Two body forces in light deformed nuclei

Nuclear Physics 12 (1959) 314----326; (~) North-Holland Publist~ing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

TWO B O D Y F O R C E S I N L I G H T D E F O R M E D N U C L E I D. M. B R I N K

Clarendon Laboratory, Oa~/ord University ? and A. K. K E R M A N

Physics Department and Laboratory /or Nuclear Science ?*, Massachusetts Institute o/ Technology, Cambridge, Mass. Received I1 March 1959 There is evidence t h a t t h e light nuclei in t h e a l u m i n i u m region h a v e rotational states. I t is well known from studies of h e a v y rotating nuclei t h a t a n independent particle model for a deformed well as used b y Nilsson is a surprisingly good a p p r o x i m a t i o n for m a n y of the properties of these collective states. However, it h a s been clear for some t i m e t h a t two b o d y correlations are i m p o r t a n t for a more complete u n d e r s t a n d i n g of t h e situation. Because the n u m b e r of single particle states for t h e light nuclei is relatively small a n d because isobaric spin is a good q u a n t u m number, we h a v e u n d e r t a k e n some studies on t h e effect of two b o d y forces on binding energies and energy levels. In particular we h a v e employed t h e m e t h o d of Bacher a n d Goudsmit to find relations a m o n g binding energies which depend only upon the existence of two body forces a n d t h e a s s u m p t i o n t h a t t h e deformed wave function coupling scheme is a good first approximation. The relations so obtained are r e m a r k a b l y well fulfilled b y the d a t a while corresponding relations obtained for the spherical shell model are not. Some of t h e possible excited states in these nuclei are also discussed and estimates m a d e of their energies.

Abstract:

1. Introduction Recently it has been suggested that the spectra of some light nuclei can be interpreted according to the rotational model of Bohr and Mottelson 1). All levels up to 4 MeV in A1s~ and Mg 2~ can be ascribed to low rotational bands 2), while it has been suggested that the spectrum of F 1. can be explained in terms of mixed rotational bands s) using Nflsson's ~) model of an ellipsoidal harmonic oscillator well. R a k a v y 4) shows that nuclei following O le should have prelate deformations, and that in nuclei of mass 18 these deformations are already as large as those in the heavy nuclei which show the characteristic rotational spectrum. According to these calculations the deformations begin to decrease again after Mg u. Beyond S 82 the deformations t This work was done in t h e spring of 1957 while D. M. Brink was with the Laboratory for Nuclear Science. tt This work is supported in p a r t t h r o u g h AEC Contract A T ( 3 0 - - 1)-2098, b y funds provided b y t h e U.S. Atomic E n e r g y Commission, t h e Office of Naval Research a n d t h e Air Force Office of Scientific Research. 314

TW O B O D Y FORCES IN L I G H T D E F O R M E D N U C L E I

315

are probably too small for the rotational model to have any validity. Besides investigating the excited states corresponding to rotational levels in these nuclei, it is of interest to try to make some statements about binding energies and particle excitations. These quantities are strongly dependent on interparticle forces. In the (?',/') coupling shell model, binding energies with two body forces have been discussed by Talmi and Thieberger 6) using the methods of Bacher and Goudsmit e) with considerable success. They succeed in expressing nuclear binding energies in terms of a few parameters, which in turn depend upon the actual nuclear forces. If the nuclear forces were known it would be possible to determine the above mentioned parameters, hence nuclear binding energies. Talmi and Thieberger 5) however, shortcut this program and use the parameters determined from one group of data to make predictions about another group. In this work we follow a similar program with the deformed nucleus shell model 7). 2. T h e P a r t i c l e C o u p l i n g S c h e m e

