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On the signs of nuclear quadrupole moments
1.E.3
I
Nuclear Physics
A99 (1967) 321--336; (~)
North-HollandPublishin# Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON T H E S I G N 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 Y I T Z H A K Y. S H A R O N t
Palmer Physical Laboratory, Princeton University tt Received 28 March 1967
Abstract: We analyse the experimental data on the signs of all the measured quadrupole moments
in the ground states of nuclei. We find positive quadrupole moments in virtually all the cases where the last major proton and neutron shells are both more than one-fifth full. For oddmass nuclei the quadrupole moment is negative (positive) when the odd nucleons occupy 1/8 or less (7/8 or more) of the last major shell. The last observation is shown to be an extension of the known rules for the signs of the quadrupole moments of odd-mass nuclei with one particle outside (or missing from) a closed shell. I. Introduction
The experimental information on the systematics of nuclear electric quadrupole moments has played an important role in the development of both the shell model i) and the collective model z, 3). Throughout the years several studies of the systematics of quadrupole moments have been carried out 4-1 o). However, the only two investigations which were carried out since 1959 have been devoted to limited regions of the periodic table 9, 10). In all these papers the signs and the magnitudes of the quadrupole moments were usually related to the number of odd nucleons, to the numbers of protons and neutrons that occupy levels in specific subsheUs or to the number of nucleons outside major closed shells. Some of the conclusions that have been reached in these aforementioned analyses of the experimental data are the following: (i) The majority of the measured quadrupole moments have a positive sign. Quadrupole moments tend to be positive in the ground states of nuclei not in proximity to closed shells in either the protons or the neutrons. (ii) For odd-mass nuclei in which the odd particle is a single nucleon outside (or missing from) a major closed shell the quadrupole moments are always negative (positive). (iii) The quadrupole moments are usually larger, and frequently very much larger, than what one might expect on the basis of single-particle models. The quadrupole moments of some rare-earth nuclei are 30 times larger than the corresponding singleparticle values. The positive quadrupole moments often have appreciably larger magnitudes than those of the typical negative quadrupole moment. t Now at Department of Physics, Northeastern University, Boston Mass. tt Work supported in part by the U.S. Atomic Energy Commission. 321
322
Y.Y.
SHARON
The purpose of this paper is to point out that somewhat more quantitative empirical observations can be made if one considers the experimental quadrupole moment data from a new point of view. 2. Analysis of the data We have performed an empirical analysis of the signs of the ground state quadrupole moments of nuclei. We include all those moments whose magnitudes and signs have both been measured experimentally up to 1965. We must exclude from consideration
SIGNS
OF
MEASURED
QUADRUPOLE 0
MOMENTS
odd proton nuclei
[] odd neutron nuclei r'-, LIJ --I _J LL
1.0 ItO ll5
74E] I
~B (9~8 (9-n I []28 ,~7o (967
£,9
0.8
I '
!
b-<[ I
66(9@261
-
0.6
El 51
+~,5
(@16
(9,os
lot® ~,oz (9,oo
~L~,43
(999
~q4
94~7
(9,3
(934
ssE~
~61 G6o
I I r~ql2 I ', ,19E](~)121
(gsl
~. 97
E]93
(9s8 1 25 Z~37
(~) 105
(91o4
E~ 14
(96
I '
E{]~6
/~8
(945
(93~ I
0.4
(91°7 (9106
~4-," (946
[(9~z7
40 ~r~41E] [
(924
,~7
(~49
~_44
! i i 4z
@64 (965 /+16s (962 ~z~
T U) Z 0 n-I-Zb ILl Z
F] 48
@09 ®TZl
- 109112
io8
.--I .._1
/x d o u b l y o d d n u c l e i
I 75 ' 'r~!J
(~)59 E]57 Q58
(992
®u ~ ' z °
F]91
il6 i ~]118
89~[~
90
~ss
~s6 Q2t
LL
o 0.2
£z}
Z 0
@m
0
[ I
79r~ 7~
•2o
E]B, do 52~s°
(gs~ (953
~
0
I
0.2 FRACTION
0.4 OF
PROTON
(932
({)54
8o,~ra ~28.
]
r:19 u_
(98s
I
d 8 I
0.6 SHELL
I(~ 19
0.8 THAT
I
I.O IS
FILLED
Fig. 1. Signs of measured quadrupole moments in the ground states of nuclei. Each nucleus is denoted by a number. The key to this numbering appears in table 1. The abscissa for each nucleus is the fraction of the last major proton shell that is actually occupied by the protons in this nucleus. The ordinate for each nucleus is the fraction of the last major neutron shell is actually occupied by the neutrons in this nucleus.
