Statistical derivation of the nuclear rotational energies

] L N U O V O CIMENTO VoL. X I I I , N. 3 1 ° Agos to 1959

Statistical Derivation of the Nuclear Rotational Energies.

K. KI:MAI~

Tara It~slilule o] Fu~tdamenlal Research - ]¢ombay (*)

(ricevuto i] 21 Aprile 1959)

Summary. - The nucleus is treated as a Fermi gas under the constraint of a given angular momentum I. I~s energy is expressed as a l)ower series in I ~, and the rotational and rotation-vibrati(m interaction terms are identified. The interpartiele interael ions do not inlluenee ~he rotal ional ener~'ies whereas they have important influen(.e on ~lle rotation-vibration energies. Several models for an~ular momentum production m'e considered. The experimenlal trend of the moments o:t' inertia is rei)rodueed t)y a model in which only nucleons outside a (,ertain (~ core)) produce the total angular molnentum. The vibrational frequency is insensitive to such models. The surface effeeis and lhe influence of velocity dependent forces are also taken into aeeouni.

1. - I n t r o d u c t i o n .

The p rob lem of t:he rot, a t ionM st,~tes in nucle i ha.s a . t t raeted cons iderab le

t t t t en t im l in r ecen t years (-*). The p rob lem has two asI)eets. The first, con-

n e c t e d with the spins, par i t ies a n d the ra t ios of the ener~'ies of the exci ted

s ta tes seems to be fa i r ly well unde r s tood . E v e n models s t a r t i n g f rom very

di f ferent ideas are in a g r e e m e n t as far as these proper t ies are concerned. The

second aspec t is connec t ed witih the abso lu te m a g n i t u d e s of the energies a nd

wi th the v i b r a t i o n a l 1o(q'turbation of the ma in r o t a l i o n a l terms. The m a i n

(') ] 'a r t of this work was done while 1he author was at |lie I)epartmen~ of ]']lysie~;, Purdue University, Lafayette, Indiana,.

(1) S. A. M()szKo~vst;l: Eucycl. -phys., 89, 411 (1957). ('-') ]). R. I~XGL~S: NUel..Phys., 8, 125 (1958). (3) L. M],ilC~IS~],m: N~cl. })hys., 8, 493 (1958). (a) H. J. ]APK]~,: Nucl..Phys., 8. 421 (1958).

5 9 2 x . XIIMA]¢

problem here is to account for the empiric~fl values of the moments of inertia.

The idea common to all previous approaches (1-~) is to find a conne(,tion bet-

tween a sheU-model type of description and the collective model. Depending

on the approximat ion and point of view a var ie ty of answers may be obtained.

The next step is to s tudy the effect of internucleonic interactions on the mo-

ment of inertia. One treats the interaction in a shell-model scheme and then

goes over to the collective description.

In the present work ,~n alterm~tive approach to the second aspect of the

problem is advanced. I t was shown by SESSL]~l~ and 170I,EY (s), is their sta-

tistical t rea tment of an atom, t lmt the requirement tha t an a tom (Fermi gas)

possess a net, angular momen tum leads to a rotat ion of the a tom as a whole.

This method a.pp]ied to nuclei gives a simple way of introducing collective

rotations. The question of the connection between shell and collective models

does not arise here. The internucleonic interactions are then taken into account

and their effect on the rotat ional and rotat ion-vibrat ion energies is discussed.

In the next section the basic method for calculating the rotat ional and

vibrat ional-rotat ional energies on the statistical model is given. I t is shown

tha t the rotationM energy (~ I ( I + 1 ) ) is independent of the s trength of two-

particle interactions while the vibrat ion-rotat ion energy (~ I ~ ( I + 1 ) ~) is con-

siderably influenced by them. The moment of inertia is found to be sensitive

only to the way in which the anguh~r m o m e n t u m I is produced by the nucleons,

whereas the vibrat ional frequency is shown to be insensitive to this. In Sec-

t ion 3 various models are discussed for the moment of inertia with 1he view o f

reproducing the observed trends. In Section 4 the effects of the finite size of the nuclens a.nd velocity dependence of nuclear forces arc taken into a(.(.ount.