Levels which are degenerate in the (i, i) coupling shell model are split in the deformed nucleus model. If the nucleus has an axially symmetric deformation, the resultant single particle levels can be classified by the component of particle angular momentum along the nuclear symmetry axis. For light nuclei where isobaric spin is a good quantum number, each level in this deformed shell model has four independent states, according as to whether the occupying particle is a proton or a neutron, with its angular momentum either parallel or antiparallel to the nuclear symmetry axis. We have a modified shell model with each level fourfold degenerate and with "closed sub-shells" occurring at 016, Ne *°, Mg *s and Si *s respectively. For nuclei with a positive deformation this scheme makes definite predictions about ground state angular momenta, since the ground state angular momentum is equal to the component K of the particle angular momentum about the nuclear s y m m e t r y axis. This scheme gives spin I = K _ 1 for odd mass nuclei in the range of mass numbers 16 < A < 20, I-----~for20
K ~--- 0,1 K---~ 1,2 K--~ 0,3

Na24: A126: A12S:

K---- 1,4 K---- 0,5 K ~--- 2,3.

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D. M. BRINK AND A. K. KERMAN

These states are degenerate in the deformed shell model without interparticle forces. In practice the states of high spin are always the ground states t, thus nuclear forces must favour the angular momentum parallel configurations. It is found that a wide range of reasonable forces reproduces this behaviour. 3.

Particle

Interactions Within

a

Sub-Shell

Model

The binding energy of one particle outside a closed sub-shell is due entirely to the interaction of this last particle with the core. Two, three or four particles in a sub-shell have a mutual or pairing energy, in addition to their interaction with the core. If we assume isobaric spin is still a good quantum number in these nuclei and subtract out Coulomb energies, we see that two particles in a sub-shell outside a closed shell can exist in three different configuarations. Corresponding to each of these two particle configurations there is a pairing energy. The states, with their pairing energies, are T----- 1, angular momenta anti-parallel, pairing energy A; T ~ 0, angular momenta anti-parallel, pairing energy B; T-----0, angular momenta parallel, pairing energy C. Provided the nuclear forces are two body forces, the interaction energy of three or four particles in a sub-shell can be expressed in terms of the parameters A, B and C. For three particles, say two protons and one neutron, we must add up the energy of pairs. The proton-proton interaction contributes a pairing energy A. The proton-neutron interaction gives a pairing energy C from the proton with its angular momentum parallel to that of the odd neutron, and a pairing {(A + B ) from the anti-parallel proton, giving a total of {(3A+B+2C). A complete shell of four particles has a mutual interaction exactly twice this, as can be seen b y counting up the energies of pairs. This result gives the possibility of making some binding energy predictions, since the binding energy of a closed shell of four particles can be predicted from the binding energy of nuclei with one and with three particles in that shell. For example in the shell 20 =< A ~ 24 we have for the binding energy of Mg4*: B(Mg~4)--B(Ne ~°) =

B(Mg~3)+B(Na~3)--B(NaZl)--B(Ne~I).

(1)

In this w a y of writing, the Coulomb corrections are already subtracted. The relation (1) has been checked in the shells ending with Ne *°, Mg 24 and Si .8 and the results given in table 1 (a) together with values of A, B and C. * If t h e s a m e t a b l e is m a d e f o r t h e o d d n u c l e i b e l o w 016 o n e s e e s t h e s a m e r e g u l a r i t y L i s, B 10, lql*).

(Li e,

TWO BODY FORCES IN LIGHT DEFORMED NUCLEI

317

TABL~ 1 (a) Binding energies of closed-shell nuclei I Predicted B.E. L

Observed B.E.

36.12

35.30

Ne2O

32.93

33.03

Mg 2'

37.74

37.60

Si2a

38.15

38.28

SS2

35.38

35.22

°,. I

A

B

C

4.5

1.6

5.5

2.5

1.6

4.3

2.3

1.6

4.3

1.9

1.2

2.7

TABL~ 1 (b) N e u t r o n binding energies for nuclei in a sub-shell B e S ( J = ½) Li'or=½)

19.32 6.75

HeSor=½) Li'(3r = ½)

C 12 Bn

18.711 11.459

Be ~ Bl°

I I

018 N 1~

15.57 10.75

C 18 N 14

J 4.88 ] 10.52

Ne 2° F 19

16.908 10.408

O l ~ o r = ½)