QUADRUPOLE MOMENTS
323
doubly even nuclei as well as odd-mass or doubly odd nuclei whose ground state spins are less than 1. The vast majority of data that we use is taken from the compilations of nuclear moments by Lindgren 11), and Fuller and Cohen az). Usually whenever one of these compilations reports a definite sign for a quadrupole moment we accept that sign as valid t,. In very few cases we supplement the information in these compilations on the basis of more up-to-date research papers or private communications (see table 1). It should be noted that our analysis covers the moments of 121 nuclei, i.e. that our investigation is based on many more data than were available to the authors of previous studies of quadrupole moments throughout the periodic table 4- s). We base our analysis on the j-j shell model. In this model the magic numbers, which correspond to closed shells in either protons or neutrons, are 2, 8, 20, 28, 50, 82, 126, 184 and so on. Successive shells of identical nucleons thus contain up to 2, 6, 12, 8, 22, 32, 44 and 58 nucleons, respectively. In fig. l each nucleus is represented as a point on a two-dimensional grid, i.e., by a pair of Cartesian coordinates. The abscissa for each nucleus is the fraction of the last proton shell that is actually occupied by the protons in this nucleus. Similarly, for each nucleus in fig. 1, the ordinate is the fraction of the last neutron shell that is actually occupied by the neutrons in this nucleus. A closed shell nucleus is considered to have its last shell totally unoccupied, so that in fig. 1 the abscissas and ordinates are always less than one and greater than or equal to zero tit. To illustrate our notation, let us recall that 12 identical particles can be placed in the 2s-ld shell. Thus 160 8 is denoted by (0, 0), 1709 by (0,~3-), 197F8 by (x~-,0), ISF 9 by (T12,TI~-), 20°Nelo by (-~,-~), 377C120 by (¼, 0) and 40 11 ~). In fig. 1 we place a plus sign at the coordinates of a nucleus with a 19Kzl by (~y, positive quadrupole moment and a minus sign at the coordinates of a nucleus with a negative quadrupole moment. Odd-proton nuclei (i.e. nuclei with an odd number of protons and an even number of neutrons) are denoted by circles, odd-neutron nuclei by squares and doubly odd nuclei by triangles. Each of the 121 nuclei in fig. 1 is denoted by a number. The key to this numbering appears in table 1. 3. Nuclei with the last proton and neutron shells both more than one-fifth filled
On the basis of fig. 1 the following observations can be made: (i) With very few exceptions, whenever the major proton and the major neutron shells are both ~ or more filled, then the quadrupole moments are positive. Out of over 60 cases there are only four exceptions to this rule. Three of these exceptions involve nuclei with 17 odd nucleons (16517, 33 17 17 35Clls and 36Cl19). These three exceptions could be accounted for if we were to assume that 16 identical nucleons form a semi-closed shell. The assumption that the ld~ single-particle level lies considerably higher than the ld~ t The author is grateful to Dr. Lindgren for a preprint of his table and for several private communications. Typographical errors which appear in ref. n) are usually corrected in ref. 4~). tt Thus, for example, we do not include the moments of l~]Pmo0(ref.44)) or l~prs3(ref. 45)). ttt The reader note that our grid should really be a closed surface rather than a square in flat space. The horizontal (or vertical) lines denoting 0 and 1 should coincide.
324
Y.Y. SHARON
TABLE 1 Nuclei wl~ose quadrupole moments are included in fig. 1. A minus sign appears before the n u m b e r designating a nucleus with a negative m o m e n t 1
~H 1
25
"6~c 21 ~ 25
49
85 37Rb48
73
137~ 55t~s82
97 b) 177bl f 72~--105
--2
6Li 3
--26 b)
47 215C26
50
87 37Rb50
74
135 56B479
98 b) 179i..1f 72---107
--3
37Li4
27 d)
,$7 • 22T125
51
87 385r49
75
137R^ 56-~t81
99
181 73Ta108
9Be 5
28 a)
4922T127
_ 5 2 a)
90 39Y51
76
1391n 57--82
100
185R^ 75-~Cl 10
51 23V28
--53
93 --77 41Nb52
141p_ 59-'82
101
187R 75 e112
4 ~) 5
l°B 5
29
6
lIB 6
--30
7 e)
1~C 5
8
174N7
--9
53 24Cr29
54
99 43Tc56
--78 b) 143Nd 60- "-83
102
189~ 7 6 u s i 13
31
55 25Mn30
55 d) l°5pcl 46--59
--79 b) 145Ncl 60"~85
103
191177~ti14
32
59 27 Co32
56
107Cd 48---59
80 e) 1 4 8 p ~ 61 ~ul87
104
1931" 77-'116
170 9
- - 3 3 13) 63 29Cu34
57
109Ccl --81 a) 147g I_ 48---61 62-11185
105
197A" 79" ~Ul 18
10
121Nell
--34 b) 6295Cu36
58
1°9I4.9-t160
82 a) 1 62~u187 49~_
t06
2011480--~121
11
23Na12
--35
65 30Zn35
59
1111~ 49-1162
83 b) 1 5 3 ~ _ 62~n191
107 f) 203i_1_ 80-~123
12
225Mg13
36
67 3021137
60
113149-1'64
84
151w63~u88
--108
2°31q: 83~'120
13
27 13Al14
37
67 31Ga36
61
115I49-~66
85
1531~ " 63~u90
--109
204~. 831~1121
--14
33 16517
38
69 31Ga38
--62
12 l,~h 51--70
86
155G,t 64~91
--110
2°6ri: 83-'123
15
365519
39
71 31Ga40
63
122~h 51~71
87
157Gcl 64~93
--111
2°9R: 83-t126
--16
135C118
40
72 31Ga41
--64
123 515672
88 a) 156Th 65~91
112
205p^ 84-u121
--17
376C119
--41 b)
73Ge41
--65
125I 53-72
89 a) 160Th 65--~95
113
207084--ut23
--18
37C120
42
75 33As42
--66
127I 53-74
90 a) 161r~. 66~Y95
114
227A~ 89--~138
19
39 19K20
43
75 345e41
--67
1291 53"76
91 a) 16315. 66~Y97
_ 1 1 5 a) 229Th 90--'139
--20
40 19K21
44
79 345245
--68
131[ 53-78
92
I6514~ 67"-t)98
_ 1 1 6 b) 233p. 91-~142
1491K22
45
79 35Br44 81 35Br46
--69 b) 1331 53-80 --70 13t 54Xe77
93
173Vh 70---103 1751" 71~Ulo4
117 b) 23311 92--141 118 b) 23511 92~143
21 a)
_ 2 2 b) 43 215c22 23 b) --24
46
94
44 215c23
47
83 36Kr47
--71
133 55Cs78
95
17671LU105
119 ~) 2 4 1 p . 94--u147
45 218C24
48
85 36Kr49
72
135~ 55~'~$80
96
1771" 7 1 - u l 06
120
M o s t of the data appear in b o t h refs. 11,12). a) Data are from ref. 11) only. b) Data are from ref. 12) only. e) Ref. ~1). d) Ref. 32). e) Ref. 33). r) Ref. 34). The author is grateful to Dr. Redi for this information.
