22.101 Applied Nuclear Physics (Fall 2006)

Lecture 10 (10/18/06) Nuclear Shell Model

References :

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), Chap.2.

P. Marm ier and E. Sheldon, Physics of Nuclei and Particles (Academ ic Press, New York, 1969), vol. II, Chap.15.2.

Bernard L. Cohen, Concepts o f Nuclear Phys ics (McGraw-Hill, New York, 1971).

There ar e sim ilaritie s be tween the e l ectron i c s t ru cture of ato m s and nuclear structure. A t om ic electrons are arranged in or bits (energy states) subject to the laws of quantum m e chanics. The dist ribu tio n of electron s in thes e states follows the Pauli exclus ion princip l e. Ato m ic electron s can be excited up to norm ally unoccupied states, or they can be rem oved com p letely from the atom. From such phenom e n a one can deduce the structure of atom s. In nuclei there are two group s of like particles, protons and neutrons. Each group is separately dist ributed over certain energy states sub j ec t also to the Pau li exclus ion princip l e. Nuclei h a ve excited states, and nucleons can be added to or rem oved from a nucleus.

Electrons and nucleons have intrinsic angul ar m o m e nta called intrinsic spins. The total angular m o m e ntum of a sy stem of interactin g particles reflects the details of the forces between particles. For exam ple, from the coupling of electron angular m o m e ntum in atom s we inf e r an in te raction betw een the spin and the orbital m o tion of an electron in the field of the nucleus (the spin-orbit coup ling). In nuclei there is also a coupling between the orbital m o tion of a nucleon and its intr insic spin (but of di fferent origin). In addition, nuclear forces between two nucleons depend strongly on the relative orientation of their sp in s.

The structure of nuclei is m o re complex than that of atom s. In an atom the nucleus provides a com m on center of attraction for all th e electrons and inter-electron ic forces generally play a sm all role. The predom inant force (C oulom b) is well understood.

1

Nuclei, on the other hand, have no center of at traction; the nucleons are held together by their m u tual interactions which are m u ch m o re complicated th an Coulom b interactions.

All atom ic electrons are alike, wherea s there are two kinds of nucleons. T his allows a rich er variety of structures. Notice that there are ~ 100 types of atom s, but more than 1000 different nuclides. Neither atom ic nor nuclear structures can be understood without quantum m echa n ics.

Experimenta l Basis

There exists considerable experim e nt al evid ence pointing to the shell-like structure of nuclei, each nucleus bein g an assem b ly of nucleo ns. Each sh ell c a n be filled with a given num b er of nucleons of each ki nd. These num bers are called m a gic numbers; they are 2, 8, 20, 28, 50, 82, and 126 . (For the as yet undiscovered superheavy nuclei the

m a gic num b ers are expected to be N = 184, 196, (272), 318, and Z = 114, (126), 164 [Marm i er and Sheldon, p. 1262].) Nuclei with magic num ber of neutr ons or protons, or both, are found to be particularly stable, as can be seen from t he following data.

(i) Fig. 9.1 shows the abundance of stable is otones (sam e N) is particularly large for nuclei with m a gic neutron num bers.

20 28

50

82

10

8

6

4

2

0

0 20

40 60 80 100

Neutr on Number (N)

Figure b y MIT OC W . Adapted f rom Meyerhof.

Fig. 9.1. Histogram of s t able isotones show ing nuclides with neutron num bers 20, 28, 50, and 82 are m ore abundant by 5 to 7 tim es than those with non-m a gic neutron numbers [fro m Meye rhof].

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(ii) Fig. 9.2 shows that the neutron separation energy S n is par ticu l ar ly low f o r nuclei with one m o re neutron than the m a gic numbers, where

S  [ M ( A  1 , Z )  M  M ( A , Z )] c 2 (9.1)

This m eans that nuclei w ith m a gic neutron num bers are m o re tightly bound.

14

10

6

2

-2

2 6 10

14 18 22 26 30

Neutr on Number (N) of Final Nucleus

Figure b y MIT OC W . Adapted f rom Meyerhof .

