65421

321 ,,,

2

)(,,

2

)(SSS

SSS

SS −+



Parameter After action

of 41

After action of

m[100]

After complete

translation

p q p ±p

q p q ±q

3

2

1

P

P

P

−

3

1

2

P

P

P

−

3

2

1

P

P

P

3

2

1

P

P

P

−

+

6

5

4

21

3

21

2

)(

2

)(

S

S

S

SSS

SS

−

−

−−

+

6

4

5

21

3

21

2

)(

2

)(

S

S

S

SSS

SS

−

−

−

+

6

5

4

21

3

21

2

)(

2

)(

S

S

S

SSS

SS

−

+

6

5

4

21

3

21

2

)(

2

)(

S

S

S

SSS

SS



Now the FEE can be finally compiled:

FEE = FOP + FP + FS + FOP-P + FOP-S with

FOP = A(p2+q2) + B(p4+q4) + Cp2q2 + (3)

FP = D(P12+P2

2) + EP32 +

FS = F(S1+S2)2 + GS3

2 + H(S42+S5

2) + IS62 + J(S1-S2)

2 +

FOP-P = K(P12+P2

2)(p2+q2) + L(P12+P2

2)p2q2 + MP32(p2+q2) + NP3

2p2q2 +

O(P12-P2

2)(p2-q2) +

FOP-S = P(S1+S2)(p2+q2) + Q(S1+S2)p

2q2 + RS3(p2+q2) + TS3p

2q2 +

US62(p2+q2) + VS6

2p2q2 + W(S42+S5

2)(p2+q2) + X(S42+S5

2)p2q2 +

Y(S42-S5

2)(p2-q2) + Z(S1-S2)(p2-q2) + a(S1-S2)

2(p2+q2) +

b(S1-S2)2p2q2 +

FS-P (piezoelectric coupling terms), FP-S-OP and other higher order energy terms have

been omitted.

The term characteristic for an improper ferroelastic transition is the bold-marked

expression in eq. (3). The elastic coefficient related to the spontaneous strain that

appears at the transition point is )(4

11211oo cc J −=



As shown in another article regarding improper transitions within this compilation,

essential features are provided here without calculation:

• Spontaneous strain (S1-S2) (p2-q2) (4)

• The transition temperature is unchanged To=TC

• All macroscopic state parameters and coefficients are independent of

temperature in the paraelastic phase (if ordinary thermal expansion and

pyroelectric effect is excluded from the model)

• The elastic coefficient related to the spontaneous strain looks like:

Tc=To

oo cc 1211 −

Tc=To

21 SS −



As discussed in another article on the Landau Theory, the Order Parameter

part of FEE FOP has got different solutions, namely:

Phase 0: (p=0, q=0) Paraelastic Phase

Phase 1: (p≠0, q=0) or (p=0, q≠0) Phase 1, Ferroelastic Phase

Phase 2: (p=q, q) Phase 2

Phase 3: (p=-q, q) Phase 3

Inspecting in eq. (3) the coupling term Z(S1-S2)(p2-q2), it turns immediately out,

that no spontaneous strain (ref. to eq. (4)) can appear in the Ferro Phases 2 and 3.

Only Ferro Phase 1 can induce such strain, whereby the two options reflect two

possible domains.

The solution (p≠0, q=0) characterizes the ferroelastic domain doubled along y-

direction, and (p=0, q≠0) the other one doubled along x-direction. Along z-direction

no doubling is taken into consideration within this model (star K1).

Be reminded, that the number of macroscopically distinguishable domains calculates

as the ratio of the orders of the point group of the paraelastic phase (4mm, order 8)

and of the ferroelastic phase (mm2, order 4) which yields 2.



From geometrical considerations it becomes obvious (and can be easily proven by

application of the formalism to calculate the orientation of domain walls in

ferroelastics (see e.g. /12/)), that the walls are fixed ones, crossing x- and y-

directions under 45° and run parallel to the z-direction. Pairs of walls stand

perpendicular to each other.

As discussed by several authors (see e.g. /14/), in case of improper transitions

(which are always accompanied by an increase of the primitive unit cell in the ferroic

phase compared to the para phase), the number of different ferroic “terrains”

calculates as product of the number of macroscopic domains (here 2) and the factor

which represents the enlargement of the ferroic unit cell (here 2). Consequently there

are 2*2=4 terrains in general which differ in the (microscopic) order parameter. Those

are:

(-p, q=0), (+p, q=0), (p=0, -q), (p=0, +q)

Each ferroleastic domain is composed of 2 terrains which are separated from each

other by so called anti-phase boundaries.



On the other hand, the solutions (p=q, q) and (p=-q, q), i.e. the simultaneous

appearance of p and q at the transition point, does not induce any orthorhombic

distortion (see eq. (4)), thus the tetragonality is preserved and no ferroelasticity can

be observed (i.e. just 1 macroscopic domain exists). Nevertheless, the volume of the

primitive unit cell is doubled along x- and y-directions, meaning an increase of the

unit cell by a factor of 4.

The number of terrains is therefore 4, which differ in the order parameter as follows:

(p=q, q=+|q|), (p=q, q=-|q|), (p=-q, q=+|q|), (p=-q, q=-|q|)

In case the star K4 is considered, the above made considerations are fully applicable

with the only difference that there is an additional doubling along the z-direction of

the unit cells in the ferro phase. The behavior with regard to the phase transition in

general, the order parameter(s), the state parameters and the physical coefficients

does not show differences.



z

x

y

paraphase,

p=q=0

⚫⚫

⚫⚫

ferroelastic phase

domain 2, p=0, q≠0

Doubling along y-axis

⚫ ⚫ ⚫

⚫ ⚫⚫

ferro-

elastic

phase

domain 1,

p≠0,

q=0

Doubling along x-axis

a

b

⚫ ⚫

⚫

⚫

⚫

⚫

Lattice

parameters:

a=2 atetra (1+S1)

b=atetra (1+S2)

c=ctetra (1+S3)

90°

domain walls

Lattice

parameters:

a=atetra

b=atetra

c=ctetra



z

x

y

paraphase,

p=q=0

⚫⚫

⚫⚫

(p=q, q)

and

(p=-q, q)

Doubling along y-axis

a

b

⚫

⚫

⚫

⚫

⚫

Lattice parameters:

a=2 atetra (1+S1)

b= a=2 atetra (1+S1)

c=ctetra (1+S3)

⚫

⚫

⚫

⚫

Dou

blin

g a

lon

g x

-axis

Lattice parameters:

a=atetra

b=atetra

c=ctetra





/03/ Mroz, B., et al., J. Phys. Cond. Matter, 1, 4425(1989)

/04/ Aizu, K., J. Phys. Soc. of Japan, 28, 706(1970)

/05/ Hempel, H., et al., phys. stat. sol. a(110), 459(1988)

/06/ Zúñiga, F. J., et al., Acta Cryst., C46, 1199(1990)

/07/ Mroz, B., et al., Phys. Rev. B55, 11174(1997)

/08/ Kovalev, O. V., “Representation of Crystallographic Space Groups”, Taylor &

Francis Ltd., 1993







/11/ Janovec, V., et al., Czech J. Phys., B25, 1362(1975)

/12/ Sapriel, J., Phys. Rev. B12, 5128 (1975

/13/ Bulou, A., et al., Key Engineering Materials 68, 133(1992)

/14/ Wadhawan, V. K., Phase Transitions, 3, 3-103(1982)

Phase Transition & Landau Theory of (NH4)2CuCl4 • 2H20

Page 1 of 42Hempel, 2007 & 2016


Table of Contents

0. Abstract

1. About Non-Primary Ferroics

2. About Higher Order Ferroic Phase Transitions

3. Observability of Effects

4. Case (NH4)2CuCl4•2H2O – Ferroelastoelectrics

4.1. Symmetry Change at 200 K

4.2. Landau Modelling

4.3. Calculation of Temperature Dependence of State

Parameters and Material Coefficients

5. Comparison with Experimental Findings

6. Summary

7. Literature


Page 2 of 42Hempel, 2007 & 2016

0. Abstract

(NH4)2CuCl4•2H2O (shortly ACC) seems to be a candidate for a ferroelastoelectric

phase transition according to Aizu’s Species .

Basing on symmetry considerations and the comprehensive usage of the Landau

Theory, the temperature dependences of State Parameters as well as of essential

Material Coefficients have been derived. All experimental data measured so far by

others are in good agreement with these results.

Moreover, is was theoretically investigated, how the crystal should behave under the

action of “external forces” (like electrical fields, mechanical stresses) imposed to the

crystal.

It is shown that the character of the phase transition should change from

ferroelastoelectric to ferroelastic, respectively to ferroelectric, or even to another

type of ferroelastoelectric behaviour – accompanied by all typical properties and

temperature dependences. Four respective scenarios have been discussed in detail.

mmmmF 24/4


Page 3 of 42Hempel, 2007 & 2016

1. About Non-Primary Ferroics

• Non-Primary (or higher order) Ferroics are crystals that exhibit a certain

domain structure – usually as result of a ferroic Phase Transition from a

Prototype- or Paraphase (which are identical denominations)

• Order Parameters (OP) have the symmetry of macroscopic tensors (rank

greater than 2), e.g.

➢ Elastic coefficients (rank 4) – Ferrobielasticity

➢ Piezoelectric coefficients (rank 3) – Ferroelastoelectricity

➢ a.s.o /1/, /2/

• At a non-primary Phase Transition the crystal usually lowers its symmetry,

the Point Group (PG) changes, but the Crystal System remains (if the hexagonal and rhombohedral system are treated as one system)

• The number of non-primary ferroic domains equals the quotient of the

order of the Point Group of the Paraphase divided by the order of the

Ferroic Phase


Page 4 of 42Hempel, 2007 & 2016

• Switching from one to another domain state should be possible in

principle if suitable external Forces are applied, e.g. at

• Ferroelastoelectrics - simultaneously mechanical stress and

an electric field

• Ferrobielastics - simultaneously mechanical stresses

2. About Higher Order Ferroic Phase Transitions /1/, /2/

• Proper Phase Transitions are those where the translational symmetry is

not changed (also called: equitranslational or ferrodistortive) transitions

• In general, proper transitions are related to the - Point ( =0) of the

Brillouin Zone of the Prototype Phase and the Order Parameter has the

symmetry of macroscopic tensor components

• Landau-Modelling is in principle feasible, provided there is a Group-

Subgroup relation between the two phases, the transition is of 2nd kind,

and the Landau- as well as the Lifshitz- Condition is satisfied. (But also

transitions of 1st kind and cases where the Landau-/Lifshitz – Conditions

are violated can be described by Landau’s theory satisfactorily.)

k


Page 5 of 42Hempel, 2007 & 2016

• To describe proper ferroic transitions, consideration of the respective

Point Group Symmetries is sufficient

• In contrast, improper transitions change the translational symmetry. They

are also called antiferrodistortive

• Improper transitions are related to other points of the Brillouin Zone than

the -Point, and the Order Parameter is necessarily of microscopic nature

and

• Landau-Modelling requires to consider the symmetry changes of the

respective Space Groups (SG)

3. Observability of Effects

• In case a Primary- and a Non-Primary Ferroic behaviour occurs

simultaneously in a crystal, the Primary Ferroicity dominates

• Higher Order Ferroic Effects are in principle difficult to detect

• The energy input needed for higher order ferroics to switch among the

domain states is generally much higher than for primary ferroics


Page 6 of 42Hempel, 2007 & 2016

• Therefore the investigation of switching effects of Non-Primary Ferroics

should be preferably undertaken in the vicinity of the Phase Transition

Temperature

4. Case (NH4)2CuCl4•2H2O – Ferroelastoelectrics

4.1. Symmetry Change at 200 K

• (NH4)2CuCl4•2H2O (abbreviated ACC) single crystals have been subject of

several experimental investigations /3/, /4/, /8/ as well as of theoretical

considerations /5/

• Firstly, Suga et al. /3/ discovered a phase transition at about 200 K, the

heat capacity shows a “-type” behaviour, thus indicating a transition of

2nd kind, which was also confirmed by Slaboszewska et al. /8/

• Bansal et al. /4/ carried out structural investigations and found a structural

change from P42/mnm ( ) to P421m ( ) at 200 K (at cooling). In

notation named after Aizu /9/, the species is called

14

4hD3

2dD


Page 7 of 42Hempel, 2007 & 2016

4/mmmF42m (within the framework of Point Groups, “F” stands here for

“Ferroelastoelectric”)

• According to Toledano /2/ and Kovalev /7/, 4/mmmF42m can only occur at

the -Point of 4/mmm, which represents the prototypic phase

• -Point means that the transition is equitranslational ( =0), and of proper

respectively of pseudoproper nature

• If a transition is equitranslational than it is sufficient to describe it within

the framework of Point Groups /2/

• The Prototypic Phase 4/mmm (Symmetry Go) consists of the following

symmetry elements (gk) :

• The Generators of 4/mmm are:

• The Order of 4/mmm is |Go| =16 (total number of symmetry elements gk)

• If looking just at the Point Groups, generally there are two options (refer to

cases A, B below) how the transition can occur:

31

]110[]110[]001[]010[]100[

1

]110[

1

]110[

1132

,

1 4,4,,,,,,1,2,2,2,2,44,4,1 zzyxzzz mmmmm −−

1,2,4 11

xz

k


Page 8 of 42Hempel, 2007 & 2016

Generators: Generators:

Order: |G1|=8 Order: |G1|=8

Case A:

42m ( ), Z=24...1

2dD

Case B:

42m ( ), Z=28...5

2dD

4/mmm ( ), Z=214

4hD

x

z y

]110[]110[

211131

,

),4(2,2,2,4,4,1

mm

zzyxzz

−

=

]010[]100[

211

]110[

1

]110[

31

,

),4(2,2,2,4,4,1

mm

zzzz =−

11 2,4 xz ]100[

1 ,4 mz

Note: Throughout this

entire paper one and

the same fixed Coor-

dinate System (x,y,z)

is used. It may differ

from usually applied

systems for the Point

Groups.