2"1A~ 95 "-lu 146 121 b) 2 4 3 A ~ 95-~1u148
QUADRUPOLE MOMENTS
325
and 2s~ levels is not in disagreement with the usual interpretation of the spectrum of 1~O 9 . Furthermore, Talmi and Unna 13) were able to account for the binding energies of nuclei at the end of the 2s-ld shell by treating the ld~ subshell as a distinct entity t The fourth apparent exception to rule (i) 233n 911"a142, is really an exception that proves 31n " i 233n the rule. The ground states of the odd isotopes of Pa, rt 2 91t'a140 ana 91ra142), have angular momenta of 3 and are the only nuclear ground states that are usually interpreted as the 3 states of rotational bands with K = ½ (refs. 15, 26)). For such a state the intrinsic quadrupole moment has an opposite sign from the measured one tt Thus the measured negative quadrupole moment indicates that 233r,, 91r'a142 has a prolate deformation ttt In the region of fig. 1 to which rule (i) does not apply we find approximately equal numbers of positive and negative quadrupole moments (out of 60 values of Q, 27 are positive and 33 are negative). On the other hand, out of the 61 nuclei to which this empirical rule is germane, not fewer than 57 possess positivequadrupole moments*. This result is especially impressive in view of the fact that our analysis involves nuclei from all the parts of the periodic table with atomic weights that range from 2 to 243. The nuclei which we consider - doubly odd, odd-neutron and odd-proton - have ground state spins which range from I to 7. Some of these nuclei are slightly deformed and others are strongly deformed. Yet, in spite of this great diversity, the specific details of the shell-model configurations for the ground-state wave functions do not seem to matter greatly in the region that is covered** by rule (i). Also, with the aforenoted exception of the ld~ subshell, one can apparently ignore the details of the subshells in the major shell structure when one predicts the signs of the nuclear quadrupole moments. Moszkowski and Townes 6) have shown that some positive quadrupole moments (as well as ground state spins and other properties) could be t However, the usual interpretation of some of the experimental data in the 2 s - l d shell seems to suggest a semi-closed shell at 14 identical nucleons rather than 16 (see refs. 1~.,9)). The existence of a semi-closed shell at 14, but not at 16, nucleons is also strongly suggested by the large binding energy of ~sSi~. All three exceptions with 17 odd nucleons have ground states with angular m o m e n t a 3 It is unlikely that we are dealing with K ½ bands whose lowest state has an angular m o m e n t u m of 2. However, if this were the case, then the negative quadrupole moments would be indicative of prolate equilibrium shapes. ** Consider a rotational band that is characterized by the quantum number K. The measured quadrupole moment Q in the state with angular m o m e n t u m I is related to the intrinsic quadrupole m o m e n t Qo by the equation Q = [(3K2-1(Iq-1))/(l+l)(21+3)]Qo. W e note in passing that this formula helps to account for the negative quadrupole moments that have been measured for the 2 + (first-excited) states of some even-even nuclei c o m m o n l y believed to be prolately deformed. *tt For 2~lPa~40, where the ground state is again characterized by K = ~ and I = ~ (ref. 17)), the sign of Q has not been measured. We expect it to be negative. Dr. R. Marrus has called the author's attention to another nucleus, 2~Am147, in which the measured quadrupole moment in the ground state (I = 1, K = 0) is negative but the actual deformation is probably prolate (ref. ~8)). This doubly odd nucleus does not appear in table 1 of this paper because it is not included in ref. 11) and because its Q is reported with a ± sign in ref. ~ ) . ++ A b o u t 20 of these lie in the rare-earth region. The linear trend of points 8 2 - - 1 0 7 , 41-49 and 114-121 correspond to portions of the stability line of neutrons with protons. ~:,t These details should be considered more carefully when one studies also the magnitudes of the qnadrupole moments and not only their signs.
326
Y.Y. SHARON
explained on the basis of a subshell structure in which subshells were not completely filled t. It seems to us, however, that the success of rule (i) in about 95 ~ of the relevant cases indicates the presence of a deformed core. Consequently, the subshell structure plays only a minor role in so far as the signs of the quadrupole mements are concerned. At one time two experimental groups ~9,2o) reported a negative value for the quadrupole moment of 237 93N P144, a nucleus whose coordinates in our scheme would be (0.25, 0.31). This would have been a significant exception ,t to rule (i). However, further experiments have now been carried out 2 3), and the best present information indicates strongly that the quadrupole moment of 237Np is indeed positive t tt
4. "One-particle" and "one-hole" rules Let us next consider an odd-proton nucleus which consists of closed major shells of nucleons plus an additional even number of neutrons and one proton. Let us also make the simplifying assumption that the neutrons couple so that their net angular momentum is zero. It has been shown long ago that the quadrupole moment of such a nucleus should be negative if this moment is due only to the effect of the last proton. The above result follows from the fact that the expectation value of the quadrupole moment operator has to be evaluated in the state with M = I, i.e. with the particle moving in an equatorial orbit. By an analogous argument, we also expect to find a negative quadrupole moment for an odd-neutron nucleus with one valence neutron outside a closed major neutron shell. This latter result follows from the argument presented above if we assume that the neutron polarizes the proton charge distribution or possesses a positive effective electric charge. From shell model considerations it can similarly be shown (see, for example, ref. t)) that the quadrupole moment should be positive for any odd-neutron nucleus where one neutron is missing from a major shell, and also for any odd-proton nucleus where one proton is missing from a major shell. These results are reasonable in view of our previous arguments and in view of the spherical charge distribution which characterizes a closed shell. The argument runs as follows. We consider a closed shell as consisting of one particle plus the "rest of the shell". The quadrupole moment of a closed shell is equal to zero. Since the quadrupole moment of the rest of the shell is evaluated in the t T h e point m a d e in ref. 6) is that prolate configurations are energetically favoured. Moszkowski a n d T o w n e s 6) suggest that this effect is large e n o u g h to prevail over the extra stability that is associated with closed subshells. In this way they explain the positive q u a d r u p o l e m o m e n t s o f the o d d - p r o t o n nuclei beyond Cu a n d also the g r o u n d state spins o f doubly o d d nuclei with N = Z. Moszkowski a n d Townes also a c c o u n t for s o m e observed g r o u n d state spins that are in disagreement with the predictions o f the simple shell m o d e l o f M a y e r a n d Jensen. ~t It would also be an exception to the systematics o f the T o w n e s - F o l e y - L o w plot (see ref. 5)) a n d also to the results o f Hill a n d Wheeler 31) on alpha decay. F u r t h e r m o r e , since the g r o u n d state spin o f ~37Np is ~+, we should expect to find a prolate d e f o r m a t i o n o n the basis o f the Nilsson level
diagram for this region (see refs. 15,22)). ttt A private communication from Dr. J. W. T. Dabbs, to whom the author is grateful for information on 2~Np144.