Fig. 9.2. Variation of neutron sepa ration energy with neutron nu m b er of the final nucleus M(A,Z) [from Meyerhof].

(iii) The first excited s t ates of even-even n u cl ei have higher than usual energies at the m agic num b ers, indicating that th e m agic nuclei are m o re tightly bound (see Fig. 9.3).

4

3

Proton

number

Z

2

N=Z

1

0 Z=20 Z=50

Z=82

0 20

50 82 126

Neutr on Number (N)

Figure b y MIT OC W . Adapted f rom Meyerhof.

Fig. 9.3. First excited s t ate energies of even-even nuclei [from Meyerhof].

	50		82			126

(iv) The neutron capture cross sections fo r m agic nuclei are sm all, indicating a wider spacing of the energy levels just beyond a closed shell, as shown in Fig. 9.4.

100

10

1

10 30

50 70 90 1 10 130

Neutr on number (N)

Figure b y MIT OCW . Adapted f rom Meyerhof.

Fig. 9.4. Cross sections for capture at 1 Mev [from Meyerhof].

Simple Shell Model

The basic a s sum p tion of the she ll m o del is that th e ef f ects of internuc lea r interactions can be rep r esente d by a single-particle potential. One m i ght think that with very high density and s t rong forces, the nuc leons would be collid ing all th e tim e and therefore cannot m a intain a singl e-particle o r bit. But, because of Pauli exclusion th e nucleons are restricted to only a lim ited num be r of allowed or bits. A typ i cal she l l-m o del potential is

V ( r )  

V o ( 9 . 1 )

1  exp[( r  R ) / a ]

where typical valu es for the param e ters are V o ~ 57 Mev, R ~ 1.25A 1/3 F, a ~ 0.65 F. In addition one can consider corrections to the well depth arising fr om (i) symmetry energy from an unequal num ber of neutrons and protons , with a neutron being able to interact with a proton in m ore ways than n-n or p-p (therefore n-p force is stronger than n-n and p-p), and (ii) Coulom b repulsion. For a gi ven sph eric ally sym m etric poten tia l V(r), on e

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can exam ine the bound-state energy levels that can be calculated from radial wave equation for a particular orbital angular m o m e ntum ℓ ,

ħ d 2 u

 ℓ 

 ℓ ( ℓ  1 ) ħ 2

 V ( r )  u ( r )  Eu

( r ) (9.2)

2 m dr 2

 2 mr 2

 ℓ ℓ



Fig. 9.5 shows the energy levels of the nuc leons for an infinite spherical well and a harm onic oscillator potential, V ( r )  m  2 r 2 / 2 . W h ile no sim p le for m ulas can be given f o r the f o rm er, f o r the la tte r one has the expr ession

E   ħ  (   3 / 2 )  ħ  ( n x  n y  n z  3 / 2 ) (9.3)

where = 0, 1, 2, …, and n x , n y , n z = 0, 1, 2, … are quantum num bers. One should notice the degeneracy in the oscillato r energy lev e ls. The quantum number can be divided into radial quantum num b er n (1, 2, …) and orbital q u antum numbers ℓ (0, 1,

…) as show n in Fig. 9.5. One can see from these results th at a central force poten tial is able to acco unt for the first th ree m agic num bers , 2, 8, 20, but not th e rem a ining four, 28, 50, 82, 126. This situation does not change when m ore rounded potential form s are used. The im plication is that som e thing very funda m e ntal about the single- particle in teraction picture is m i ssing in the description.

40

30

20

10

0

1 h 2 f 3 p 1 1 2

1 g 2 d 3 s

70

1 f 2 p

40

1 d 2 s

20

h 

1 p 8

1 s 2

1 i 2 g 3 d 4 s 168

 = 6

3 p  = 5

1 i

2 f  = 4

3 s

1 h  = 3

2 d

1 g  = 2

2 p

1 f  = 1

2 s

1 d  = 0

1 p 1 s

Figure b y MIT OCW . Adapted from Meyerhof. 6

Fig. 9.5. Energy levels of nucleons in (a) infin ite spherical w e ll (range R = 8F) and (b) a parabolic potential well. In the spectroscopic notation (n, ℓ ), n refers to the num b er of tim es the orbital angul ar m o m e ntum state ℓ has appeared. Also shown at certain levels are the cumulative number of nucleons that can be put in to a ll th e leve ls up to the indicated level [from M e yerhof].