Page 9 of 42Hempel, 2007 & 2016

• As stated before, in case of ACC it will be sufficient to undertake the

theoretical investigations within the framework of the Point Groups

• Moreover it was found (Bansal et al. /4/) that the symmetry G1 of the

ferroelastoelectric phase shall be , which corresponds to “case A”

• Note: Within the framework of Point Groups, cases A and B are equivalent

– all following results can be transferred from one to the other case by a

45°-Rotation around the z-axis. We are going to deal with case A only

• The number of possible ferroelastoelectric domains equals

• These two domains differ (at the lowest rank) in the matrix of the

Piezoelectric Constants hijk as well as in the Gyration Tensor’s

Components gkl, which appear at the transition temperature (so called

“morphic” coefficients). Above the transition temperature all hijk and gkl are

zero (centrosymmetric prototypical phase)

3

2dD

28

16

42

/4==

mofOrder

mmmofOrder


Page 10 of 42Hempel, 2007 & 2016

• Piezoelectric Coefficients’ Tensor (Voigt’s Notation), e.g. Sirotin et al. /10/

Domain 1 Domain 2

• Gyration Coefficients’ Tensor (Voigt’s Notation), e.g. /10/

Domain 1 Domain 2

• The Symmetry elements that map domain 1 into domain 2 are all which

were lost at the transition, i.e.:

36

14

14

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

h

h

h

−

−

−

36

14

14

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

h

h

h

−

000

00

00

11

11

g

g

−

000

00

00

11

11

g

g

]001[]010[]100[1

]110[1

]110[31 ,,,1,2,2,4,4 mmmzz −

Note: both domains differ by sign in

their optical activity along x- resp. y-

axis. This property is also called

„Ferrogyrotropy“

Note: both domains differ by sign in

their piezolectric modules. Switching

between the domain states should be

possible via simultaneous action of

stresses and electr. fields like: S4/E1

or S5/E2 or S6/E3


Page 11 of 42Hempel, 2007 & 2016

4.2. Landau Modelling

As listed at Kovalev /7/ there are 10 Irreducible Representations (IDs) of

Point Group 4/mmm (at the -Point) - Table of Characters ( )

0002-2000-210=Eu

000-22000-29=Eg

-111-1-11-1-118=B2u

1-1-1111-1-117=B2g

11-1-1-1-1-1116=B1u

-1-1111-1-1115=B1g

1-11-1-1-11-114=A2u

-11-111-11-113=A2g

-1-1-1-1-111112=A1u

1111111111=A1g

24 z

12,

12 yx

34,

14 zz

1]110[2

,1

]110[2 −1 ]001[m

]010[

,]100[

m

m

34,14 zz ]110[

,]110[

m

m −

ID

)(gi

gk


Page 12 of 42Hempel, 2007 & 2016

• As it can be easily shown (e.g. Kocinski /11/) that in case of equitranslational

phase transitions ( =0) only those IDs will lead to a certain low symmetry

phase (subgroup of 4/mmm) where the characters of the symmetry elements of

the subgroup (here ) are either:

=1 for one-dimensional IDs

or

= 0, 2 for two-dimensional IDs

• → Only IDs 6 and 8 are able to induce the - PG Symmetry

Note: 6 corresponds to case A as mentioned above and 8 to case B

• Now it will be checked that 6 really can induce a ferroelastoelectric phase. To

this end it is necessary to calculate the respective reduction coefficients (e.g.

/11/):

m24

m24

)()(1 )(

ggG

mi

oGg

vP

oE =

)(gi

)(gi

k


Page 13 of 42Hempel, 2007 & 2016

ID Ferroelastoelectricity Ferroeleasticity Ferroelectricity

1=A1gNo (mE=0) Not checked for

2=A1uYes (mE=1) No (mE=0) No (mE=0)

3=A2gNo (mE=0) Not checked for

4=A2uYes (mE=3) No (mE=0) Yes (mE=1)

5=B1gNo (mE=0) Not checked for

6=B1uYes (mE=2) No (mE=0) No (mE=0)

7=B2gNo (mE=0) Not checked for

8=B2uYes (mE=2) No (mE=0) No (mE=0)

9=EgNo (mE=0) Not checked for

10=EuYes (mE=5) No (mE=0) Yes (mE=1)


Page 14 of 42Hempel, 2007 & 2016

• Next task is to check for the Landau- and Lifshitz Conditions:

• The Lifshitz Condition is automatically fulfilled because the transition is related

to =0

• Fulfilment of Landau’s Condition is checked by calculating the expression /11/:

with

(1+1+1+1-1-1-1-1-1-1-1-1+1+1+1+1)+

(1+1+1+1-1-1-1-1-1-1-1-1+1+1+1+1)+

(1+1+1+1-1-1-1-1-1-1-1-1+1+1+1+1) = 0

• The Landau Condition is also fulfilled → ID 6 is an “active” representation

0)(1 ?3

6=

gG

oGgo

3233 ))((6

1)()(

2

1)(

3

1)(

66666ggggg ++=

3

1)(3

6=g

6

12

1

k


Page 15 of 42Hempel, 2007 & 2016

• Next step is to decompose the Strain Tensor S and Polarization Vector P

into irreducible parts with regard to PG 4/mmm

• To this end the ordinary representation of the symmetry elements T(g)

(with regard to the Cartesian coordinate system (x=1, y=2, z=3) and the

Projection Operator will be utilized:

with = S resp. P

=

3

2

1

P

P

P

P

=

332313

232212

131211

SSS

SSS

SSS

S

)()()()( xfgTgG

dxf

oGg

ijo

ij

=

ij

)(xf


Page 16 of 42Hempel, 2007 & 2016

• This exercise yields the following irreducible parts which form the “basis

functions” for the Free Energy Expansion (FEE):

Strain S:

½(S11+S22), S33, ½(S11-S22), S23, S13, S12 respectively

(in Voigt’s Notation /10/ and normalized)

Polarization P:

P1, P2, P3

• Next step is to figure out the terms that will constitute the FEE. Each term

entering FEE has to be invariant under all symmetry operations of 4/mmm. But

it is sufficient to ensure the invariance just for the generating symmetry

elements (Generators)

• Following table shows how the Order Parameter Q and the State Parameters S

and P behave if the Generators act on them. For S and P this has to be done in

the Cartesian Space (x, y, z) and for Q the corresponding ID 6 has to be

applied

65421

321 ,,,

2

)(,,

2

)(SSS

SSS

SS −+


Page 17 of 42Hempel, 2007 & 2016

1

Initial Irreducible Parts,

written as normalized

Vectorsafter action of after action of after action of

−

+

6

5

4

21

21

2

)(32

)(

S

S

S

SSS

SS

3

2

1

P

P

P

12x14z

−

−

−−

+

6

4

5

21

21

2

)(32

)(

S

S

S

SSS

SS

−

−

−

+

6

5

4

21

21

2

)(32

)(

S

S

S

SSS

SS

−

+

6

5

4

21

21

2

)(32

)(

S

S

S

SSS

SS

−

−

−

3

2

1

P

P

P

−

−

3

2

1

P

P

P

−

3

1

2

P

P

P

Q−QQ−Q


Page 18 of 42Hempel, 2007 & 2016

• The FEE describes the dynamics and can now be written (ref. also to /5/):

F=F(Q,S,P)=FOP+FS+FP+FP_S+FOP_P+FOP_S+FOP_S_P

with

FOP = AQ2+BQ4

FS = C(S1-S2)2+D(S1+S2)

2+ES32+F(S4

2+S52)+GS6

2+H(S1+S2)S3+

r(S1+S2)S62+xS3S6

2

FP = J(P12+P2

2)+KP32

FP_S = L(P12+P2

2)(S1+S2)+M(P12+P2

2)S3+NP32 (S1+S2)+OP3

2 S3+ (1)

RP1P2S6+ g(P12-P2

2)(S1-S2) +h(P2P3S4+P1P3S5)

FOP_P = tQP1P2P3+UQ2(P12+P2

2)+VQ2P32

FOP_S = WQ2(S1+S2)+XQ2S3+YQ2(S42+S5

2)+ZQ2(S1-S2)2+bQ2S6

2

FOP_S_P = dQP3S6+eQ(P1S4+P2S5)+ fQP3S4S5

whereby all term-factors are assumed to be constant, except A=Ao(T-To).

Factor B has to be positive B>0.


Page 19 of 42Hempel, 2007 & 2016

• The transformation matrix within the “Voigt’s Space” that maps

into has got the form:

and can be equally used to calculate the irreducible parts of the elastic

coefficients, entering the FEE

• This yields:

(2)

• The upper index “o” means that the elastic stiffnesses are those at constant Q

and P and the impermeabilities are those at constant Q and S.

−

+

6

5

4

21

21

2

)(32

)(

S

S

S

SSS

SS

6

5

4

3

2

1

S

S

S

S

S

S

−

100000

010000

001000

00002

1

2

1000100

00002

1

2

1

oooooo

ooooooo

KJccHcG

ccFcEccDccC

332211231366

5544;3312111211

2

1;

2

1

2

1;;

2

1

2

1

2

1

2

1);(

4

1);(

4

1

======

===+=−=


Page 20 of 42Hempel, 2007 & 2016

4.3. Calculation of Temperature Dependences of State Parameters and

Material Coefficients

(3)

k

k

i

i

EP

F

S

F

=

=

• Solving this system of equations if non or certain values of „external forces“ Ti

(mechnical stress) resp. Ek (electric field) are applied yield following 4 selected

scenarios. Inserting the calculated expressions for Si and Pk into FEE gives

the Free Energy just as a function of the OP Q.

• The actual transition temperature Tc and the temperature dependence of Q

can be derived. Moreover, according to the Curie Principle (ref. e.g. /11/) the

symmetry of the Para- as well as of the Ferroic Phase reduces if certain

“external forces” are applied. Resulting Aizu’s Species are denominated.


Page 21 of 42Hempel, 2007 & 2016

Applied External Forces Symmetry Change (Species)

(based on Case A)

Kind of Ferroicity

Ek=0, Ti=0 Ferroelastoelectric (Scenario 1)

E30 Ti=0 Ferroelastic (Scenario 2)

E10 Ti=0 Ferroelastic

E20 Ti=0 Ferroelastic

Field along 45° between x,y

(→ E1=E20, E3=0 Ti=0)

Ferroelastic

E1E20, E3=0 Ti=0 Ferroelastic

T10, T2=0 (or T1T20) Ferroelastoelectric (Scenario 4)

Stress |2T| along 45° between x,y

(→ T1=T2=T6=T)

Ferroelectric (Scenario 3)

Stress along arbitrary angle bet-

ween x,y, but 45° (→ T1T2T60)

Ferroelectric

T3 0, Ek=0 → no symmetry

breaking

Ferroelastoelectric

mmmmF 24/4

zyxxymmmFm 24

xxzy Fmm 22

yyzx Fmm 22

yxxyzyxFmmm 2

1Fmz

zyxyzx Fmmm 222

zyxxyzyxxy mFmmmm 2

zz mF2/2

mmmmF 24/4


Page 22 of 42Hempel, 2007 & 2016Page 22 of 42

Scenario 1: Ek=0, Ti=0, , ferroelastoelectric

- The behaviour in the Paraphase is

included when putting Q0 (4)

- Since the term-factor A in FEE is not

renormalized, the transition temperature

where Q appears, is unchanged Tc=To

mmmmF 24/4

2

2

21

2

23

654321

4

2

2

1

4

2

0

SQDEH

HXEWS

QDEH

HWDXS

SSSPPP

=

−

−=

−

−=

======


Page 23 of 42Hempel, 2007 & 2016

Scenario 2: E30, Ti=0, , ferroelastic

- Higher order terms regarding Q

have been neglected


included when putting Q0

- Since S6 is proportional to Q, the

transition can be classified to be

of pseudo-proper ferroelastic nature

(5)

• Through the terms dependent on E3 the term-factor A in FEE renormalizes and

the actual transition temperature Tc does not coincide with To.

