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Analysis of nuclear quadrupole moments on the basis of a kinematical model
Nuclear Physics 2 (1986/57) 4 ~
~75; North-Holland Publishing Co., A m s t e r d a m
A N A L Y S I S OF N U C L E A R Q U A D R U P O L E M O M E N T S ON T H E B A S I S OF A K I N E M A T I C A L MODEL R. K. O S B O R N A N D E. D. K L E M A t
Oak Ridge National Laboratory Oak Ridge, Tennessee Received 7 J u l y 1956 A b s t r a c t : T h e k i n e m a t i c a l m o d e l u s e d p r e v i o u s l y in t h e correlation of n u c l e a r m a g n e t i c m o m e n t s h a s been applied to t h e a n a l y s i s of n u c l e a r q u a d r u p o l e m o m e n t s . T h e e m p i r i c a l rules f o u n d in t h e p r e v i o u s w o r k are in g e n e r a l conf i r m e d ; however, t h e d3; t nuclei f o r m a n e x c e p t i o n a l g r o u p a n d axe d i s c u s s e d in detail. A possible e x p l a n a t i o n of t h e a n o m a l o u s m a g n e t i c m o m e n t of W ~*s is given. S h a p e factors are o b t a i n e d for a n u m b e r of o d d m a s s nuclei on t h e basis of t h e p r e s e n t model.
1. T h e K i n e m a t i c a l Model
In a previous paper 1), referred to as I, it was shown that primarily upon the basis of kinematical considerations the concept of a rotating core plus odd nucleon adequately correlates practically all nuclear magnetic moments. The correlation is essentially expressed in a set of empirical rules which distinguish fundamentally among various core-plus-particle configurations depending upon whether the single particle in question is characterized by/" = l+½ or ] = l--½. The present investigation is an extension of the kinematical model to a consideration of nuclear quadrupole moments and shapes. The crux of the model, discussed in detail in I, m a y be briefly summarized. The system (a given nucleus) is to be regarded as being primarily defined by a limited set of dynamical variables and the data-selected components of the state vector they generate. However, for concreteness as well as for the purpose of enabling comparatively straightforward and exact calculations of relevant quantities, a simple physical system is conceived which is compatible with our choice of defining dynamical variables. But it is to be emphasized that the conceptual physical system is itself far less general than the dynamical variables, and hence literal inferences drawn essentially from the nature of the conceptual system must be interpreted with caution. In fact, our intent is precisely the converse; namely, to attempt to determine whether nuclear data taken t N o w a t t h e U n i v e r s i t y of M i c h i g a n . A n n Arbor, M i c h i g a n . 454
As~vs~s
oF NUCLEAR Q U A D R ~ L S
MOMENTS
4~6
in conjunction with a relatively simple model can be employed in the determination of the nature of the model itself. Accordingly, we assume that a given nucleus m a y be regarded as consisting of a rigid core if A and Z are even; a rigid core plus one nucleon if A is odd; a rigid core plus two nucleons if A is even and Z odd. For reasons discussed in I, nuclei with even A and odd Z will be given no further consideration in the present work; and since we are concerned here with the implications for the model of data on nuclear quadrupole moments, we shall restrict our attention in the following exclusively to odd mass nuclei. The rigid core is presumed to be characterized by an axiallysymmetric spheroidal deformation. The dynamical variables appropriate to such a system of deformed-core-plus-nucleon are:
1 ) S 2, the square of the intrinsic spin moment of the extra-core nucleon; 2) L ~, the square of the orbital angular momentum of the nucleon; 3) j2= (L~_S)2, the square of the total nucleon angular morn entum; 4) .~2, the square of the core angular momentum; 5) J'~ = (.~e ~_j)~ the square of the total angular momentum of the system; 6) J , , the projection of the system angular momentum on the z-axis of a space-fixed coordinate system; 7) .~',, the projection of the core angular momentum on the body symmetry axis (taken to be along the z-axis of a coordinate system fixed in the body); 8) J~, the projection of the total nucleon angular momentum along the body-symmetry axis; 9) ~',, the projection of the total angular momentum of the system along the body-symmetry axis; and 10) 11, the parity of the total system. Not all of these operators correspond to constants of motion for the system. In fact, the only ones which m a y be presumed to be constants of the motion are j 2 , j r , .~',, and /7; the first two because of the invariance of the system as a whole under arbitrary rotations of the space coordinate system; . ~ because of the invariance of the core configuration under rotations about the bodysymmetry axis; a n d / 7 because of system reflection symmetry. It should be pointed out perhaps that although J~ would be a constant
456
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of the motion for a single nucleon moving in a static axially symmetric potential s), it cannot in general be expected to be so if the well is rotating 3. 4. 5). In fact, both the hydrodynamical model and the rigid rotator model lead to a kinetic coupling of the particle motion to that of the core of the form j r , . j , (with which neither J~ nor J~ commutes) if it is assumed that the unconstrained degrees of freedom of the system are those for collective core rotation and single particle motion relative to the body coordinate system. Furthermore, if the particle and core degrees of freedom cannot be regarded as unconstrained, it is probable that it is precisely this kinetic coupling which will be most influenced by the constraints. It is for this reason, as well as for others elucidated in I, that we have chosen to construct our base vectors with which to construct nuclear ground states as follows: Since the kinematical properties of the system will be independent of whether the particle coordinates are expressed in a body -- or space-fixed coordinate system, we choose a representation for the nucleon which diagonalizes L 2, S 2, J2, and J,; i.e., zT, =
Z
c ( V i ; 3, , ~ - ~ ) z ~ r ? -~.
(1)
2* To this is adjoined the core representation Du,K, which diagonalizes .~*, .Z',, and .Z'~ with eigenvalues 2(2+1), #, and K, respectively, to form the base vectors
-~-i-
C(j~ff; m, M---m) Z~ DM-,,,,o, '~ even.
(2)
This representation diagonalizes the following operators with corresponding eigenvalues: j2 ~ I(I+1)
(3a)
.W* m ),(2+1)
(3b)
J' ~ L~ ~ S~ ~ J,, ~ ..W', ~ H ~
(3c) (3d) (3e) (3f) (3g) (~h)
i6+1) l(1+1) 3/4 M 0 (-~),.
A discussion of the significance of the restriction on the representation characterized by (3g) is presented below.
ANALYSIS O F N U C L E A R Q U A D R U P O L E MOMENTS
457
Thus, in general, the ground state of a given nucleus, characterized by the parity of the extra-core nucleon and a specified total angular momentum I, would be • 'f =
a jz
(4)
J4~
all even l's if parity even and all odd l's if parity odd. In accordance with the intent to determine whether nuclear data might delineate some simple regularity in the description of the ground states, and hence provide implications for the model itself, it was assumed that two of the components of the state vector (4) were dominant and that the ground state might be represented by
--a )
(5)
It was found from the analysis of magnetic moment data in I that a simple set of rules might be postulated for the prescription of the quantum numbers appropriate to a given nuclear ground state; i.e., 1) I = ½ nuclei; the state consists of the components ~ 0 , , and ~12,',' where (fl) are taken from the single-nucleon shell model and (fl') are nearby single-particle states of the same parity; 2) I > • nuclei for which i = l+½; the components are either ~0~ and ~2~ or ~12j~ and ~14~, where again the particle quantum numbers are dictated by the single-nucleon shell model; and 3) I > ½ nuclei for which ~"= l--½; the components are ~ i ~ and ~12j',', where (i'l') are nearby proper parity states. In all cases it is presumed that the mixing states lie in the same major shell. It is observed below that the application of rule (3) to ds/2 single-particle configurations must be modified. It is to be noted that the core configuration characterized by ~t = 2 appears to be of predominant importance in the description of nuclear ground states. This predominance is peculiar neither to the rigid rotator model nor to the present choice of representation, but rather appears to be common to collective models in general. For example, if in the strong-coupling limit of the hydrodynamical
458
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model a) one assumes that the total angular momentum = the total single-particle angular momentum = the projection of the single-particle angular momentum along the b o d y - s y m m e t r y axis in the ground state; then for an I = 3/2 nucleus the corresponding core states are ~ = 0 and 2 in equal amounts. It is clear at the outset that if we are to avoid the question of dynamics, one or more parameters must be introduced in order to characterize the shape of the core. Furthermore, these parameters must be treated empirically. This proliferation of empirical parameters is actually less objectionable than it might seem at first sight, since the results of m a n y quite different types of experiments are expected to depend more or less sensitively upon nuclear shape; for example, quadrupole moments, rotational energy levels, and scattering cross sections. Here also we shall extend our considerations to light as well as heavy nuclei, in spite of the rapidly growing success of sophisticated applications of the shell model in the region A ~ 40. We feel justified in this extension on the ground that overlapping success or failure of apparently dissimilar points of view can be meaningful in its own right. Since in the following, nuclear shapes are to be considered only in a most elementary sense, it is both feasible and desirable to introduce easily defined shape parameters and to treat them exactly throughout. Thus, at the risk of oversimplification, the burden of successful correlation is placed upon the relevance of the basic tenets of the model itself, rather than upon the resolution of ambiguities of approximation. Consequently, subsequent considerations rest upon the following: a) The assumptions in I summarized above which provide a prescription for two-component wave functions consistent with observed nuclear magnetic moments. b) The assumption that the nuclear core is characterizable in terms of the static moments of inertia of a spheroid; i.e. A; = A; =# A; where the primes imply that these quantities are defined in the principal axis system of the body. The actual computational parameters of assumption (b) are the semi-axes of the spheroid, which are related to the moments of inertia as follows:
ANALYSIS
OF NUCLEAR
QUADRUPOLE
4~9
MOMENTS
A; = A',, = ~ MR~(y'q--cd)
(6a)
A~ ~ - 2, - -~,2 ,,2 6 ff//XO~ ,
(6b)
and where ? and a are the semi-axes in units of R o along and perpendicular to the symmetry axis, respectively, and M is the mass of the core. If now the wave function is taken to be (5), the expression for the quadrupole m o m e n t of the nucleus is
Q = (1-a2)(IRjllQ°oplZRfl) + 2a(1 -a')~ (IRfllQ°oplZ2'/' l ') + a a ( / a ' j ' l'lQ°oplI,Vi'l'), M = 1.
(7)
An appropriate expression for Qop m a y be derived from the definition /16xe\½ z _ ,~ Qo~ - = / - - / ~ e,~Y2 (¢,), \ 5/ ~ol
(8)
by assuming that all nucleons but the last odd one belong to the rotating core. Then
\
5
/
cor,
= k--if-] J¢or.O(e)r'Yr (~)dSrq-[--ff -) ep4Y2" (¢p) 16~ ½
16~ {
= (--if-) ~ D~-,(o)f~,q(e)r'*YT"(~')dSr'+(~
(9) m
- ) % ~ Y 2 (fp)"
The symbol 0 stands for the Euler angles of a rotation from the space to the body-fixed principal axis coordinate system. Hence the primed integration runs over the static shape of the core. If we define a shape factor Q0 by (16:r1½ fj ~ o ( e ) r , , y o ( f ) d a r '
Q0 = \ ~ - /
then in terms of the shape parameter defined above we have for a uniform charge distribution Qo = ~ZeR~(y'--~').
(10)
The expectation value for the quadrupole m o m e n t m a y now be expressed as
460
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O = Q0 [(1 --a2)P.(Ij;
~l)+a 2 P.(Ij'; ,~' 2') + 2a (1 --a2)½p.(I]; +[(1--a2)Pp(IX; ]l]l)+a2Pp(IX'; i' l' ]' l') -t-2a(1--a2)~Pp(Ia; izi'l')a,~,] = (2.+Qp.
M')t,,,]
The expression implies that an appropriate radial function for the last odd nucleon has been adjoined to the kinematical wave function. The quantities P . and Pp are the core and particle projection factors, respectively, and essentially measure the distortion of the charge distribution due to the core rotation. They are defined b y
P,(Ii; Lt') = (IaillD~olla'il) = (--
I (I-t-l)(~l-t-~)
(12a) c(a'2a; 0, 0)w(a'mi; 2i) J
and Pp(I~.;
/
,
i z i ' r ) = @ i l I ~-U!
