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Shell Model V - Nuclear Moments and Collective Model

发布时间:2013-12-04 15:02:18  

The Shell Model of the Nucleus

5. Nuclear moments.

The Collective Model of the Nucleus

The Bohr-Mottelson Collective Model of the Nucleus

James Rainwater

(1917 –1986)

U.S.AAage Bohr(1922 –DenmarkBen Mottelson(1926 –USA / DenmarkBetween about 1950 –1955, Bohr and Mottelson followed the idea

on suggested by Rainwater. All three

received the 1975 Nobel Prize –“For their work on the connection of with single particle motion”

Electric nuclear momentsThe electrostatic energy associated

with charge distribution ?(r) in electric

potential ?(r) is:??3?U???(r)?(r)dr?, 3eZ???(r).dr

ELECTRIC CHARGE

+3?P???(r)r.drELECTRIC DIPOLE223?Q???(r)3z?r.drELECTRIC QUADRUPOLE+ -

Electric and Magnetic nuclear moments“For the of the nucleus (as well as all other static multipole moments with odd parity; for example –magnetic monopole, magnetic quadrupole or electric octupole) .

WHY? Because the both the strong and the electric forces are

invarient under the parity operation –which means we should never get different nuclear properties under space inversion

The nuclear Quadrupole moment Q

PROLATE“Cigar”

Q?0

Q?0

Unlike atoms –nuclei can easily distort from spherical state. Unlike atoms the potential the nucleons move in is formed by the nucleons themselves. [In an atom it is the potential coming from the nucleus that dominates]

The nuclear Quadrupole moment QQQuadrupole Moment (Barns)

Magic Numbers

The regions of sphericity

Quadrupole moment on the Shell modelQ?0Q?0An extra proton or neutron added to a

closed shell configuration makes an

OBLATEellipsoid A proton or neutron removed from a closed shell configuration make a PROLATEellipsoid

Quadrupole moment on the Shell modelmissing

Odd protonextra?223?Q???(r)(3z?r).dr ? ?nlj*22?(r)(3z?r)?nlj(r)dr3 ? -missing

extra22/3?r??R?RoA552

Deformation effects shell states

Nilsson Modela definite direction is defined –the direction 3 along the axis. The total angular momentumj is projected along the 3rd±j, ±(j-1), ±(j-2) etc. This phenomenon also splits the

degeneracy previously seen for spherical (closed shell) nuclei

The Nilsson Shell Model Plot for Deformed Nuclei

Irrotational collective rotation of the The rotation is called irrotational

because the nucleus is not quite

solid –It is largely the “skin” of

“outer shape” of the nucles that

is rotatingNuclear collective rotation occurs around an axis perpendicular to the symmetry axis “3”

Collective rotational motion

Consider energy of a rigid rotator

?

and J?I?

2

E?1

2I?2 ?J

2I

Eqn :1

2IJ?2??E?

E? ?2J(J?1)

J2I,

?J,m?YJ,m(?,?)Schroding

The nucleus can also vibrateBreathing mode –

First observed in 1977 –very

high energy

Requires nuclear fluid

compressibilityQuadrupole DeformationRotating wave with Sherical Harmonic Wavefunction –circulates –or vibrates the nucleus.

Spin = 2, Parity = +

Nuclear vibrations are bosonic106

46Pd60

The giant dipole resonancep

?High energy photon

PHOTON CROSS SECTION (mb)nEE14MeV???.c?.3x1023Fs?1??c197MeV.F22?1 ?2x10s

PHOTON ENERGY (MeV)

Nuclear magnetic moments

l

The magnetic moment of the nucleus comes about because

(1)We have charged particles –protons

moving around the center of the

nucleus (i.e. p, d, f, g etc states)

(2)Both protons and neutrons have their

own INTRINSIC magnetic moments.

?glp?N

????p?gsp?N(s/?)?5.586?N(s/?)????n?gsn?N(s/?)??3.826?N(s/?)?For the PROTON we must add the mag mom due to ORBITAL motion to get the full mag. Mom.??l?gsp?N

?sglp?1

For the NEUTRON we can write down a similar equation but define that gln=0After time averaging rapid motion about j then ?l(l?1)???i??N?(gli?gsi)j?(gli?gsi)?j?1??2gln?0

See SEC 6.6

Magnetic dipole momentsMAGNETIC MOMENT (nuc magnetons)JACKNIFEODD NEUTRONSTRETCH

NUCLEAR SPIN JShown are the

“Schmidt lines”

Magnetic dipole momentsShown are the “Schmidt Lines” named after German physicist T. Schmidt who discovered these lines empirically in 1937.

MAGNETIC MOMENT (nuc magnetons)ODD PROTONSTRETCHODD NEUTRONJACKNIFE

NUCLEAR SPIN J

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