01. sifat nuklir 1

SIFAT NUKLIRKuliah Fisika Nuklir

XZ

A

Satuan• Energi - electron-volt– 1 electron-volt = energi kinetik dari suatu elektron ketika bergerak

melalui beda potensial 1 Volt;• 1 eV = 1.6 × 10-19 Joules• 1 kW•hr = 3.6 × 106 Joules = 2.25 × 1025 eV• 1 MeV = 106 eV, 1 GeV= 109 eV, 1 TeV = 1012 eV

• massa - eV/c2

• 1 eV/c2 = 1.78 × 10-36 kg• massa electron = 0.511 MeV/c2

• Massa proton = 938 MeV/c2 = 0.938 GeV/ c2

• massa neutron = 939.6 MeV/c2

• momentum - eV/c: • 1 eV/c = 5.3 × 10-28 kg m/s• momentum baseball saat 80 mi/hr 5.29 kgm/s 9.9 × 1027 eV/c

• jarak• 1 femtometer (“Fermi”) = 10-15 m

Sifat Nuklir

Sifat Statik• Muatan listrik• Radius inti• Massa• Energi Ikat• Momentum angular• Paritas• Momen dipol magnetik dan

kuadrupol listrik• Energi eksitasi

Sifat Dinamik• Peluruhan• Probabilitas reaksi

Ilustrasi inti

Radius Inti

Rapat inti tidak berubah, shg

R=roA1/3

Radius inti• Volume inti sebanding dengan massa nuklirnya, maka

semua nuklida memiliki densitas yang sama• ro~1.1 to 1.6 fm• Jari-jari nuklir dapat diperngaruhi oleh gaya nuklir,

distribusi muatan, dan distribusi massa

Radius Gaya IntiRadius medan gaya nuklir harus lebih kecil dari pada jarak terdekat (do)

T’=T-2Ze2/dd = jarak dari pusat ke nukleus

T’ = energi kinetik partikel

T = energi kinetik mula-mula partikel

do = jarak terdekat – mencapai pada tumbukan pada T’=0

T

Zedo

22

do~10-20 fm tuk Cu and 30-60 fm tuk U

Setiap muatan yang mengarah pada gaya nuklir dapat digunakan untuk probe jarak dari pusat suatu nukleus dalam nuklir dimana gaya nuklir (atraktif) menjadi relatif penting dibandingkan Couloomb (gaya tolak)

Karena neutron tidak kena gaya Coulomb, percobaan hamburan neutron dan penyerapan diharapkan lebih mudah untuk diterangkan, namun neutron harus berenergi cukup tinggi bila memiliki panjang gelombang de Broglie yang lebih kecil dibandingkan dengan dimensi nuklir, tapi pada energi tinggi, inti menjadi cukup transparan untuk neutron.

Potential Square-Well dan Woods-Saxon

Friedlander & Kennedy, p.32

ARro

e

VV

/)(1

Vo=potensial pada pusat nukleus

a = konstanta ~ 0.5 fm

R = jarak dari pudat dimana V=0.5Vo (untuk setengah radius) atau V=0.9Vo dan V=0.1Vo untuk drop-off dari 90 hingga 10% dari potensial penuh of the full potential

ro~1.35 hingga 1.6 fm fontuk Square-Well,

ro~1.25 fm untuk Woods-Saxon dengan radius paro potensial,

ro~2.2 fm untuk Woods-Saxon dengan drop-off dari 90 hingga 10% -- tebal kulit – dari potensial penuh

Percobaan hamburan yang menghasilkan kesepatan perkiraan terhadap poternsial Square-Well; persamaan Woods-Saxon sesuai dengan data yang lebih baik

Hamburan Electron

• Using moderate energies of electrons, data is compatible with nuclei being spheres of uniformly distributed charges

• High energy electrons yield more detailed information about the charge distribution (no longer uniformly charged spheres)

• Radii distinctly smaller than indicated by the methods that determine nuclear force radii

• Re (half-density radius)~1.07 fm

• de (“skin thickness”)~2.4 fm

Bentuk Fermi

]/)[(1)(

eee aRro

er


Nuclear Skin• Although charge density results give information on how protons are distributed in

the nuclei, no experimental techniques exist for determining the total nucleon distribution– it is generally assumed that neutrons are distributed in roughly the same

way as protons– nuclear-potential radii are about 0.2 fm larger than the radii of the charge

distributionsNucleus Fraction of nucleons in the “skin”12C 0.9024Mg 0.7956Fe 0.65107Ag 0.55139Ba 0.51208Pb 0.46238U 0.44

Massa Inti

• Nukleus tersusun dari proton dan neutron• Proton dan Neutron lebih masif 1840 kali dari

elektron• Jumlah total nukleon adalah A• Jumlah proton dalam nukleus sama dengan

bilangan atom Z• Jumlah netron adalah A - Z

Massa Inti

• 1 satuan massa atom (u) didefinisikan massa isotop • 1 mol atom =12 g, • 6,02 1023 atom =12 10-3 kg• Massa 1 atom = 12 10-3 kg/(6,02 1023) = 1,99 10-

26 kg.• Sesuai dengan definisi 1 u sama dengan 1/12 massa

aisotop 1 u = 1,99 10-26 kg/12 = 1,66 10-27 kg.