~ (~p) I~4' l,]
,., rz(2I-~)(2l+J)]½ (2]+ 1)½W(if II;
(12b)
2~.),
where (gl1211/'~") =
(_~)½(_l)r+r_,[(2l,+l)(2],+l)]iC(l,2l;
O, O) W(ljl, f ; ½2)"
(12c)
Since y/a > 1 for prolate nuclei and y/a < 1 for oblate nuclei, it is clear that Q0 is positive for prolate and negative for oblate nuclei. In order to evaluate the particle contribution to the quadrupole moment, we have taken= 3/SeR~ and have used the value R 0 = 1.44× 10-aaAt era. In general it is found that, as expected, the core contribution predominates in the determination of the sign and magnitude of the moment. The fact that the core contribution does indeed generally determine the moment raises the interesting and not entirely trivial question of the distinction between the static and dynamic nuclear shapes; i.e. between the shape seen b y an observer in the rotating principal axis system and that seen b y an observer in a spacefixed coordinate system t. As the momental tensor is not irre? T h e a u t h o r s are i n d e b t e d to Dr. L. C. B i e d e n h a r n for he l pful d i s c u s s i o n s on t h i s p oint.
ANALYSIS OF NUCLEAR QUADRUPOLE MOMENTS
461
ducible, b u t in fact includes an important scalar component, it is not to be expected that its expectation value as computed b y the space-fixed observer will be simply proportional to the quadrupole expectation value. However, the sign of the dynamic shape does exhibit such a simple relationship. Explicitly, if the momental tensor, Ju,, is expressed in that system of coordinates in which the components of the position vector are r~: 1 = =V (2)-~(x+iy); r o = z, then the relation between the momental tensor in the spacefixed and rotating coordinate systems is L" J~,(e) = 2~ C(IlL;/z, v ) C ( I 1 L ; / d , vt )Du,+,,~,+cyu, ,, (e'). (13) L#' u"
Since J is a symmetric tensor and since C(11L;/~, v ) = (--1) L C ( l l L ; v,/~), we see that the irreducible components of J rotate according to D O and D z, respectively. If now we define the expectation value of a space Cartesian component of f in a pure state to be ( f , ) = (Ij~l[J~j(e)lI2]l), we find that for it = 0 and axial symmetry in the body-fixed system of coordinates,
2A'~,+A',
(l,~) = (l,,) = (l~) = - - , 3
(14)
and for arbitrary it,
(J,,)-(l,,)
= P,(I]; itit)[A~--A',].
(15)
Thus for positive core projection factors the sign of the dynamic shape and of the static shape will be the same. A further point of interest here is the question of the axis of core rotation. The assumption that the projection of the core angular momentum, K, in the b o d y system is zero requires the core to rotate about an axis perpendicular to the axis of symmetry. The above expression for the dynamic shape factor is based upon that assumption. The converse assumption that the core rotate about the body symmetry axis would imply it = K. In this case the dynamic shape factor would be
(J~®)~.~-(Juba,,K =
2).-- 1
--
it+l
P,(Ii; /2)[A',--A',].
(16)
Thus for a given it the dynamic shape will be of opposite sign for K = i t and K = O. If now the very considerable correlation between the signs of
46~
R. K. OSBORN A N D E. D. KLEMA
observed quadrupole moments and the state of fullness of corresponding subshells 6) is not to be completely destroyed by the concept of the rotating core, it is clear that the dynamic shape must be of the same sign as the static shape -- as it is the latter which is presumably determined b y the number of particles in unfilled subshells. At least this must be the case whenever the core contribution to the quadrupole moment dominates the particle contribution. If it is assumed that the core deformations in the ground state are volume preserving, then a simple relationship exists between the semi-axes; viz. ~ = 1/~ ~. Accordingly, the expression for the quadrupole moment shape factor becomes Qo --
•
Since the cubic ~8_h~_
1 = 0,
h --
5Q°
ZeR '
(xs)
has only one positive root, we see that assumptions (a) and (b) plus knowledge of nuclear magnetic and quadrupole moments suffice to determine a nuclear shape parameter, 7. 2. A p p l i c a t i o n 2.1 THE
/" = l + ~
of Model
NUCLEI
The signs and magnitudes of the distortions computed on the basis of the present model for the 39/" = l + ½ nuclei which have measured quadrupole moments are in the main quite reasonable and consistent. The situation relative to Li Tis quite unsatisfactory, but also probably not very meaningful. The measured quadrupole moment for Pr TM is so closely accounted for b y the odd proton that the computed core shape is spherical -- which might be regarded as an anomalously small distortion. Conversely, the enormous distortions computed for Lu lv6 and Ta ~8~ seem anomalously large. The single-particle configuration assumed in I for the nucleus Yb ~78 seems to require modification. It was taken to be fs/2, which presumably admixed with the fT/a state, 2 = 2 being a good quantum number. So far as the magnetic moment is concerned, this admixture is quite satisfactory. Here there are two possible values for the mixing ratio a (since this is a case of spin-orbit partner mixing) -- one near unity and the other very small. The
ANALYSIS O F N U C L E A R Q U A D R U P O L E MOMENTS
463
state with the small ratio, however, predicts an anomalously large distortion whereas the large ratio implies that the single-particle configuration is more nearly f~/, than f6/m. If this implication is exploited, as suggested b y Moszkowski and Townes '), then we would assign Yb *Tsthe ground-state admixture (fT/z, 2 = 2, I = 5/2) + (f~/2, 2 = 4, I = 5/2). As seen in table I, such a configuration correlates the magnetic moment quite satisfactorily and leads to a prediction of an entirely reasonable value of the distortion. The quadrupole and magnetic moments and nuclear shapes of the remaining nuclei in this class all seem consistently correlated within the context of the model. Further, there is some reinforcement of the conclusion previously drawn that the appropriate core state admixture is always )t and 2 + 2 . Among the ds/z, fT/s, and g912 nuclei there were magnetic moments which required the 2 ---- 4 core component. But whether the basic core component was to be that with 2 ---- 0 or that with I = 2 was ambiguous so far as the magnetic moments alone were concerned -- although the concept of interacting core states was invoked to provide a justification for 2 ---- 2 rather than 1 ---- 0. However, the necessity for the presence of a large 2---- 2 component in the core configurations for these nuclei can be inferred from consideration of the core contribution to their quadrupole moments. The assumption of the (0, 4) admixture leads to the contribution
(2. = Q,oa' P,(II; 44)
(19)
whereas the admixture (2, 4) provides
Q, = Qo [(1--ai)P"(II; 2 2 ) + 2 a ( 1 - - a i ) t P,(II; 24) +a2 P,(II; 44)].
(20)
Now
P,(II;
44) = - - 0 . 1 0 1 2 , ---- + 0 . 0 1 7 3 , ---- + 0 . 0 8 2 6 ,
I = I = I =
5/2 7/2 9/2.
Thus the (0, 4) admixture leads to opposite-signed static and dynamic shapes for I = 5/2; and for I ---- 7/2 and 9/2 to such a severe decrease in the dynamic asymmetry that only abnormally large static deformations could account for the observed quadrupole moments. Conversely, an examination of table I reveals that the (2, 4) admixture predicts signs of the distortions of these nuclei
464
R. K. OSBOR.M AND E. D. KLEMA
which are quite consistent with expectation 6), and magnitudes which are in general reasonable. 2.2 T H E
j = 1--~, I >
3/21NUCLEI
In the case of j = l--½ nuclei, which according to I admix single-particle states, one is led immediately to reinforcement of the conclusion that 2 tends to be a constant of the motion-provided that I > 3/2. In accordance with the discussion in I it is expected that a two-state description of the Is/s, gT/,, and hg/s nuclei (except for E u ass, which will be discussed later) would consist of one or the other of two alternative admixtures; either
(I,l=I,2=O)+ ( I , i = I - 1 , 2 = 2 ) or ( I , / = I , 2 = 2 ) + ( I , / ' = I - 1 , 2 = 2 ) . In the first instance the core contribution to the quadrupole moment is 1--5 Qs = Qo a' 7 ( I + ~ j ' (21) and in the second instance it is 41(1+1)--15
Qs = Qo7(i+l)(2i+3)
I
1-a s
(21+5)(1+3)--1514i(i+1)_15 _1"
(22)
Thus the former configuration is characterized b y opposite-signed static and dynamic shapes whereas the latter is not; hence it is expected that the latter configuration should predominate in the ground state in accordance with the previous remarks. Furthermore, it is to be noted that the core contribution to the quadrupole moment for these nuclei is independent of the sign of a, the mixing coefficient determined b y the magnetic moments. The nucleus Eu 1~ was assigned in I the ground state of a d s / a odd particle with core states 2 = 2 and 4 admixed. This configuration, however, contains an unusually large amount of the 2 = 4 component, and also predicts a very large distortion. For this nucleus Moszkowski and Townes 6) have suggested the shell configuration (g~ls)~12(dsis)6(hal/~)~. In accordance with this suggestion, we would regard Eu xr~ as a gT/~ odd particle coupled to a core in the state 2 = 2 to give a total spin I = 5/2, and admixing the state (d5/2, 2 = 2, I = 5/2). This latter assignment does indeed seem quite preferable in that only a small amount of the (dt/2, 2 = 2) component is required to correlate the magnetic moment. Furthermore, though the distortion remains large @/~ = 1.57, see table I), it
ANALYSIS OF NUCLEAR QUADRUPOLE MOMENTS
465
is considerably less than that predicted by the previous configuration 0,1~ = 1.73). 2.3. T H E
7" = l - t ,
I = 312 N U C L E I
The situation relative to the ds/z nuclei is, however, no longer satisfactory if the rules advocated in I are strictly enforced. For purposes of magnetic moment correlation the ds/s nuclei were classed with all other j = l--½ nuclei and presumed therefore to be characterized by mixed single-particle configurations rather than mixed core configurations. It was found that all of the ds/2 magnetic moments, with the exception of those for C1~, C1s~, S ~, and Xe TM, which apparently required admixture of the d6/2 state, could be accounted for by admixture of the nearby Sl/s state with ~ a constant of the motion and equal to 2. But for such a state the core contribution to the quadrupole moment is Qs = Qoa~Ps({ ½2 2) = - - 0 . 2 Qo a2.
(23)
Thus in accordance with the suggestion above that the static and dynamic shapes should be of the same character (both oblate or both prolate), we feel that these configurations for the da/~ nuclei are unacceptable. Now it is perhaps significant that only for this group of ] = l--½ nuclei is it possible to account for magnetic moments on the assumption of core state mixing alone, with the same exceptions as mentioned above for the ds/~--sl/2 configuration. The admixed core states would be those characterized by ~ = 0 and 2; hence the core contribution to the quadrupole moment would be
Qs -= 2Qoa(1--a')~ Ps (zla'~, 02) = 0.4 Q0a(1--a') ½.
(24)
This is the same as the corresponding quantity for Pa/z cases, and in general leads to a reasonable description of the shape of the d~/z (when applicable) and Ps/2 nuclei. The magnetic moments of C1as, ClaL S =, and Xe TM require admixture of the (ds/2, ;t = 2) state. The apparent difference between the confiuration assigned to these cases and that assigned to the other members of their class is probably mainly a consequence of the oversimplification inherent in the assumption of only two components in the wave function. In actuality, one would expect that
466
R.K.
O S B O R N A N D E.