Shell Mode l with Spin- O rbit Coupling

It rem a ins for M. G. Mayer and independently Haxel, Jensen, and Suess to show (1949) that an essential m i ssing piece is an attractive interacti on between the orbital angular m o m e ntum and the intrinsic spin a ngular m o m e ntum of the nucleon. To take into account this in teraction we add a term to the Ha m iltonian H,

p 2

H  2 m  V ( r )  V so ( r ) s  L (9.4)

where V so is another central potentia l (known to be attractive). This m odification m eans that the in teraction is no l onger spherically symmetric; th e Ham iltonian now depends on the re lative orien t ation o f the spin and orbital angular m o m e nta. It is beyond the scope of this class to go into th e b ound-state calcula tion s for this Ham iltonian. In o r der to understand the m eaning of the results of such calculations (eigenvalues and eigenfunctions) we need to di gress som e what to discuss the addition of tw o angular mom e ntum operators.

The presenc e of the spin -orbit coupling term in (9.4) m eans that we will have a different set of eigenfunctions and eig e nvalues for the new des c rip tion. W h at are thes e new quantities relative to the eigenfunctions and eigenvalues we had for the problem without the spin-orbit co upling interaction? W e first obse rve that in labeling the energy levels in Fig. 9.5 we had already taken into accou n t the fact th at the nu cleo n has an orbital angular m o m e ntum (it is in a state with a specified ℓ ), and that it h a s an in trin sic spin of ½ (in unit of ħ ). For this reaso n the num ber of nucleons that we can put into each level has been counted correctly. For exam pl e, in the 1s ground state one can put two nucleons, for zero orbital angular m o m e ntum an d two spin orientations (up and down).

The student can verify that for a state of given ℓ , the num ber of nucleons tha t can go in to that s t ate is 2(2 ℓ +1). This com e s about becaus e the eigenfun ctions we are using to describ e the system is a represen tatio n that diagonalizes the square of th e orbital angular mom e ntum operator L 2 , its z - com p onent, L z , the square of the intrinsic spin angular mom e ntum operator S 2 , and its z-component S z . Let us use the following notation to label these eigenfunctions (or representation),

ℓ , m , s , m  Y m ℓ  m s (9.5)

ℓ s ℓ s

where Y m ℓ is the spherical harm onic we fi rst enc ountered in Lec4, a nd we know it is the

eigenfunction of the square of the orbital angular m o m e ntum operator L 2 ( it is also the

eigenfunction of L z ). The function  m s

is the spin eig e nf unction with the exp ected

properties,

S 2  m s  s ( s  1 ) ħ 2  m s , s=1/2 (9.6)

s s

S  m s  m ħ  m s ,  s  m  s (9.7)

z s s s s

The properties of  m s with respect to operations by S 2 and S z com p letely m i rror the

properties of Y m ℓ

with respe c t to L 2 and L z . Going back to our representation (9.5) we

see that the eigenfunction is a “ket” with indices whic h are the good quantum numbers for the problem , na m e ly, the orbital angular m o mentum and its projection (som etim es called the m a gnetic quantum number m, but here we use a sub s crip t to den o te tha t it go es with the o r bital angu lar m o m e ntum ), the spin (which has the fixed value of ½) and its projection (which can be +1/2 or -1/2).

The representation given in (9.5) is no longer a good representation when the spin-orb it co upling term is added to th e Ha m ilton i an. It turn s out that the good repres entation is jus t a linear com b ination of the old represen tation. It is sufficient for

our purpose to just know this, with ou t going in to the details o f how to construc t the lin ear

com b ination. To understand the properties of the new representation we now discuss angular m o m e ntum addition.