Mg 24 N a ~a

16.580 12.418

Ne ~1 N a 2~

6.756 11.042

23.336 23.460

Si 28 A1*T

17.191 13.1

Mg 25 A1.6

[ 7.324 { 11.3

24.515 24.4

S 82 psi

15.0 12.372

Si 2~ pao

[ 8.473 I 11.278

23.5 23.650

F18( J : ½)

--3.55 8.01 1,666 8.436

3.271 9.669

15.77 t 14.76 20.377 ?t 19.895 20.45 21.27 20.179 tit 20.077

¢ The n o t a t i o n Or ~ ~) beside Be 8, Li T, H e 5 and Li s m e a n s t h a t we h a v e used n e u t r o n binding energies t a k e n f r o m t h e J = ~ excited s t a t e s of H e 6, Li 5, Li ~ and Be T, which are a t 2.6, 2.5, 0.48 and 0.43 MeV a b o v e t h e g r o u n d states. These would be the states corresponding to the J = ½ level if a deformed s t r u c t u r e is a p p r o p r i a t e for these nuclei. The shell model predicts t h a t the s u m s should be equal for J ~ ~ states, b u t t h e y are off b y 6.3 MeV. This is n o t surprising because it is well k n o w n t h a t t h e ]-] coupling shell model does n o t work in this region. tt The s a m e s u m relation can be s h o w n to hold for t h e first four and t h e last four nucleons in a i ~ ~ shell, and therefore these are included here for comparison. t?t The n o t a t i o n ( J = ½ ) beside O 1T m e a n s t h a t we have used the n e u t r o n binding in the J = ½, 872 keV excited s t a t e of O 1T. The s a m e n o t a t i o n beside F ]8 m e a n s t h a t t h e binding of t h e last n e u t r o n in F TM is t a k e n with respect to t h e J : ½, 510 keV excited s t a t e of F ~T. The shell model predicts t h a t t h e s u m s should be equal for t h e J = ~ states, b u t t h e y are off b y 1.4 MeV.

318

V.

M. BRINK

AND

A.

K. KRRMAN

The relation (1) can be written in a more compact way in terms of neutron binding energies. Thus, for example, if each .symbol stands for a neutron binding energy we have Mg24+Ne21 = Na23+Na ~.

(la)

In table 1 (b) the neutron binding energies of the interesting nuclei are given. Relation (la) means t h a t the sums of the rows in each section of the table must be equal. The above binding energy relation is a property of the four particle structure of the levels in the deformed nucleus shell model, and also holds for 7"= ½ shells in the (?', I") coupling model. Such shells occur between C1. and O 1~ and between Si*s and S82. Figures for these shells are also given in table 1.

4. Zero P o i n t E n e r g y of P r e c e s s i o n At this stage two corrections should be made to the remarks of section 3. The first concerns the zero point energy of precession of the deformed nucleus. An even deformed nucleus in its lowest state has spin zero and its deformation is oriented at random, giving no zero point energy of precession. On the other hand an odd-mass nucleus has a resultant particle angular momentum which tends to line up the nuclear deformation in the direction of the total angular momentum of the nucleus. This partial specification of the orientation of the nucleus gives rise to a zero point precession, with an associated zero point energy 1 where Q is the precession angular momentum and v¢ is the moment of inertia of the deformed nucleus about an axis perpendicular to its s y m m e t r y axis. The precession angular momentum is associated with the collective motion of the nucleus and is equal to the difference between the total angular momentum I and the particle angular momentum j. In the lowest state of a rotational band I = I~ = 1", = K and Q, = 0. Hence E =

=

1

1

V ( I + 1 ) _ K S _ 2 ( I , i , + I , i,)

(2)

1 = ---) [ K - - 2 ( I , j , + I , j , ) + j , 2 + i , ' ] in the lowest state when K = I. In the above the total angular m o m e n t u m I is resolved along the z axis, which is the symmetry axis of the deformed