D. K L E M A
the ds/s state is slightly admixed in all of the da/2 configurations, but to such a small extent that only for the four cases above is it impossible to ignore it completely so far as magnetic and quadrupole moments are concerned. There is some further justification for this point of view in that even with the admixture of the d6/~ state, 2 does not appear to be a good quantum number. That is, if one assumes a configuration consisting of the states (dais, ;t = 2) and (ds/s, 2---- 2), one finds, because of the cross term in the expression for the magnetic moment, that two possibilities exist for the mixing ratio a, one close to unity and the other extremely small. Neither of these admixtures is satisfactory in that the former suggests that the appropriate single-particle configuration should be ds/2 instead of d3/2, whereas the latter implies that the static deformations are so nearly averaged out b y the core motion that only for unreasonable deformations could the core contribute significantly to the quadrupole moment. Conversely, the two-component state (ds/2, 2 = 0 ) + (ds/,, 2---- 2) correlates the magnetic moments for relatively small amounts of admixed state while at the same time predicting reasonable distortions. Thus it appears that 2 is not a good quantum number for the d3/a nuclei, although/" m a y be so considered except for these four cases. In table I the mixing coefficients and nuclear distortions are presented for these four nuclei on the assumption of the (d3/~, 2 = 0)+(ds/2, 2 = 2) configuration. 3. D i s c u s s i o n It would appear that the experimental information on nuclear quadrupole moments is not only consistent with the basic assumptions of the kinematical model developed in I, b u t actually provides additional support in several instances. In particular it seems to strengthen the basic assumption that ~"= l+½ and ] = l--½ nuclei are fundamentally differentiated with respect to the nature of the core-particle degrees of freedom which are more nearly constants of the motion in the ground state. Actually, of course, consistent correlation of both magnetic and quadrupole moments within the context of the model has largely destroyed this differentiation so far as the ds/2 nuclei are concerned. But the fact that it is for only these / = l--½ nuclei that the distinction disappears merely suggests the possibility that the distinguishing mechanism is significantly less operative if I = 3]2 than otherwise. It is clear that to a large
ANALYSIS O F N U C L E A R Q U A D R U P O L E MOMENTS
467
extent this differentiation is an assumed consequence of the shell model, in that the /"----ld-~ levels frequently seem to be more isolated from other favorable parity single-particle levels than is the case for the /" = l--½ levels. Thus the factor of foremost importance here is the same one that has led to the gross level ordering scheme on the ]'--j coupling model which provides a reasonable accounting for the magic numbers and for most nuclear spins and parities. A second mechanism which quite consistently operates to favor this differentiation is the pairing energy. It is observed that the ]" :- l--½ single-particle states fs/s, gT/s, and h,/2 appear to admix predominantly the nearby Pa/2, ds/2, and fv/s states, respectively. Thus whenever the admixed level is non-empty, the change in the pairing energy favors the admixture. If the admixed level is empty, there is, of course, no pairing energy change. Conversely, exactly the same effect tends to oppose the inverse admixture. However, these considerations are considerably weakened in their application to the I = 3/2 nuclei, since both the Ps/s and da/s states are characterized by nearby lower spin states of favorable parity; i.e. Pl/s and sl/s, respectively. But in spite of this fact, practically all nuclei characterized by the single-particle assignments Ps/s or ds/z are observed (see table I) to admix core states rather than single-particle states. This apparent preference for core-state admixture stems directly from the assumption that the static and dynamic core shapes should be of the same sign. Since PB(II-- 1; 2 2) = (I-- 5)/7 (I-+- 1), it is clear that the contribution to the core shape from the ]" = I - - 1 , it' =- 2 state that is admixed in the case of fs/a, gT/~ and ha/~ nuclei is always negative; i.e., such admixtures invariably tend to change the character of the dynamic shape. But inspection of table I shows that in these cases the favorable shape contribution from the 1"= I, it ---- 2 state considerably exceeds the unfavorable contribution without exception; so that the sign of the resultant dynamic shape is always the same as that of the static one. Hence the net result of this type of admixture for I > 3/2 nuclei is simply to reduce somewhat the dynamic asymmetry of the configuration. This situation for the I = 3/2 nuclei is, however, completely altered by the fact that the pure states ~"----I = 3/2, it = 0 or 2 are spherically symmetric. Thus here the unfavorable shape contribution from the 1"----I--1, i t ' = 2 configuration is the total contribution; and if the assumption concerning the correspondance
468
R.
K.
OSBORN
AND
E.
D.
KLEMA
between static and dynamic shapes remains valid, such admixtures must be regarded as unlikely. Conversely, it is seen from eq. (24) that the static and dynamic shapes are of the same sign for core state admixtures with j ' = 1"= I = 3/2. Furthermore, the four d3/~ nuclei discussed above which require the ds/~, 2 = 2 state to account for their magnetic moments are also characterized by proper shape correspondence since
P~(I,I+1;22)
--
(21-- l ) ( I + 6) 7 ( I + 1 ) ( 2 I + 3)
> 0,
I >½.
(25)
The significance of this seems to be t h a t the configurations of the I = 3/2 nuclei are more sensitive to core dynamics than are those of the I > 3/2 cases. Finally, it is to be pointed out t h a t the correlation of the quadrupole moment data provides some corroboration of the assumption that the core rotation is such that its angular m o m e n t u m has zero projection upon the body s y m m e t r y axis; i.e., the assumption that K = 0. From the kinematical point of view, and so far as magnetic moments are concerned, this assumption is largely arbitrary -- its only effect being to restrict 2 to even values if the parity of the nucleus is to be t h a t of the single particle. The assumption receives some justification from the analysis by Bohr and Mottelson ~) of the hydrodynamical model of the core, but it would be desirable to establish it empirically. First of all it is to be expected that for the axially symmetric system the magnitude of K would be a good quantum number. That is, axial s y m m e t r y implies that the state of the system should be invariant under an infinitesmal rotation about the body z'-axis, and hence an eigenfunction of .~,,. But the further invariance of the system under rotations of 180 ° about the axis perpendicular to the s y m m e t r y axis defines an operator which does not commute with -~r, but does with .~,~,; and hence if the state is chosen to be an eigenfunction of this operator, then it is the magnitude of K which is expected to be a good. quantum number. Therefore, those nuclear states which contain a core component characterized by ~t = 0 and consequently K = 0 should have other core components with K = 0 and even values of ;t only 1). It is to be noted that K need not be restricted to even values, but rather that 2 + K must be even. Since the magnetic moments are independent of K (except
ANALYSIS O F N U C L E A R Q U A D R U P O L R MOMENTS
469
insofar as the evenness or oddness of K determines the evenness or oddness of ,1), one finds that a consistent scheme for the correlation of magnetic moments can largely be worked out employing odd values of ~. However, one finds further that the dynamic shapes computed from these core-plus-particle configurations are either of the wrong sign, as is almost invariably the case for the I ---- 3/2 nuclei, or so much more spherically symmetric than those computed from the K = 0 configurations that the consequent distortions are considerably larger -- often unreasonably so. Hence we feel t h a t the data, taken in conjunction with the assumptions of the model, provide adequate justification for continuing with the assignment of K ---- 0 to nuclear ground-state configurations. It is clear from the above that the empirical determination of the K = 0 assignment for odd-mass nuclei is deduced primarily from the evidence on quadrupole moments. However, one finds that the assignement K :/: 0 for I = 1/2 nuclei is actually ruled out on the basis of magnetic moment evidence alone; except for the cases s) W 18s and Os 187, whose moments are ~ 0.12 and cannot be correlated by any K = 0 configuration. These are satisfactorily accounted for by the core-plus-particle configuration (Pl/z, ~t = l, K = 1 ) ~ (fu2, ~" 3, K 1). The results of the analysis discussed above are presented in table I. The odd-particle configurations listed are those given by Mayer and Jensen 9) unless indicated by an asterisk. The cases so indicated are discussed in detail in the text. The quantum numbers which define the ground states are given in accordance with the convention (1 -- a ~)i (l]kI) -k a (l' j' ~' I). The magnetic moments used in calculating the a's are taken from the recent compilation of Walchli lo). The observed quadrupole moments are taken from Ramsey 11) or from Gordy, Smith and Trambarulo I~), unless noted by a letter. In these cases the literature references are cited. In each case the value used in the computation and listed as Qobs is a weighted average of the data indicated above. It is interesting to compare the deformations calculated in the present work with those presented in Blatt and Weisskopf is), calculated for a static, axially symmetric spheroid. It is found, as is to be expected, that the shape parameters in table I lead to deformations which are in general of the order of five times as great as those obtained for non-rotating nuclei. Large static deformations were also predicted within the context of the hydrodynamical =
=
470
R. K. OSBORN AND E. D. KLEMA
model 8). Such large deformations tend to strengthen the requirement employed above that the character of the static and dynamic shapes be the same 6). References l) 2 3 4 5 6 7
8 9 10) ll) 12) 13)
R. K. Osborn and E. D. Klema, Phys. Rev. 100 (1955) 822 S. G. Nilsson, Dan. ~Mat.-fys. Medd. 29 (1955) No. 16 A. Bohr, Dan. Mat.-fys. Medd. 26 (1952) No. 14 K e n n e t h W. Ford, Phys. Rev. 90 (1953) 29 A. K. Kerman, Dan. Mat.-fys. Medd. 30 (1956) No. 15 S. A. Moszkowski and C. H. Townes, Phys. Rev. 93 (1954) 306 A . B o h r a n d B . R . M o t t e l s o n , Dan. Mat.-fys. Medd. 27 (1953) No. 16 K. Murakawa, Phys. Rev. 98 1285 (1955) Maria Goeppert Mayer and J. Hans D. Jensen, Elementary Theory of Nuclear Shell Structure (John Wiley and Sons, New York, 1955) H . E . Walchli, A Table of Nuclear Moment Data, ORNL-1469 (Supplement II) Norman F. Ramsey, Nuclear Moments (John Wiley and Sons, New York, 1953) Waiter F. Gordy, William V. Smith, and Ralph F. Trambarulo, Microwave Spectroscopy (John Wiley and Sons, New York, 1953) J o h n M. B l a t t and Victor F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, New York, 1952)
471
ANALYSIS OF NUCLEAR QUADRUPOLE MOMENTS
TABLE I Analysis of nuclear quadrupole moment data. The odd-paxticle configurations are given in the second column and the assumed ground states in the third. The mixing coefficients axe given in the fourth column. The experimental values of the quadrupole moments axe listed in the fifth column. The calculated shape parameters axe given in the last three columns. Spin 3/2 Nuclei
Odd Particle Configuration
Nucleus
Assumed Ground State Q u a n t u m Numbers, (lj2I)
a
Qo~
>'/~
7
P$/|
O d d Proton
aLi4 sBs mCust
nCUu uGau a~Gat0 uAs.a uBr.. ~Br~ a7Rbso
,
(Isvs)S(Ips/m)l (Isl/s)S(Ip~/s)s (2ps/s)I (2pw) l (2psls) s (2pvl) a (2ps/s)S(lfv2) 2 (2ps/s)S(l fi/t) 4 (2pvs)* (lf6/i) 4 (2ps!j)S(lfs/i) s
(1 (1 (1 (1 (1 (1 (1 (1 (1 (I
3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2
0 0 0 0 0 0 0 0 0 0
3/2) 3/2)
3/2)
(l (1 (1 (1 (1 (l (I (1
3/2)
(1
3/2) 3/2)
3/2) 3/2) 3/2)
3/2)
3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2
2 2 2 2 2 2 2 2 2
3/2) 3/2) 3/2) 3/2)
3/2) 3/2) 3/2)
3/2)
3/2) (1 3/2 2 3/2)
0.463 0.665 0.792 0.751 0.844 0.702 0.972 0.822 0.781 0.546
1.656 0.02 0.040~) 1.373 ---0.15b, c ) i 0.932 ---0.15 b ) 0.939 1.093 0.2318 0.1461 1.062 1.188 0.3 0.33 d) 1.107 0.28 d) 1.088 0.15e) 1.051
0.777 0.853 1.036 1.032 0.956 0.970 0.917 0.950 0.958 0.975
2.13 1.61 0.90 0.91 1.14 1.09 1.30 1.16 1.14 1.08
1.241 1.038
0.898 0.982
1.38 1.06
Odd Neutron .Bes soHgm
(lsl/s)s(lps/s) I (2fT/s)S(lh./I)l°(3Ps/j)3 (2f6/I)'(lils/I)11
(1,3/2 0 3/2) (1 3/2 0 3/2)
(1 3/2 2 3/2) (1 3/2 2 3/2)
0.595 0.792
0.02 ~ )
o.5og)
i
i a)
b)
c) d)
e) f) g)
G. Wessel, Phys. Rev. 92 (1953) 1581 H. D. Dehmelt, Z. Physik 133 (1952) 528; 134 (1953) 642 A. Bassompierre, C. R. Acad. Sci. 237 (1953) 1224 B. Bleaney, K. D. Bowers and M. H. L. Pryce, Proc. Roy. Soc. A 228 (1955) 166 B. Bleaney, K. D. Bowers and R. S. Trensm, Proc. Phys. Soc. A 66 (1953) 410 H. Kopfermann, A. Steudel, S. Wagner and W. Walcher, Nachr. Akad. Wiss. C,Sttingen, Math-Physik KI I I a No. 1 (1953) 1 J . G . King and V. Jaccaxino, Phys. Rev. 94 (1954) 1810; 91 (1953) 209A W. Gordy, J. Chem. Phys. 19 (1951) 792 H. G. Dehmelt, Z. Physik 130 (1951) 480 B. Senitzky, I. I. Rabi and M. L. Perl, Phys. Rev. 98 (1955) 1537A H. Krtiger and U. Meyer-Berkhout, Naturwissenschaften 42 (1955) 94 W . D . Knight, Phys. Rev. 92 (1953) 539A K. Murakawa, Phys. Rev. 98 (1955) 1285 H. G. Dehmelt, H. G. Robinson and W. Gordy, Phys. Rev. 93 (1954) 480
472
R. K. OSBORN AND E. D. KLEMA
TABLE I (continued) Spin 3/2 Nuclei Odd Particle Configuration
Nucleus
Assumed Ground State Q u a n t u m Numbers, (I]M)
a
Qobs
?