Massa inti

• 1 u = 1,99 10-26 kg/12 = 1,66 10-27 kg• 1 u setara dengan energi 931,502 MeV

Proton Netron

Muatan +1,6 10-19C 0C

Massa Diam 1,67252 10-27 kg938,256 MeV1,007277 u

1,67482 10-27kg939,550 MeV1,008665 u

Spin ½ ½

Masses• Atomic masses

– Nuclei and electrons• Nuclear mass

– m0 is electron rest mass, Be (Z) is the total binding energy of all the electrons

– Be(Z) is small compared to total mass• Consider beta decay of 14C

– 14C14N+ + β- +antinuetrino + energy• Energy = mass 14C – mass 14N

• Positron decay

Masses• For a general reaction

Terms• Binding energy

– Difference between mass of nucleus and constituent nucleons• Energy released if nucleons formed nucleus

• average binding energy per nucleon

– Measures relative stability• Mass excess (in energy units)

– M(A,Z)-A» Useful when A remains constant

Binding Energies

http://www.lbl.gov/abc/wallchart/chapters/02/3.html

Binding energy• Binding Energy of an even-A nucleus is generally higher than the

average of the values for the adjacent odd-A nuclei– this even-odd effect is more pronounced in graphing A vs.

the binding energy from the addition of one more nucleon• The very exothermic nature of the fusion of H atoms to form He--the

process that gives rise to the sun’s radiant energy--follows from the very large binding energy of 4He

• Energy released from fission of the heaviest nuclei is large because nuclei near the middle of the periodic table have higher binding energies per nucleon

• The maximum in the nuclear stability curve in the iron-nickel region (A~56 through 59) is thought to be responsible for the abnormally high natural abundances of these elements

• Mass excess==M-A

Stable NucleiN even odd even oddZ even even odd oddNumber 160 53 49 4

• As Z increases the line of stability moves from N=Z to N/Z ~ 1.5 – influence of the Coulomb force. For odd A nuclei– only one stable isobar is found while for even A nuclei– no stable odd-odd nuclei

Terms• Binding can be used to determine energetics for reaction using mass

excess– Energy need to separate neutron from 236U and 239U

Binding-Energy

• Volume of nuclei are nearly proportional to the number of nucleons present– nuclear matter is quite incompressible

• Total binding energies of nuclei are nearly proportional to the numbers of nucleons present– saturation character• a nucleon in a nucleus can apparently interact with only a

small number of other nucleons

–liquid-drop model of nucleus

Liquid-Drop Binding Energy:

• c1=15.677 MeV, c2=18.56 MeV, c3=0.717 MeV, c4=1.211 MeV, k=1.79 and =11/A1/2

• 1st Term: Volume Energy– dominant term• in first approximation, binding energy is

proportional to the number of nucleons– (N-Z)2/A represents symmetry energy• binding E due to nuclear forces is greatest for the

nucleus with equal numbers of neutrons and protons

124

3/123

23/2

2

2

1 11 AZcAZcA

ZNkAc

A

ZNkAcEB

• 2nd Term: Surface Energy– Nucleons at surface of nucleus have unsaturated forces– decreasing importance with increasing nuclear size

• 3rd and 4thTerms: Coulomb Energy– 3rd term represents the electrostatic energy that arises from the Coulomb

repulsion between the protons• lowers binding energy

– 4th term represents correction term for charge distribution with diffuse boundary

• term: Pairing Energy– binding energies for a given A depend on whether N and Z are even or odd• even-even nuclei, where =11/A1/2, are the stablest

– two like particles tend to complete an energy level by pairing opposite spins

Mass Parabolas

• For odd A there is only one -stable nuclide– nearest the minimum of the parabola

• For even A there are usually two or three possible -stable isobars– all of the even-even type


Magic Numbers

• Certain values of N and Z--2, 8, 20, 28, 50, 82, and 126 --exhibit unusual stability– evidence from masses, binding energies, elemental and

isotopic abundances, numbers of species with given N or Z, and -particle energies

– accounted for by concept of closed shells in nuclei


Single-Particle Shell Model

• Collisions between nucleons in a nucleus are suppressed by the Pauli exclusion principle– only accounts for magic numbers 2-20

• Strong effect of spin-orbit interactions – if orbital angular momentum (l) and spin of nucleon

interact in such a way that total angular momentum=l+1/2 lies at a lower energy level than that with l-1/2, large energy gaps occur above magic numbers 28-126

• Ground states of closed-shell nuclei have spin=0 and even parity

R=roA1/3

Nuclear Shapes: Radii

• Nuclear volumes are about proportional to nuclear masses, thus all nuclei have approximately the same density

• Although nuclear densities are high compared to ordinary matter, nuclei are not densely packed with nucleons

• ro~1.1 to 1.6 fm• Nuclear radii can mean different things, whether they are

defined by nuclear force field, distribution of charges, or nuclear mass distribution

Nuclear-Force RadiiThe radius of the nuclear force field must be less than the distance of closest approach (do)

T’=T-2Ze2/dd = distance from center of nucleus

T’ = particle’s kinetic energy

T = particle’s initial kinetic energy

do = distance of closest approach--reached in a head on collision when T’=0

T

Zedo

22

do~10-20 fm for Cu and 30-60 fm for U

Any positively charged particle subject to nuclear forces can be used to probe the distance from the center of a nucleus within which the nuclear (attractive) forces become significant relative to the Coulombic (repulsive force).