TWO

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NUCLEI

319

nucleus. In the expression (2), K / 2 , f is the zero point precession energy associated with a rigid body with total angular momentum K, and component K along the s y m m e t r y axis. It effectively reduces the binding energy in states with K ~ 0. Provided, however, the moment of inertia is the same in nuclei with one and with three particles outside a closed shell, the effect of the zero point energy cancels in the binding energy relation (1). In calculating the constants A, B and C, however, the zero point precession energy has been subtracted out. The second term in (2) is the Coriolis force acting on a nucleon in the nucleus, but is has diagonal elements only in the K -----½ case, 2(I

=

½ 1 I d . , + I , , i , I I = ½) = a,

where a is the "decoupling factor" 1). The third term in (2) can be expanded in the form 1

= X

1

(iL+i,*,)+

1

(3)

The first term in this expression is a sum over one-body operators and can be included in the interaction with the core; while the second term is a sum over two-body operators, hence is an effective two-body interaction. 5. D e f o r m a t i o n E n e r g y

The second correction concerns the deformation energy of the nucleus. R a k a v y estimates that the binding energy of deformation can change as much as one MeV from nucleus to nucleus as a result of the change of deformation. Energies of this order are far greater than the energy differences in table 1, hence demand investigation. Suppose we have a closed sub-sheU nucleus with an equilibrium deformation/9o. The energy of this core for a deformation fl near fl can be represented approximately by =

1

Eo+~El(fl--flo)

•

The energy of a single particle outside the closed sub-shell will also depend upon the deformation and for small deviations from the deformation ~o can be expressed as ,(~) = ,0+,l(~--flo). With n particles in the shell outside the core the binding energy is E = E(fl)+ne(~)

= Eo+~E1 (fl--flo)2+neo+n*l (fl--flo). Minimising this expression gives a new equilibrium deformation

(4)

320

D.M.

B R I N K AI~D A. K. KERMAN

and equilibrium energy

E = Eo+n,o-- ½n%1~ Et __Eo+n(eo el"]_½n(n_l) ex'. 2E1/ -~1

(5)

The part --e1~/2E1 in the binding energy (5) appears as an additional interaction with the core while the part --½n (n--1)e1*/E1 is proportional to the number ½n (n-- 1) of pairs of particles in the last shell, hence is an effective two-body interaction independent of spin and isobaric spin. This pairing energy is estimated to be 1.2 MeV in the 16 ~ A _< 20 shell, 0.2 MeV in the 20 _< A <= 24 shell and 0.7 MeV in the 24 ~ A ~ 28 shell, an appreciable fraction of the total interactions as given in table 1. 6. I n t e r a c t i o n s of P a r t i c l e s in Different S u b - S h e l l s In section 3 pairing energies have been investigated in states where the odd nucleons are all in the same level. In order to discuss excited states of these nuclei, or ground states of nuclei in which the odd nucleons are filling different sub-shells, it is necessary to study pairing interactions between nucleons in different levels. If nuclear forces conserve angular momentum and isobaric spin, two nucleons, one in level x 1 and another in level x 2, can exist in four different states with isobaric spin T = 0 or T = 1, and with angular momenta along the nuclear symmetry axis parallel or anti-parallel. With a two-body interaction the pairing energy of nl particles in level x 1 and nz particles in level xz can be expressed in terms of the pairing energies in these four basic states. In calculating this pairing energy we make use of the fact t h a t a two-body operator is determined by its matrix elements in two-body states. If V and V 1 are two-body potentials which have the same matrix elements in each of the four basic states of two nucleons, one in level x 1 and the other level x2, then the two potentials have equal matrix elements between general states with n 1 nucleons in level x~ and % nucleons in level x~. Let V be the given two-body potential and put 4

=

a

V,,

(6)

*~1

where V~ are four independent two-body operators. If the four coefficients a~ are determined so that V and V 1 have the same matrix elements in each of the four basic states of two nucleons, V x has the same matrix elements as V between m a n y particle states of nucleons in levels x 1 and x 2 . The state of a particle in a level is specified by giving the isobaric spin and the component of