Tc depends on the absolute value of E3.

Note: S1=S2 because of orientation of the mirror planes in Ferrophase (ref. to slide 8, case A).

zyxxymmmFm 24

3

2

3

36

5421

22

1

4

0

EGK

bQ

KP

EQGK

dS

SSPP

+=

−=

====

2

2

3222

2

21

2

3222

2

23

164

2

2

1

4

2

2

1

164

2

4

2

SEDEKHK

HNENQ

DEH

HXEWS

EDEKHK

HNDNQ

DEH

HWDXS

=

−

−+

−

−=

−

−+

−

−=


Page 24 of 42Hempel, 2007 & 2016

Scenario 3: T1=T2=T6=Ti0, Ek=0, , ferroelectric


have been neglected



- P3 is proportional to Q, and the

transition can be classified to be

of pseudo-proper ferroelectric nature

(6)

• Through the terms dependent on Ti the term-factor A in FEE renormalizes and


Tc depends on applied mechanical stress Ti.

Note: S1=S2 because of orientation of the mirror planes in Ferrophase (ref. to slide 8, case A).

zyxxyzyxxy mFmmmm 2

i

i

QGK

dP

KG

Qd

GS

SSPP

−=

+=

====

4

82

1

0

3

2

22

6

5421

22

2

21

2

2

23

44

2

2

1

44

2

SEDH

EQ

EDH

HXEWS

EDH

HQ

EDH

HWDXS

i

i

=

−−

−

−=

−+

−

−=


Page 25 of 42Hempel, 2007 & 2016

Scenario 4: T1=Ti0, Ek=0, , ferroelastoelectric


have been neglected



(7)

• Through the terms dependent on Ti the term-factor A in FEE renormalizes and


Tc depends on applied mechanical stress Ti.

zyxyzx Fmmm 222

0

0

654

321

===

===

SSS

PPP

i

i

i

TEDH

E

CQ

EDH

HXEWS

TEDH

E

CQ

EDH

HXEWS

TEDH

HQ

EDH

HWDXS

−+−

−

−=

−−+

−

−=

−+

−

−=

44

1

2

1

4

2

2

1

44

1

2

1

4

2

2

1

42

1

4

2

2

2

22

2

2

21

2

2

23


Page 26 of 42Hempel, 2007 & 2016

• Slonczewski et al. /12/ derived a compact formalism to calculated the

temperature dependence of the elastic stiffness coefficients cij resulting from

the motion of the order parameter, and at constant external electric fields (Ek).

• A very similar formalism is valid for the description of dielectric impermea-

bilities kl at constant external mechanical stresses (Ti).

• The piezoelectric modules hmn are those measured at freely moving OP.

• In the following the isothermal behaviour of material coefficients will be derived,

taking into account that:

- the OP Q can freely move

- there might be additional contributions, originating from the piezo-

electric effect if allowed by symmetry


Page 27 of 42Hempel, 2007 & 2016

vn

2

kvvk, km

2

n

2

m

2

nm

2

mn PS

FR

PS

F

QS

F

QQ

F

QS

F

SS

Fc

−

−

=

−12

kvk jk

vj RPP

F with

=

2

vn

kvvk kmnmnm

mn SP

FR

SP

F

QP

F

QQ

F

QP

F

PP

F

−

−

=

− 2

,

221222

kvk jk

vj RSS

Fwith

=

2

QS

F

QQ

F

QP

F

SP

Fh

nmnm

mn

−

=

− 21222

(8)

• As result for the 4 scenarios the temperature dependences of the material

coefficients are explicitly provided on the slides 30-34. The appearing constants

are the same as summarized on slide 19 (eq. (2)).

• The equilibrium state of ACC is obtained if the state parameters regarding the 4

scenarios (ref. to eqs. (4-7)) are introduced in FEE (eq. (1).


Page 28 of 42Hempel, 2007 & 2016

• As result the FEE (eq. (1)) of ACC under defined external forces reads like:

Scenario 1: F=AQ2 + B’Q4 with A=Ao(T-To) (9)

Scenarios 2-4: F=A’Q2 + B’Q4 with A’=Ao(T-Tc) (10)

• From this the equilibrium OP-expressions can be calculated:

Scenario 1: Q2=-A/(2B’) (11)

Scenarios 2-4: Q2=-A’/(2B’) (12)

• As depicted in eqs. (8), for the temperature dependences of the material

coefficients also the 2nd derivatives are needed. From FEE (eq. (1)) we get:

Scenario 1: FQQ=2A+12BQ2+2W(S1+S2)+2XS3 (13)

Scenario 2: FQQ=2A+12BQ2+2VP32+2W(S1+S2)+2XS3+2bS6

2 (14)

Scenario 3: FQQ=2A+12BQ2+2VP32+2W(S1+S2)+2XS3+2bS6

2 (15)

Scenario 4: FQQ=2A+12BQ2+2W(S1+S2)+2XS3+2Z(S1-S2)2 (16)

• For the individual scenarios the explicit expressions for the state parameters

(ref. to eqs. (4-7)) have to be introduced into eqs. (13-16), as well as eqs.

(11,12). The paraphase is described if put Q=0.


Page 29 of 42Hempel, 2007 & 2016

• Finally the OP-expressions (eqs. (11,12)), the state parameters (eqs. (4-7)) as

well as the 2nd derivatives- expressions (eqs. (13-16)) are the ones which need

to be introduced into the formulas in eq. (8) and in the resulting expressions on

slides 30-34 in order to describe the temperature dependences of the

coefficients.

• Furthermore following additional abbreviations are used throughout the paper

to describe the piezoelectric contributions (acc. to eq. (8)):

( ) ( )

( ) ( )

( ) ( ) 213121133

13322331132

13121133

121133

213121133

212

211

12332

132112

213121133

212

211

2131133

2211

22

2

2

oooo

o

oooo

oo

oooooo

ooo

oooooo

ooo

cccc

cRRRR

cccc

ccR

cccccc

cccRR

cccccc

cccRR

−+

−====

−+

+=

−−−

−==

−−−

−==


Page 30 of 42Hempel, 2007 & 2016

Scenario 1: Ek=0, Ti=0, , ferroelastoelectric

QQ

QQ

QQ

QQ

F

XQWQHc

F

WQZQDCc

K

dQxSSSrbQGc

cc

J

eQYQFc

F

XQEc

cc

F

WQZQDCc

)2)(2(

)2(222

2

)(2)(222

2

)(22

)2(2

)2(222

13

22

12

2

321

2

66

4455

22

44

2

33

1122

22

11

−=

−−+−=

−++++=

=

−+=

−=

=

−++=

dQh

eQhh

G

dQVQOSSSNK

F

eQUQMSSSLJ

=

==

−++++=

=

=

−++++=

36

2514

22

32133

12

1122

22

32111

2

)(22)(22

0

2

)(22)(22

mmmmF 24/4


Page 31 of 42Hempel, 2007 & 2016

Scenario 2: E30, Ti=0, , ferroelastic

K

NOP

F

XQWQHc

K

NP

F

WQZQDCc

K

dQ

F

dPbQS

xSSSrbQGc

cc

J

hPeQYQFc

K

OP

F

XQEc

cc

K

NP

F

WQZQDCc

QQ

QQ

QQ

QQ

QQ

2

4)2)(2(

2

)2()2(222

2

)()4(

2)(222

2

)()(22

2

)2()2(2

2

)2()2(222

23

13

23

22

12

2236

3212

66

4455

23

22

44

23

2

33

1122

23

22

11

−−=

−−−+−=

−+

−

++++=

=

+−+=

−−=

=

−−++=

QQ

QQ

QQ

QQ

F

dSVQPdPbQSdQh

hh

F

dSVQPWQNPh

F

dSVQPXQOPh

hPhh

eQhh

G

dQROPRNOPRRNP

F

dSVQPVQOSSSNK

F

ehQPtQPRS

F

hPeQUQMSSSLJ

)4)(4(

)4(22

)4(22

2

)()2(16)()2(2

)4(22)(22

2

2

2

)()(22)(22

633636

3132

63331

63333

32415

2514

2

33

2

313

2

31211

2

3

2

632

32133

33612

1122

2

3

22

32111

++−=

=

+−=

+−=

==

==

++++−

+−++++=

−+=

=

+−++++=

zyxxymmmFm 24

Circled term induces

temperature dependence

already in Paraphase


Page 32 of 42Hempel, 2007 & 2016

Scenario 3: T1=T2=T60, Ek=0, , ferroelectriczyxxyzyxxy mFmmmm 2

K

NOP

F

XQWQHc

K

NP

F

WQZQDCc

K

dQ

F

dPbQS

xSSSrbQGc

cc

J

hPeQYQFc

K

OP

F

XQEc

cc

K

NP

F

WQZQDCc

QQ

QQ

QQ

QQ

QQ

2

4)2)(2(

2

)2()2(222

2

)()4(

2)(222

2

)()(22

2

)2()2(2

2

)2()2(222

23

13

23

22

12

2236

3212

66

4455

23

22

44

23

2

33

1122

23

22

11

−−=

−−−+−=

−+

−

++++=

=

+−+=

−−=

=

−−++=

QQ

QQ

QQ

QQ

F

dSVQPdPbQSdQh

hh

F

dSVQPWQNPh

F

dSVQPXQOPh

hPhh

eQhh

G

dQROPRNOPRRNP

F

dSVQPVQOSSSNK

F

ehQPtQPRS

F

hPeQUQMSSSLJ

)4)(4(

)4(22

)4(22

2

)()2(16)()2(2

)4(22)(22

2

2

2

)()(22)(22

633636

3132

63331

63333

32415

2514

2

33

2

313

2

31211

2

3

2

632

32133

33612

1122

2

3

22

32111

++−=

=

+−=

+−=

==

==

++++−

+−++++=

−+=

=

+−++++=

Circled term induces

temperature dependence

already in Paraphase


Page 33 of 42Hempel, 2007 & 2016

Scenario 4: T10, Ek=0, , ferroelastoelectric

• In order to properly describe this scenario – in particular to prove that c11c22,

c44c55 and h14h25 - the following additional (higher order) terms*) have to be

introduced into the FEE (ref. to eq. (1)):

m(S1-S2)2(S1+S2)

2 and n(S42-S5

2)(S1-S2) and p(P1S4-P2S5) (S1-S2)Q

QQ

QQQQ

QQ

QQ

F

SSZQWQSSZQWQZQSmSDCc

J

eQYQSSnFc

J

eQYQSSnFc

K

dQxSSSrbQGc

F

XQSSZQWQHc

F

XQEc

F

SSZQWQZQSSmDCc

F

SSZQWQZQSSmDCc

))(42))((42(2822

2

)(2)(22

2

)(2)(22

2

)(2)(222

)2))((42()2(2

))(42(2)412(22

))(42(2)412(22

21212

2112

22

2155

22

2144

2

321

2

66

2113

2

33

2

2122

1

2

222

2

2122

2

2

111

−−−+−−−+−=

−+−−=−+−+=

−++++=

−+−=−=

−−−+−++=

−+−+−++=

zyxyzx Fmmm 222

*) these terms are not needed

to describe scenarios 1-3


Page 34 of 42Hempel, 2007 & 2016

Scenario 4: T10, Ek=0, , ferroelastoelectric, continued

dQh

QSSpeQh

QSSpeQh

G

dQVQOSSSNK

F

QSSpeUQSSgMSSSLJ

F

QSSpeUQSSgMSSSLJ

=

−−=

−+=

−++++=

=

−−−+−−+++=

−+−+−++++=

36

2125

2114

22

32133

12

22

212

2132122

22

212

2132111

)(

)(

2

)(22)(22

0

2

))((2)(22)(22

2

))((2)(22)(22

zyxyzx Fmmm 222


Page 35 of 42Hempel, 2007 & 2016

S1=S2, S3

h14=h25, h36

(morphic)

c11=c22, c12

TcTo

S1=S2, S3, P3, h15=h24

c44=c55, 11=22

S6, h14=h25, h36, 12

(morphic)

c11=c22, c33, c12, c13, 33

h31=h32, h33

Tc=To

c33, c13

c44=c55, c66, 11=22, 33

Scenario 1: , ferroelastoelectricmmmmF 24/4 Scenario 2: , ferroelasticzyxxymmmFm 24

Qualitatively, the temperature dependences can now be depicted, taking not into

account the ordinary pyroelectric and thermal expansion effect, as well as the

multidomain state in the ferroic phases:

c66

11, 22, (12)

12, (11, 22)


Page 36 of 42Hempel, 2007 & 2016

S1=S2, S3, S6

c44=c55, 11=22, 12

P3, h14=h25, h15=h24

(morphic)

c11=c22, c33, c66, c13, c12

33

h33, h31=h32, h36

(morphic)

TcTo

S1S2, S3

c44c55, c66, 1122, 33

h14 h25, h36

(morphic)

c11 c22, c12

Tc≠To

c33, c13

Scenario 3: , ferroelectriczyxxyzyxxy mFmmmm 2Scenario 4: , ferroelastoelectriczyxyzx Fmmm 222


Page 37 of 42Hempel, 2007 & 2016

5. Comparison with Experimental Findings

Up to now no investigations of ACC are known where a constant electric field or

mechanical stress was applied when passing through the Phase Transition Point.