QUADRUPOLE MOMENTS
327
state with M = / , the particle must be in the state with M = - L It can be shown that the particle again has a negative quadrupole moment, of the same magnitude that it would have in the state with M = L Consequently, the rest of the shell must have a positive moment of the same magnitude. Moszkowski and Townes 6) applied these predictions of the shell model (which we shall designate as the "one-particle" and "one-hole" rules) to the limited amount of data that was available in 1954. They found these rules to be verified in every case that was relevant. With the present, more extensive data, we still obtain perfect agreement with these predictions for the signs of the pertinent quadrupole moments. There are 13 one-particle nuclei for which quadrupole moments have been measured; ~Li4 , ,~09, ~Sc22 ' 45. 47 53 63 65 12,q.~, 123q~, 21Sc24, 21SCz6, 2~Cr29, 29Cu34, 29Cu36, 51~70, 51~v72, a433xTA60~,,.,83,203n-83~112° and 209n-83161a26. In every case the quadrupole moment is negative. It is interesting to note that none of these 13 nuclei lies in that region of fig. 1 where rule (i) is applicable; thus there is never any conflict between rule (i) and the oneparticle rule. There are 12 other cases which represent one-hole nuclei and in all 35 39 41 of these instances we find positive quadrupole moments: 16819, 19K20, 19K22, 49 • 59 85 87 109r lllr ll3[ l15r and 1 3 7 ~ 22T127, 27C032, 36Kr49, 38Sr49, 491n60, 491n62, 49 n64, 491n66 56tJa81 • Nine of these one-hole nuclei fall in the region of fig. 1 to which rule (i) applies. In these cases the positive quadrupole moments can be explained either on the basis of rule (i) or on the basis of the one-hole rule. Rule (i), however, could not have been used to predict the positive quadrupole moments of 137n56tsa81, 27Co3259 and 3919K2o. The one-particle rule explains why, with the hypothetical semi-closed shell at 16 identical nucleons, the three nuclei 35 17Cl18, 37 17C120 and 33 ~6S17 could be expected to have negative quadrupole moments. This explanation would have to assume that the one-particle rule takes precedence over our empirical rule (i) in the very unusual cases where both of these rules apply. We have seen that the one-particle and one-hole rules account correctly for the signs ot the quadrupole moments of all 25 nuclei to which these rules can be applied. At first glance this success would seem to suggest that the simple spherical shell picture (with a positive effective charge for the odd neutron) is fully able to account for the quadrupole moments of one-particle and one-hole nuclei. However, as was first noted by Townes, Foley and Low 5), the more detailed shell-model predictions often fail even in these simple cases. To show this inadequacy of the simple shell picture we present in table 2 the calculated and the measured magnitudes of the quadrupole moments O in the ground states of one-particle and one-hole nuclei. Most of the experimental data are taken from ref. 11). The calculated values have been found by applying the single-particle formula Q _ _T_2J- 1 s Ag;s-v2 e 2 j + l 3--- - 0 x 1 0 - 2 b both to odd-proton nuclei and to odd-neutron nuclei (see for example ref. ~0)). Here j is the angular momentum of the ground state, A the atomic weight and the
328
Y.
Y.
S H A R O N
TABLE 2 Quadrupole moments of one-particle or one-hole nuclei Nucleus
J
ro
Q calculated (b. e) (single-particle value)
Q measured (b. e)
Q calculated Q measured
A. Odd-proton one-particle nuclei 7Li 4
~--
1.91
--
,*3 21Sc22 45 21Sc2¢ 47 218c26
~--
1.30
--0.093
--0.05 ~)
1.86
~--7_
1.30 1.30
--0.096
--0.22
0.44
--0.099
--0.22 ~)
0.45
23Cu34
~--
1.29
--0.079
--0.18 ~)
0.44
265Cu36
~--
1.29
--0.083
--0.19
0.44
121Sh 51~70 123Rh 51-~72 20~Bi120
~+
1.20
--0.141
--0.29 b)
0.47
~+
1.20
--0.160
--0.37 ~)
0.43
,~--
1.18
--0.231
--0.64
0.36
1.18
--0.235
--0.37 e)
0.64
B. Odd-proton one-hole nuclei 39 19K20 ,~+ 1.32 41 19K22 3+ 1.32
+0.060
+0.09
0.67
209Bi126
~9Co32
.~--
~--
0.040
-- 0.045
0.89
40.062
+0.11
0.56
1.27
+0.110
+0.404
0.27
+0.155
+1.20
0.129
1O991n60
~-+
1.19
111149-n62 113r49-"64 llsl49~tJ66
,~+
1.19
+0.157
+1.18
0.133
~9 +
1.19
+0.159
+1.14
0.139
-~-+
1.19
+0.161
+1.16
0.139
C. Odd-neutron one-particle nuclei
1~0 9 53 24Cr29 143Nd 60~ "-83
--
0.048
-- 0.027
1.78
1.25
--
0.066
-- 0.03
2.20
1.20
--0.177
--0.48 a)
0.37
+0.054
1.00
~+
1.35
~-~--
D. Odd-neutron one-hole nuclei 36SS~9
{+
1.30
+0.054
22T12749 •
~--
1.25
+0.094
+0.24 a)
0.39
8~Kr49
~+
1.20
+0.134
+0.45
0.30
87Sr49
2 ,
1.20
+0.136
40.36
0.38
137R~ 50-,~81
~+
1.20
+0.115
+0.28
0.41
The data for this column, unless otherwise indicated, are taken directly from reLn). The exceptions are: a) Data from ref. 35). e) Average of several reported values. b) Data from ref. 12). a) One of several reported values.