I t is shown t.hat the velocity dependence does not give as large a factor to the

moment of inertia as one obtains from simple subst i tut ion of the effective for

the actual nucleon mass in the final expressions. When all such effects are

taken into account and an appropriate model for the mechanism of angular mo-

men tum product ion is introduced, agreement with cxperimentM trends is ob-

tained. One does not expect a st~tistical model to allow a determinat ion of

spins and parities of the states. The present investigation takes these things

for granted bu t provides a new way of looking at the energies.

2. - R o t a t i o n a l e n e r g y a n d v i b r a t i o n - r o t a t i o n i n t e r a c t i o n .

Following SESSLE1¢ and J~'()],EY (5) w e observe tha t the center of the Fermi

sphere in the momentum space has to be displaced from the origin by an

amount D n in order to produce a net angular momentunl /~ of a Fermi ~'as

(5) A. M. S~'SSL1~R and H. M. FOLd:Y: t)hys. Ret,., 96, 366 (1954).

S T A T I S T I C A l , D E R I V A T I O N OF q ' ] [E N U C L E A I ¢ ]~.OTAT]()NAL E N E R G I E S 593

:,t zero t empera tu re , s:ly of neutrons. I t is given by tile reh~tion

(la) In =f(r X Ds(r)) 2N(k)d3k d'~r,

fl(r X Dn) on(r) (lb) =. ,

where _Y(k) is tile densi ty of m o m e n t u m states and ~.(r) tile neu t ron den-

sity (*). Tile protons in the nucleus are t r ea ted as another independent Fermi gas.

A similar expression for its angular m o m e n t u m , I , can also be wri t ten. Tile

to ta l angular m o m e n t u m of the nucleus is now wri t ten as

(,_,)

Eqs. (1) and (2) are based on the assumpt ion t ha t a simple addit ion of tlle

nngular m o m e n t a arising f rom different par t s of the nucleus takes place, The significance of D mus t be noted. SESSLER and FOLEY (5) have pointed out

t h a t if one follows the usual procedure of maximiz ing the probabi l i ty of distri- bu t ion with the const ra in t of constant angular m o m e n t u m together wi th the

usual constraints of to ta l energy and number of particles, one gets a m o m e n t u m distr ibut ion which is equivalent to a Fermi m o m e n t u m sphere displaced f rom

the origin. The co-ordinates of the centre of the m o m e n t u m sphere are given

by tile Lagr~nge pa rame te r s associated with the constra int of angular mo-

men tum. These :n'e denoted b y D . Then the equ. (~) and (2) are wr i t ten down following SESSLE[{ ,rod FOLEY. D's are s<) htr quite arbi t rary . F r o m a different point of view also one m a y expect t h a t s tar t ing f rom "/ knowledge

of the interact ions it will be possible to express tile to ta l angular m o m e n t u m of a nucleus (or an a tom) as an integral over the densi ty as done in (t). Then D ' s will no longer be a rb i t r a ry bu t will be derived f rom a knowledge of

interactions. This is of course not possible a t present. One therefore prescribes some way of determining D's . SESSLEn and FOLEY (5) have used a var ia t ional

prineiple (v.p.) in which the energy is minimized to determine D. This is a

reasonable procedure and we shall adop t it in the present work. I t is quite

clear, however, th:lt wha t we achieve b y equ. (l) and (2) and the v.p. for D

is only a eertain model for angular m o m e n t u m coupling. I t is possible to

(,onstruct other models b y supplement ing tile v.p. for D with certain other

restrict ions on its form.

(') Wherever the variables of integration are omitted they are to be understood as the three-dimensional volume element d~r or d3r '.