The experimental data available so far /13,14/ are all measured according to the

Scenario 1 (see page 35), under not defined domain conditions in the Ferrophase.

iiii /1=

Predicted behaviorThe temperature dependence of the dielectric

impermeabilities (which are within Scenario 1

exactly the inverse of the permittivities) is predicted

to show no jump at the phase transition point, but a

different slope in the Ferroic Phase. This coincides

well with the data measured.


Page 38 of 42Hempel, 2007 & 2016

Despite the constant (and comparable) slope in

the Para- and -Ferroic Phase, c33 is well

described by the predicted behavior (although

the jump is somehow smeared).

c11 = c22 is less in coincidence with the

predictions but similar. The jump (if at all) is

even more smeared and the slope in the

phases are different - as it is predicted.

Predicted behavior

c11=c22, c12

c33, c13


Page 39 of 42Hempel, 2007 & 2016

Predicted behavior

c44=c55, c66

h14=h25, h36

c44 = c55 and c66 show now jump but just a weak

change of slope at the transition point. This is in

line with the predictions, and can be explained

with the smallness of respective coefficients in the

FEE influencing the stiffnesses.

h36 is in agreement with the predicted behavior. It

reflects the temperature dependence of the order

parameter. As expected, above the transition point

h36 is identical to zero (‘morphic coefficient’).


Page 40 of 42Hempel, 2007 & 2016

5. Summary

The transition of (NH4)2CuCl4•2H2O at 200 K is a candidate of a classical

ferroelastoelectric phase transition.

Moreover, if certain constant external field (e.g. electric field, mechanical

stress) is applied to the crystal throughout the entire temperature range, the

type of Ferroicity should change. It has been shown that the transition can then

be either ferroelastic or ferroelectric, accompanied by all typical dependences

(incl. domain structures).

Especially interesting is the predicted temperature dependence of c66 in the

Paraphase, if an electric field E3 is applied (scenario 2). But without that field

(e.g. “switched off” at a defined temperature) c66 should return to the

temperature-independent value as described by scenario 1.

A similar behaviour is predicted for 33 within scenario 3, where a constant

mechanical stress is applied, which could also be “switched off”.


Page 41 of 42Hempel, 2007 & 2016

6. Literature

/1/ Wadhawan, V.K., Phase Transitions, 3, 3 (1982)

/2/ Toledano, P., et al., Phys. Rev. B16, 386 (1977)

/3/ Suga, H., et al., Bull. Chem. Soc. Japan, 38, 1007 (1965)

/4/ Bansal, M.L., et al., J. Phys. Chem. Solids (GB), 40, 109 (1979)

/5/ Hempel, H., et al., Ferroelectrics, 104, 361 (1990)

/6/ Janovec, V., et al., Czech. J. Phys., B25, 1362 (1975)

/7/ Kovalev, O.V., “Irreducible and Induced Representations and Corepresentations of

Space Groups”, Moscow, Nauka, 1986

/8/ Slaboszewska, M., et al., Ferroelectrics, 302, 55 (2004)

/9/ Aizu, K., J. Phys. Soc. of Japan, 27, 387 (1969)

/10/Sirotin, Yu.I., Shaskolskaya, M.P., “Fundamentals of Crystal Physics”,


/11/Kocinsky, J., “Theory of Symmetry Changes at Continuous Phase Transitions”,


/12/Slonczewski, J.C., et al., Phys. Rev. B1, 3599 (1970)


Page 42 of 42Hempel, 2007 & 2016

/13/Slaboszewska M., Tylczynski Z., Pietraszko A., Z., Karaev A., Ferroelectrics 302,

55-58(2004)

/14/Tylczynski Z., Wiesner M., Materials Chemistry and Physics 149-150, 566-573(2015)

Hempel, 2008-2015 Page 1 of 57

Phase Sequence and Landau Theory of LiKSO4


Table of Contents

1. Introduction

2. Basics

3. Symmetry Considerations (Phase II omitted)

4. Landau Modelling

5. Phase Diagram and Stability

6. Temperature Dependences of Material

Coefficients

7. Discussion of Results and Comparison

with Experimental Findings

8. Conclusion

9. Literature



1. Introduction

LiKSO4 (abbreviated LPS) exhibits a series of Phase Transitions (PTs), if temperature

(or/and pressure) is changed.

Numerous investigations have been made so far, but there is still no consistent view how

this crystal behaves.

The most recent summary regarding the phase sequence is as follows:

P63/mmc

or

P63mcP63P31cCc?????

Pmcn

or

Pc21n

Temperature (K)

IIIIIVVVIVIIVIIIIXX II

no. of formulae units in primitive elementary cell Z = 2 2 2 4 2

708205

(260 heating)

190

(195 heating)

165135806030 943

Fig. 1 Phase Transition Sequence acc. to /1/, /2/, /3/ (and references therein)



The essential findings can be summarized:

• Phase I shall exhibit Point Group (PG) symmetry 6/mmm or 6mm (see /2/, /3/)

• Symmetry of phase II seems to be Pmcn (centro-symmetric PG: mmm) or Pc21n

(polar PG: mm2) (see /2/, /3/)

• No ferroelectricity nor ferroelastoelectricity was experimentally found in the entire

temperature interval (see e.g. /4/)

• Ferroelasticity was observed in phases II and V (see e.g. /4/)

• All PTs observed are of 1st Order, PT III→IV shall be reconstructive /1/

• Phases IV and V are not „pure“ but „mixed“ phases, depending on the crystal‘s

„individuality“, i.e. depending on growth- and thermal treatment conditions (see /5/)

• LPS exhibits certain twinning in phases III and IV (see /5/, /6/)

• Depending on the a.m. „individuality“, and actual twinning composition,

measurements‘ results are often less reproducible (or comparable), and the PT

between phases IV and V is sluggish (see e.g. /5/)

• The few theoretical models which have been established to describe the ferro-

elastic IV–V PT are not comprehensive enough to describe LPS‘ behaviour in this

temperature region satisfactorily (see /7-13/). The most advanced model was

proposed by Quirion et al. /14/

• Also, no attempt has been undertaken yet to model the (unusual)

series of PTs III–IV-V



The aim of the present paper is to undertake an extended approach to model the phase

sequence I-III–IV-V, and to compare it with existing data and temperature dependences.

2. Basics

The following basic assumptions have been made:

1. The Landau Theory will be consequently applied

2. Phase I is assumed to exhibit 6mm symmetry (because of the 1st and 2nd finding

above, and because of the fact that PG 6mm is the common supergroup of the

phases III, IV, and V (see e.g. /15/). Therefore 6mm (to be precise - Space Group

P63mc) will be considered to represent the Prototypic Phase, as it was also

assumed by Quirion et al. /14/ and Willis et al. /11/.

In this context, there is no need to argue about reconstructive transition(s) as

suggested in /1/, despite the fact that the atomic displacements accompanied with

the III → IV PT shall be significant (see e.g. /5/).

3. Phase II will not be considered (but „skipped“) – in order to reduce by far the

complexity of the calculations.



4. All phases of the targeted sequence I–III–IV–V do have the same translational

symmetry, thus the PTs are all of the „ferrodistortive“ type where the translational

symmetry is preserved, and the modelling can be done within the framework of

PGs.

5. No „mixture“ of phases will be taken into account, only „pure“ phases.

6. No kind of twinning will be taken into account for the Landau Modelling – with other

words – the results of the following theoretical considerations are valid for the case

that each PT leads to a defined single orientation state of the crystal (ref. to

following slide).

Note:

Modelling of the mentioned phase sequence under the assumption that the Prototypic Phase

exhibits 6/mmm instead of 6mm symmetry should be possible as well, but then the concept of

PTs, induced by „reducible representations“, needs to be applied (see e.g. /13/, /17/).

The author believes that in this case the complexity of the calculations is not reasonably

manageable, and the expected additional information is assumed to be marginal and will

primarily be related to the (theoretical) possibility of polarization reversal along the 6-fold axis

(i.e. ferroelectricity) within certain phases (which was not observed experimentally!).



3. Symmetry Considerations (Phase II omitted)

Throughout all the phases one fixed rectangular system of coordinates x(1), y(2), z(3) is

used (mind the orientation of the symmetry elements in the stereographs below)

I (6mm) III (6) IV (3m) V (m)

Symmetry elements gi:

(see e.g. /16/)

(PG-Generators in red)

Order G of group: 12 6 6 2

Number of orientation

states n=12/G6,3m,m 1 2 2 6

Kind of expected none ferroelastoelectri- ferroelastoelectri- ferroelasticity,

ferroicity with regard (prototypic phase) city (and ferrogyro city and ferrobi- ferroelectricity

to the prototypic phase tropy*)) elasticity

see e.g. /17/, /18/):

*) Ferrogyrotropy represents an implicit form of ferroicity (see /19/)

y(2)

x(1)

z(3)

+ +

1, C61, C6

2, C63, C6

4, C65,

m[010], m[-110], m[100],

m[210], m[110], m[120]

1, C61, C6

2, C63, C6

4, C65 1, C6

2, C64 , m[210], m[-110],

m[120](=my)1, m[120](=my)



III (6) IV (3m) V (m)

Number of orientation

states n=12/G6,3m,m 2 2 6

Lost symmetry elements

of PG 6mm - being the

twin operations between

the orientation states

Tensor components in Piezoelectric Piezoelectric Strain Tensor Components

which the orientation coefficients coefficients (see next slide)

states differ

Gyration Elastic coefficients Polarization Vector Components

(see next slide)

kind of expected ferroelastoelectricity ferroelastoelectricity ferroelasticity and

ferroicity: (and ferrogyrotropy) and ferrobielasticity ferroelectricity

)1()2(

ijij GG −=

)1(

14

)2(

14 hh −=)1(

11

)2(

11 hh −=

)1(

15

)2(

15 cc −=

m[010], m[-110], m[100],

m[210], m[110], m[120]

C61, C6

3, C65,

m[010], m[100], m[110]

C61, C6

2, C63, C6

4, C65,

m[010], m[-110], m[100], m[210], m[110]

(upper index defines the

orientation state)



Application of the lost symmetry elements yield following 6 orientation states for phase V.

Depicted are the spontaneous values only, calculated e.g. after Aizu /20/:

(1) (2) (3) (4) (5) (6)

a) Polarisation

b) Deformation

Notes: 1. In the prototypic phase I (6mm) all spontaneous components (a, c, P1) are zero.

2. The abbreviations a=½(S11-S22), c=S13 are used.

3. Orientational states (1), (3) and (5), and the related spontaneous strains and polarizations

represent the a „stand alone“ PT 3m → m.

0

0

1P

−

02

32

1

1

1

P

P

−

−

02

32

1

1

1

P

P

−

02

32

1

1

1

P

P

−

0

0

1P

02

32

1

1

1

P

P

−

00

00

0

c

a

ca

−

−−

−−

02

3

2

2

3

22

322

3

2

cc

ca

a

ca

a

−−

−

−−

02

3

2

2

3

22

322

3

2

cc

ca

a

ca

a

−

−

−−

−

02

3

2

2

3

22

322

3

2

cc

ca

a

ca

a

−

−

−

00

00

0

c

a

ca

−

02

3

2

2

3

22

322

3

2

cc

ca

a

ca

a



Sapriel /21/ calculated the permissible domain wall orientations for all ferroelastic species.

For phase V (PG m) it turns out that there are 19 different walls allowed if we refer to the

prototypic phase I (PG 6mm).

(a and c represent the spontaneous tensor components as outlined on the previous slide,

x, y, z the coordinates)

z=0 x=0 y=0

x=3 y x=-3 y x=y/3

y=cz/a3 y=-cz/a3 x=-y/3

x=cz/a x=-cz/a (3x+3 y)+2cz/a=0

(3x+3 y)-2cz/a=0 (3x-3 y)+2cz/a=0 (3x-3 y)-2cz/a=0

(x+3 y)+2cz/a=0 (x+3 y)-2cz/a=0 (x-3 y)+2cz/a=0

(x-3 y)-2cz/a=0

Note:

The expressions marked in purple describe the permissible domain walls regarding a “stand

alone ferroelastic PT” from a phase with symmetry 3m to m only. This particular PT constitutes a

subset of the complete series of transitions I-III-IV-V. In the limit a>>c they coincide with the

green equations which were found experimentally (see e.g. /4/).