QUADRUPOLE MOMENTS
329
e - p a r t i c l e nuclei. The value of r 0 the constant of proportionality -T- sign refers to o none-ho~e in the law relating the nuclear radius (in fm) and A s, is taken from Hofstadter's tables 24) or extrapolated from them. The quantity e designates the effective charge of the odd particle. In table 2 a value of unity was always arbitrarily assumed for the effective charge e in all the cases (for odd-neutron nuclei as well as for odd-proton nuclei). Admittedly there is a considerable amount of uncertainty that is associated with the experimental measurements of quadrupole moments and also with the interpretation of, and the shielding corrections to, such measurements. Just the same, however, it is clear from table 2 that the one-particle model, which had such outstanding success in accounting for the signs of the quadrupole moments of oneparticle and one-hole nuclei, cannot account for the magnitudes of these moments. The marked change that we find in the quadrupole moments of the several odd-mass isotopes of Sc (or of Bi) and the difference in the ground state spins of 1211Sb70 and ~23~h all show that in some cases the even group of nucleons cannot be ignored. We also see that (except in the cases of ~s709 , 24Cr29 53 and 43 21Scz2 ) the one-particle model predicts moments that are much too small. We cannot account for the experimental data without invoking effective proton and neutron charges which vary from case to case *. Thus, even in these simple cases, core deformation details must be taken into account.
5. An alternative to the one-particle and one-hole rules
Since the one-particle and the one-hole rules should account for the magnitudes of the relevant quadrupole moments but fail to do so, we would like to propose a possible rule to supplement them. This rule, which we shall designate as rule (ii), is strictly an empirical one. It is similar to rule (i) in its emphasis on the fraction of the last shell that is filled, rather than on the number of nucleons outside closed shells. Rule (ii) makes reference only to the signs of quadrupole moments and has nothing to say about their magnitudes. Rule (ii) can be stated as follows: ~Or odd neutron odd-proton nuclei whose last major neutron shell is ~ or less filled (~ or more filled) proton
protons the quadrupole moments are always negative (positive). The number of even neut ....
[or
the fraction of the last neutron proton shell that is actually filled] does not matter. This rule is verified by the signs of the quadrupole moments of 24 out of the 25 one-particle and one-hole nuclei; it does not apply to the twenty-fifth case (~Li4). In addition this rule is also verified by the negative quadrupole moments of the five odd-mass iodine isotopes, of 14S6oNd85and of ~7Sm85 (all with three valence protons or neutrons). Rule (ii) also predicts correctly the positive quadrupole moments of seven 5n nuclei with three or five neutron or proton holes each. In three of these cases t/ 1 3561Ja79, 205~ 207r~ ~ 83 s4vo12x and s4vo123), rule fi) does not apply; in the other four cases (36K47, t We note, however, that in table 2 the values in the last column are constant over a part (Sc-Sb) of section A. The values are also approximately constant over a part (Ti-Ba) of section B.
330 197--
Y.Y. 201T'r
SHARON
203
79,~ulls, sortg121 and soHgla3) rule (i) is relevant. Rules (ii) and (i) are never in conflict, and when both apply (for example in the cases of nine one-hole nuclei) both predict positive quadrupole moments. We saw that rule (ii) makes definite predictions for the signs of the moments of some nuclei with three or five identical odd valence nucleons (or holes). In other cases, where there are three or five valence particles (holes) but where rule (ii) does not apply, the quadrupole moments may have other signs. Thus 23Na~2, ~Zo~Net t and the odd 31Ga isotopes, all have three identical odd valence particles and yet all possess positive quadrupole moments. Rule (ii) predicts correctly the signs of the quadrupole moments of 39 nuclei (including 12 cases in which it overlaps with rule (i)). The positive quadrupole moment of 14962Sm87 (with the last neutron shell O. 113 filled) is the only exception that we could find to this rule *. 6. O t h e r c a s e s
Quadrupole moments have been measured for ten doubly odd nuclei which do not fall into the region of fig. 1 where both coordinates are larger than one-fifth. The ten quadrupole moments, which are covered neither by any of our rules nor by the oneparticle and the one-hole rules, seem to depend strongly on the details of the coupling schemes *t. The moments of these doubly odd nuclei do not fall into a general pattern that is easily identified. Moszkowski and Townes 6) have shown that the ground state spins of doubly odd nuclei with N = Z are usually such as to avoid negative quadrupole moments. It would be interesting to study the quadrupole moments of doubly odd nuclei more carefully in order to account for the relationships between the moments of odd-mass and doubly odd isotopes of the same element. For 3Li, 5B, 17C1, 3tGa, 71Lu or 83Bi all the isotopes of any one element have quadrupole moments of one sign. However, for 19K, 21Sc or 51Sb, the doubly odd isotopes have quadrupole moments of the opposite signs from the odd ones. Doubly odd nuclei with one proton or one neutron outside (or missing from) major closed shells should be especially interesting. Quadrupole moments have been measured for nine singly magic nuclei. The negative signs for the quadrupole moments of ~s709 and of 209-.s3t~1126follow from the fact that in each of these cases there is only one nucleon outside doubly magic closed shells. If we assume that 16 identical nucleons form a closed shell, then we can likewise explain why the singly magic nucleus 37VC120has a negative quadrupole moment. Of the remaining six singly magic nuclei, all of which are magic in the neutrons and lie at the bottom of fig. 1, five nuclei possess quadrupole moments that are positive. The sixth case is that of 14L~ 591~r82. This nucleus has a negative quadrupole moment which t The recent work by Woodgate 36) seems to provide further evidence that the moment of X~SmsT is indeed positive. ** In ref. 7), Townes found this to be true for all doubly odd nuclei. We, however, can account for the signs of the quadrupole moments of those odd-odd nuclei to which rule (i) applies (with ~C119 being the only possible exception).