s¢
7'/a
--0.0789 k) --0.0626 k)
1.251)
0.774 0.865 1.139
1.136 1.075 0.937
0.68 0.80 1.22
d3/t Odd Proton
I~C118 1rClso vTIrll4 77]rH8
79AuH8
(ld~/,)e (2sl/,)~ (I d3,,a)l (ld6/z)' (2sl/,)*( I ds;,)l (lg~/,)s (2d6,,,)6 (l h n/,) I° (2d3/,) s (lgT/s)8 (2ds/2)6 (I hws) z° (2d~2) s (1 gT/~)e(2ds/t)e (l htl/t) 12 (2da/2) s
(2 3i2 0 3/2) (2 3/2 0 3/2) (2 3/2 0 3/2)
(2 5/2 2 3/2) (2 512 2 3i2) (2 3/2 2 3/2)
0.534 0.479 0.426
(2 3/2 0 3/2)
(2 3/2 2 3/2)
0.337
1.2 l)
1.164
0.927
1.26
(2 3/2 0 3/2)
(2 3/2 2 3'2)
0.213
0.5cm )
1.129
0.941
1.20
--0.065n, o) 0.589 0.045 ° ) 1.054 --0.14P) 0.875
1.303 0.974 1.069
0.45 1.08 0.82
Odd Neutron 16S17 15S19
.Xe~v
(lds/2)6 (2sl/~)2 (ld3/z) 1 (1 ds/~)6(2sl/t)~ (Ida/s, s (2ds/2)6 (ig7/I)8 (3sl/g)z (lhn/s)x°(2d~/~) 1
k) 1)
(2 312 0 312) (2 3/2 0 3/2)
(2 5/2 2 3/2) (2 3/2 2 3/2)
(2 3/2 0 3/2)
(2 5/2 2 3/2)
0.497 0.628 0.473
G. P. Koster, Phys. Rev. 86 (1952) 148 W . v . Siemens, Ann. Phys. 13 (1953) 136 K. Murakawa and S. Suwa, Phys. Rev. 87 (1952 l 1048 m) W. v. Siemens, Ann. Phys. 13 (1953) 158 W. v. Siemens, Naturwissenschazften 38 (1951) 455 n) G. R. Bird and C. H. Townes, Phys. Rev. 94 (1954) 1203 o) C . A . Burrus, Jr. and W. Gordy, Phys. Rev. 92 (1953) 274 H. G. Dehmelt, Phys. Rev. 91 (1953) 313 p) E. Brun, J. Oeser, H. H. Staub and C. G. Telschow, Phys. Rev. 93 (1954) 904 A. Bohr, J. Koch and E. Rasmussen, Ark. Fys. 4 (1952) 455
ANALYSIS
OF
NUCLWAR
QUADRUPOLE
473
MOM]gNTS
TABL~Z I (continued) Spin 5/2 Nuclei Odd Particle Configuration
Nucleus
.Assumed Ground State Quantum Numbers, (1/~I)
a
Qobs
7
0t
7/0t
0.297 0.959 0.248 0.499 0.844 0.976 0.352
0.11 q) 0.153 r ) --0.65 s) ---0.819 t) --0.054 u ) 1.2 2.6
1.241 1.232 0.877 0.8{}9
0.124
---0.004 v )
0.955
1.023
0.93
0.438
0.30 w)
1.302
0.876
1.48
d&/l Odd Proton itNal.
lsAllt uSb~o 6sI:t ~Prn
(2 5/2 2 3/2) (2 512 0 512)
(Id6/t)s (Ids/s)~ (2d5/I)z
(lgT/s)J(2d6:l) 1 (Ig,/t)S(2dvl) t
76Re~tt
(lg:/s)S(2dvs) 6 ( lgT/s)s(2d6/t)6 (1 ht t:t) t2
80,
(Idu,) i
#sEuss
(2 (2 (2 (2 (2 (2
5/2 2 512 2 5/2 0 5/2 0 5/2 2
(2 (2 (2 (2 (2 (2 (2
5/2) 5/2) 5/2) 5/2) 5/2)
5/2 5/2 5/2 5/2 5/2 5/2 5/2
4 2 4 4 2 2 4
3/2) 5/2) 5/2) 5/2) 5/2) 5/2) 5/2)
0.898 0.901 1.068 1.072 1.001 0.999 1.130 0.941 1.232 0.901
1.38 1.37
0.82 0.81 1.00 1.20 1.37
Odd neutron
5/2 0 5/'2)
(2 5/2 2 5,2) fS!I
Odd Proton ~Rbu
(2Ps/t)' (Ifvj)
q) r) s)
t)
(3 5/2 2 5/2)
(I 3/2 2 5/2)
M . L . Perl, I. I. Rabi and B. Senitzky, Phys. Rev. 98 (1955) 611; 97 (1955) 388 P. L. Sagalyn, Phys. Rev. 94 (1954) 885 H. Lew and G. Wessel, Phys. Rev. 90 (1953) I G. F. Koster, Phys. Rev. 86 (1952) 148 A . W . Jache, G. S. Blevins and W. Gordy, Phys. Rev. 97 (1955) 680 K. Murakawa, Phys. Rev. 93 (1954) 1232 G. Sprague and D. H. Tomboulian, Phys. Rev. 92 (1953) 105 K. Murakawa and S. Suwa, Z. Phys. 137 (1954) 575 V. Jaccaxino, J. G. King and H. H. Stroke, quoted by R. Livingston, B. M. Benjamin, J. T. Cox and W. Gordy, Phys. Rev. 92 (I953) 1271 T. Kamei, J. Phys. Soc. Japan 7 (1952) .649 H. Lew, Phys. Rev. 91 (1953) 619
u) v) G. R. Bird and C. H. Townes, Phys. Rev. 94 (1954) 1203 calculated from data of S. Geschwind, G. R. Gunther-Mohr and G. Silvey, Phys. Rev. 85
(1952) 474
w) H. Krfiger and U. Meyer-Berkhout, Naturwissenschatten 42 (1955} 94 B. Senitzky, I. I. Rabi and M. L. Perl, Phys. Rev. 98 (1955) 1537A
474
R. K. OSBORN AND E. D. IKL]~MA TABLR I (continued) Spin 7/2 Nuclei
Odd Particle Configuration
Nucleus
Assumed Ground State Q u a n t u m Numbers, (l],~l)
a
Qot~
7
ot
7/,v
Odd Proton
lil~V|8 sTCols
(lfv,) a (lfvl) 6 (lfvt)'
(3 7/2 0 7/2) (3 7/2 2 5/2) (3 7/2 2 7/2)
7oYbtos
(2f71t)7(11q/I)t°(2p~t)'*
(3 7/2 2 5/2)
ssMnso
(3 7/2 2 7/2) (3 7/2 4 5/2) (3 7/2 4 7/2)
0.897 0.591 0.427
0.3 z ) 0.55Y, z) 0.5 z)
1.126 1.177 1.177
0.942 0.922 0.922
1.20 1.28 1.28
0.657
3.9
1.206
0.911
1.32
0.715 0.890 1.043 1.314 1.350 2.287
1.182 1.060 0.979 0.872 0.861 0.661 0.747
0.60 0.84
Odd Neutron (3 7/2 4 5/2)
gT/i
Odd Proton 5tSbn
5ai75 ~CsTs ~TLass esEu~ ?lLUt04 vsTa~o0
(lgv/I) 1 (Zgvl)* (Zg7/i) 6 (lgvs) 7 (IgTjl)v (2d6/I) a* (lg~/I)v (2d6/s)* (1 hit/I) B (1 gv/I)T(2ds/s)* (1 hll/I) l°
y)
z) aa)
ab) ac) ad)
ae)
(4 (4 (4 (4 (4 (4
7/2 7/2 7/2 7/2 7/2 7/2 (4 7/2
2 2 2 2 2 2 2
7/2) 7/2) 7/2) 7/2) 5/2) 7/2) 7/2)
(2 (2 (2 (2 (2 (2 (2
5/2 2 5/2 2 5/2 2 5/2 2 5/2 2 5/2 2 5/2 2
7/2) 7/2) 7/2) 7/2) 5/2) 7/2) 7/2)
0.543 0.565 0.554 0.612 0.400 0.646 0.376
- - 1 . 0 aa) --0.47 ab) --0.003 a¢) 0.8 ad) 2.5 5.9 6.2 ae)
1.791
A. J a v a n and A. Engelbrecht, Phys. Rev. 96 (1954) 649 A. Javan, G. Silvey, C. H. Townes and A. V. Grosse, Phys. Rev. 91 (1953) 222A K. Murakawa and T. Kamei, Phys. Rev. 92 (1953) 325 A. W. Jache, G. S. Blevins and W. Gordy, Phys. Rev. 97 (1955) 680 K. Murakawa, Phys. Rev. 93 (1954) 1232 G. Sprague and D. H. Tomboulian, Phys. Rev. 92 (1953) 105 W. Gordy, J. Chem. Phys. 19 (1951) 792 calculated from data of R. Livingston, O. R. Gilliam and W. Gordy, Phys. Rev. 76 (1949) 149 K. H. Althoff and H. Kriiger, Naturwissenschaften 41 /1954) 368 K. Murakawa, Phys. Rev. 98 (1955) 1285 K. Murakawa, J. Phys. Soc. J a p a n 9 (1954) 391 K. Murakawa and T. Kamei, Phys. Rev. 92 (1953) 325 B. M. Brown and D. H. Tomboulian, Phys. Rev. 88 (1952) 1158
1.06
1.51 1.57 3.46 2.40
ANALYSIS OF NUCLEAR QUADRUWLE MOMENTS
476
TABLE I (continued) Spin 9/Z Nuclei Odd
Nucleus
Assumed
Particle
Quantum
Configuration
Ground
State
Numbers,
(1jAI)
Qobs
a
/
Y
)
a
/
Y/a
.%/a
Odd ,J%
(2P,/*)‘(lf~/,)e(2P1/*
‘1
Proton
(4 9/2
2 9/2)
(4 9/2
4 9/2)
0.179
rsTc,
(lg./*)’ (2PJ,)4(lf,/,)a(2PI/*)p
(4 9/2
2 9/2)
(4 9/2
4 9/2)
0.629
,Jn,,
(lg,/,)s (2P*/,)‘(lf,/*)~
(4 9/2
2 9/2)
(4 9/2
4 9/2)
,Jn,,
(2P,/,)*(lg,/*)B (2P*/*)‘(lf,/*)@
(4 9/2
2 9/2)
(4 9/2
-0.2af)
0.986
1.008
0.98
0.34 as)
1.046
0.978
0.716
0.98 ah)
1.096
0.966
1.15
4 9/2)
0.710
l.ocah)
1.096
0.966
1.16
4 9/2)
0.720
0.961
1.020
0.94
1
1.09
(2P,/*)‘(lg,/*)g Odd Neutron ,rGe,r
(2P~ll)‘(lfs/*)e(2PlI,)p
(4 912 2 9/Z)
(4 9/2
&e,r
(lg*/,)r (2P,/,)‘(lf*/,)s(~g,/,)’
(4 912 2 712)
=Kr,r
(2Ps/n)“(lf,/*)~(2P,,,)’
(4 912 2 9/2)
(4 912 4 712) (4 912 4 9/2)
-0.21
0.748
1.05”i)
1.160
0.928
1.26
0.664
0.15
1.022
0.989
1.03
0.978
1.011
0.97
(lg&’ he/s Odd Proton S3BiIr,
(lh&
(6 9/2
Phys.
2 9/2)
K.
Murakawa,
ag)
K.