TWO BODY FORCES IN LIGHT D]~IrORMED NUCLEI

321

angular momentum g2 ~-- -blg2] along the nuclear s y m m e t r y axis. In the following it is convenient to introduce an operator ~b = D/IQ[ instead of g2. It takes the value 1 when the angular momentum is parallel to the nuclear symmetry axis and --1 when anti-parallel. In the following the suffix i refers to nucleons in level x I and the suffix/" to nucleons in level x~. Put P l = 2~ ibm,

P~ = 2~ P~;

the sums are taken over particles in the levels xx and x 2 respectively; T~ is the isobaric spin operator for the i th particle and p, is the reduced angular momentum operator introduced above. A complete set of states for n 1 particles in level xx and n~ particles in level x 2 is specified by giving T1, T~, P1, P~ and the total isobaric spin T. We choose for the V~ V1 =

n lu 2 ,

V~ = ~ p~pj = P I P 2 , 0 V a ~-- 4 ~

l" t • lr t =

V4 = 4 Y

(7)

4T 1 • T~, --

4Q

.

where Q1 = X~P,v~ and 03 = ~.jpjv~. It m a y be shown t h a t for diagonal matrix elements Q1 m a y be replaced by Q1 = P1T1 Q1 = 0

for n 1 = 1 for n 1 = 2

Q1 = P1TI Ql=0

for n 1 ---- 3 for n 1 = 4 ;

(8)

similar results hold for O3. Finally eq. (6) for V 1 m a y be written V 1 -~ anln2--l-bP 1 P 2 + 4 c T 1 • T2--~4dQ 1 • Q~.

(9)

Putting P = P I + P 2 and T : T I + T 2 the basic particle states can be labelled by P and T, and the corresponding pairing energy by Vet. The V~T can be expressed in terms of the parameters of eq. (9):

Voo =

a - - b - - 3 c + 3d,

Vlo : a + b - - 3 c - - 3 d , Vol = a - - b + c - - d ,

(10)

V n = a+b+c+d.

The binding energies of a further set of nuclei can be correlated with the help of expression (9) for the pairing energy between particles in different levels. First we consider the ground states of three nuclei with one neutron in

322

D. M. BRINK AND A. K. KERMAN

level x2, two n e u t r o n s in level x 1, and zero, one, or two p r o t o n s in level x l . These states h a v e isobaric spins T ' ~, 1 and ½ respectively, a n d pairing energies between the levels x 1 and x~ as follows: T

=

3. -~.

E

----- 2 a + 2 c ;

T = 1: E = 3 a + c ! ( b - - d ) ( : : t : according as P = 2 or 0); (11) T = "2"" 1. E=4a. In obtaining these results from (9) we m a k e use of the relation 2 T 1 • T~ = T 2 - - T 1 2 - - T u 2. T h e g r o u n d states of the set of t h r e e nuclei with four neutrons, four n e u t r o n s and one proton, a n d four n e u t r o n s a n d two p r o t o n s f o r m a second group whose binding energies can be correlated b y calculating pairing forces between sub-shells. The states h a v e isobaric spins T = 2, T = a and T = 1 respectively, with pairing energies between the levels x 1 a n d x~: T = 2: T ---- 3:

E = 4a+4c; E = 6a+2c;

T=I

E=8a.

(12)

Making use of the relations (11) a n d (12) along with the results of section 3 we can p r o v e the following relation between neutron binding energies (assuming the m o m e n t s of inertia are all the same) O2O+Ne21

= O19+Ne 2~ = F ~ 0 + F 21,

Ne~4+Mg ~s = Ne~3+Mg ~6 = N a 2 4 + N a 9~, Mg~s+Si29 __--Mg~7+Si 3o = A12S+AI ~9.

(13)

These binding energies are listed in table 2 along with the sums of eq. (13). TABLE 2

Pairing energies b e t w e e n sub-shells 019

3.96 6.60 6.75

Ne II F zl O ~°

(10.36) (7.73) (7.57)

14.32 ---

NaS4 Mg ~6

5.19 6.96 7.33

Mg 2s N a s5 Ne 24

11.12 9.24 8.97

16.31 16.20 16.10

Mga 7 Alas Sis.