Hempel, 2008-2015 Page 10 of 57


4. Landau Modelling

The Landau Theory can describe the PTs that shall be permitted form a high symmetry

group. The maximal subgroups in which the respective Irreducible Representations (IRs)

exhibit identity representations, represent the allowed low symmetry groups. The low

symmetry phases are characterized by the presence of certain Order Parameters (OPs) /17/.

The Landau Theory imposes to look up the „active“ or „acceptable“ IRs only, which can lead

to a PT of second order between strictly periodic crystalline structures /22/. This means:

(a) The anti-symmetrized square, noted {}2, of n(k*) must not have any IR in common with

the vector representation of Go (k* specifies the relevant star of k-vectors) – Lifshitz

Condition (absence of asymmetric gradient terms of OP).

(b) The symmetrized third power of n(k*), noted []3, must not contain the totally

symmetric IR of Go – Landau Condition (absence of cubical OP terms).

Nevertheless is has been proven that many 1st order PTs can be successfully described

within the Landau Theory (see /22/). If the Landau Condition is violated, cubical OP-term(s)

are allowed and lead necessarily to a PT of 1st order. If the Landau Condition is obeyed and

a 1st order PT shall be described, as it is the case of LPS (will be shown later), the Free

Energy Expression (FEE) has to be expanded up to the 6th power of the order parameter(s).

To get started the Irreducible Representations of PG 6mm need to be inspected.

Hempel, 2008-2015 Page 11 of 57


Irreducible Representations of PG 6mm acc. to /23/, /24/:

Note: The IRs 5 and 6 are originally complex ones. They have been transformed into real, physical

IRs by a suitable unitary transformation (e.g. ref. to /25/ and /26/):

IRreal= U+ IR U with U= 1/2 and U+=1/2

IR

acc. to

/21/

IR

acc. to

/15/

1 C61 C6

2 C63 C6

4 C65 m[010] m[-110] m[100] m[210] m[110] m[120]

1 A1 1 1 1 1 1 1 1 1 1 1 1 1

2 A2 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1

3 B1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1

4 B2 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1

5 E1

6 E2

−2

1

2

32

3

2

1

−2

1

2

32

3

2

1

−−

−

2

1

2

32

3

2

1

−

10

01

−

−−

2

1

2

32

3

2

1

−

2

1

2

32

3

2

1

−10

01

−−

−

2

1

2

32

3

2

1

−

−

10

01

−

−−

2

1

2

32

3

2

1

10

01

−

−−

2

1

2

32

3

2

1

−

2

1

2

32

3

2

1

−10

01

−

−−

2

1

2

32

3

2

1

−−

−

2

1

2

32

3

2

1

−10

01

−−

−

2

1

2

32

3

2

1

10

01

−

−−

2

1

2

32

3

2

1

10

01

−

−−

2

1

2

32

3

2

1

−

i

i

0

1

− ii

11

−

2

1

2

32

3

2

1

−

2

1

2

32

3

2

1

Hempel, 2008-2015 Page 12 of 57


• Since 1-4 are one-dimensional they give rise to one-dimensional Basis Functions (BF)

(one-dimensional OPs), whereas 5 and 6 are interrelated to two-dimensional BFs

(two-dimensional OPs).

• The identification of the PG-symmetries induced by the individual IRs can be easily

derived by checking under which symmetry elements of PG 6mm the BFs are invariant.

This can be conveniently done in the space where the OPs move. Those elements

constitute the PGs of the respective low symmetry phases.

• Corresponding investigations lead inter alia to:

• By checking the respective reduction coefficients (see e.g. /26/) it can be shown that the

IRs 2, 4 and 5 fulfil both, the Landau- and the Lifshitz- conditions. Therefore, neither

gradient- nor cubical OP terms are allowed to enter the FEE.

IR Basis Function (BF),

written as OP-vector

BF are invariant under action of symmetry

elements gi of PG 6mm

Induced low temperature

symmetry (PG)

2 Q6 6

4 Q3m 3my

5 my

1, C61, C6

2, C63, C6

4, C65

1, C62, C6

4 ,

m[210], m[-110], m[120] (=my)

=

0

0

2

1

m

m

Q

Q1, m[120] (=my)

Hempel, 2008-2015 Page 13 of 57


• Next step towards the FEE is to decompose the (ordinary) strain tensor and (ordinary)

polarisation vector into their irreducible parts regarding PG 6mm.

• Utilisation of the projection operator (see e.g. /26/)

with = P resp. S and

– operator related to symmetry

element gk

yield the following results:

Strain S: ½(S11+S22), S33, ½(S11-S22), S12, S13, S23 respectively

(in Voigt’s Notation /16/ and normalized)

Polarisation P: P1, P2, P3

=

3

2

1

P

P

P

P

=

332313

232212

131211

SSS

SSS

SSS

S

)()()()( xfgTgG

dxf

Gg

kijkij

k

=)(xf

65421

321 ,,,

2

)(,,

2

)(SSS

SSS

SS −+

)( kgT

Hempel, 2008-2015 Page 14 of 57


• Each term of the FEE has to be invariant under the action of all the symmetry elements

of PG 6mm

• It is sufficient to prove this invariance under the action of the “generators” of PG 6mm,

namely C61 and m[100]

• To this end the transformation properties of Si, Pk, Q6, Q3m, Qm1,2 have been

compiled:

irreducible part after action of C61 after action of m[100]

Q6 Q6 -Q6

Q3m -Q3m -Q3m

2

1

m

m

Q

Q

+−

+

21

21

2

1

2

32

3

2

1

mm

mm

QQ

QQ

− 2

1

m

m

Q

Q

−

+

6

5

4

21

3

21

)(2

1

)(2

1

S

S

S

SS

S

SS

3

2

1

P

P

P

+−

+

3

21

21

22

32

3

2

1

P

PP

PP

−

3

2

1

P

P

P

+−−

+

−

−

−−

+

621

45

54

621

3

21

)(32

1

)3(2

1

)3(2

1

2

3

2

)(

2

1

)(2

1

SSS

SS

SS

SSS

S

SS

−

−

−

+

6

5

4

21

3

21

)(2

1

)(2

1

S

S

S

SS

S

SS

Hempel, 2008-2015 Page 15 of 57


• The FEE, reflecting the full dynamics of the system, can now be written as:

F = FOP+FOP-OP-OP+FS+FP+FS-P+FS-P-OP+FOP-P+FOP-S with

FOP = A1Q62 + A2Q6

4 + A3Q66 + B1Q3m

2 + B2Q3m4 + B3Q3m

6 + C1(Qm12+Qm2

2) +

C2(Qm12+Qm2

2)2 + C3(Qm13-3Qm2

2Qm1)2 + C4(Qm1

2+Qm22)3

FOP-OP-OP = 1Q62Q3m

2 + 2Q62(Qm1

2 + Qm22) + 3Q3m

2(Qm12+Qm2

2)

FS = D[(S1-S2)2+S6

2] + E(S1+S2)2 + FS3

2 + G(S1+S2)S3 + H(S42+S5

2)

FP = I(P12+P2

2) + JP32

FS-P = K(S1+S2)P3 + LS3P3 + M(P1S5+P2S4)

FS-P-OP = N(S1+S2)P3Q62 + OS3P3Q6

2 + P(P1S5+P2S4)Q62 + Q(S1+S2)P3Q3m

2 + RS3P3Q3m2 +

S(P1S5+P2S4)Q3m2 + U(S1+S2)P3(Qm1

2+Qm22) + VS3P3(Qm1

2+Qm22) +

W(P1S5+P2S4)(Qm12+Qm2

2) + xQ6(P1S4-P2S5) + yQ3m[(S1-S2)P1-S6P2]

FOP-P = XQ62P3 + YQ3m

2P3 + Z(Qm12+Qm2

2)P3 + b(P12+P2

2)Q62 + d(P1

2+P22)Q3m

2 +

e(P12+P2

2)(Qm12+Qm2

2) + f(P1Qm2-P2Qm1)

FOP-S = hQ62[(S1-S2)

2+S62] + iQ6

2S3 + jQ62(S1+S2) + kQ6

2(S42+S5

2) + lQ3m2[(S1-S2)

2+S62] +

mQ3m2S3 + nQ3m

2(S1+S2) + oQ3m2(S4

2+S52) + p(Qm1

2+Qm22)[(S1-S2)

2+S62] +

q(Qm12+Qm2

2)S3 + r(Qm12+Qm2

2)(S1+S2) + s(Qm12+Qm2

2)(S42+S5

2) + t(Qm2S5-Qm1S4) +

u[(S1-S2)(Qm22-Qm1

2)-2S6Qm1Qm2)] + v[(S1-S2)S5-S4S6]Q3m + w[(S1-S2)S5-S4S6]Qm1Qm2

Note:

The OP-terms have been provided up to the power of six to reflect that all PTs are of 1st order

(1)

Hempel, 2008-2015 Page 16 of 57


whereby all term-factors, as usual in Landau’s Theory, are assumed to be constant,

except A1=Ao(T-T6), B1=Bo(T-T3m), C1=Co(T-Tm1m2).

• The transformation matrix within the “Voigt’s Space” that mapped

into looks like

and can be equally used to calculate e.g. the irreducible (strain) parts of the elastic

coefficients, resp. of the piezoelectric coefficients. The results are summarized below.

• The following equalities regarding eq. (1) hold:

( ) ( )

( )

ooo

o

o

o

o

ooooo

ooooo

hMhLhKJI

cHcGcFccE

CrystalshexagonalincccnoteccD

153331

33

33

11

11

4413331211

1211661211

,,,1

,1

2

1,,

2

1,

4

1

holds2

1:

4

1

=======

===+=

−=−=

−

+

6

5

4

21

21

2

)(32

)(

S

S

S

SSS

SS

6

5

4

3

2

1

S

S

S

S

S

S

−

100000

010000

001000

00002

1

2

1000100

00002

1

2

1

Hempel, 2008-2015 Page 17 of 57


• The elastic stiffnesses coij in the FEE (eq. (1)) are those defined at constant polarisation

and at constant OPs.

• The dielectric impermeabilities oij in eq. (1) are those at constant strains and at

constant OPs.

• The piezoelectric coefficients hokm in eq. (1) are those at constant OPs.

• Moreover, it must be kept in mind that the ordinary thermal expansion / contraction

(i.e. the linear temperature dependence of the symmetry allowed strains as well as the

ordinary pyroelectric effect (linear temperature dependence of the polarisation) is not

covered by the present model. This must be taken into consideration when comparing

the theoretical results with experimental data.

• Next step is the solving of the following system of 9 state equations for the „free“ crystal:

(2)

0

0

==

==

k

k

i

i

EP

F

S

Fi=1…6

k=1…3

Hempel, 2008-2015 Page 18 of 57


• At no „external forces“ Ti (mechanical stresses) nor Ek (electric fields) the OP-

dependences of Si and Pk are obtained.

• Neglecting higher orders of the OPs we get:

(3)

(4)

(5)

(6)

(7)

(8)

)]))(2((

))2(())2([(1

)(

2

)(

2

1

)(

2

)(

2

1

22

21463

23462

26461

543

4

22

213

232

261

21

22

2

4

22

213

232

261

21

22

1

mm

m

mmmmm

mmmmm

QQJqLZKKK

QJmLYKKKQJiLXKKKKK

S

K

QQKQKQK

D

QQuS

K

QQKQKQK

D

QQuS

+−+−

+−+−+−+−

++++

−+

++++

−−

)(

4

2

4

2

2

2

2

1

216

225

124

mm

mm

m

m

QQpD

QuQS

QMHI

tIfMS

QMHI

fMtIS

++

−

−

−

−

Hempel, 2008-2015 Page 19 of 57


)]))(2()((

))2()((

))2()([(2

1

4

2

4

2

2

2

2

1463435

2

3462425

2

6461415

54

3

122

221

mm

m

m

m

QQJqLZKKKZKKKK

QJmLYKKKYKKKK

QJiLXKKKXKKKKKJK

P

QMHI

MtfHP

QMHI

fHMtP

+−+−+

+−+−+

+−+−+−

−

−

−

−

• The following abbreviations have been used:

Note:

Inserting the equilibrium values of the OPs for the individual phases (ref. to eqs. (16-19)) into

eqs. (3-11) give the respective dependences – ultimately on the temperature.

It should be noted that S5, (S4), P1 and (P2) show „pseudoproper-ferroelastic“- resp.

„pseudoproper-ferroelectric“ behaviour, because they depend linearly on the OPs.