G. Kessler
ah) ai)
G. F. Koster, Phys. Rev. 86 (1962) 148 G. R. Bird and C. H. Townes, Phys. Rev.
and R. E.
data of W. A. Hardy,
98
(1966)
2 9/2)
af)
berg, G. W.
Rev.
(3 7/2
Trees,
Phys.
Cohen,
-0.4
1285 Rev.
92 94
G. Silvey, C. H. Townes,
Parker and V. W.
0.686
Phys.
(1963) (1964)
303 1203
calculated
from
B. F. Burke, M. W. P. StrandRev.
92
(1963)
1632
~75; North-Holland Publishing Co., A m s t e r d a m
A N A L Y S I S OF N U C L E A R Q U A D R U P O L E M O M E N T S ON T H E B A S I S OF A K I N E M A T I C A L MODEL R. K. O S B O R N A N D E. D. K L E M A t
Oak Ridge National Laboratory Oak Ridge, Tennessee Received 7 J u l y 1956 A b s t r a c t : T h e k i n e m a t i c a l m o d e l u s e d p r e v i o u s l y in t h e correlation of n u c l e a r m a g n e t i c m o m e n t s h a s been applied to t h e a n a l y s i s of n u c l e a r q u a d r u p o l e m o m e n t s . T h e e m p i r i c a l rules f o u n d in t h e p r e v i o u s w o r k are in g e n e r a l conf i r m e d ; however, t h e d3; t nuclei f o r m a n e x c e p t i o n a l g r o u p a n d axe d i s c u s s e d in detail. A possible e x p l a n a t i o n of t h e a n o m a l o u s m a g n e t i c m o m e n t of W ~*s is given. S h a p e factors are o b t a i n e d for a n u m b e r of o d d m a s s nuclei on t h e basis of t h e p r e s e n t model.
1. T h e K i n e m a t i c a l Model
In a previous paper 1), referred to as I, it was shown that primarily upon the basis of kinematical considerations the concept of a rotating core plus odd nucleon adequately correlates practically all nuclear magnetic moments. The correlation is essentially expressed in a set of empirical rules which distinguish fundamentally among various core-plus-particle configurations depending upon whether the single particle in question is characterized by/" = l+½ or ] = l--½. The present investigation is an extension of the kinematical model to a consideration of nuclear quadrupole moments and shapes. The crux of the model, discussed in detail in I, m a y be briefly summarized. The system (a given nucleus) is to be regarded as being primarily defined by a limited set of dynamical variables and the data-selected components of the state vector they generate. However, for concreteness as well as for the purpose of enabling comparatively straightforward and exact calculations of relevant quantities, a simple physical system is conceived which is compatible with our choice of defining dynamical variables. But it is to be emphasized that the conceptual physical system is itself far less general than the dynamical variables, and hence literal inferences drawn essentially from the nature of the conceptual system must be interpreted with caution. In fact, our intent is precisely the converse; namely, to attempt to determine whether nuclear data taken t N o w a t t h e U n i v e r s i t y of M i c h i g a n . A n n Arbor, M i c h i g a n . 454
As~vs~s
oF NUCLEAR Q U A D R ~ L S
MOMENTS
4~6
in conjunction with a relatively simple model can be employed in the determination of the nature of the model itself. Accordingly, we assume that a given nucleus m a y be regarded as consisting of a rigid core if A and Z are even; a rigid core plus one nucleon if A is odd; a rigid core plus two nucleons if A is even and Z odd. For reasons discussed in I, nuclei with even A and odd Z will be given no further consideration in the present work; and since we are concerned here with the implications for the model of data on nuclear quadrupole moments, we shall restrict our attention in the following exclusively to odd mass nuclei. The rigid core is presumed to be characterized by an axiallysymmetric spheroidal deformation. The dynamical variables appropriate to such a system of deformed-core-plus-nucleon are:
1 ) S 2, the square of the intrinsic spin moment of the extra-core nucleon; 2) L ~, the square of the orbital angular momentum of the nucleon; 3) j2= (L~_S)2, the square of the total nucleon angular morn entum; 4) .~2, the square of the core angular momentum; 5) J'~ = (.~e ~_j)~ the square of the total angular momentum of the system; 6) J , , the projection of the system angular momentum on the z-axis of a space-fixed coordinate system; 7) .~',, the projection of the core angular momentum on the body symmetry axis (taken to be along the z-axis of a coordinate system fixed in the body); 8) J~, the projection of the total nucleon angular momentum along the body-symmetry axis; 9) ~',, the projection of the total angular momentum of the system along the body-symmetry axis; and 10) 11, the parity of the total system. Not all of these operators correspond to constants of motion for the system. In fact, the only ones which m a y be presumed to be constants of the motion are j 2 , j r , .~',, and /7; the first two because of the invariance of the system as a whole under arbitrary rotations of the space coordinate system; . ~ because of the invariance of the core configuration under rotations about the bodysymmetry axis; a n d / 7 because of system reflection symmetry. It should be pointed out perhaps that although J~ would be a constant
456
R.
K.
OSBORN
AND
E.
D.
KLEMA
of the motion for a single nucleon moving in a static axially symmetric potential s), it cannot in general be expected to be so if the well is rotating 3. 4. 5). In fact, both the hydrodynamical model and the rigid rotator model lead to a kinetic coupling of the particle motion to that of the core of the form j r , . j , (with which neither J~ nor J~ commutes) if it is assumed that the unconstrained degrees of freedom of the system are those for collective core rotation and single particle motion relative to the body coordinate system. Furthermore, if the particle and core degrees of freedom cannot be regarded as unconstrained, it is probable that it is precisely this kinetic coupling which will be most influenced by the constraints. It is for this reason, as well as for others elucidated in I, that we have chosen to construct our base vectors with which to construct nuclear ground states as follows: Since the kinematical properties of the system will be independent of whether the particle coordinates are expressed in a body -- or space-fixed coordinate system, we choose a representation for the nucleon which diagonalizes L 2, S 2, J2, and J,; i.e., zT, =
Z
c ( V i ; 3, , ~ - ~ ) z ~ r ? -~.
(1)
2* To this is adjoined the core representation Du,K, which diagonalizes .~*, .Z',, and .Z'~ with eigenvalues 2(2+1), #, and K, respectively, to form the base vectors
-~-i-
C(j~ff; m, M---m) Z~ DM-,,,,o, '~ even.
(2)
This representation diagonalizes the following operators with corresponding eigenvalues: j2 ~ I(I+1)
(3a)
.W* m ),(2+1)
(3b)
J' ~ L~ ~ S~ ~ J,, ~ ..W', ~ H ~
(3c) (3d) (3e) (3f) (3g) (~h)
i6+1) l(1+1) 3/4 M 0 (-~),.
A discussion of the significance of the restriction on the representation characterized by (3g) is presented below.
ANALYSIS O F N U C L E A R Q U A D R U P O L E MOMENTS
457
Thus, in general, the ground state of a given nucleus, characterized by the parity of the extra-core nucleon and a specified total angular momentum I, would be • 'f =
a jz
(4)
J4~
all even l's if parity even and all odd l's if parity odd. In accordance with the intent to determine whether nuclear data might delineate some simple regularity in the description of the ground states, and hence provide implications for the model itself, it was assumed that two of the components of the state vector (4) were dominant and that the ground state might be represented by
--a )
(5)
It was found from the analysis of magnetic moment data in I that a simple set of rules might be postulated for the prescription of the quantum numbers appropriate to a given nuclear ground state; i.e., 1) I = ½ nuclei; the state consists of the components ~ 0 , , and ~12,',' where (fl) are taken from the single-nucleon shell model and (fl') are nearby single-particle states of the same parity; 2) I > • nuclei for which i = l+½; the components are either ~0~ and ~2~ or ~12j~ and ~14~, where again the particle quantum numbers are dictated by the single-nucleon shell model; and 3) I > ½ nuclei for which ~"= l--½; the components are ~ i ~ and ~12j',', where (i'l') are nearby proper parity states. In all cases it is presumed that the mixing states lie in the same major shell. It is observed below that the application of rule (3) to ds/2 single-particle configurations must be modified. It is to be noted that the core configuration characterized by ~t = 2 appears to be of predominant importance in the description of nuclear ground states. This predominance is peculiar neither to the rigid rotator model nor to the present choice of representation, but rather appears to be common to collective models in general. For example, if in the strong-coupling limit of the hydrodynamical
458
R.
K.
OSBORN
AND
E.
D.
KLEMA
model a) one assumes that the total angular momentum = the total single-particle angular momentum = the projection of the single-particle angular momentum along the b o d y - s y m m e t r y axis in the ground state; then for an I = 3/2 nucleus the corresponding core states are ~ = 0 and 2 in equal amounts. It is clear at the outset that if we are to avoid the question of dynamics, one or more parameters must be introduced in order to characterize the shape of the core. Furthermore, these parameters must be treated empirically. This proliferation of empirical parameters is actually less objectionable than it might seem at first sight, since the results of m a n y quite different types of experiments are expected to depend more or less sensitively upon nuclear shape; for example, quadrupole moments, rotational energy levels, and scattering cross sections. Here also we shall extend our considerations to light as well as heavy nuclei, in spite of the rapidly growing success of sophisticated applications of the shell model in the region A ~ 40. We feel justified in this extension on the ground that overlapping success or failure of apparently dissimilar points of view can be meaningful in its own right. Since in the following, nuclear shapes are to be considered only in a most elementary sense, it is both feasible and desirable to introduce easily defined shape parameters and to treat them exactly throughout. Thus, at the risk of oversimplification, the burden of successful correlation is placed upon the relevance of the basic tenets of the model itself, rather than upon the resolution of ambiguities of approximation. Consequently, subsequent considerations rest upon the following: a) The assumptions in I summarized above which provide a prescription for two-component wave functions consistent with observed nuclear magnetic moments. b) The assumption that the nuclear core is characterizable in terms of the static moments of inertia of a spheroid; i.e. A; = A; =# A; where the primes imply that these quantities are defined in the principal axis system of the body. The actual computational parameters of assumption (b) are the semi-axes of the spheroid, which are related to the moments of inertia as follows:
ANALYSIS
OF NUCLEAR
QUADRUPOLE
4~9
MOMENTS
A; = A',, = ~ MR~(y'q--cd)
(6a)
A~ ~ - 2, - -~,2 ,,2 6 ff//XO~ ,
(6b)
and where ? and a are the semi-axes in units of R o along and perpendicular to the symmetry axis, respectively, and M is the mass of the core. If now the wave function is taken to be (5), the expression for the quadrupole m o m e n t of the nucleus is
Q = (1-a2)(IRjllQ°oplZRfl) + 2a(1 -a')~ (IRfllQ°oplZ2'/' l ') + a a ( / a ' j ' l'lQ°oplI,Vi'l'), M = 1.
(7)
An appropriate expression for Qop m a y be derived from the definition /16xe\½ z _ ,~ Qo~ - = / - - / ~ e,~Y2 (¢,), \ 5/ ~ol
(8)
by assuming that all nucleons but the last odd one belong to the rotating core. Then
\
5
/
cor,
= k--if-] J¢or.O(e)r'Yr (~)dSrq-[--ff -) ep4Y2" (¢p) 16~ ½
16~ {
= (--if-) ~ D~-,(o)f~,q(e)r'*YT"(~')dSr'+(~
(9) m
- ) % ~ Y 2 (fp)"
The symbol 0 stands for the Euler angles of a rotation from the space to the body-fixed principal axis coordinate system. Hence the primed integration runs over the static shape of the core. If we define a shape factor Q0 by (16:r1½ fj ~ o ( e ) r , , y o ( f ) d a r '
Q0 = \ ~ - /
then in terms of the shape parameter defined above we have for a uniform charge distribution Qo = ~ZeR~(y'--~').
(10)
The expectation value for the quadrupole m o m e n t m a y now be expressed as
460
R.
K. OSBORN
AND
E.
D.
KLEMA
O = Q0 [(1 --a2)P.(Ij;
~l)+a 2 P.(Ij'; ,~' 2') + 2a (1 --a2)½p.(I]; +
M')t,,,]
The expression
P,(Ii; Lt') = (IaillD~olla'il) = (--
I (I-t-l)(~l-t-~)
(12a) c(a'2a; 0, 0)w(a'mi; 2i) J
and Pp(I~.;
/
,
i z i ' r ) = @ i l I ~-U!