QUADRUPOLE MOMENTS
331
is apparently due to subshell details [the probable proton configuration for its ground state being (g~)8 d~_]. Finally, there are about eleven odd-proton nuclei* and seven odd-neutron** nuclei which lie in those regions of fig. 1 to which no rule applies. For these 18 nuclei subshell 93 details can be important. (For example both 73 32Ge41 and ~1Nb52 have negative quadrupole moments; this is probably due to the existence of a semi-closed shell of 40 identical nucleons. We noted earlier, however, that in those cases[such as 37~Se41] where rule (i) applies the effect of the semi-closed shell of 40 nucleons is not evident.) The moments of the odd-proton nuclei in this group tend strongly to be positive, while the moments of the odd-neutron nuclei show no such tendency. In some of the 18 cases the plot of Townes (see ref. 7), p. 450) can be very useful; often, however, these cases lie in that portion of Townes' plot where there is a transition from negative to positive quadrupole moments and no definite information can be obtained.
7. Concluding remarks We note that out of the 121 quadrupole moments that we consider, 84 (or 69.4 %) are positive and 37 are negative. These totals can be broken down into subcategories as follows: In the group of 17 doubly odd nuclei, 11 nuclei (64.7 %) have positive quadrupole moments and six have negative moments. In the group of 61 oddproton nuclei, 40 nuclei (65.6 %) have positive moments and 21 have negative moments. Finally, from among the 43 odd-neutron nuclei whose moments we analyse, 33 nuclei (76.7 %) have positive moments and 10 have negative moments. We do not believe that much significance should be attached to the somewhat larger percentage of positive moments for odd-neutron nuclei. This result occurs partly because in the region of fig. 1 to which rule (i) applies we find 27 of our 43 odd-neutron cases, but only seven of the 17 doubly odd cases and 27 of the 61 odd-proton cases. It is also due to the fact that of the 13 one-particle nuclei for which quadrupole moments (negative in every case) have been measured, 10 are odd-proton nuclei and only 3 are odd-neutron. Throughout our work we have found a surprisingly complete symmetry between the general effects that protons and neutrons have on the signs of the quadrupole moments. It had already been noted in earlier works 7,8, lo) that the quadrupole moments of odd-neutron and of odd-proton nuclei are usually of about the same order of magnitude. For odd-neutron or odd-proton nuclei we do not find any empirical correlations between the spins of the ground states and the signs of the measured quadrupole moments. The success of rule (i) suggests a possible connection between the proton-neutron force and the preponderance of positive nuclear quadrupole moments that is observed in nature. It has been noted that the positive quadrupole moments often have appreciably larger 5 t ~sMn3o ' alGaa5 ,8, atGaa8 , 8 8 8tGa4o , - / 1 4iNb52 ,98 43Tc56 ,99 1~53Cs78,I3~Cs80,161Eu88,I~aaEuso,2e'A89clas.
332
Y.Y. SHARON
magnitudes than those of the typical negative quadrupole moment 7). F r o m the diagram on page 450 of Townes' Handbuch article 7) it is clear that the average positive quadrupole moment is at least three times larger than the average negative quadrupole moment. However, it is precisely when rule (i) is applicable that we encounter very large positive quadrupole moments, moments that often are an order of magnitude larger than the single-particle estimates. This fact is illustrated when we consider the quantity [Ore. . . . . . a/A~] for the 60 quadrupole moments (27 positive, 33 negative) to which rule (i) does not apply. The average value of this quantity for the 27 positive quadrupole moments is 0.0163 b; the average value for the 33 negative quadrupole moments is -0.0139 b, i.e. just 15 ~o less in absolute value. It must be emphasized that our analysis has dealt only with the signs of quadrupole moments. It is possible, however, to extend our empirical analysis (in terms of the filled fractions of the major shells in both protons and neutrons) to include also the systematics of the magnitudes of the quadrupole moments. These magnitudes have been considered, from several other points of view, in the previous studies of quadrupole moments 4-1o). In these studies most authors found it fruitful to follow the approach of Townes 7) by concentrating on the "inferred intrinsic quadrupole moments" (see ref. 7), p. 449) rather than concentrating on the measured moments. One is then able to include in the analysis also the inferred magnitudes of the intrinsic quadrupole moments of nuclei with ground state spins of less than unity t The magnitudes of quadrupole moments increase rapidly with distance from closed shells. Consequently, the effects of the shell structure become very obvious in plots such as those that appear on p. 450 ofref. 7). The general picture that has emerged in this paper has been well known for a long time. However, our new parametrization makes possible observations that are more general and quantitative. We have already seen how our two simple empirical rules have enabled us to account for a large majority of the signs of the measured quadrupole moments. On the basis of our study we can make predictions of the signs of many quadrupole moments that have not yet been measured and whose signs sometimes could not have been predicted on the basis of the previous studies of moment systematics 4-8). We can also make an informed guess as to the signs of quadrupole moments whose magnitudes have been measured but whose signs have not been definitely established. To illustrate with several such cases from refs. 11, 12, 3o), '76 76 80 82 138T_ we predict positive quadrupole moments for 33As43,35Br41,35Br45,35Br47, 57La81, 45 • 167r~ 166~__ 170 22"1"123, 68Jc-~r99, 69111197and 69Tmlol. Our analysis can also be usefulin a case where two reported measurements of a given quadrupole moment disagree in both magnitude and sign. Thus, in ref. 11), we would have selected Q = - 0 . 2 0 b rather than Q = +0.16 b for 29Cu34, 63 Q = - 0 . 1 9 rather than Q = +0.15 for 65 29Cu36, Q = - 0 . 4 8 2 rather than Q = +0.02 for 143~A 6 0 1 " ~ 8 3 and Q = - 0 . 2 5 5 rather than Q = +0.01 * We believe that rules (i) and (ii) may also apply to the signs of the intrinsic quadrupole moments of all nuclei, with A > 40 including doubly even ones.