594 K. KU.~A]~

To illustrate the general features of the method we use in this section the

model of angular m o m e n t u m coupling in which D, and D,, are to be deter-

mined entirely by the variat ional procedure. No other restriction on the form

of D , D~ is placed. We shall calculate the energies up to the term ~ 14, the

so-called vibrat ion-rotat ion terms. We shall put 1 = t i = e = m = ] 8 3 7 m ~ - m - - ~ m . In these units

unit length = 2.88-10 -~ cm

unit energy ---- 4.998.10 -~ MeV or 1 MeV = 20 n.u.

The name nuclear units and abbreviat ion n.u. is used for these units.

The total energy ~" of the nucleus will be the sum of the following kinetic

and potential energies:

f ,. 1D2

(3b) T, = f (cg~ + ½D~9,) ,

(3c) V,~ = ½fv~p(r', r) Qp(r)Qv(r') ,

r') ~n(r) 9 , (r ' ) , (3d)

Y-,,,, ~fv~ ' (3e) = .~(r , r)9.(r)o~(r'), -j

where c=3(3x~)~/10; the subscripts p and n refer to protons and neutrons

respectively. Other notations are standard. Tile unknown functions D , D~, 9~, 9, of equation (3) are now found by

minimizing the total energy C, under the constraints of constant numbers of

particles and constant value of (I) 2 and I . I n the usual method of Lagrange

multipliers the quant i ty to be varied is

(4)

where %, an, ). and ){ "ire constants to be determined later from the equations

representing the constraints, e~ is the unit vector in the z-direction.

Variations are taken with reference to the eight unknown functions and

the requirement

(5) ~e = 0 ,

with reference to each of them determines their respective forms. The re-

STATISTICAL I).ERIVATION OF TI lE N U C L E A R ROTATIONAL E N E R G I E S ,~9,~

sults are

(6)

(7)

(~)

D = D = D~, = - - r × ( 2 I + 2 ' e ) ,

'; -rE - - ~ D , , ~ a. o ~ ( + o ( r ' ) ] , 3 e ~ ( r ) : - i ~ , V r ' ) I:.n

/ 5 0 - r r - ~ p ~ ( r ) = l D 2 ~ a " ' -~ ~ : IVan,%( ) + ] , . . % ( r ) ]

F r o m (1), (2) a.nd (6) 2, 2', and D-" are determined. I f the densities ~o. and o

a.re cylindrically symmet r i c and if the vector model is assumend for I , i .e . , if

(9)

(10)

(u)

where

(12)

and

= ½ we have

2

2, 1

D 2 = D 2,, = D~ = 2~I2r~q~(O) ,

~(0) ~,.(0) ½(l +(.os~O)

~%(r) o ( r ) + ~)~(r) .

We shall confine ourselves to even-even nuclei only. In t ha t c~se the ground

s ta te has 1 = I x = 0 and for exci ted s ta tes I z - - O . Thus 2 ' = 0 in all the eases to be t r ea ted here.

Equa.tion (9) shows t h a t 2 -1 has the form of the m o m e n t of inertia of tile nucleus. Wi th ~(0) given by (12) it, corresponds to a. rigid m o m e n t of inertia,. q~(0) will be referred to as the flow-function.

Since the equat ions (7) and (8) are of tile same form we shall ma.ke the reasonable assumpt ion t h a t on(r ) and ~%(r) are propor t ional to each o ther

and tba.t

(,3+ ÷ o ooIr'll @ ( r ,

The equations (7) and (8) are decoupled by this assumpt ion and

(~)

596 ~ . KU~rA~¢

For the ground s ta te of even-even nuclei D 2 = = 0 f rom equqtion (11). Then

the eha, nge in densi ty ~%(r) iu going f rom the ground sta,te to an exci ted

s ta te wi th tottfl a,ngular m o m e n t u m 1 is given by :

j, . 10c _~ 1D~ ~ r r' r'

(15) 9 ~)o,; (r)~(r)=~ .+~a=-~ }~( , ) ~ ) ( ) ,

where 90=(r) is the ground s ta te density a.nd 3%, the ('ha.nge in the cons tant a , is de te rmined f rom the condition

whieh in a consequence of the (.onstraint of cons tant number of pa.rtieles

f C,,(r) = N .