6.44 7.72 8.47

Si 3° A1so Mg ~s

10.61 9.03 8.50

17.05 16.75 16.97

FSO Nell 1~e2 $

Binding e n e r g y of the l a s t n e u t r o n in MeV (Predicted values in brackets). T h e last c o l u m n gives t h e s u m s of corresponding n e u t r o n binding energies (cf. ecI. (13)).

One sees t h a t the a g r e e m e n t is again v e r y good. P r e d i c t e d values for 0 30 a n d F zl n e u t r o n binding energies are in brackets.

TWO BODY FORCES IN LIGHT DEFORMED

NUCLEI

~23

More detailed examination of the energies enable one to evaluate some of the parameters. For example, a - - c = 1.1 Me¥, and b - - d is smaU and positive.

7. Particle E x c i t a t i o n s The nuclei in the region we are considering have energy levels corresponding to particle excitations in addition to the collective rotational levels. In the approximation of the present model and if we neglect configurational mixing, the positions of these levels depend on the single particle level spacings and on the pairing energies between and within sub-shells. These particle excitations are of two main types, the first corresponding to excitation of particles in the unfilled sub-shell, and the second to the excitation of particles in filled sub-shells into vacancies in unfilled sub-shells. In order to calculate the energies for particle excitations it is necessary to know values of the four parameters occurring in the expression for the interaction energy between sub-shells. From the analysis of section 6 it is possible to determine a - - c . There is also an indication that a + c and b - - d are small; but it is not possible to get values for all the parameters from experimental data. In the following we do not attempt to predict level positions exactly, but rather to get a qualitative picture of the particle excitation spectrum. We consider explicitly the nuclei from Mg.4 to Se *s and assume a = --b = --c = --d = 0.5 MeV. This choice of parameters follows if the nuclear force is spin-independent and if it is required that a + c ----- b - - d = O. 7.1. 4n NUCLEI: Mg** The lowest group of excited states corresponds to the excitation of one particle from t h e / 2 = ~ level to the f2 = { level. To break the closed D = _3 2 sub-shell requires an energy ½(3A + B + 2 C ) , plus the single particle excitation energy between the levels Y) -----~- and ~ = {. On the other hand there is a gain in binding energy due to the between-level pairing energy. The particle excitations can be classified according to the total isobaric spin T and total z-component of the angular momentum K. The levels with their excitation energies are: T = 1,

K = 4:

E ~- 3 a + c + b - - d ,

e = 10 MeV;

T ~ 0,

K = I: K = 4:

E = 3a+c--(b--d), E : 3a+b--3c+3d,

e = 10 MeV; e = 10 MeV;

K = 1:

E :

e=

3a--b--3c--3d,

5 MeV;

E are the between- level pairing energies and e are excitation energies. All two-particle and three-particle excitations lie at energies greater than 8 MeV.

324

D.

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AND

A.

K.

KERMAI~I

7.2. (4n+l) NUCLEI: A1~, Mg2a The lowest excited state is I = ½ arising from the excitation of the odd nucleon from the D = ~ level to t h e / 2 = ½ level. T h e excitation energy corresponds to the spacing between the single particle levels. T h e r e m a y be o t h e r low excited states coming from the excitation of the odd nucleon to the o t h e r unoccupied single particle levels ~2 = ½ and ,(2 = ~. Besides these single particle excitations, there are states corresponding to core-excitations where £2 = ~- nucleon is excited to the unfilled D = ½ state. Denoting the isobaric spin of the two odd nucleons in the O = ½ state b y T~, the levels with their predicted excitation energies are: T 2 = 1, T z = 0,

T = ~,a T = I ~, T = -if, x

K-2"3" K = a• K = 2_t. 2-

E -----2 a + 2 c ,

e = 7.6 MeV;

E = 2a+4c, E = 2a+2b,

e -----4.6 MeV; e = 7.6 MeV;

K =

7.

E :

e -----5.6 MeV;

K =

z.

E = 2a,

2a--2b,

e = 6.6 MeV.