QUADRUPOLE MOMENTS
333
for 14s~,t601,,~85.These selections are identical with those that were made, probably on other grounds *, by the authors of the compilation in ref. 12). 8. Possible explanations for the validity of our rules
The simple spherical shell model of Mayer and Jensen 1) predicts that there should be equal numbers of positive and negative quadrupole moments. This result also follows from calculations which minimize the total single-particle energies of particles which are placed in the orbitals of a deformed harmonic oscillator well 2s). The singleparticle spin-orbit force may change the order of levels in an oscillator well but its average effect should favour neither oblate nor prolate deformations 26). Many explanations have been advanced for the preponderance of positive quadrupole moments. Among the possible causes thus cited have been the effects of: (i) surface tension plus Coulomb repulsion, in conjunction with large core deformations 6). In a calculation along these lines Moszkowski and Townes 6) found that the quadrupole moments should be positive whenever a shell of identical nucleons is more than one-third full. However, in a slightly different calculation, Gottfried 27) showed that these effects are not a major reason for the preponderance of prolate shapes. His results, which are valid only for an oscillator well, are also valuable because they suggest that for our purposes it is really not necessary to consider shapes that are axially asymmetric. (ii) tensor Jbrces. It is commonly accepted that tensor forces are responsible for the positive quadrupole moment of the deuteron. There is some reason to suppose that these forces favour elongated shapes also in heavier nuclei tt. However, further work should be done using two-body tensor forces which have opposite signs in states with T = 0 and with T = 1. (iii) the deviations of the real single-particle well from its assumed harmonic-oscillator shape in which there is a heavy degeneracy of energy levels 25' 28, 26). The observed preponderance of positive quadrupole moments has beenlinked to the tendency of a square well to lower the energy of single-particle states with high orbital angular momenta 29) (or with low values of Iz (ref. 28)). This effect may turn a delicate balance in favour of prolate shapes. Further investigations into this effect were carried out by Lemmer and Weisskopf 26), who started by considering a deformed harmonic oscillator potential well. They found that prolate equilibrium shapes are energetically favoured in the first half of a shell and that oblate equilibrium shapes are favoured in the second half. This result is in agreement with the result that was obtained in ref. 25) but is in contradiction to the trend of the experimental data. Lemmer and Weisskopf showed, however, that the differences between the energies of the prolate and the oblate shapes are smaller at the end of a shell [when the oblate ? Negative values for the quadrupole moments of the Nd isotopes were indeed obtained in recent experiments (see ref. 42)). ?? See ref. 6), p. 306 where this point is discussed and further references are given.
334
Y.Y.
SHARON
shapes are favoured] and larger at the beginning of a shell [when the prolate shapes are favoured]. They concluded that a perturbation of the oscillator well (such as the 12 term which is used in the Nilsson model to simulate square well effects, or a r 4 term to make the well steeper) is capable of accounting for the observed preponderance of prolate nuclear shapes. It would be interesting to extend these calculations to manyparticle states which are eigenstates of the total angular momentum. Our empirical observations seem to suggest that despite the complexity of the problem it may be possible to construct a crude model which would explain quadrupole moments of nuclei over the entire periodic table. Such a model should take into account the fractions of both the last major proton and the last major neutron shells that are occupied in the case of each nucleus. It is not unreasonable to expect that the mechanism which favours prolate shapes would predominate when certain fractions of shells are filled rather than when there are a certain number of particles outside closed shells. This is because for heavier nuclei, where the closed-shell cores are larger, more valence particles would be required to achieve corresponding core deformation effects. It is worthwhile to recall here the results that were obtained by Marshalek, Person and Sheline 9). Unlike the calculations cited earlier in this section, these authors explicitly considered both the protons and the neutrons in each nucleus. Marshalek, Person and Sheline 9) carried out calculations of equilibrium deformations in several regions of the periodic table. In their calculations they considered several possible Nilsson configurations for the ground state of each doubly even nucleus in the region of interest. Subsequently, in each case, they minimized the total single particle energies, for a given number of neutrons and a given number of protons, as functions of the prolate deformation. These authors confined themselves to elongated (and spherical) shapes and did not consider oblate deformations; the deformation contours that they calculated reproduce the observed systematics of the intrinsic positive quadrupole moments in the rare-earth and transuranic nuclei. However, the results of another calculation within the same general framework are in strong disagreement with the empirical rules that we deduced here. Kumar and Baranger 3s) carried out similar calculations for neutron deficient isotopes in the vicinity of Ba. In their work they used Nilsson wave functions and pairing plus quadrupole forces, taking into account the possibility of oblate as well as prolate deformations. Kumar and Baranger predict that oblate deformations should prevail throughout most of the region where both N and Z are between 50 and 82, including isotopes with Z > 56. This prediction has not been proved or disproved experimentally, but its validity seems somewhat unlikely in view of our analysis of the nuclear quadrupole moment data. If we accept the results of Kumar and Baranger then we must conclude that the simple Nilsson picture will not be able to explain the general features of rules (i) and (ii) that were devoloped in this paper. These rules are of course purely empirical, but they do seem to have wide validity.
Note added in proof." In the last two years the signs of a few additional ground-state
QUADRUPOLE MOMENTS
335
quadrupole moments have been determined. Positive moments have been found for 147~61t,m86 (ref.40)), 152~6312,1389(ref.42)) a n d 159,vr,65xv94 (ref.41)). N e g a t i v e m o m e n t s h a v e b e e n f o u n d f o r ~17515b66 (ref.37)) a n d 137r53184 (ref.39)). W e also n o t e t h a t t h e p o s i t i v e q u a d r u p o l e m o m e n t o f 16~Er99 (refs. 11, 12)) was i n a d v e r t e n t l y o m i t t e d f r o m t a b l e 1 a n d fig. 1 o f o u r p a p e r . O f t h e s e n e w results the n e g a t i v e m o m e n t s are a c c o u n t e d f o r by r u l e (ii) a n d t h e p o s i t i v e m o m e n t s by r u l e (i), e x c e p t t h a t n e i t h e r rule applies to 9s 14~7~61t,m86. F o r t h e m o m e n t o f a2M053, w h o s e m a g n i t u d e has b e e n r e c e n t l y m e a s u r e d (ref.38)), we p r e d i c t a n e g a t i v e sign.