The two angular m o m e nta we want to add are obviously th e orbital angular mom e ntum operator L a nd the in trin sic spin ang u lar m o m e ntum operator S , since th ey are the only angular m o m e ntum ope rators in our problem . Why do we want to add them ? The reason lies in (9.4). Notice th at if we def i ne the tota l ang u lar m o m e ntum as

we can then write

j  S  L ( 9 . 8 )

S  L  ( j 2  S 2  L 2 ) / 2 (9.9)

so the problem of diagonalizing (9.4 ) is the sam e as diagonalizing j 2 , S 2 , and L 2 . This is then the b a sis for choosing our new represen tati o n . In analog y to (9.5) we will deno te the

new eigenfunctions by

jm j ℓ s , which has the properties

j 2 jm ℓ s  j ( j  1 ) ħ 2

jm j ℓ s , ℓ  s  j  ℓ  s (9.10)

j z jm j ℓ s  m j ħ jm j ℓ s ,  j  m j  j (9.11)

L 2 jm ℓ s  ℓ ( ℓ  1 ) ħ 2 jm ℓ s , ℓ = 0, 1, 2, … (9.12)

j

S 2 jm ℓ s  s ( s  1 ) ħ 2 jm ℓ s , s = ½ (9.13)

j

In (9.10) we indica te th e values that j can take f o r given ℓ and s (=1/2 in our discussion), the lower (u pper) lim it corresponds to when S and L are antip a rallel (parallel) as show n in the sk etch .

Returning n o w to the en ergy lev e ls o f the nucleo ns in the sh e ll m odel with spin-orb it coupling we can unders t and the conv entional sp ectroscop i c n o tation where the value o f j is shown as a subscript.

This is th en the nota tion in which the shell-m odel energy leve ls are d i sp la yed in Fig. 9 . 6.

V ( r )

r

v = 5

3 p 2 f

12 6

~ 10 -12 cm

3 p 1/2

2 f 5/2

1 i 13/2

3 p 3/2

1 h

3 s 2 d

1 g

82

4

50

3

2 p

1 f

28

2

2 s

1 d

20

1

1 p

8

0

1 s

2

1 h 9/2

2 f 7/2

3 s 1/2

2 d 3/2

1 h 1 1/2

2 d 5/2

1 g 7/2

1 g 9/2

2 p 1/2

1 f 5/2

2 p 3/2

1 f 7/2

1 d 3/2

2 s 1/2

1 d 5/2

1 p 1/2

1 p 3/2

1 s 1/2

40

30

20

10

0

Figure b y MIT OCW . Adapted f rom Meyerhof.

Fig. 9.6. Energy lev e ls o f nucleons in a sm oot hly varying potential well with a strong spin-orb it co upling term [ f r om Meyerhof ] .

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For a given ( n , ℓ , j ) level, the nucleon occupation num ber is 2 j +1. It would app ear th at having 2j+1 identical nucleons occupying the sam e level would violate the Pauli exclus ion princip l e. But this is not the case s i nce each nucleon would have a dis tinct

value of m j ( t his is why there are 2j+1 values of m j for a given j).

W e see in Fig. 9.6 the sh ell m odel with spin-orbit coupling gives a set of energy levels having breaks at the seven m a gic num bers . This is co nsidered a m a jor trium ph of the m odel, for which Mayer and Jensen were awarded the Noble pr ize in physics. For our purpose we will use the results of the she ll m odel to p r edict the g r oun d-sta t e sp in and parity of nuclei. Bef o re going into th is discu ssion we leave th e studen t with the following comments.

1. The shell m odel is m o st useful when app lied to closed-shell o r near closed -shell nuclei.

2. Away from closed-shell nuclei collective m odels taking into account the rotation and vibration of the nucleus are m o re appropriate.

3. Sim p le versions of the shell m odel do not take into account pairing forces, the effects of which are to m a ke two like- nucleons com b ine to give zero orbital angula m o mentum .

4. Shell m odel does not treat distortion effects (deform e d nuclei) due to the attraction between one or m o re outer nuc leons an d the clo s ed-shell core. W h en the nuclear c o re is no t sp heric a l, it can exhibit “rotational” spectrum .