(9)

(10)

(11)

(12)

KLGJKLJFK

KLJFKEJKJrKZKLZJqKK

JnKYKLYJmKKJjKXKLXJiKK

−=−=

−−−=−+−=

−+−=−+−=

2 4

)4)(4( )2()2(

)2()2( )2()2(

6

2

5

2

6

22

4563

562561

Hempel, 2008-2015 Page 20 of 57


• Inserting the expressions for Si (Q6, Q3m, Qm1, Qm2) and Pk (Q6, Q3m, Qm1, Qm2)

into the FEE (eq. (1)) gives the Free Energy Expression F‘ of the mechanically and

electrically “free” crystal just as a function of the OPs:

F‘= A1Q62 + A‘2Q6

4 + A‘3Q66 + B1Q3m

2 + B‘2Q3m4 + B‘3Q3m

6 + C‘1(Qm12+Qm2

2) +

C‘2(Qm12+Qm2

2)2 + C‘3(Qm13-3Qm2

2Qm1)2 + C‘4(Qm1

2+Qm22)3 +

‘1Q62Q3m

2 + ‘2Q62(Qm1

2 + Qm22) + ‘3Q3m

2(Qm12+Qm2

2)

• The renormalized term-factors are now: A‘2, A‘3, B‘2, B‘3, C‘1, C‘2, C‘3, C‘4,

‘1, ‘2, ‘3, but A1 and B1 are unchanged.

• This yields inter alia the following expressions for the transition temperatures

of the „free“ crystal:

OP: Q6 T6 - unchanged with regard to eq. (1)

OP: Q3m T3m - unchanged with regard to eq. (1)

OP: Qm1, Qm2 T‘m1m2 - renormalized as

(13)

(14)

22

222

2

22

21

'

21)4(

2)4( MHIC

tfMHfItM

MHIC

HfItTT

oo

mmmm−

−+−

−

++=

Hempel, 2008-2015 Page 21 of 57


• Minimizing of F’ with regard to the OPs according to the system of equations

yield the specific equilibrium values of the OPs:

Phase I (PG 6mm):

Phase III (PG 6):

Phase IV (PG 3m):

0'

0'

0'

0'

2136

=

=

=

=

mmm Q

F

Q

F

Q

F

Q

F

(16)

(17)

(18)03

133

0 212'2

'31

'3

'2

'3

'22

36 ==−−−== mmm QQ B

BB

B

B

B

BQ Q

000 2136 ==== mmm QQ Q Q

003

133

2132'2

'31

'3

'2

'3

'22

6 ===−−−= mmm QQ Q A

AA

A

A

A

AQ

(15)

Hempel, 2008-2015 Page 22 of 57


Phase V (PG m):

Notes:

The coefficients A‘3, B‘3, C‘4 have to be positive and A‘2, B‘2, C‘2 are assumed to be negative.

Other Solutions where the different OPs do exist simultaneously would imply to have PTs related

to so-called “reducible representations” which are not considered here because their induced low

symmetry phases are beyond the scope of this work.

• Next task is to calculate the 2nd derivatives of FEE (ref. to eq. (1)) which are needed

later (ref. to chapter 6., eqs. (29-31)) to derive the temperature dependences of the

material coefficients

• After creation of the derivatives, the OP-dependence of the strain- and polarization

components have to be used (eqs. (3)-(11)) and then the respective expressions for

the equilibrium OPs (ref. to eqs. (16-19) above) have to be applied as well

(19) C

CC

C

C

C

CQ Q Q Q mmm 2'

2

'41

'4

'2

'4

'22

2136

31

33000 −−−====

Hempel, 2008-2015 Page 23 of 57


• The 2nd derivatives of FFE (eq. (1)) for the individual phases calculate like:

Note:

Phase V (PG m) is characterized by equilibrium values of Qm2≠0 and Qm10 (see e.g. eq. (19)).

The expressions for the 2nd derivatives are provided up to the OP power of three. Higher order

terms have been omitted. K10-K22 are constants.

(20)

for

Para-

phasePhase PG 6 Phase PG 3m Phase PG m

Q6Q6 2A1 2A1+(12A2+K10)Q62 2A1+(2ɣ1+K14)Q3m

2 2A1+(2ɣ2+K18)Qm22

Q3mQ3m 2B1 2B1+(2ɣ1+K11)Q62 2B1+(12B2+K15)Q3m

2 2B1+(2ɣ3+K19)Qm22

Qm1Qm1 2C1 2C1+(2ɣ2+K12)Q62 2C1+(2ɣ3+K16)Q3m

2 2C1+(4C2+K20)Qm22

Qm2Qm2 2C1 2C1+(2ɣ2+K13)Q62 2C1+(2ɣ3+K17)Q3m

2 2C1+(12C2+K21)Qm22

Qm1Qm2 0 0 0 K22Qm23

ki QQ

F

2

Hempel, 2008-2015 Page 24 of 57


5. Phase Diagram and Stability

• The FEE - F’ (eq. (13)) is principally suited to investigate the possible equilibrium

states of the LPS-phase sequence, but the calculations are too complex to handle

• Particularly we are here only interested in the appearance of the phases III, IV and V

• Since ID 5 (refer to slide 12) gives rise to a two-component OP (Qm1, Qm2), but

phase V is induced by the special combination Qm1=0 and Qm20, we are going to

investigate a simplified “effective” FEE, as proposed by Gufan /27/. Also the terms

with B‘3, C‘3 and C‘4 in eq. (13) have been omitted:

F‘= A1Q62 + A‘2Q6

4 + A‘3Q66 + B1Q3m

2 + B‘2Q3m4 + C‘1Qm2

2 + C‘2Qm24 +

‘1Q62Q3m

2 + ‘2Q62Qm2

2 + ‘3Q3m2Qm2

2

• Minimizing of F’ with regard to the OPs according to the system of equations

yield:

(21)

0'

0'

0'

236

=

=

=

mm Q

F

Q

F

Q

F(22)

Hempel, 2008-2015 Page 25 of 57


Phase I (6mm):

Phase III (6):

Phase IV (3m):

Phase V (m):

The coefficients B‘2, C‘2 and A‘3 have to be positive and A‘2 is assumed to be negative.

Note:

Other Solutions where the different OPs do exist simultaneously would imply to have PTs

related to “reducible representations” which are not considered here because their induced

low symmetry phases are beyond the scope of this work.

(23)

(24)

(25)

(26)

0 2

0 21'

2

12

36 ==−== mmm QQB

BQQ

'

2

'

12

21362

0 0 0C

CQQQQ mmm −====

0 0 0 2136 ==== mmm QQQQ

0 0 3

133

2132'

2

'

31

'

3

'

2

'

3

'

22

6 ===−−−= mmm QQQA

AA

A

A

A

AQ

Hempel, 2008-2015 Page 26 of 57


The schematic temperature dependences of the OPs are depicted in Fig. 2

• To check whether a phase sequence I-III-IV-V is possible at all and the solutions, as

given above, are stable, we shall look at the phase diagram

• To this end the formalism developed by Gufan /28/ for two coupled OPs was expanded

to three coupled OPs

• First finding is that in fact the phases I-III, III-IV, III-V, IV-V are adjacent (ref. to Fig. 3).

• Determination of the “loss of stability limits” for the considered phases turns out that

they do not coincide, but do overlap. This means that all related PTs are necessarily of

1st order (see e.g. /25/) – which is well proven by all experimental findings so far

• As usual for 1st order PTs, the actual phase boundaries are determined by the equality

of the FEEs of the concerned neighboring phases. In case of three coupled OPs as in

LPS, the PTs are depicted by planes in the 3-dimensional phase diagram spanned

by A1, B1, C’1 (ref. to Fig. 3)

• Since we stick to the basics of the Landau Theory, we have in eqs. (21), (23-26)

consequently assumed a linear temperature dependence of the coefficients A1, B1, C’1and weak or no temperature dependence of the remaining coefficients

Hempel, 2008-2015 Page 27 of 57


T (K)

Q6

Q3m

Qm2

0

0

0

0Qm1

m (V)

3m (IV)

6 (III) 6mm (I)

T“m1m2=195 T“3m=260 T“6 708heating:

T“m1m2=190 T“3m=205 T“6 708cooling:

T‘m1m2

T3m

T6

Fig. 2 Qualitative temperature dependences of the Order Parameters Q6, Q3m, Qm1, Qm2

Hempel, 2008-2015 Page 28 of 57


• Moreover, A‘3, B‘2 and C‘2 in eq. (21) must be positive definite (because FEE must be

positive for large OP-values), and A‘2 shall be negative definite (1st order transition)

• The OP-coupling coefficients ‘1, ‘2 and ‘3 must be all positive, because otherwise the

phase diagram looks different and the phases of interest are not adjacent - which

contradicts the above mentioned phase sequence

• Within the phase diagram we have now to search for a thermodynamic path along a

straight line passing through the phases of the sequence I-III-IV-V, like a vector:

with A1, B1, C’1 being the coefficients from eq. (21) and

unit vectors along the axes

Note: Indeed, this path reflects the practical procedure i.e. the observation of the mentioned phase

sequence at changing temperature.

• It also turns out that such a sequence is feasible provided the path-vector proceeds not

too far from the origin of the coordinate system – namely above a Three-Phase Point.

In particular the following inequalities have to be satisfied (see also /28/):

'

111

'111 CBA eCeBeA

++axese

( ) ( ) ( )

( ) ( ) ( )2'

23'

22'

2

'

22'2'

2

2'

2

'

22'

'

1'

23'

'

2

2'

2

'

22'

1

2

2'

3'

22'

12'

2'2'

1

2'

12'

2'

12

'3'

'

1

2'

12'

2'

1

64

444

8

4

64

444

8

4

CA

CACACand

CA

CAA

BA

BABABand

BA

BAA

−+−

−

−+−

−

(27)

Hempel, 2008-2015 Page 29 of 57


B1

C‘1

A1

I-III(straight plane ||

B1-C‘1-plane)

IV-V(straight plane ||

A1-axis)

I-V(A1-B1-semi-plane)

III-IV (bent plane ||

C‘1-axis)

III-V (bent plane

|| B1-axis)

Phase I

(PG 6mm)

Phase V

(PG m)

Phase IV

(PG 3m)

Phase III

(PG 6)

I-IV(A1-C‘1-semi-plane)

Fig. 3 Section of Phase Diagram

Hempel, 2008-2015 Page 30 of 57


• If in eq. (27) the equality sign is fulfilled, the equations describe lines in the phase

diagram which are parallel to axis C’1 respectively B1 where 3 adjacent phases start

to coexist. One of the three is always accompanied by mixed different OPs (which

is not of interest here)

• Looking again at the path vector and Fig 3. It turns out that the sequence I-III-IV-V

materializes if T6>T3m>T’m1m2. Interestingly, the sequence I-IV-III-V is theoretically also

possible if T6>T’m1m2>T3m. However, series like I-V-III-IV or I-V-IV-III are not possible.

• When coming from high temperatures and moving along the path-vector we shall

calculate those temperatures where the vector crosses the planes which represent the

phase boundaries of the relevant adjacent phases in the phase diagram. These

actual transition temperatures shall be equivalent to those which are observed

experimentally. In fair approximation one obtains:

I(PG 6mm) → III(PG 6)

III(PG 6) → IV(PG 3m) with A’20 (28)

oAA

ATT

'3

2'2

6"

64

+=

oo

moo

m

BA

BA

TBTA

BA

T

−−

−−

2'

2'2

362'

2'2

"3

3

3

Hempel, 2008-2015 Page 31 of 57


• Finally the predicted temperature behaviour of the OPs, basing on the above made

stability considerations, can be summarized as follows (see once more Fig. 2):

➢ Coming from high temperatures, at T”6 Q6 suddenly appears with a jump

(just like for an ordinary PT of 1st order)

➢ On further cooling at T”3m Q6 suddenly disappears and at the same

temperature Q3m appears with a jump, although Q3m could (theoretically)

exist already above T”3m (see dashed line in Fig. 2, which shows the

typical 2nd order behavior reflected by eq. (25)), if this were not forbidden

by stability reasons

➢ Qm2 behaves similar like Q3m. The equilibrium value of Qm1 is always

equal to zero

IV(PG 3m) → V(PG m) (14) eq. from Twith

CB

BC

TCB

BCT

T mm

o

o

mo

omm

mm'

21

2'2

2'2

32'2

2'2'

21

"21

1−

−

=

Hempel, 2008-2015 Page 32 of 57


6. Temperature Dependences of Material Coefficients

• The FEE as provided in eq. (1) does have dynamical significance and is suited to derive

the temperature dependences of the material coefficients).