~ (~p) I~4' l,]
,., rz(2I-~)(2l+J)]½ (2]+ 1)½W(if II;
(12b)
2~.),
where (gl1211/'~") =
(_~)½(_l)r+r_,[(2l,+l)(2],+l)]iC(l,2l;
O, O) W(ljl, f ; ½2)"
(12c)
Since y/a > 1 for prolate nuclei and y/a < 1 for oblate nuclei, it is clear that Q0 is positive for prolate and negative for oblate nuclei. In order to evaluate the particle contribution to the quadrupole moment, we have taken
ANALYSIS OF NUCLEAR QUADRUPOLE MOMENTS
461
ducible, b u t in fact includes an important scalar component, it is not to be expected that its expectation value as computed b y the space-fixed observer will be simply proportional to the quadrupole expectation value. However, the sign of the dynamic shape does exhibit such a simple relationship. Explicitly, if the momental tensor, Ju,, is expressed in that system of coordinates in which the components of the position vector are r~: 1 = =V (2)-~(x+iy); r o = z, then the relation between the momental tensor in the spacefixed and rotating coordinate systems is L" J~,(e) = 2~ C(IlL;/z, v ) C ( I 1 L ; / d , vt )Du,+,,~,+cyu, ,, (e'). (13) L#' u"
Since J is a symmetric tensor and since C(11L;/~, v ) = (--1) L C ( l l L ; v,/~), we see that the irreducible components of J rotate according to D O and D z, respectively. If now we define the expectation value of a space Cartesian component of f in a pure state to be ( f , ) = (Ij~l[J~j(e)lI2]l), we find that for it = 0 and axial symmetry in the body-fixed system of coordinates,
2A'~,+A',
(l,~) = (l,,) = (l~) = - - , 3
(14)
and for arbitrary it,
(J,,)-(l,,)
= P,(I]; itit)[A~--A',].
(15)
Thus for positive core projection factors the sign of the dynamic shape and of the static shape will be the same. A further point of interest here is the question of the axis of core rotation. The assumption that the projection of the core angular momentum, K, in the b o d y system is zero requires the core to rotate about an axis perpendicular to the axis of symmetry. The above expression for the dynamic shape factor is based upon that assumption. The converse assumption that the core rotate about the body symmetry axis would imply it = K. In this case the dynamic shape factor would be
(J~®)~.~-(Juba,,K =
2).-- 1
--
it+l
P,(Ii; /2)[A',--A',].
(16)
Thus for a given it the dynamic shape will be of opposite sign for K = i t and K = O. If now the very considerable correlation between the signs of
46~
R. K. OSBORN A N D E. D. KLEMA
observed quadrupole moments and the state of fullness of corresponding subshells 6) is not to be completely destroyed by the concept of the rotating core, it is clear that the dynamic shape must be of the same sign as the static shape -- as it is the latter which is presumably determined b y the number of particles in unfilled subshells. At least this must be the case whenever the core contribution to the quadrupole moment dominates the particle contribution. If it is assumed that the core deformations in the ground state are volume preserving, then a simple relationship exists between the semi-axes; viz. ~ = 1/~ ~. Accordingly, the expression for the quadrupole moment shape factor becomes Qo --
•
Since the cubic ~8_h~_
1 = 0,
h --
5Q°
ZeR '
(xs)
has only one positive root, we see that assumptions (a) and (b) plus knowledge of nuclear magnetic and quadrupole moments suffice to determine a nuclear shape parameter, 7. 2. A p p l i c a t i o n 2.1 THE
/" = l + ~
of Model
NUCLEI
The signs and magnitudes of the distortions computed on the basis of the present model for the 39/" = l + ½ nuclei which have measured quadrupole moments are in the main quite reasonable and consistent. The situation relative to Li Tis quite unsatisfactory, but also probably not very meaningful. The measured quadrupole moment for Pr TM is so closely accounted for b y the odd proton that the computed core shape is spherical -- which might be regarded as an anomalously small distortion. Conversely, the enormous distortions computed for Lu lv6 and Ta ~8~ seem anomalously large. The single-particle configuration assumed in I for the nucleus Yb ~78 seems to require modification. It was taken to be fs/2, which presumably admixed with the fT/a state, 2 = 2 being a good quantum number. So far as the magnetic moment is concerned, this admixture is quite satisfactory. Here there are two possible values for the mixing ratio a (since this is a case of spin-orbit partner mixing) -- one near unity and the other very small. The
ANALYSIS O F N U C L E A R Q U A D R U P O L E MOMENTS
463
state with the small ratio, however, predicts an anomalously large distortion whereas the large ratio implies that the single-particle configuration is more nearly f~/, than f6/m. If this implication is exploited, as suggested b y Moszkowski and Townes '), then we would assign Yb *Tsthe ground-state admixture (fT/z, 2 = 2, I = 5/2) + (f~/2, 2 = 4, I = 5/2). As seen in table I, such a configuration correlates the magnetic moment quite satisfactorily and leads to a prediction of an entirely reasonable value of the distortion. The quadrupole and magnetic moments and nuclear shapes of the remaining nuclei in this class all seem consistently correlated within the context of the model. Further, there is some reinforcement of the conclusion previously drawn that the appropriate core state admixture is always )t and 2 + 2 . Among the ds/z, fT/s, and g912 nuclei there were magnetic moments which required the 2 ---- 4 core component. But whether the basic core component was to be that with 2 ---- 0 or that with I = 2 was ambiguous so far as the magnetic moments alone were concerned -- although the concept of interacting core states was invoked to provide a justification for 2 ---- 2 rather than 1 ---- 0. However, the necessity for the presence of a large 2---- 2 component in the core configurations for these nuclei can be inferred from consideration of the core contribution to their quadrupole moments. The assumption of the (0, 4) admixture leads to the contribution
(2. = Q,oa' P,(II; 44)
(19)
whereas the admixture (2, 4) provides
Q, = Qo [(1--ai)P"(II; 2 2 ) + 2 a ( 1 - - a i ) t P,(II; 24) +a2 P,(II; 44)].
(20)
Now
P,(II;
44) = - - 0 . 1 0 1 2 , ---- + 0 . 0 1 7 3 , ---- + 0 . 0 8 2 6 ,
I = I = I =
5/2 7/2 9/2.
Thus the (0, 4) admixture leads to opposite-signed static and dynamic shapes for I = 5/2; and for I ---- 7/2 and 9/2 to such a severe decrease in the dynamic asymmetry that only abnormally large static deformations could account for the observed quadrupole moments. Conversely, an examination of table I reveals that the (2, 4) admixture predicts signs of the distortions of these nuclei
464
R. K. OSBOR.M AND E. D. KLEMA
which are quite consistent with expectation 6), and magnitudes which are in general reasonable. 2.2 T H E
j = 1--~, I >
3/21NUCLEI
In the case of j = l--½ nuclei, which according to I admix single-particle states, one is led immediately to reinforcement of the conclusion that 2 tends to be a constant of the motion-provided that I > 3/2. In accordance with the discussion in I it is expected that a two-state description of the Is/s, gT/,, and hg/s nuclei (except for E u ass, which will be discussed later) would consist of one or the other of two alternative admixtures; either
(I,l=I,2=O)+ ( I , i = I - 1 , 2 = 2 ) or ( I , / = I , 2 = 2 ) + ( I , / ' = I - 1 , 2 = 2 ) . In the first instance the core contribution to the quadrupole moment is 1--5 Qs = Qo a' 7 ( I + ~ j ' (21) and in the second instance it is 41(1+1)--15
Qs = Qo7(i+l)(2i+3)
I
1-a s
(21+5)(1+3)--1514i(i+1)_15 _1"
(22)
Thus the former configuration is characterized b y opposite-signed static and dynamic shapes whereas the latter is not; hence it is expected that the latter configuration should predominate in the ground state in accordance with the previous remarks. Furthermore, it is to be noted that the core contribution to the quadrupole moment for these nuclei is independent of the sign of a, the mixing coefficient determined b y the magnetic moments. The nucleus Eu 1~ was assigned in I the ground state of a d s / a odd particle with core states 2 = 2 and 4 admixed. This configuration, however, contains an unusually large amount of the 2 = 4 component, and also predicts a very large distortion. For this nucleus Moszkowski and Townes 6) have suggested the shell configuration (g~ls)~12(dsis)6(hal/~)~. In accordance with this suggestion, we would regard Eu xr~ as a gT/~ odd particle coupled to a core in the state 2 = 2 to give a total spin I = 5/2, and admixing the state (d5/2, 2 = 2, I = 5/2). This latter assignment does indeed seem quite preferable in that only a small amount of the (dt/2, 2 = 2) component is required to correlate the magnetic moment. Furthermore, though the distortion remains large @/~ = 1.57, see table I), it
ANALYSIS OF NUCLEAR QUADRUPOLE MOMENTS
465
is considerably less than that predicted by the previous configuration 0,1~ = 1.73). 2.3. T H E
7" = l - t ,
I = 312 N U C L E I
The situation relative to the ds/z nuclei is, however, no longer satisfactory if the rules advocated in I are strictly enforced. For purposes of magnetic moment correlation the ds/s nuclei were classed with all other j = l--½ nuclei and presumed therefore to be characterized by mixed single-particle configurations rather than mixed core configurations. It was found that all of the ds/2 magnetic moments, with the exception of those for C1~, C1s~, S ~, and Xe TM, which apparently required admixture of the d6/2 state, could be accounted for by admixture of the nearby Sl/s state with ~ a constant of the motion and equal to 2. But for such a state the core contribution to the quadrupole moment is Qs = Qoa~Ps({ ½2 2) = - - 0 . 2 Qo a2.
(23)
Thus in accordance with the suggestion above that the static and dynamic shapes should be of the same character (both oblate or both prolate), we feel that these configurations for the da/~ nuclei are unacceptable. Now it is perhaps significant that only for this group of ] = l--½ nuclei is it possible to account for magnetic moments on the assumption of core state mixing alone, with the same exceptions as mentioned above for the ds/~--sl/2 configuration. The admixed core states would be those characterized by ~ = 0 and 2; hence the core contribution to the quadrupole moment would be
Qs -= 2Qoa(1--a')~ Ps (zla'~, 02) = 0.4 Q0a(1--a') ½.
(24)
This is the same as the corresponding quantity for Pa/z cases, and in general leads to a reasonable description of the shape of the d~/z (when applicable) and Ps/2 nuclei. The magnetic moments of C1as, ClaL S =, and Xe TM require admixture of the (ds/2, ;t = 2) state. The apparent difference between the confiuration assigned to these cases and that assigned to the other members of their class is probably mainly a consequence of the oversimplification inherent in the assumption of only two components in the wave function. In actuality, one would expect that
466
R.K.
O S B O R N A N D E.