Since neutrons are not subject to Coulomb forces, one might expect neutron scattering and absorption experiments to be easier to interpret, however the neutrons must be of sufficiently high energy to have de Broglie wavelengths small compared to nuclear dimensions, but at high energies, nuclei become quite transparent to neutrons.

Square-Well and Woods-Saxon Potentials


ARro

e

VV

/)(1

Vo=potential at center of nucleus

a=constant~0.5 fm

R=distance from center at which V=0.5Vo (for half-potential radii) or V=0.9Vo and V=0.1Vo for a drop-off from 90 to 10% of the full potential

ro~1.35 to 1.6 fm for Square-Well, ro~1.25 fm for Woods-Saxon with half-potential radii, ro~2.2 fm for Woods-Saxon with drop-off from 90 to 10%--the “skin thickness”--of the full potential

Scattering experiments lead to only approximate agreement with the Square-Well potential; the Woods-Saxon equation fits the data better.

Electron Scattering

• Using moderate energies of electrons, data is compatible with nuclei being spheres of uniformly distributed charges

• High energy electrons yield more detailed information about the charge distribution (no longer uniformly charged spheres)

• Radii distinctly smaller than indicated by the methods that determine nuclear force radii

• Re (half-density radius)~1.07 fm

• de (“skin thickness”)~2.4 fm

Fermi Shape

]/)[(1)(

eee aRro

er


Nuclear Skin• Although charge density results give information on how protons are distributed

in the nuclei, no experimental techniques exist for determining the total nucleon distribution– it is generally assumed that neutrons are distributed in roughly the

same way as protons– nuclear-potential radii are about 0.2 fm larger than the radii of the

charge distributionsNucleus Fraction of nucleons in the “skin”12C 0.9024Mg 0.7956Fe 0.65107Ag 0.55139Ba 0.51208Pb 0.46238U 0.44

Spin• Nuclei possess angular momenta Ih/2– I is an integral or half-integral number known as the

nuclear spin• Protons and neutrons have I=1/2• Nucleons in the nucleus, like electrons in an atom, contribute

both orbital angular momentum (integral multiple of h/2 ) and their intrinsic spins (1/2)

• Therefore spin of even-A nucleus is zero or integral and spin of odd-A nucleus is half-integral

• All nuclei of even A and even Z have I=0 in ground state

Magnetic Moments• Nuclei with nonzero angular momenta have magnetic moments• Bme/Mp is used as the unit of nuclear magnetic moments and called a

nuclear magneton• Magnetic moment results from a distribution of charges in the

neutron, with negative charge concentrated near the periphery and overbalancing the effect of an equal positive charge nearer the center

• Magnetic moments are often expressed in terms of gyromagnetic ratios – g*I nuclear magnetons, where g is + or - depending upon

whether spin and magnetic moment are in the same direction

•Only nuclei with I1/2 have quadrupole moments

•Interactions of nuclear quadrupole moments with the electric fields produced by electrons in atoms and molecules give rise to abnormal hyperfine splittings in spectra

• Methods of measurement: optical spectroscopy, microwave spectroscopy, nuclear resonance absorption, and modified molecular-beam techniques

Methods of Measurement1) Hyperfine structure in atomic spectra

2) Atomic Beam method

split into 2I+1 components

3) Resonance techniques

2I+1 different orientations

Quadrupole Moments: q=(2/5)Z(a2-b2)

Statistics• If all the coordinates describing a particle in a system are

interchanged with those describing another particle in the system the absolute magnitude of the wave function representing the system must remaining the same, but it may change sign– Fermi-Dirac (sign change)• each completely specified quantum state can be

occupied by only one particle (Pauli exclusion principle)– Bose-Einstein (no sign change)• no restrictions such as Pauli exclusion principle apply

• A nucleus will obey Bose or Fermi statistics, depending on whether it contains an even or odd number of nucleons

Parity• Depending on whether the system’s wave function changes sign

when the signs of all the space coordinates are changed, a system has odd or even parity

• Parity is conserved• even+odd=odd, even+even=even, odd+odd=odd

– allowed transitions in atoms occur only between an atomic state of even and one of odd parity

• Parity is connected with the angular-momentum quantum number l– states with even l have even parity– states with odd l have odd parity

01. sifat nuklir 1

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