T h e r e is a strong m a t r i x element, with value 6d, of the pairing potential, mixing the two T = y, x /2----- ~- states, and with the above e n e r g y spacings the lowest T = ½, K = a state should h a v e an excitation e n e r g y of e = 2.5 MeV. 7.3. (4n+3) NUCLEI: A1~7, Ses7 There is a considerable analogy between the core excitations of ( 4 n + 3 ) nuclei and the particle excitations of ( 4 n + 1) nuclei and vice-versa. T h e r e are particle excitations of AW and Se ~7 which correspond to lifting a nucleon from the c l o s e d / 2 = { level into the £2 ~ { level. The pairing lost in the /2 = { level is regained in the ~ level, while the pairing between t h e levels remains unchanged. T h u s the excitation e n e r g y is r o u g h l y the spacing between the single particle levels, i.e. 1.5 MeV. On the o t h e r h a n d a particle in the £2 = { level m a y be excited corresponding to the various values for the isobaric spin T 1 of the two nucleons remaining in t h e / 2 = ~ state. T h e levels with the predicted energies are: T-t

Tx

=

=

1,

O,

T = {,

K -~

12 ".

E = 2a+2c,

e=6.5

T = {,

K=

!2. .

E = 2a--4c,

T = {,

K =

121 ."

E = 2a+2b,

e = 3 . 5 MeV; e = 6.5 MeV;

s2 ".

E = 2a--2b,

e = 4.5 MeV;

±" 2.

E=2a,

e=5.5

K

T=½,

--

K=

MeV;

MeV.

Again t h e r e is a strong m a t r i x element mixing the two T = ½, K = ½ states, a n d one s t a t e arising from this mixing is predicted to h a v e an e x c i t a t i o n of E ~ 1.4 MeV. This s t a t e m a y give rise to the I = ½ level and I = • level at

TWO

BODY

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325

0.8 a n d 1.01 MeV e x c i t a t i o n e n e r g y in A127. R a k a v y predicts a close s i m i l a r i t y b e t w e e n t h e s p e c t r a of ( 4 n + I ) a n d ( 4 n + 3 ) nuclei, b u t t h e basis of his prediction neglects pairing energies. As seen above, t h e details of t h e particle e x c i t a t i o n s p e c t r u m is sensitive to the s t r e n g t h of t h e pairing forces a n d R a k a v y ' s prediction does n o t necessarily follow. 7.4. ( 4 n + 2 ) N U C L E I : Mg 26, A1~6, Se ~e

T h e nucleus A12~ h a s t h r e e s t a t e s w i t h t w o particles in t h e / 2 = { level. T h e y are as discussed in section 3: T = 1, K = 0; T -= 0, K ~ 0; T = 0, K -----5. T h e r e are r o t a t i o n a l levels built on each of these single particle states a n d in p a r t i c u l a r t h e r e is a sequence I -----0, 2, 4 . . . . built on the T----- 1, K : 0 1 e v e l a n d asequenceI= 1, 3, 5, . . . built on t h e T = 0 , K = 0 level. T h e odd r o t a t i o n a l s t a t e s are excluded in the first case because t h e i n t e r n a l w a v e function is s y m m e t r i c for r o t a t i o n s t h r o u g h 180 ° a b o u t a n axis p e r p e n d i c u l a r to t h e s y m m e t r y axis. I n t h e second case e v e n r o t a t i o n a l s t a t e s are excluded b e c a u s e t h e i n t e r n a l w a v e function is odd u n d e r such rotation. E x c i t a t i o n of a single particle f r o m the ~2 = { to t h e D = ½ level produces the states: T = 1,

K -~ 3: K -~ 2:

E --- a - - b + c - - d ,

e ~ 3.5 MeV;

T ~- 0,

K -----3:

E --- a + b + 3 c - - 3 d ,

~ = 1.5 MeV;

K ---- 2:

E ---- a - - b - - 3 c + 3 d ,

e = 3.5 MeV.