D. K L E M A
the ds/s state is slightly admixed in all of the da/2 configurations, but to such a small extent that only for the four cases above is it impossible to ignore it completely so far as magnetic and quadrupole moments are concerned. There is some further justification for this point of view in that even with the admixture of the d6/~ state, 2 does not appear to be a good quantum number. That is, if one assumes a configuration consisting of the states (dais, ;t = 2) and (ds/s, 2---- 2), one finds, because of the cross term in the expression for the magnetic moment, that two possibilities exist for the mixing ratio a, one close to unity and the other extremely small. Neither of these admixtures is satisfactory in that the former suggests that the appropriate single-particle configuration should be ds/2 instead of d3/2, whereas the latter implies that the static deformations are so nearly averaged out b y the core motion that only for unreasonable deformations could the core contribute significantly to the quadrupole moment. Conversely, the two-component state (ds/2, 2 = 0 ) + (ds/,, 2---- 2) correlates the magnetic moments for relatively small amounts of admixed state while at the same time predicting reasonable distortions. Thus it appears that 2 is not a good quantum number for the d3/a nuclei, although/" m a y be so considered except for these four cases. In table I the mixing coefficients and nuclear distortions are presented for these four nuclei on the assumption of the (d3/~, 2 = 0)+(ds/2, 2 = 2) configuration. 3. D i s c u s s i o n It would appear that the experimental information on nuclear quadrupole moments is not only consistent with the basic assumptions of the kinematical model developed in I, b u t actually provides additional support in several instances. In particular it seems to strengthen the basic assumption that ~"= l+½ and ] = l--½ nuclei are fundamentally differentiated with respect to the nature of the core-particle degrees of freedom which are more nearly constants of the motion in the ground state. Actually, of course, consistent correlation of both magnetic and quadrupole moments within the context of the model has largely destroyed this differentiation so far as the ds/2 nuclei are concerned. But the fact that it is for only these / = l--½ nuclei that the distinction disappears merely suggests the possibility that the distinguishing mechanism is significantly less operative if I = 3]2 than otherwise. It is clear that to a large
ANALYSIS O F N U C L E A R Q U A D R U P O L E MOMENTS
467
extent this differentiation is an assumed consequence of the shell model, in that the /"----ld-~ levels frequently seem to be more isolated from other favorable parity single-particle levels than is the case for the /" = l--½ levels. Thus the factor of foremost importance here is the same one that has led to the gross level ordering scheme on the ]'--j coupling model which provides a reasonable accounting for the magic numbers and for most nuclear spins and parities. A second mechanism which quite consistently operates to favor this differentiation is the pairing energy. It is observed that the ]" :- l--½ single-particle states fs/s, gT/s, and h,/2 appear to admix predominantly the nearby Pa/2, ds/2, and fv/s states, respectively. Thus whenever the admixed level is non-empty, the change in the pairing energy favors the admixture. If the admixed level is empty, there is, of course, no pairing energy change. Conversely, exactly the same effect tends to oppose the inverse admixture. However, these considerations are considerably weakened in their application to the I = 3/2 nuclei, since both the Ps/s and da/s states are characterized by nearby lower spin states of favorable parity; i.e. Pl/s and sl/s, respectively. But in spite of this fact, practically all nuclei characterized by the single-particle assignments Ps/s or ds/z are observed (see table I) to admix core states rather than single-particle states. This apparent preference for core-state admixture stems directly from the assumption that the static and dynamic core shapes should be of the same sign. Since PB(II-- 1; 2 2) = (I-- 5)/7 (I-+- 1), it is clear that the contribution to the core shape from the ]" = I - - 1 , it' =- 2 state that is admixed in the case of fs/a, gT/~ and ha/~ nuclei is always negative; i.e., such admixtures invariably tend to change the character of the dynamic shape. But inspection of table I shows that in these cases the favorable shape contribution from the 1"= I, it ---- 2 state considerably exceeds the unfavorable contribution without exception; so that the sign of the resultant dynamic shape is always the same as that of the static one. Hence the net result of this type of admixture for I > 3/2 nuclei is simply to reduce somewhat the dynamic asymmetry of the configuration. This situation for the I = 3/2 nuclei is, however, completely altered by the fact that the pure states ~"----I = 3/2, it = 0 or 2 are spherically symmetric. Thus here the unfavorable shape contribution from the 1"----I--1, i t ' = 2 configuration is the total contribution; and if the assumption concerning the correspondance
468
R.
K.
OSBORN
AND
E.
D.
KLEMA
between static and dynamic shapes remains valid, such admixtures must be regarded as unlikely. Conversely, it is seen from eq. (24) that the static and dynamic shapes are of the same sign for core state admixtures with j ' = 1"= I = 3/2. Furthermore, the four d3/~ nuclei discussed above which require the ds/~, 2 = 2 state to account for their magnetic moments are also characterized by proper shape correspondence since
P~(I,I+1;22)
--
(21-- l ) ( I + 6) 7 ( I + 1 ) ( 2 I + 3)
> 0,
I >½.
(25)
The significance of this seems to be t h a t the configurations of the I = 3/2 nuclei are more sensitive to core dynamics than are those of the I > 3/2 cases. Finally, it is to be pointed out t h a t the correlation of the quadrupole moment data provides some corroboration of the assumption that the core rotation is such that its angular m o m e n t u m has zero projection upon the body s y m m e t r y axis; i.e., the assumption that K = 0. From the kinematical point of view, and so far as magnetic moments are concerned, this assumption is largely arbitrary -- its only effect being to restrict 2 to even values if the parity of the nucleus is to be t h a t of the single particle. The assumption receives some justification from the analysis by Bohr and Mottelson ~) of the hydrodynamical model of the core, but it would be desirable to establish it empirically. First of all it is to be expected that for the axially symmetric system the magnitude of K would be a good quantum number. That is, axial s y m m e t r y implies that the state of the system should be invariant under an infinitesmal rotation about the body z'-axis, and hence an eigenfunction of .~,,. But the further invariance of the system under rotations of 180 ° about the axis perpendicular to the s y m m e t r y axis defines an operator which does not commute with -~r, but does with .~,~,; and hence if the state is chosen to be an eigenfunction of this operator, then it is the magnitude of K which is expected to be a good. quantum number. Therefore, those nuclear states which contain a core component characterized by ~t = 0 and consequently K = 0 should have other core components with K = 0 and even values of ;t only 1). It is to be noted that K need not be restricted to even values, but rather that 2 + K must be even. Since the magnetic moments are independent of K (except
ANALYSIS O F N U C L E A R Q U A D R U P O L R MOMENTS
469
insofar as the evenness or oddness of K determines the evenness or oddness of ,1), one finds that a consistent scheme for the correlation of magnetic moments can largely be worked out employing odd values of ~. However, one finds further that the dynamic shapes computed from these core-plus-particle configurations are either of the wrong sign, as is almost invariably the case for the I ---- 3/2 nuclei, or so much more spherically symmetric than those computed from the K = 0 configurations that the consequent distortions are considerably larger -- often unreasonably so. Hence we feel t h a t the data, taken in conjunction with the assumptions of the model, provide adequate justification for continuing with the assignment of K ---- 0 to nuclear ground-state configurations. It is clear from the above that the empirical determination of the K = 0 assignment for odd-mass nuclei is deduced primarily from the evidence on quadrupole moments. However, one finds that the assignement K :/: 0 for I = 1/2 nuclei is actually ruled out on the basis of magnetic moment evidence alone; except for the cases s) W 18s and Os 187, whose moments are ~ 0.12 and cannot be correlated by any K = 0 configuration. These are satisfactorily accounted for by the core-plus-particle configuration (Pl/z, ~t = l, K = 1 ) ~ (fu2, ~" 3, K 1). The results of the analysis discussed above are presented in table I. The odd-particle configurations listed are those given by Mayer and Jensen 9) unless indicated by an asterisk. The cases so indicated are discussed in detail in the text. The quantum numbers which define the ground states are given in accordance with the convention (1 -- a ~)i (l]kI) -k a (l' j' ~' I). The magnetic moments used in calculating the a's are taken from the recent compilation of Walchli lo). The observed quadrupole moments are taken from Ramsey 11) or from Gordy, Smith and Trambarulo I~), unless noted by a letter. In these cases the literature references are cited. In each case the value used in the computation and listed as Qobs is a weighted average of the data indicated above. It is interesting to compare the deformations calculated in the present work with those presented in Blatt and Weisskopf is), calculated for a static, axially symmetric spheroid. It is found, as is to be expected, that the shape parameters in table I lead to deformations which are in general of the order of five times as great as those obtained for non-rotating nuclei. Large static deformations were also predicted within the context of the hydrodynamical =
=
470
R. K. OSBORN AND E. D. KLEMA
model 8). Such large deformations tend to strengthen the requirement employed above that the character of the static and dynamic shapes be the same 6). References l) 2 3 4 5 6 7
8 9 10) ll) 12) 13)
R. K. Osborn and E. D. Klema, Phys. Rev. 100 (1955) 822 S. G. Nilsson, Dan. ~Mat.-fys. Medd. 29 (1955) No. 16 A. Bohr, Dan. Mat.-fys. Medd. 26 (1952) No. 14 K e n n e t h W. Ford, Phys. Rev. 90 (1953) 29 A. K. Kerman, Dan. Mat.-fys. Medd. 30 (1956) No. 15 S. A. Moszkowski and C. H. Townes, Phys. Rev. 93 (1954) 306 A . B o h r a n d B . R . M o t t e l s o n , Dan. Mat.-fys. Medd. 27 (1953) No. 16 K. Murakawa, Phys. Rev. 98 1285 (1955) Maria Goeppert Mayer and J. Hans D. Jensen, Elementary Theory of Nuclear Shell Structure (John Wiley and Sons, New York, 1955) H . E . Walchli, A Table of Nuclear Moment Data, ORNL-1469 (Supplement II) Norman F. Ramsey, Nuclear Moments (John Wiley and Sons, New York, 1953) Waiter F. Gordy, William V. Smith, and Ralph F. Trambarulo, Microwave Spectroscopy (John Wiley and Sons, New York, 1953) J o h n M. B l a t t and Victor F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, New York, 1952)
471
ANALYSIS OF NUCLEAR QUADRUPOLE MOMENTS
TABLE I Analysis of nuclear quadrupole moment data. The odd-paxticle configurations are given in the second column and the assumed ground states in the third. The mixing coefficients axe given in the fourth column. The experimental values of the quadrupole moments axe listed in the fifth column. The calculated shape parameters axe given in the last three columns. Spin 3/2 Nuclei
Odd Particle Configuration
Nucleus
Assumed Ground State Q u a n t u m Numbers, (lj2I)
a
Qo~
>'/~
7
P$/|
O d d Proton
aLi4 sBs mCust
nCUu uGau a~Gat0 uAs.a uBr.. ~Br~ a7Rbso
,
(Isvs)S(Ips/m)l (Isl/s)S(Ip~/s)s (2ps/s)I (2pw) l (2psls) s (2pvl) a (2ps/s)S(lfv2) 2 (2ps/s)S(l fi/t) 4 (2pvs)* (lf6/i) 4 (2ps!j)S(lfs/i) s
(1 (1 (1 (1 (1 (1 (1 (1 (1 (I
3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2
0 0 0 0 0 0 0 0 0 0
3/2) 3/2)
3/2)
(l (1 (1 (1 (1 (l (I (1
3/2)
(1
3/2) 3/2)
3/2) 3/2) 3/2)
3/2)
3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2
2 2 2 2 2 2 2 2 2
3/2) 3/2) 3/2) 3/2)
3/2) 3/2) 3/2)
3/2)
3/2) (1 3/2 2 3/2)
0.463 0.665 0.792 0.751 0.844 0.702 0.972 0.822 0.781 0.546
1.656 0.02 0.040~) 1.373 ---0.15b, c ) i 0.932 ---0.15 b ) 0.939 1.093 0.2318 0.1461 1.062 1.188 0.3 0.33 d) 1.107 0.28 d) 1.088 0.15e) 1.051
0.777 0.853 1.036 1.032 0.956 0.970 0.917 0.950 0.958 0.975
2.13 1.61 0.90 0.91 1.14 1.09 1.30 1.16 1.14 1.08
1.241 1.038
0.898 0.982
1.38 1.06
Odd Neutron .Bes soHgm
(lsl/s)s(lps/s) I (2fT/s)S(lh./I)l°(3Ps/j)3 (2f6/I)'(lils/I)11
(1,3/2 0 3/2) (1 3/2 0 3/2)
(1 3/2 2 3/2) (1 3/2 2 3/2)
0.595 0.792
0.02 ~ )
o.5og)
i
i a)
b)
c) d)
e) f) g)
G. Wessel, Phys. Rev. 92 (1953) 1581 H. D. Dehmelt, Z. Physik 133 (1952) 528; 134 (1953) 642 A. Bassompierre, C. R. Acad. Sci. 237 (1953) 1224 B. Bleaney, K. D. Bowers and M. H. L. Pryce, Proc. Roy. Soc. A 228 (1955) 166 B. Bleaney, K. D. Bowers and R. S. Trensm, Proc. Phys. Soc. A 66 (1953) 410 H. Kopfermann, A. Steudel, S. Wagner and W. Walcher, Nachr. Akad. Wiss. C,Sttingen, Math-Physik KI I I a No. 1 (1953) 1 J . G . King and V. Jaccaxino, Phys. Rev. 94 (1954) 1810; 91 (1953) 209A W. Gordy, J. Chem. Phys. 19 (1951) 792 H. G. Dehmelt, Z. Physik 130 (1951) 480 B. Senitzky, I. I. Rabi and M. L. Perl, Phys. Rev. 98 (1955) 1537A H. Krtiger and U. Meyer-Berkhout, Naturwissenschaften 42 (1955) 94 W . D . Knight, Phys. Rev. 92 (1953) 539A K. Murakawa, Phys. Rev. 98 (1955) 1285 H. G. Dehmelt, H. G. Robinson and W. Gordy, Phys. Rev. 93 (1954) 480
472
R. K. OSBORN AND E. D. KLEMA
TABLE I (continued) Spin 3/2 Nuclei Odd Particle Configuration
Nucleus
Assumed Ground State Q u a n t u m Numbers, (I]M)
a
Qobs
?