• Basing on the work of Slonczewski /29/ and Rehwald /30/, the material coefficients

calculate from FFE (eq. (1)) as follows:

Pkv

k jk

vjQik

i ji

kj RPP

FR

QQ

Fwith

=

=

=

3

1

22

vn

Pkv

vk kmkn

Qik

ki imnm

Emn

PS

FR

PS

F

QS

FR

QS

F

SS

Fc

−

−

=

2

,

22

,

22

(29)

Hempel, 2008-2015 Page 33 of 57


(31)kn

Qik

ki imnm

mnQS

FR

QP

F

SP

Fh

−

=

2

,

22

Qik

i ji

kj RQQ

Fwith

=

2

Notes:

1. The coefficients cEmn,

Tmn to be determined according to the equation above are those at constant

electric field, and at constant mechanical stress, respectively. These boundary conditions are usually

realized within practical measurements.

2. In equations (29-31) “Qi”, “Qk” stand symbolically for Q6, Q3m, Qm1, Qm2 (to shorten the expressions).

(30)vn

Skv

vk kmkn

Qik

ki imnm

Tmn

SP

FR

SP

F

QP

FR

QP

F

PP

F

−

−

=

2

,

22

,

22

Skv

k jk

vjQik

i ji

kj RSS

FR

QQ

Fwith

=

=

=

6

1

22

Hempel, 2008-2015 Page 34 of 57


For phase V (PG m) the following relations hold for which govern the OPs’ influence on

the material coefficients via eqs. (29-31). Approximations for small OP-values are provided

in form of truncated Taylor Series:

For the phases III (PG 6) and IV (PG 3m) yield:

The expressions for FQiQi are collected in the table (eq. (20)) on page 23.

Now, first the stiffnesses at constant polarisation and the dielectric impermeabilities at

constant strain as well as the piezoelectric coefficients are calculated and presented on the

following 4 slides.

Note: In the following tables the “Qi” are those from eqs. (16)-(19) and the expressions for the state

parameters are those taken from eqs. (3)-(11)

+−

−= 2

2

1

202211

2

41

212211

22

m

1QQQQQQ

QQQ QC

KC

2C

1

FFF

FR

mmmmmm

mm

( ) ( ) 1

3

1

6 3366

−−==

mmQQQ

mPGQQQPG FR ,FR

(32)

+−

−= 2

2

1

212222

2

121

212211

11

m

1QQQQQQ

QQQ QC

KC

2C

1

FFF

FR

mmmmmm

mm

322

1

2222112

2212211

21

m

QQQQQQ

QQQQ QC

K

FFF

FRR

mmmmmm

mm −

==

QikR

Hempel, 2008-2015 Page 35 of 57


Phase I

(6mm)Phase III (6) Phase IV (3m) Phase V (m)

c11= co11

c22= co11

c33= co33

c44= co44

c55= co44

66

26

232

6

)(42

QQF

QjNP hQ

+−

mmQQ

mm

F

QnQP lQ

33

23

232

3

)(42

+−

++−++

++−++−

Qm

Qm

Qm

mRQurSS pUP wS

RQurSS pUP RQwSpQ

122

22135

222

22

213112

2522

))(2(4

))(2(4)(2

66

26

232

6

)(42

QQF

QjNP hQ

+−

mmQQ

mm

F

QnQP IQ

33

23

232

3

)(42

+−

66

26

23 )(4

QQF

QiOP +−

mmQQ

m

F

QmRP

33

23

23 )(4 +

−Q

m RQqVP 222

22

3 )(4 +−

262kQ 2

32 moQQ

m RtsQ 1222

22 −

262kQ

232 moQ ( )

++−−

+++−−

Qmmm

Qmm

Qm

mRtQsSQWP QSS w

RtQsSQWP RQSS wsQ

122521221

222

2521112

2

2

2122

)42()(2

)42()(2

−+−−−

−+−−+−−

Qm

Qm

Qm

mRQurSS pUP wS

RQurSS pUP RQwSpQ

122

22135

222

22

213112

2522

))(2(4

))(2(4)(2

Hempel, 2008-2015 Page 36 of 57


Phase I


c66= co66

c12= co12

c13= co13

c23= co13

c15= 00 vQ3m

(ferrobielastic)

66

26

232

6

)(42

QQF

QjNP hQ

+−−

232 mlQ

Qmm RQupQ 112

222

2 42 −262hQ

mmQQ

mm

F

QnQP IQ

33

23

232

3

)(42

+−− ( )

+−−

+−−+−

−

−−

Qm

Qm

Qm

m

RQuSS p wS

RQuSS p rUP

RQSw

pQ

122

2215

222

22

212

3

112

225

2

22

))(2(4

))(2(4)(42

( )

66

2633 )(4

QQF

QiOPjNP ++− ( )

mmQQ

m

F

QmRPnQP

33

2333 )(4 ++

−( )

++

+++−+−

Qm

Qm

RQqVP wS

RQqVPurSS pUP

122

235

222

23213

2

)())(2(4

( )

66

2633 )(4

QQF

QiOPjNP ++−

( )

mmQQ

m

F

QmRPnQP

33

2333 )(4 ++

−

( )

( )

( )

( )

++−+−+

+++

++

++−++−

−

Q

m

mmm

Qmmm

Qm

RQurSS pUPSS w

QtQsSQWPwS

RtQsSQWPQ

urSS pUPRQSS Sw

122221321

225215

2225212

213112

22152

)(2)(2

42

42

)(22)(

( )

+−

+−+−−−

Qm

Qm

RQqVP wS

RQqVPurSS pUP

122

235

222

23213

2

)())(2(4

Hempel, 2008-2015 Page 37 of 57


Phase I


11= o11

22= o11

33= o33

h31= ho31

h32= ho31

h33= ho33

262bQ 2

32 mdQ

66

26

2321 ))((2

QQF

QXOSSS N +++−

mmQQ

m

F

QYRSSS Q

33

23

2321 ))((2 +++

−Q

m RQZVSSS U 222

22

321 ))((2 +++−

( )

66

263321

26

))((4

QQF

QjNPXOSSS N

NQ

++++

−

( )

mmQQ

m

m

F

QnQPYRSSS Q

QQ

33

233321

23

))((4 ++++

−

( )

( )

++++

++−++++

−

Qm

Qm

m

RQZVSSS UwS

RQurSS pUPZVSSS U

UQ

122

23215

222

2213321

22

)(2

)(2))((4

( )

66

263321

26

))((4

QQF

QiOPXOSSS N

OQ

++++

−

( )

mmQQ

m

m

F

QmRPYRSSS Q

RQ

33

233321

23

)2)((4 ++++

−

( ) Qm

m

RQqVPZVSSS U

VQ

222

23321

22

))((4 ++++

−

262bQ 2

32 mdQ

Qm RfeQ 11

2222 −

Qmmm RfQePQWS eQ 22

22125

22 )42(2 ++−

( )

66

263321

26

))((4

QQF

QjNPXOSSS N

NQ

++++

−

( )

mmQQ

m

m

F

QnQPYRSSS Q

QQ

33

233321

23

))((4 ++++

−

( )

( )

+++−

−+−−+++

−

Qm

Qm

m

RQZVSSS U wS

RQurSS pUPZVSSS U

UQ

122

23215

222

2213321

22

)(2

)(2))((4

Hempel, 2008-2015 Page 38 of 57


Phase I


h15= ho15

h24= ho15

h14= 0xQ6

(ferroelastoelectric)0

h11= 0 0yQ3m

(ferroelastoelectric)

26PQ

23mSQ ( )

−+++

++++

−

Qmmm

Qmmmm

m

RQSS wfQePQWS

RtQsSQWPfQePQWS

WQ

122212125

2225212125

22

)()42(

42)42(

2

6PQ 23mSQ

Qm RtfWQ 112

2 −

• The terms surrounded by circles give rise to expressions that are present in the

entire temperature region – from phase I through V, i.e. even there where the

OPs are identical to zero (in particular there holds acc. to eq. (32): ).

• Subsequently the stiffnesses at constant electric fields and the dielectric

impermeabilities at constant mechanical stresses are calculated

( )

( )

+++

++−+++−

Qmm

Qmmm

RfQePQWSwS

RQurSS pUPfQePQWS

1221255

2222132125

42

)(2)42(2

( ) Qmm RfQePQWSt 122125 42 +++

( ) 1

12211 2 −== CRR QQ

Hempel, 2008-2015 Page 39 of 57


To this end the terms standing most right in equations (29) and (30) are

calculated and result in*):

and*)

(33)

Note: *) OP-related terms have been considered to be small and are therefore omitted.

( ) 1

112211 2−

= oPP RR

( ) 1

3333 2−

oPR

( )( ) ( )( )oooooo

oooSS

cccccc

cccRR

3312112

131211

2133311

22112 +−−

−−=

( ) ( )( )oooooo

oooSS

cccccc

cccRR

3312112

131211

2133312

21122 +−−

−=

( ) oooo

oSSSS

cccc

cRRRR

3312112

13

1332233113

2 +−===

( )( ) oooo

ooS

cccc

ccR

3312112

13

121133

2 +−

+−

024421441544552255115 ========== SSSSSSSSSS RRRRRRRRRR

( ) 1

445544

−= oSS cRR ( ) ( ) 1

1211

1

6666 2−−

−= oooS cccR

Hempel, 2008-2015 Page 40 of 57


stiffnesses at

constant

polarisation

„electromechanical“ contributions in the individual phases

Phase I


cE11= c11

cE22= c22

cE33= c33

cE44= c44

cE55= c55

Pm RVQL 33

222 )( +−

Pm RWQM 22

222 )( +−

Pm RUQK 33

222 )( +−

Pm RWQM 11

222 )( +−

Pm RUQK 33

222 )( +− PRK 33

2− PRNQK 3322

6 )( +−

++−

Pm

Pm

RQQK

RyQ

3322

3

112

3

)(

)(

PRK 332− PRNQK 33

226 )( +−

++

−−

Pm

Pm

RQQK

RyQ

3322

3

222

3

)(

)(

PRL 332− PROQL 33

226 )( +− P

m RRQL 3322

3 )( +−

PRM 222−

++−

P

P

RPQM

RxQ

2222

6

112

6

)(

)( P

m RSQM 2222

3 )( +−

PRM 112−

−+

+−

P

P

RxQ

RPQM

222

6

1122

6

)(

)( P

m RSQM 1122

3 )( +−

Hempel, 2008-2015 Page 41 of 57


stiffnesses at

constant

polarisation


Phase I


cE66= c66

0 0 0

cE12= c12

cE13= c13

cE23= c23

cE15= c15 0 0 0

Pm RUQK 33

222 )( +−

( )

+

+−

Pm

m

RVQL

UQK

332

2

22 )(

( )

+

−

Pm

m

RSQM

yQ

1123

3 )(

Pm RyQ 22

23 )(−

PRK 332− PRNQK 33

226 )( +−

++

−−

Pm

Pm

RQQK

RQy

3322

3

1123

2

)(

PKLR 33− ( )

+

+−

PROQL

NQK

3326

26 )(

( )

+

+−

Pm

m

RRQL

QQK

3323

23 )(

PKLR 33− ( )

+

+−

PROQL

NQK

3326

26 )(

( )

+

+−

Pm

m

RRQL

QQK

3323

23 )(

( )

+

+−

Pm

m

RVQL

UQK

332

2

22 )(

Hempel, 2008-2015 Page 42 of 57


Impermea-

bilities

at

constant

strain


Phase I


T11= 11

T22= 22

T33= 33

Sm RWQM 55

222 )( +−

( ) Sm RWQM 44

222+−

( )

( )( )

+

+

++

++

++

+

−

SS

m

m

Sm

SSS

m

RR

VQL

UQK

RVQL

RRR

UQK

2313

22

22

3322

2

122211

222

)(2

)(

2

)(

SRM 552−

++−

S

S

RPQM

RxQ

5522

6

442

6

)(

)( ( )

++

−+

−

Sm

SSS

m

RSQM

RRR

yQ

5522

3

122211

23

)(

2

)(

SRM 442−

( )

+

+−

S

S

RxQ

RPQM

552

6

44

226

)(

( )

+

+−

Sm

Sm

RyQ

RSQM

662

3

44

223

)(

( )

( )

++

+

++

−

SS

S

SSS

RRKL

RL

RRR

K

2313

332

122211

2

2

2( )

( )( )

++

++

++

++

+

−

SS

S

SSS

RROQL

NQK

ROQL

RRR

NQK

231326

26

3322

6

122211

226

)(2

)(

2

)(( )

( )( )

+

+

++

++

++

+

−

SS

m

m

Sm

SSS

m

RR

RQL

QQK

RRQL

RRR

QQK

2313

23

23

3322

3

122211

223

)(2

)(

2

)(

Hempel, 2008-2015 Page 43 of 57


7. Discussion of Results and Comparison with Experimental Findings

• In this chapter the qualitative temperature behaviour of the state parameters and

material coefficients will be depicted and compared with experimental findings

• In order to do this into the expressions of the previous slides the precise dependences

have to be introduced, i.e. for strains eqs. (3)-(8), for polarisations eqs. (9)-(11), for

eqs. (32), for and eq. (33). Everywhere where the OPs appear, their temperature

dependences have to be inserted from eqs. (16)-(19).