E = a+b+c+d,

e = 5.5 MeV;

I n A1~e t h e r e is a low e -~ 0.5 MeV, I ----- 3 s t a t e which m a y be the T ~- 0, K ---- 3 state. I n addition there is a set of s t a t e s coming f r o m the e x c i t a t i o n of one nucleon f r o m t h e / 2 = { level to t h e / 2 ~ { level. T h e s e are s t a t e s w i t h t h r e e particles in the /2 = { level a n d t h r e e particles in the /2 = { level. T h e y h a v e a similar classification to the s t a t e s c o m i n g f r o m exciting one nucleon f r o m t h e / 2 = { level to t h e / 2 -----{ level b u t w i t h K = 4 a n d K ----- 1 a n d w i t h e x c i t a t i o n energies a b o u t 1 MeV higher.

8. Configuration Mixing T h e question of configuration m i x i n g h a s b e e n c o m p l e t e l y neglected in t h e preceding discussion. I t m a y therefore s e e m s o m e w h a t surprising t h a t the pairing relation (1) of section 3 is so a c c u r a t e l y verified. T h e discussion of single particle e x c i t a t i o n s in section 7 provides s o m e reason for this. F i r s t l y t h e 4n nuclei h a v e a v e r y low d e n s i t y of excited states. T h e lowest I = 0 excited s t a t e is p r e d i c t e d to occur at an e x c i t a t i o n of 9 MeV. This low d e n s i t y of excited s t a t e s should m e a n t h a t the configuration m i x i n g in the g r o u n d s t a t e is small. T h e m a j o r f a c t o r c o n t r i b u t i n g to configurational

~26

D. M. BRINK AND A. K. K~RMAN

mixing in the ( 4 n + l ) and (4n+3) nuclei is the Coriolis force or rotation particle coupling s), and it has been shown by Rakavy 4) and Paul 3) that this force modifies the energy spectrum of light odd-mass nuclei in an important way. This work shows, however, that the energy shift of the ground state due to rotation particle coupling is always rather small, for it almost never happens that a low excited state has the same angular moment u m as the ground state. We have argued that the configuration mixing should be small in the ground state of 4n, ( 4 n + l ) and (47,+3) nuclei, and thus provide a reason for the validity of eq. (1). On the other hand (4n+2) nuclei have a much higher density of excited states, and it is possible that here configuration mixing is much stronger. In particular the value of {(3A + B + 2 C ) obtained from the binding of the lowest states of (4n+2) nuclei may easily be larger than the value found from relation (1). This is because the first approximation to the energy in the projected wave functions only involves matrix elements of one and two body operators in the deformed well wave functions. Recent detailed work by Redlich 11) and Levenson 12) shows that the projected wave functions in fact agree with the detailed shell model wave functions obtained by configuration mixing. Finally, it may be worth remarking that the above theory fits perfectly s) into the framework of the projection theory for rotational states considered by Wheeler, Wigner, Peierls and others 10). References 1) A. Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) 2) Litherland et al., Phys. Rev. 102 (1956) 208 3) E. B. Paul, Phil. Mag. 2 (1957) 311; E. B. Paul and J. H. Montague, Nuclear Physics 8 (1958) 61 4) G. l~acavy, Nuclear Physics 4 (1957) 375 5) I. Talmi and 1R. Thieberger, Phys. Rev. 103 (1956) 6) R. F. Bacher and S. Goudsmit, Phys. ]Rev. 46 (1934) 948 7) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955); K. Gottfried, Phys. Rev. 103 (1956) 1017; S. A. Moszkowski, Phys. Rev. 99 (1955) 803 8) A. K. Kerman, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 15 (1956) 9) A. K. Kerman, Nuclear Reactions, Vol. I, edited by P. M. E n d t and M. Demeur (North Holland Publishing Co., Amsterdam 1959) Ch. X, section 15 I0) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102; M. Redlich and E. P. Wigner, Phys. Rev. 95 (1954) 122; R. E. Peierls and J. Yoccoz, Proc. Phys. Soc. A 70 (1957) 381 11) M. Redlich, Phys. Rev. 110 (1958) 468 12) C. A. Levinson, private communication