16.451 Lecture 23: Applications of the Shell Model 27/11/2003 1

Generic pattern of single particle states solved in a Woods-Saxon (rounded square well) potential model with appropriate spin-orbit interaction to reproduce the observed “magic number” pattern:

State labels:

where n labels the order of occurrence of a given l value, and the state labels for orbital angular momentum are:

jn

ihgfdps

6543210

Each state can hold (2j +1) neutrons and (2j+1) protons, corresponding to 2(2j+1) distinct configurations of identical nucleons (mt, mj) to satisfy the Pauli exclusion principle

0, ii mMjJ

0, ii mMjJ

Quantum numbers for a nucleus:

First of all, consider a “closed shell”, which corresponds to a completely filled single-particle state, e.g. 1s1/2, 1p3/2, etc... containing (2j+1) protons or neutrons:

The total angular momentum is:

all have the same j in a given shell

each m value is different, ranging from –j to +j

There is effectively only one configuration here, with total z-projection M = 0.Therefore, the total angular momentum of a closed shell must be J = 0!

The total parity is:

12)1()1(

j

12)1()1(

j

Since (2j+1) is always even, the parity of a closed shell is always positive.

2

Valence Nucleons:

For a closed shell + n nucleons, the angular momentum and parity is determined bythe n “valence” nucleons, since the closed shell contributes J = 0+ :

nnjJn

ii )1()1(,

1

The parity is uniquely determined, but there may be several different values of Jthat are consistent with angular momentum coupling rules. Residual interactionsbetween the valence nucleons in principle determine which of the allowed J has thelowest energy – we can’t predict this a priori but can learn from experiment.

“Holes” -- for a state that is almost full, it is simpler to consider angular momentum coupling for the missing nucleons than for the ones that are present:

12

)1(1

0j

nii

n

ii jj

result fora closed shell

magnitudes of the two partial sums have to be the same, and M values opposite.

Total angular momentum for a collection of missing particles, i.e. holes, is the same as for that same collection of particles in a given shell model state.

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Magnetic Moments:

As in lecture 19, we can write:

)1(

1

JJJg NJ

)1(

1

JJJg NJ

Predictions are only simple in the limit of one valence nucleon, and then we have:

Nps sg

,

Nns sg

,

valence proton:

valence neutron:

Simplifying the results (assignment 4), we find:

Ns

Ns

jjgjjjgj

gjgj

)(2/)/()(

/)(

112/32/1

22/12/1

Ns

Ns

jjgjjjgj

gjgj

)(2/)/()(

/)(

112/32/1

22/12/1

with 0,1,83.3,58.5 npsnsp gggg

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Application with one valence nucleon: 17O

E (

MeV

)

There are 8 protons and 9 neutrons, so we only need the low lying states in the shell model spectrumto understand the energy levels:

Ground state:full to here plusone neutron

valence n

Ground state quantum numbers should be those of the valence neutron in the 1d5/2 state:

J = 5/2+

Magnetic moment prediction: j = l + ½, odd neutron = neutron = -1.91 N

measured value: -1.89 N excellent agreement!

5

Excited states of 17O can be understood by promoting the valence neutron:

E (

MeV

)

ground state full to here

½ + state

First excited state: J = ½ +

6

Excited states of 17O:

E (

MeV

)

Next excited state: J = ½ -

explained by promoting a neutron from the filled 1 p1/2 level to the 1d5/2 level

½ - neutron hole

0+ pair

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Excited states of 17O

E (

MeV

)

The 5/2- state is not so easy: to have negative parity, there must be an odd nucleon in a p state (or f state, but that is higher)

J = ?

5,4,3,2,1,0,2/52/5 JJ

But two neutrons are required to have differentvalues of mj by the Pauli principle. Writing outthe allowed configurations only J = 0, 2, 4 are allowed! (J = 2 will work here)

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Two valence nucleons: (enough already!)

This problem is much more complicated! The inner “core” nucleons couple to J = 0+

but in general there is more than one possibility for the angular momentum couplingof the valence particles.

a) (pp) or (nn) case: Z and N are both even in these cases, so we know that the ground state configuration will be 0+ no matter what

shell model state they are in. Excited states will have higher angular momentum, with possibilities restricted by the

Pauli principle.

b) (np) case: Z and N are both odd in this case. Only 6 examples in the whole nuclear chart!!! In isolation, (np) prefers to form a bound state – the deuteron – with J = 1+.

np pair: no restrictions on total J. possibilities are 0, 1, 2, 3, 4, 5

Ground state of 18F is 1+ i.e. deuteron quantum numbers!

Example: 18F, Z = 9, N = 9

N.B. valence nucleons interact with each other, orall J values would be degenerate!

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Magnetic moments revisited: (see Krane, Fig. 5.9)

Ns

Ns

jjgjjjgj

gjgj

)(2/)/()(

/)(

112/32/1

22/12/1

Ns

Ns

jjgjjjgj

gjgj

)(2/)/()(

/)(

112/32/1

22/12/1

Single particle predictions:

Data for odd-proton nuclei:

(lines are calculated from the formulae)

Agreement is notterrific, but thevalues lie within thetwo “Schmidt lines”

j

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Odd neutron nuclei:

What is wrong?• the single particle model is too simple – nucleons interact with each other• configurations may be mixed, i.e. linear combinations of different shell model states• magnetic moments of bound nucleons may not be the same as those of free nucleons...

17O example: “perfect”

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Title page of a research monograph, Oxford, 1990:

From the preface:

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Even the most sophisticated nuclear models are not completely successful!

from p. 53

Indium isotopes, Z = 49

data, for those withJ = 9/2+ ground states

theory, with residualinteractions, and g-factorsreduced by 50% comparedto free nucleons!

long-lived excitedstates with J = ½-

so, don’t be disappointedif your homework questiondidn’t work out perfectly!

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Collective Excitations in nuclei – a new class of models (Krane, Sec. 5.2)

Introduction:

• Over half the known nuclei have configurations (Z,N) even, J = 0+

• Recall that an empirical pairing term is included the semi-empirical mass formula to account for their unusual stability.

N.B. The pairing term is not accounted for in the shell model, which ignores all interactions between particles!!!

• It costs too much energy to break a pair of nucleons and populate higher single particlestates, so the excitations of even-even nuclei tend to be of a collective nature

the nuclear matter distribution as a whole exhibits quantized vibrations in some cases and rotations in others, with characteristic frequency patterns.

• Vibrational spectra are seen in nuclei that have an intrinsic spherical shape

• Rotational excitations tend to occur in nuclei with permanent quadrupole deformations

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Vibrational states:

Model: quantized oscillations of a liquid droplet at constant density (why? repulsive short-distance behaviour of the N-N force!)

Consider oscillations about a spherical equilibrium shape, with a time-dependentboundary surface expressed as a linear combination of spherical harmonic functions:

),()(1),,( YtRtR o

expansion describes any shape at all,given appropriate coefficients. Eachcontribution can in principle oscillate ata different frequency....

Normal modes of the system correspond to excitations with a particular value of and , and these will occur at characteristic frequencies.

Application to nuclei: 1. restriction to axial symmetry, i.e. = 0

2. vibrations are quantized: En = n ħ

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Illustration: time sequence of oscillating nuclear shapes

= 0

= 1

= 2

= 3

forbidden – density change!

forbidden – CM moves!

OK...

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