The neutron cont r ibut ion to tlle energy of the excited sta.te m a y now be e~deulated f rom equat ion (3), using (7), (8), (11), (13) and (15). In t e rms of

~0~ and ~o0L ~ and D of the exci ted st~te we h~ve, to second order

o

Adding the neu t ron and pro ton contributions, we have, because of the linear

dependence on 3ff :rod ~, the to ta l energy

where 8~o..,-: ~) .+~Cp. ~Note t lmt f rom (9) and (11)

where 2o~=fr~cf(O)oo.~(r), the <~moment of Inert, ia~ of the nucleus in the

ground state. Sinee 8[)A~2212 we have up to te rms in 14:

(20) E , = ~l~2012fr~cf(0)O~oA(r ) __ ~1 X~J~fr~?(o)~e~(r). .

(*) I t is easy to show that in ibis method a one body potential energy of the form T?:fQ_)=(r)~.(r) will not contribufe to ~.he energy of the excited state.

STATISTICAL DEI¢IVATION O1," TItE NUCLEAR ROTATIONAL I':NERGIES 597

Rei)la('in,o" 12 by its q u a n t u m meehani( 'al value / ( / d - l ) we get the rota-

t ional energy f rom tit(' first te rm of (20)

I ( ~ + ] ) (21) (El)r°t - 2~r2q~(0)Oo~(r) "

I t is seen t h a t the ro ta t ional ener~T is determined entirely by the ground state de~mity and the flow-function of(0). This ~(0) is de termined by the a n ~ ' l l ] a r

m o m e n t u m eoupling (Eq. (11)). Fu r the r Eq. (21) does not eontaill ~OA in il~ a.nd is there, fore independent, of the int.era(,tion V(r, r ' ) . I t is impor t an t to

note t lmt here the ro ta t ional energy is determined by the angular m o m e n t u m

eoui)lino" s(,heme a, nd is otherwise independent of the two p a r t M e interact ion

stren~'th. No su('h clear ('ul. sepa=ralion of these lwo causes o('eurs in other t r ea lments of this problem and eoml)arisons may not. be justified. However ,

i t seems to point towards a s('heme in whi(.h 1,he angular n n ) m e n l ; 1 H q l is i ) r o -

duee(t t)y the (,ombined effect of only q. relat ively small number of nucleons.

The present; model, in whi('h D is de termined ent irely by the var ia t ional pvin-

,.iple, gives rise to ri~'id votati(m and is thus mm('( 'eptabh,. Before introdu('in~ models tihat give be t t e r atzreement wilh experinlental data we analyse the

l)ehaviour of tim se(.,m(l order terms. The se(.ond order te rms in equat ion (20) represent the vibrat ion-rota t ion

interact.ion energy :

~' 2~I'~(I ~ ~l )2 *2q)(O ) (SOn ~_ 81)p) , ( 2 2 1 = .

80, , is degermined f rom equat ion (15) and (16) and there are (,orre~ponding" equat ions for gffv- To i l lustrate the l)ehaviour ,)f ghi~ terni we solw~ tim equ'.~-

t ion (151 for the ease where

,(23) V(r, r ' ) = V'o 3(r - - r ' ) ,

where 1~'o, is a ( 'onstant of the dimensions energy densi |y. The ,~;olution of (15)

and (16) Ul) to terms in 2~1" is

(24)

w h e r e

(:~5)

Substit, uting in ( ' 2 2 ) w e lmve

9 /~412 (/" ~_ ] 12 !/r40Q2~o A (26) (E,)ro~.,, ~ S0c

8e'~(r) = - - 20c z ° l ~" / - - ~ S - - r2q~(0)] '

9 . . . . ]-~

(h' >oA)21

39 - I I A'!~9~,o C i m e M o .