Prediction of Ground-State Spin and Parity

There are three general rule s for using the shell model to predict the total angular mom e ntum (spin) and parity of a nucleus in the ground state. These do not always work, especially away from the m a jor shell breaks.

1. Angular m o m e ntum of odd-A nuclei is determ ined by the angular m o m e ntum of the last nucleon in th e species (neutron or proton) that is odd.

2. Even-even nuclei have zero ground-state spin, because the net angular m o m e ntum associated w ith even N and even Z is zero, and ev en parity.

3. In odd-odd nuclei the last neutron couples to the last proton with their intrinsic spins in pa ra lle l orienta t ion.

To illus t ra te how these ru les work, we cons ider an exam ple for each cas e. Consider the odd-A nuclide Be 9 which has 4 protons and 5 neutr ons. W ith the last nucleon being the fifth neutron, we see in Fig. 9. 6 that this nucleon goes into the state 1 p 3 / 2 ( ℓ =1, j=3/2). Thus we would predict the spin and parity of this nuclide to be 3/2 - . For an even- even nuclide we can take A 36 , with 18 protons and neutrons, or Ca 40 , with 20 protons and neutrons. F o r both cases we woul d predict spin and parity of 0 + . For an odd-odd nuclide we take Cl 38 , which has 17 protons and 21 neutrons. In Fig. 9 . 6 we see th at the 17 th proton goes into the state 1 d 3 / 2 ( ℓ =2, j=3 / 2), while the 2 1 st neutron g o es into the state

1 f 7 / 2 ( ℓ =3, j=7 / 2). From the ℓ and j values we know that fo r the last proton the orbital and spin ang u lar m o m e nta are po intin g in opposite direction (because j is equal to ℓ -1/2 ). For the last neutron the two m o m e nt a ar e po intin g in the s a m e dire ction (j = ℓ +1/2).

Now the rule tells us tha t the two spin m o m e nta are parallel, theref ore the orbital angular mom e ntum of the odd proton is pointing in the oppos ite direction from the orbital angular mom e ntum of the odd neutron, wi th the latter in the sam e di rection as the two spins.

Adding up the four angular m o m e nta, we have +3+1/2+1/2-2 = 2. T hus the total angular mom e ntum (nuclear spin) is 2. W h at about the parity? The parity of the nuclide is the product of the two parities, one for the last pr oton and the other for the last neutron.

Recall that the parity of a state is d e term ined by the orbital angular m o m e ntum quantum num b er ℓ ,   (  1 ) ℓ . So with the p r oton in a s t a t e with ℓ = 2, its pa rity is even, while the neutron in a state with ℓ = 3 has odd parity. The parity of the nucleus is therefore odd. Our prediction for Cl 38 is then 2 - . The studen t can verify, using for ex am ple the Nuclide Chart, th e foreg o ing pred ictions are in ag reem ent with experim e nt.

Potentia l W e lls for Neutrons and Protons

W e summ ar ize th e qualitative features of the potentia l well s for neutrons and protons. If we exclude the Coulom b inter action for the m o m e nt, then the well for a proton is known to be deeper than that for a neutron. The reason is that in a given nucleus usually there are m o re neutrons than protons, esp ecially for th e heavy nuclei, and th e n - p interactions can occur in m o re ways than either the n-n or p-p interactions on account of

the Pauli ex clusion p r in ciple. The difference in well depth  V s is called the s y mm etry energy; it is approxim ately given by

 V s

  27 ( N  Z ) Mev (9.14)

A

where the (+) and (-) signs are for protons and neutrons respectively. If we now consider the Coulomb repulsion between prot ons, its ef f ect is to ra ise the poten tia l f o r a proton . In other words, the Coulom b ef f ect is a positiv e con t ribu tion to the nuclea r p o tential which is larger at the cente r than at the surface.

Com bining the symm etr y and the Coulom b effects we have a sketch of the potential for a neutron and a pr oton as indicated in Fig. 9.7. One can als o estim ate th e

Net Neutron Potential

E Net Proton Potential V C

Fermi Level E F

Symmetry Ef fect

Coulomb Ef fect

Figure b y MIT OCW . Adapted f rom Marmier and Sheldon.