s¢
7'/a
--0.0789 k) --0.0626 k)
1.251)
0.774 0.865 1.139
1.136 1.075 0.937
0.68 0.80 1.22
d3/t Odd Proton
I~C118 1rClso vTIrll4 77]rH8
79AuH8
(ld~/,)e (2sl/,)~ (I d3,,a)l (ld6/z)' (2sl/,)*( I ds;,)l (lg~/,)s (2d6,,,)6 (l h n/,) I° (2d3/,) s (lgT/s)8 (2ds/2)6 (I hws) z° (2d~2) s (1 gT/~)e(2ds/t)e (l htl/t) 12 (2da/2) s
(2 3i2 0 3/2) (2 3/2 0 3/2) (2 3/2 0 3/2)
(2 5/2 2 3/2) (2 512 2 3i2) (2 3/2 2 3/2)
0.534 0.479 0.426
(2 3/2 0 3/2)
(2 3/2 2 3/2)
0.337
1.2 l)
1.164
0.927
1.26
(2 3/2 0 3/2)
(2 3/2 2 3'2)
0.213
0.5cm )
1.129
0.941
1.20
--0.065n, o) 0.589 0.045 ° ) 1.054 --0.14P) 0.875
1.303 0.974 1.069
0.45 1.08 0.82
Odd Neutron 16S17 15S19
.Xe~v
(lds/2)6 (2sl/~)2 (ld3/z) 1 (1 ds/~)6(2sl/t)~ (Ida/s, s (2ds/2)6 (ig7/I)8 (3sl/g)z (lhn/s)x°(2d~/~) 1
k) 1)
(2 312 0 312) (2 3/2 0 3/2)
(2 5/2 2 3/2) (2 3/2 2 3/2)
(2 3/2 0 3/2)
(2 5/2 2 3/2)
0.497 0.628 0.473
G. P. Koster, Phys. Rev. 86 (1952) 148 W . v . Siemens, Ann. Phys. 13 (1953) 136 K. Murakawa and S. Suwa, Phys. Rev. 87 (1952 l 1048 m) W. v. Siemens, Ann. Phys. 13 (1953) 158 W. v. Siemens, Naturwissenschazften 38 (1951) 455 n) G. R. Bird and C. H. Townes, Phys. Rev. 94 (1954) 1203 o) C . A . Burrus, Jr. and W. Gordy, Phys. Rev. 92 (1953) 274 H. G. Dehmelt, Phys. Rev. 91 (1953) 313 p) E. Brun, J. Oeser, H. H. Staub and C. G. Telschow, Phys. Rev. 93 (1954) 904 A. Bohr, J. Koch and E. Rasmussen, Ark. Fys. 4 (1952) 455
ANALYSIS
OF
NUCLWAR
QUADRUPOLE
473
MOM]gNTS
TABL~Z I (continued) Spin 5/2 Nuclei Odd Particle Configuration
Nucleus
.Assumed Ground State Quantum Numbers, (1/~I)
a
Qobs
7
0t
7/0t
0.297 0.959 0.248 0.499 0.844 0.976 0.352
0.11 q) 0.153 r ) --0.65 s) ---0.819 t) --0.054 u ) 1.2 2.6
1.241 1.232 0.877 0.8{}9
0.124
---0.004 v )
0.955
1.023
0.93
0.438
0.30 w)
1.302
0.876
1.48
d&/l Odd Proton itNal.
lsAllt uSb~o 6sI:t ~Prn
(2 5/2 2 3/2) (2 512 0 512)
(Id6/t)s (Ids/s)~ (2d5/I)z
(lgT/s)J(2d6:l) 1 (Ig,/t)S(2dvl) t
76Re~tt
(lg:/s)S(2dvs) 6 ( lgT/s)s(2d6/t)6 (1 ht t:t) t2
80,
(Idu,) i
#sEuss
(2 (2 (2 (2 (2 (2
5/2 2 512 2 5/2 0 5/2 0 5/2 2
(2 (2 (2 (2 (2 (2 (2
5/2) 5/2) 5/2) 5/2) 5/2)
5/2 5/2 5/2 5/2 5/2 5/2 5/2
4 2 4 4 2 2 4
3/2) 5/2) 5/2) 5/2) 5/2) 5/2) 5/2)
0.898 0.901 1.068 1.072 1.001 0.999 1.130 0.941 1.232 0.901
1.38 1.37
0.82 0.81 1.00 1.20 1.37
Odd neutron
5/2 0 5/'2)
(2 5/2 2 5,2) fS!I
Odd Proton ~Rbu
(2Ps/t)' (Ifvj)
q) r) s)
t)
(3 5/2 2 5/2)
(I 3/2 2 5/2)
M . L . Perl, I. I. Rabi and B. Senitzky, Phys. Rev. 98 (1955) 611; 97 (1955) 388 P. L. Sagalyn, Phys. Rev. 94 (1954) 885 H. Lew and G. Wessel, Phys. Rev. 90 (1953) I G. F. Koster, Phys. Rev. 86 (1952) 148 A . W . Jache, G. S. Blevins and W. Gordy, Phys. Rev. 97 (1955) 680 K. Murakawa, Phys. Rev. 93 (1954) 1232 G. Sprague and D. H. Tomboulian, Phys. Rev. 92 (1953) 105 K. Murakawa and S. Suwa, Z. Phys. 137 (1954) 575 V. Jaccaxino, J. G. King and H. H. Stroke, quoted by R. Livingston, B. M. Benjamin, J. T. Cox and W. Gordy, Phys. Rev. 92 (I953) 1271 T. Kamei, J. Phys. Soc. Japan 7 (1952) .649 H. Lew, Phys. Rev. 91 (1953) 619
u) v) G. R. Bird and C. H. Townes, Phys. Rev. 94 (1954) 1203 calculated from data of S. Geschwind, G. R. Gunther-Mohr and G. Silvey, Phys. Rev. 85
(1952) 474
w) H. Krfiger and U. Meyer-Berkhout, Naturwissenschatten 42 (1955} 94 B. Senitzky, I. I. Rabi and M. L. Perl, Phys. Rev. 98 (1955) 1537A
474
R. K. OSBORN AND E. D. IKL]~MA TABLR I (continued) Spin 7/2 Nuclei
Odd Particle Configuration
Nucleus
Assumed Ground State Q u a n t u m Numbers, (l],~l)
a
Qot~
7
ot
7/,v
Odd Proton
lil~V|8 sTCols
(lfv,) a (lfvl) 6 (lfvt)'
(3 7/2 0 7/2) (3 7/2 2 5/2) (3 7/2 2 7/2)
7oYbtos
(2f71t)7(11q/I)t°(2p~t)'*
(3 7/2 2 5/2)
ssMnso
(3 7/2 2 7/2) (3 7/2 4 5/2) (3 7/2 4 7/2)
0.897 0.591 0.427
0.3 z ) 0.55Y, z) 0.5 z)
1.126 1.177 1.177
0.942 0.922 0.922
1.20 1.28 1.28
0.657
3.9
1.206
0.911
1.32
0.715 0.890 1.043 1.314 1.350 2.287
1.182 1.060 0.979 0.872 0.861 0.661 0.747
0.60 0.84
Odd Neutron (3 7/2 4 5/2)
gT/i
Odd Proton 5tSbn
5ai75 ~CsTs ~TLass esEu~ ?lLUt04 vsTa~o0
(lgv/I) 1 (Zgvl)* (Zg7/i) 6 (lgvs) 7 (IgTjl)v (2d6/I) a* (lg~/I)v (2d6/s)* (1 hit/I) B (1 gv/I)T(2ds/s)* (1 hll/I) l°
y)
z) aa)
ab) ac) ad)
ae)
(4 (4 (4 (4 (4 (4
7/2 7/2 7/2 7/2 7/2 7/2 (4 7/2
2 2 2 2 2 2 2
7/2) 7/2) 7/2) 7/2) 5/2) 7/2) 7/2)
(2 (2 (2 (2 (2 (2 (2
5/2 2 5/2 2 5/2 2 5/2 2 5/2 2 5/2 2 5/2 2
7/2) 7/2) 7/2) 7/2) 5/2) 7/2) 7/2)
0.543 0.565 0.554 0.612 0.400 0.646 0.376
- - 1 . 0 aa) --0.47 ab) --0.003 a¢) 0.8 ad) 2.5 5.9 6.2 ae)
1.791
A. J a v a n and A. Engelbrecht, Phys. Rev. 96 (1954) 649 A. Javan, G. Silvey, C. H. Townes and A. V. Grosse, Phys. Rev. 91 (1953) 222A K. Murakawa and T. Kamei, Phys. Rev. 92 (1953) 325 A. W. Jache, G. S. Blevins and W. Gordy, Phys. Rev. 97 (1955) 680 K. Murakawa, Phys. Rev. 93 (1954) 1232 G. Sprague and D. H. Tomboulian, Phys. Rev. 92 (1953) 105 W. Gordy, J. Chem. Phys. 19 (1951) 792 calculated from data of R. Livingston, O. R. Gilliam and W. Gordy, Phys. Rev. 76 (1949) 149 K. H. Althoff and H. Kriiger, Naturwissenschaften 41 /1954) 368 K. Murakawa, Phys. Rev. 98 (1955) 1285 K. Murakawa, J. Phys. Soc. J a p a n 9 (1954) 391 K. Murakawa and T. Kamei, Phys. Rev. 92 (1953) 325 B. M. Brown and D. H. Tomboulian, Phys. Rev. 88 (1952) 1158
1.06
1.51 1.57 3.46 2.40
ANALYSIS OF NUCLEAR QUADRUWLE MOMENTS
476
TABLE I (continued) Spin 9/Z Nuclei Odd
Nucleus
Assumed
Particle
Quantum
Configuration
Ground
State
Numbers,
(1jAI)
Qobs
a
/
Y
)
a
/
Y/a
.%/a
Odd ,J%
(2P,/*)‘(lf~/,)e(2P1/*
‘1
Proton
(4 9/2
2 9/2)
(4 9/2
4 9/2)
0.179
rsTc,
(lg./*)’ (2PJ,)4(lf,/,)a(2PI/*)p
(4 9/2
2 9/2)
(4 9/2
4 9/2)
0.629
,Jn,,
(lg,/,)s (2P*/,)‘(lf,/*)~
(4 9/2
2 9/2)
(4 9/2
4 9/2)
,Jn,,
(2P,/,)*(lg,/*)B (2P*/*)‘(lf,/*)@
(4 9/2
2 9/2)
(4 9/2
-0.2af)
0.986
1.008
0.98
0.34 as)
1.046
0.978
0.716
0.98 ah)
1.096
0.966
1.15
4 9/2)
0.710
l.ocah)
1.096
0.966
1.16
4 9/2)
0.720
0.961
1.020
0.94
1
1.09
(2P,/*)‘(lg,/*)g Odd Neutron ,rGe,r
(2P~ll)‘(lfs/*)e(2PlI,)p
(4 912 2 9/Z)
(4 9/2
&e,r
(lg*/,)r (2P,/,)‘(lf*/,)s(~g,/,)’
(4 912 2 712)
=Kr,r
(2Ps/n)“(lf,/*)~(2P,,,)’
(4 912 2 9/2)
(4 912 4 712) (4 912 4 9/2)
-0.21
0.748
1.05”i)
1.160
0.928
1.26
0.664
0.15
1.022
0.989
1.03
0.978
1.011
0.97
(lg&’ he/s Odd Proton S3BiIr,
(lh&
(6 9/2
Phys.
2 9/2)
K.
Murakawa,
ag)
K.
G. Kessler
ah) ai)
G. F. Koster, Phys. Rev. 86 (1962) 148 G. R. Bird and C. H. Townes, Phys. Rev.
and R. E.
data of W. A. Hardy,
98
(1966)
2 9/2)
af)
berg, G. W.
Rev.
(3 7/2
Trees,
Phys.
Cohen,
-0.4
1285 Rev.
92 94
G. Silvey, C. H. Townes,
Parker and V. W.
0.686
Phys.
(1963) (1964)
303 1203
calculated
from
B. F. Burke, M. W. P. StrandRev.
92
(1963)
1632