5 9 8 K. K U M A R

where we have assumed tha t ~o:, = 2~o.= 2~)o,- Actually, the non-linear dependence of 3~0A on ~on and ~o0p introduces ~ dependence on neut ron excess bu t we ignore those terms.

F r o m (25) and (26) we see t h a t the v ibra t ion- ro ta t ion energy depends on

the residual two part icle interact ion. Since V'o. has been defined to be posi-

t ive the effect is to increase the v ibra t ion- ro ta t ion contr ibut ion to the energy.

To fur ther i l lustrate this point we adop t a constant density model for the

nucleus. The conclusions are found not to change much when more realistic,

smooth densi ty dis tr ibut ions are taken. In this case the analogue of (25) which

will be used in (26) is given b y

(27) ~ L : d~,[1 - o . o 3 h ] -~ ,

where V o is an average potent ia l energy of residual two part icle in teract ions in MeV.

For cons tan t densi ty inside a spheroid of bounda.ry given b y

(28)

(26) reduces to

(29) (Ez):o~.~ --

where

(3o)

,r = Ro] = Ro(1 + ~P~(cos 0 ) ) ,

+1

( f f f " > =f ~.(O)fm(O) d(cos 0 ) . - 1

Comparing with the expression in te rms of the f requency of v ib ra t ion oJ (6),

(31) (E,)ro~.,b = ---2 \ E l \~s) I s ( f + 1) .

One obtains for the v ibra t iona l f requency oJ, which is an average of [ and

y-v ibra t ions

160c 7 (ffAo~t ((p]5} (32) ~,2 = ~ " ~ \ 5 - 1 n°~(1 - 0.03vo)

(~2/,) 25 { IU 1

(~) A. BOaR and B.R. MOTTELSON: Mat. Fys. Medd. Dan. Vid. SeIsk., 26, no. 16 (1953); A. BonR: Rotational States o] Atomic Nuclei (Copenhagen, 1954).

STATISTICAL DERIVATION OF THE NUCLEAR ROTATIONAL ENERGIES 599

Pu t t i ng in numerical values for c2. A and R o = I . 2 A "10 era,

(33) ~ , ~ 350A ~(1 - - 0.031~,) t .

The dependence of (o on the deformat ion p a r a m e t e r is ve ry weak due to the fac t t h a t i t appears only in angular integrals which in turn occur only as

ra t io in the expression for ~o, Eq. (32). Wi thou t interactions, i.e. with V o = 0,

a value ~o~62 MeV is obta ined. I t is abou t 10 to 60 t imes larger than the

empir ical values ob ta ined b y fi t t ing the exper imenta l ro ta t ional energies to

a form (1,6):

I ( I + 1) 2i~( i+ ~)~ (34) E , = - 2~ - - ~ a - "

I t is quite possible to t ry to explain the coefficients in the two te rms independent ly bu t t ha t p~ocedure will not take into account the physical con-

nect ion between the two terms. F rom (33) it is seen t ha t in order to bring ~o to its observed value the inter-

act ion potent ia l 1 ~ 3 0 MeV it needed. This is a reasonable value. I t has thus been shown t h a t interact ions are very i m p o r t a n t for v ibra t ion- ro ta t ion

interact ions. Their effect is to make the nucleus (, soft ,) for vibrations. On

the other hand the f requem T of vii)ration, ~,), is a lmost independent of defor-

ma t ion and the angular m o m e n t u m coupling scheme (i.e. of the flow funct ion

F(0)). We base this r emark on explicit caleulations for several models to be repor ted below with a uniform densi ty distr ibution. The reason for insensiti-

v i ty to the densi ty distr ibution is pa r t l y the following. The density distri-

but ion for a spheroidal nucleus is obta ined f rom t h a t of a spherical nucleus b y repla(.ing r/Ro in the la t te r b y r/R, where R--RoJ(O ). Any integral in- volving the funct ion 5 ~ of 0(r) and ~(0) ('an be expressed as follows:

(35)

where x is a dimensionless var iable wi th the range c~ ~ x ~ 0. Unless there

is something capricious abou t ~ (x ) the cqu. (29) will continue to give a good

approx ima t ion to (26).