• It must be noted here that the curves shall be correct for phases III and IV, because

there are no ferroelectric nor ferroelastic domains, but care has to be taken because

ferroelastocelectricity respectively ferrobielasticity are present which do have an effect

on certain material coefficients

• experimentally found behaviour for phase V is influenced by the always existing

multidomain state and can therefore hardly be compared with theoretical predictions.

Only exemptions are the “longitudinal properties” along the c-axis, because the domains

behave all equally in this direction.

• Nevertheless the predicted behaviour in phase V is shown as well – but under the

assumption, that the crystal undergoes the transition IV → V to a (defined) single

domain state as stipulated earlier (see slide 6)

QijR

SijR P

ijR

Hempel, 2008-2015 Page 44 of 57


• The temperature dependences around the I→III PT are not shown, because phase II

has been omitted, and therefore no such PT (i.e. skipping of phase II) can materialize

in practical experiments.

• Wherever possible, experimental data measured at increasing temperatures have been

chosen, because those data are more reproducible then in cooling runs (see e.g. /5/)

• Some more remarks have to be made:

- Since some coupling terms of the general form X*(OP)2 appear in FEE (see

eq. (1)), the affected material coefficients (cij, kl, hmn) show jumps downwards

at the transitions temperatures (T“6, T“3m, T“m1m2) where the respective OPs

appear suddenly (e.g. on cooling).

- Also the opposite is true: If at a transition temperature an OP becomes

instable (i.e. disappears) the jump of the material coefficient is directed

upwards.

- It has been shown that exactly at T“6 Q6 disappears and Q3m appears

(the same applies at T“m1m2 for Q3m and Qm2). The accompanied jumps of

the coefficients compete and yield a “resulting” jump which can be positive or

negative or even zero (latter in case of exact compensation of contributing

jumps)

- Precise predictions would required to know the signs and values of the

relevant term-factors of FEE (see eq. (1))

Hempel, 2008-2015 Page 45 of 57


• Another comment has to be made if stiffnesses cij are compared with compliances sij,

The relation sii=1/cii holds only if all cij with ij are equal zero (or very small). The

relation between impermeabilities and permittivities is analogue.

• The relation between the piezocoefficients hmj and the usually measured dki reads

like /31/:

(34)

• As mentioned earlier the present calculations do not cover the ordinary thermal

dilatation/contraction nor the ordinary pyroelectric effect.

• Since in particular the signs of the majority of the term-factors of FEE (see eq. (1))

are not known, different temperature dependences - depending on the signs chosen –

are possible. Wherever reliable experimental data are available, only those signs have

been considered, where the curve fit is the best.

Snm

Eijminj

Smn

Eijminj cdhrespshd == .

Hempel, 2008-2015 Page 46 of 57


orthorhombic

setting, b=3 a

T“m1m2T“3m

S1

S2 S1=S2

T“m1m2T“3m

S3

T“m1m2T“3m

S4=S6=0

T“m1m2T“3m

S5

Taken from /32/

• There is a good agreement between predicted and experimental data (see also /5/ and /33/)

• Since strain S5 linearly depends on OP Qm2 (see eq. (13)), pseudoproper ferroelastic behaviour should

be observed in phase V

• As found in /5/, LPS is monoclinic, but pseudo-orthorhombic in phase V. The angle between the polar

c-axis and axes a (and b) amounts to 90° → therefore strain S5 is in fact zero (but can theoretically be

unequal zero as schematically depicted above).

• As an explanation from the phenomenological point of view it can be assumed that the responsible

coupling coefficients t and f in eq. (1) and in eq. (7) are very small.

Hempel, 2008-2015 Page 47 of 57


T“m1m2T“3m

P1

T“m1m2T“3m

P2=0

Taken from /34/

• There is a good agreement between predicted and experimental data

• Since polarisation P1 linearly depends on OP Qm2 (see eq. (9)), pseudoproper ferroelectric behaviour

should be observed in phase V as well

• As found experimentally in /12/, /35/, LPS doesn’t exhibit any spontaneous polarisation component P1

nor a P1- (or P3-) polarisation switching in phase V

• Nevertheless, P1 is allowed by group theory (similar like for S5 on the previous slide)

• As an explanation from the phenomenological point of view it can be assumed that the coupling

coefficients t and f in eq. (1) and in eq. (9) are very small.

T“m1m2T“3m

P3

Hempel, 2008-2015 Page 48 of 57


Taken from /12/

Taken from /8/

• Relation ii=1/ii holds exactly for phases III,

IV, but in phase V only if ij <<1 for iǂj

• There is a good agreement regarding 33, but

the experimentally found slope in phase IV is

not predicted. Experiments by others (see /35/, /12/) revealed no slope in phase IV nor any

singularity at the PT IV III.

• There is some principal agreement regarding 11 and 22, but the predicted temperature

dependence (with negative overall slope) above (!) T‘‘m1m2 has never been observed experimentally.

• A possible explanation is that coefficient f in eq. (1) and in the respective expressions on slide no. 37

is very small, as already assumed to explain the observed behaviour of P1

T“m1m2T“3m

33

T“m1m2T“3m

11=22

11≠22

Hempel, 2008-2015 Page 49 of 57


• Depicted curves for the hij coefficients are just possible examples. Depending on the magnitude

and sign of the relevant term-factors (which are not known from theory) other curve-forms are

possible as well (ref. to comments on slides nos. 44 and 45).

• Also it must be noted that usually the piezocoefficients dij are practically measured, which cannot be

directly compared with the hij (see eq. (34)).

• Interesting is the fact, that h15 and h24 should be temperature dependent (with a positive overall

slope) already in phases (I), III and IV, i.e. above T‘‘m1m2.

• No measurable slope would again imply that the coupling coefficients f and t in eq. (1) are very

small.

• Experimentally measured piezocoefficients dij can be found e.g. in /12/, /36/.

T“m1m2T“3m

h31=h32,h33

h31≠h32,h33

T“m1m2T“3m

h15=h24

h15≠h24

Hempel, 2008-2015 Page 50 of 57


Taken from /37/

Taken from /14/

• There is a good agreement for c11, c22 –

meaning that the observed dependences can be

described by the formulas derived earlier.

• c33 shows at T‘‘3m a jump which can be explained

if, on e.g. heating, at T‘‘3m Q6 appears accompa-

nied by a jump downwards and Q3m disappears

accompanied by a jump upwards. Obviously the

first dominates and the resulting jump is directed

downwards (see also explanation on slide no. 44)

• The same explanation holds in principle for c13

and c23.

T“m1m2T“3m

c11=c22

c11≠c22

T“m1m2T“3m

c33,

c13=c23

c33,

c13≠c23

Hempel, 2008-2015 Page 51 of 57


Taken from /37/

• There is less agreement for c44, c55 which should

show a positive overall slope in phases (I), III and

IV, i.e. above T‘‘m1m2. Only possibility to explain

the negative slope is that coefficient t is very small,

and the terms with Q2 dominate, i.e. they provide

negative (slope) contributions – with coefficients

k, o (see eq. (1) being positive.

• The direction of the experimentally found jump at

T‘‘3m can be explained similarly as for c33.

• The predicted jump at T‘‘m1m2 was experimentally

not observed, indicating small values of coeffici-

ents t, f and opposite signs of coefficients o and s.

T“m1m2T“3m

c44≠c55

c44=c55

T“m1m2T“3m

c66

• c66 is in fair agreement with measured data

• The negative slope above T‘‘3m (i.e. in phase

III) is explainable within the model if coeffi-

cient h is positive definite

Hempel, 2008-2015 Page 52 of 57


• The depencence of c12 is closely related to those of c11, c22, c66 (ref. to the expressions derived

earlier)

• c15 is a „morphic“ coefficient, i.e. it is zero above T‘‘3m. It fits well with RUS measurements /11/

but it must be noted that another coordinate system (rotated by 30° around c-axis) was used. This

mean that our c15 coincides with c14 of /11/.

T“m1m2T“3m

c12

T“m1m2T“3m

c15

Note: There are numerous experimental data regarding the elastic stiffnesses of LPS available in the

literature, e.g. /38/-/41/, but many of them are contradictorily.

Hempel, 2008-2015 Page 53 of 57


8. Conclusion

Basing on Symmetry Considerations, the Landau Theory of Phase Transitions and the

related Phase Diagram an attempt has been made to theoretically describe the phase

sequence of LPS.

It turned out that the majority of experimental data found up to now can be satisfactorily

explained and even reasonably fitted qualitatively.

The predicted appearances of S5 and P1 in phase V are not evidenced by experiments.

Also the expected overall temperature dependences of the material the coefficients 11,

22, c44, c55 in phases I, III and IV have not been experimentally observed.

These substantial deviations can be explained qualitatively well with the model if certain

coefficients in the Free Energy Expansion are assumed to be very small.

Hempel, 2008-2015 Page 54 of 57


9. Literature

/1/ Perpetuo G. J. et al., Phys. Rev. B45, 5163 (1992)

/2/ Scherf, Ch., Dissertation, RWTH Aachen, 2000

/3/ Park, H. M., et al., Acta Cryst. A61, C373 (2005)

/4/ Sorge, G. et al., phys. stat sol. (a)97, 431 (1986)

/5/ Bhakay-Tamhane, S. et al., Phase Transitions 35, 75 (1991)

/6/ Klapper, H. et al., Acta Cryst. B43, 147 (1987)

/7/ Zeks B., et al., Phys. Stat. Sol. b122, 399 (1984)

/8/ Fujimoto S., et al., J. Phys. D: Appl. Physics 18, 1878 (1985)

/9/ An Tu et al., Solid State Commun., 61, 1 (1987)

/10/ Mroz B., et al., J. Phys.: Condensed Matter 1, 5965 (1989)

/11/ Willis F., et al., Phys. Rev. B54, 9077 (1996)

/12/ Sorge, G., Hempel, H., Ferroelectrics 81, 167 (1988)

/13/ Hempel, H., phys. stat. sol. (b)163, K77 (1991)

/14/ Quirion G., J. Phys. C: Condensed Matter 15, 4979 (2003)

Hempel, 2008-2015 Page 55 of 57


/15/ Boyle L. L., et al. Acta Cryst. A28, 485 (1972)



/17/ Janovec V., et al., Czech J. Phys. B25, 1362 (1975)

/18/ Toledano P., Toledano J.-C., Phys. Rev. B16, 386 (1977)

/19/ Wadhawan, V. K., Phase Transitions 3, 3 (1982)

/20/ Aizu, K., J. Phys. Soc. Japan 28, 706 (1970)

/21/ Sapriel, J., Phys. Rev. B12, 5128 (1975)

/22/ Toledano P., Toledano J.-C., “The Landau Theory of Phase Transitions“, World

Scientific Publishing, 1987

/23/ Kovalev, O. V., “Representation of Crystallographic Space Groups”, Taylor & Francis

Ltd., 1993

/24/ Cracknell, C. J., Bradley, A. P., “The Mathematical Theory of Symmetry in Solids”,

Clarendon Press, Oxford, 1972



Hempel, 2008-2015 Page 56 of 57




/27/ Gufan et al., Sov. Solid State Phys. 22, 951 (1980)

/28/ Gufan et al., Sov. Solid State Phys. 22, 270 (1980)

/29/ Slonczewski, J. C. and Thomas, H., Phys. Rev. B1, 3599 (1970)

/30/ Rehwald, W., Adv. In Pysics, Vol. 22, 721 (1973)

/31/ Xu, Y., „Ferroelectric Materials and Their Applications“, Elsevier Science

Publishers, 1991

/32/ Tomaszewski, P. E., et al., Phase Transitions 4, 37 (1983)

/33/ Desert, A. et al., J. Phys. Cond. Matter 7, 8445 (1995)

/34/ Breczewski, T., et al., Ferroelectrics 33, 9 (1981)

/35/ Cach, R., et al., Ferrolectrics 53, 337 (1984)

/36/ Mroz, B., et al., Ferroelectrics 42, 71 (1982)

/37/ Kabelka, H., et al., Ferrolelectrics 88, 93 (1988)

/38/ Borisov, B. F., et al., phys. stat. sol. (b)199, 51 (1997)

Hempel, 2008-2015 Page 57 of 57


/39/ Drozdowski, M., et al. Ferroelectrics 77, 47 (1988)

/40/ Ganot, F., et al., J. Phys. C: Solid State Phys. 20, 491 (1987)

/41/ Kassem, E. M., et al., Acta Physica Polonica A63, 449 (1983)

structural phase transitions and landau's theory

Science

structural phase transitions

landau theory

ferroelastics

ferroelastoelectrics

symmetry change

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nh4 2cucl4

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