Fig. 9.7. Schem a tic showing the effects of symm etry and Coulom b interactions on the potential for a neutron and a pr oton [from Mar m ier and Sheldon].

well depth in each cas e using the Ferm i Gas m o de l. One ass u m e s the nu cleons of a fixed kind behave like a fully degenerate gas of fer m ions (degeneracy here m eans that the states are filled continuo usly starting fr om the lowest energy state and there are no unoccupied states below the occup i ed ones), so that the num ber of states occupied is equal to the num b er of nucleons in the particular nucleus. Th is ca lcula tio n is car ried out separately f or neutrons and protons. The highe s t energy s t ate that is occupied is called the Fermi level , and the m a gnitude of the differen ce between this st ate and the ground

state is c alle d the Fermi energy E F . I t turns ou t th at E F is prop ortion al to n 2/3 , where n is the num ber of nucleons of a given kind, therefore E F (neutron ) > E F (pro to n). The sum of E F and the separation energy of the last nucleon provides an estim ate of the well depth. (The separation energy f o r a neutron or prot on is about 8 Mev for m a ny nuclei.) Based on these con sideration s o n e obtain s th e results sho w n in Fig. 9.8.

a

8 MeV

Fermi- level

8 MeV

b

8 MeV

8 MeV

Ferm leve

37 MeV

43 MeV

i-

l

26 MeV

23 MeV

(n) (p)

(n) (p)

Figure b y MIT OCW . Adapted f rom Marmier and Sheldon.

Fig. 9.8. Nuclear potential wells for neutrons and protons accordi ng to th e Ferm i-gas model, assum i ng the m e an binding energy per nu cleon to b e 8 Mev, the m ean relativ e nucleon adm i xture to be N/A ~ 1/1.8m Z/A ~ 1/ 2.2, and a range of 1.4 F (a) and 1.1 F (b) [from Mar m ier and Sheldon].

We have so far considered only a spheri c ally sy m m etric nuc lear po tentia l well.

W e know there is in add ition a cen trifugal contribution of the for m ℓ ( ℓ  1 ) ħ 2 / 2 mr 2 and a spin-orb it co ntribu tion. As a result o f the f o rm er the well b ec o m es narro wer and shallower for the higher orbital angular m o m e ntum states. Since the spin-orbit coupling is attra ctiv e, its ef f ect depends on whether S is pa rallel o r anti-para lle l to L . The effects are illustrated in Figs. 9.9 and 9.10. Notice that for ℓ = 0 both are absent.

We conclude this chapter with the rem a rk that in addition to the bound states in the nuclear potential we ll th ere ex ist also vi rtua l states ( l evels) which ar e positiv e ene r gy states in which the wave function is large w ithin the potential well. This can happen if the deBroglie wavelength is such that appr oxim ately standing waves are form ed within the well. (Correspondingly, the reflection coefficient at the edge of the potential is large.)

A virtual level is therefore not a bound state; on the other ha nd, there is a non-negligible probability that ins i de th e nucleus a nucleon can be found in such a state. See Fig. 9.11.

l ( l +1)

1 x 2 h 2 2 M r 2

E 4 p E 3 p

E 2 p

E 1 p

u 1 p (r)

u 2 p (r)

u 3 p (r)

2 x 3 h 2 2 M r 2

r

E 3 d

E 2 d E 1 d

u 1 d (r)

u 2 d (r)

u 3 d (r)

2 M r 2

E 3 f r

E 2 f

E 1 f

u 1 f (r)

u 2 f (r)

u 3 f (r)

3 x 4 h 2

r

E 4 s

E 3 s

E 2 s E 1 s

R

u 1 s (r)

u 2 s (r)

u 3 s (r)

V 0

r

l = 0 ( s ) l = 1 ( p ) l = 2 ( d ) l = 3 ( f )

F i g u r e b y M I T O C W . Adapted f rom Cohen.

Fig. 9.9. Energy levels and wave func tions for a square well for ℓ = 0, 1, 2, and 3 [from Cohen].