I t m a y be recalled t ha t in the usual theory (1,.7)

(36) 09 \B2] '

(7) G. M. TEMPER and N. P. HEYDENBERG: Phys. Rev., 104, 967 (1956).

600 K. KU MAf¢

where C2 and B,: are constants occuring in the collective Hami l ton ian

C., is the so-called surface tension and B2 has the significance of ~ mass.

These two te rms have been found to be quite sensitive to the shell s t ruc ture (1,7).

Na tu ra l ly one c,~nnot explain this shell dependent behavior f rom a stat ist ical

t r ea tmen t . However , the st~tistic~d m e t h o d m a y apply to (,) in vir tue of

Equ. (36) ~md the fac t t h a t bo th C2 mid Be seem to have similar dependence

on the shell s t ructure.

3. - Models for the m o m e n t of iner t ia .

I t has been shown al)ove t h a t the momen t of inertia, whii 'h is given b y

(38) a = )~o 1 = ? 2 ~ ( 0 ) ~ o A ( r ) ,

depends only on the angular m o m e n t u m coupling. The model considered above gives rigid rotat ion, l lowever , empir ical ly the momen t s of inerti:~ are

more sensitive to deformat ion nnd are sigIdfieantly less than the rigid wtlues.

Because of this it has long been suspe(.ted (8) tha t the ro ta t iona l st~ttes arise due to the ('o-opera.tion of a. reh~tively sma.ll numbe r of p'~rticles. This can be given a. concrete expression in the present model by writing, instead of equ. (2)

(4o) i @r × D ) ~ ( r ) o.~(r) ,

where ,~]~(r) is ~t funct ion which picks out the par t s of ¢~(r) t ha t ( ,ontribute to the angub~r moment . A full specification of ~j~(r) a, nd the prescr ipt ion for

de termining D const i tutes a model. The subscr ipt of ~]~ distinguishes ~/'s of

different models. Following' models ma5 be ment ioned

a) D to be de te rmined b y v.p. and '~]r---- 1 h'~s been tree,ted above. I t

gives rigid rotat ion.

b) D to be de te rmined by v.p. and

(41) ~]l, = :l r > Ro(1 - - ~b) ,

is) K. W. FORD and D. L. HILL: Ann. ~ev. Nuc. So., 5, 25 (1955).

S T A T I S T I C A L ] ) E R I V A T I O N O F T H E N U C L ] ~ A R I ¢ O T A T I O N A L E N E R G I E S 601

where G~c~/2, corresl)onds to rigid ro ta t ion of m a t t e r outside a, spherica.1

inert core (prol~te shapes only).

c) D to be de te rmined by v.p. and

f % = 0 r < R u ~ , / , ( 4 2 ) 3

[ '~], = 1 r>~ R o C ] ,

where c , < 1, corresponds to ri~'id ro ta t ion of m a t t e r outside a spheroidal inert

core of same shape as the nucleus.

d) D := De=, D rio be del, ermined by v.p. and ,~],,= I corresponds to a

qu,~si-laminar flow with all the nucleus p~rtieipatin}a'.

e) D = D e z , D to be de te rmined by v.p. and

[ ~]~- 0 r < . R o ( l - - O . 5 a - - s ~ . 2 ) , (43) {

[ ~]o = 1 r : . R 0 ( 1 - - 0 .5c~ - - z ~ '~) ,

which corresponds to a flow of the same type :is in d) bu t only the. m a t t e r outside the core defined b y (43) takes pa r t in the flow.

For :ill these models using (40) in Equ. (4) and the v.p.

( 4 4 ) D - - r x [21 -F- 2 ' e : ) , ~ ,

(45) D e _ 2el"r2%(O),,/~.

For t.he models a), b) and c)

' (1 + cos- ' O) (46) %(0) : c;~(O) = ,,

and for the models el) and c)

(47) %(0) -- 9,,(0) := shl~ 0 .

The m o m e n t (if inertia, for the models is then wri t ten schemat ical ly

o 5

(48) .% = 3~ ;~(0)~-~! > 4 9 , / 0 ) P > '

~ , - - ~ r and hence model a) is ruled out.

~0 cannot be made to fit emperieal da ta wha teve r form one m a y choose for ~'~ as funct ion of ~ or otherwise.

602 x. ~:UMAR

~c can be m a d e to fit t he d a t a s o m e w h a t b y p u t t i n g ec = (1 - - 0.5:¢)" 0.84

(See Fig. i curve (i) a nd also Equ . (54)). This is one of the m a n y

th ings one m a y t r y and is no t sa t i s fac tory .

~ d - ~ 3 , (1 - - 1.5 a) a nd is ru led out .

~ gives the bes t fit to the d a t a if s~= 2. See fig. 1 curve (ii).

~ o d e l s d) a n d e) give rise to "~ pecul iar flow, equ. (47). P e r h a p s one should

no t t ake it l i terally. Bu t , the t rue phys ica l p ic ture , if it is a t all possible, m a y

be e x p e c t e d to be s o m e w h a t c loser to this r a t h e r t h a n to the r igid flow of

E q u . (48) in v iew of the success of this model . The conclusion one can be

more sure of is the necessi ty of an iner t core as far as the angu la r m o m e n t u m

0.5

0.4

0.3

0.2

o.1

°o o:z o4

is concerned.

I n c o n t r a s t to tile i m p o r t a n t dif-

ference t h a t these models m a k e to

the r o t a t i o n a l energy t h e y give prac-

t ica l ly indis t iguishable resul ts for vi-

b r a t i on - ro t a t i on energy. The finer de-

tails of de fo rma t ion dependence of (29)

are indeed different b u t t h e y are also

u n i m p o r t a n t .

Fig. 1. The moment of inertia on statistical model. Curve (i): model (c), with effective mass and surface corrections, s~--(1--0.Se). Curve (ii): model (e), without

effective mass and surface corrections, e~=2. Curve (iii): model (e) with surface and effective mass corrections, e~:3. The circles are obtained using the experimental data

given in rcf. (11) ~ ~ 0.63fl(1 - - 0.16fl + ...).

4. - Surface and effective mass effects.

Surface effects m a y be p'~rtly t a k e n care of b y using for the dens i ty of

m o m e n t u m s ta tes (9)

(49) y ( k ) d3k = d3k (1 - - \ k . ] '

(9) D. L. HILL and J. A. WHEELER: Phys. Rev., 89, 1102, (1953); K. A. BRUECK- •ER: Phys. Rev., 96, 508 (1954).

STATISTICAL DERIVATION OF ThE NUCLEAR ROTATIONAL ENERGIES 603

where k0= 3~/4Ro for the cuse of the nucleus. The effect on the kinet ic

energy is to mul t ip ly the expressions (3a) and (3b) b y a fac tor

( 5 0 )

and the expressions of the

b y a fac tor

(51)

angular m o m e n t u m (16) and (40) are mult ipl ied

~ '- (1~- k0/2kr) ;

in all o ther expressions ~' occurs in exact ly the same way as ~ ( r ) does. The

Fe rmi m o m e n t u m k F m n y be assumed constant inside the nucleus, hence

(52) ko/k F ~ 3z/4Ro(3~2~) ~ ~: ].23 N - t .

Effect ive muss corre('.tion can be taken into account b y taking another fac tor ~ with the kinet ic energy expressions (3a) and (3b). To be correct this

should be a funct ion of r and occur inside the integral sign (~0). The proce

dure for calculation of energies goes through wi thout much change. Finally,

(53) @~ = j r 2 % ( 0 ) ~ ( r ) ~1'~., ~,, ~oA (r) .

This m a y be expressed to a good approx imat ion b y

(54)

is a cons tant de te rmined by tak ing appropr ia te averages. Fo r ~ the potent ia l model of F r a h n and L e m m e r was used and ~ value of ~ ~ 0.84

was found. ~ is not a free p a r a m e t e r but is de te rmined f rom the density

of tile nuclear m a t t e r Eq. (52), and the ana.lysis of the m a n y body nuclear problem (~o). When this fac tor is used then in model e) with e~ == 3 gives the

curve (iii) in Fig. ] . This we consider to be the best, curve. The s~me factor

has ~tlso been used with other models. These corrections do not affect the v ibra t ion- ro ta t ion energy very much.

(lo) H. A. BETHE: Phys. Rev., 103, 1353 (1956); W. E. FRAHN and R. It. LEMMER: ~¥UOVO Cimento, 5, 1565 {1957); 6, 1221 {1957).

(11) K. ALDER and others: Rev. Mod. Phys., 28, 432 (1956), Table V2.

604 K. KU_MAR

5 . - C o n c l u s i o n s .

A st~Ltistical m o d e l is d e v e l o p e d w h i c h g ives a c ons i s t e n t a c c o u n t of t h e

r o t a t i o m f l s p e c t r a of nucle i i n c l u d i n g t h e v i b r a t i o n - r o g a t i o n i n t e r a c t i o n effects .

I t is shown t h a t in o r d e r to o b t a i n co r r ec t m o m e n t s of i n e r t i a i t is n e c e s s a r y

to a s s u m e t h e p resence of an ( ( iner t core )) in t h e nuc leus w h i c h does not,

p a r t i c i p a t e in p r o d u c i n g t h e a n g u l a r m o m e n t u m of t h e r o t a t i o n a l s t a t e s . The

short~ r~mge i n t e r a c t i o n s a m o n g t h e nuc le i h a v e no effe(,t on t h e i r m o m e n t s

of i ne r t im F u r t h e r it, seems t h a t t h e flow in t h e a c t i v e o u t e r reg ions m u s t

d e v i ~ t e s ign i f i can t ly f rom a r i g id flow. On the o t h e r h a n d v ibr :~ t ional fre-

q u e n c y is p r a c t i c a l l y i n d e p e n d e n t of t h e f lou r p a t t e r n a n d d e f o r m a t i o n . I t is,

however , s t r o n g l y d e p e n d e n t on t h e i n t e r a c t i o n s t r e n g t h . I n t h e d - func t ion

l i m i t f~ s t r e n g t h 1 'o= 30 3IeV gives co r r ec t resu l t s .

I t is a. p l e a s u r e to a .cknowledge t h e very he lp fu l d i scuss ions w i t h Dr , D. C,

PEASLEE on x, a r ious a spec t s of th i s work .

R I A S S U N T O (*)

Si t r a t t a il nucleo come un gas di Fermi con l ' imposizione di un dato impulso a n g o l a r e / . Si esprime la sua energia come una serie di potenze in /2 e si identificano i termini d ' interazione rot~zionali e vibrorotazionMi. Le interazioni fra part icel le non influiscono sulle energie rotazionali, ment re agiscono in modo ri levante sulle energie vibrorotazionMi. Si considerano vari modelli per la produzione degli impulsi angolari. La tendenza dei momenti d ' inerzia r isul tante sperimentahnente 5 r iprodot t~ da un modello in cui solo i nucleoni esterni a un determinato ~ core ~ produc(mo l ' impulso angolare totale. La frequenza vibr~zionale non 5 influenzata da tale modello. Si t iene anche conte degli effetti superficiali e dell ' influenza delle forze dipendenti dalla velocit£.

(*) Traduz ione a cara della ]~'cdaziotte.

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Statistical derivation of the nuclear rotational energies