References 1) M. G. Mayer and J. H. D. Jensen, Elementary theory of nuclear shell structure (John Wiley and Sons, New York, 1955) 2) J. Rainwater, Phys. Rev. 79 (1950) 432 3) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk, 26, No. 14 (1952) 4) W. Gordy, Phys. Rev. 76 (1949) 139 5) C. H. Townes, H. M. Foley and W. Low, Phys. Rev. 76 (1949) 1415 6) S. A. Moszkowski and C. H. Townes, Phys. Rev. 93 (1954) 306 7) C. H. Townes, in Handbuch der Physik, Vol. 38, ed. by S. Fltigge (Springer-Verlag, Berlin, 1957) p. 377 8) H. Kopferrnann, Nuclear moments (Academic Press, New York, 1959) p. 399 9) E. Marshalek, L. W. Person and R. K. Sheline, Revs. Mod. Phys. 35 (1963) 108 10) F. W. Byron et al., Phys. Rev. 136 (1964) B1654 11) Ingvar Lindgren, in Alpha-, beta- and gamma-ray spectroscopy, ed. by K. Siegbahn (NorthHolland Publ. Co., Amsterdam, 1965) p. 1621 12) G. H. Fuller and V. W. Cohen, Nuclear Data Sheets (May 1965), appendix 1 13) I. Talmi and I. Unna, Ann. Rev. Nucl. Sci. 10 (1960) 353 14) H. E. Gove, Proc. Kingston Int. Conf. on Nuclear Structure, ed. by D. A. Bromley and E. Vogt (University of Toronto Press, Toronto, 1963) p. 438 15) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1, No. 8 (1959) 16) R. Marrus, W. A. Nierenberg and J. Winocur, Nuclear Physics 23 (1961) 90 17) S. G. Nilsson, private communication 18) R. Marrus and J. Winocur, Phys. Rev. 124 (1961) 1904 19) L. D. Roberts, J. W. T. Dabbs, G. W. Parker and R. D. Ellison, Bull. Am. Phys. Soc. 1 (1956) 207 20) M. H. L. Pryce, Phys. Rev. Lett. 3 (1959) 375 21) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 22) S. G. Nilsson, Mat. Fys. Medd. Dan. Vial. Selsk. 29, No. 16 (1955) 23) S. H. Hanauer, J. W. T. Dabbs, L. D. Roberts and G. W. Parker, Phys. Rev. 124 (1961) 1512 24) R. Herman and R. Hofstadter, High energy electron scattering tables (Stanford University Press, Stanford, California, 1960) 25) S. A. Moszkowski in Handbuch der Physik, Vol. 39, ed. by S. Fltigge (Springer-Verlag, Berlin, 1957) 26) R. H. Lemmer and V. Weisskopf, Nuclear Physics 25 (1961) 624 27) K. Gottfried, Phys. Rev. 103 (1956) 1017 28) S. Moszkowski, Phys. Rev. 99 (1955) 803 29) J. p. Elliott, Lectures on Collective Motion in Nuclei, University of Rochester (1958) p. 130 30) R. G. Cornwall and J. D. McCullen, to be published 31) R. A. Haberstroh, W. J. Kossler, O. Ames and D. R. Hamilton, Phys. Rev. 136 (1964) B932 32) K. H. Channapa and J. M. Pendlebury, Proc. Phys. Soc. 86 (1965) 1145 33) D. Ali, I. Maleh and R. Marrus, Phys. Rev. 138 (1965) B1356 34) O. Redi, private communication; Quarterly Progress Report No. 74, Research Lab. of Electronics, M.I.T. (July 1964) p. 43
336 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46)
Y.Y.
SHARON
K. Kumar and M. Baranger, Phys. Rev. Lett. 12 (1964) 73 G. K. Woodgate, private communication A. D. Jackson, private communication A. Narath and D. W. Alderman, Phys. Rev. 143 (1966) 328 K. Murakawa, J. Phys. Soc. Japan 19 (1964) 1539 J. Reader, Phys. Rev. 141 (1966) 1123 C. Arnoult and S. Gerstenkorn, J. Opt. Soc. America 56 (1966) 177 L. Gabla, Acta Phys. Polon. 25 (1964) 617 K. F. Smith and P. J. Unsworth, Proc. Phys. Soc. 86 (1965) 1249 B. Budick and R. Marrus, Phys. Rev. 132 (1963) 723 A. Y. Cabezas et al., Phys. Rev. 126 (1962) 1004 I. Lindgren, in Perturbed angular correlations, ed. by E. Karlsson et al., (North-Holland Publ. Co., Amsterdam, 1964) p. 379
I
Nuclear Physics
A99 (1967) 321--336; (~)
North-HollandPublishin# Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON T H E S I G N 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 Y I T Z H A K Y. S H A R O N t
Palmer Physical Laboratory, Princeton University tt Received 28 March 1967
Abstract: We analyse the experimental data on the signs of all the measured quadrupole moments
in the ground states of nuclei. We find positive quadrupole moments in virtually all the cases where the last major proton and neutron shells are both more than one-fifth full. For oddmass nuclei the quadrupole moment is negative (positive) when the odd nucleons occupy 1/8 or less (7/8 or more) of the last major shell. The last observation is shown to be an extension of the known rules for the signs of the quadrupole moments of odd-mass nuclei with one particle outside (or missing from) a closed shell. I. Introduction
The experimental information on the systematics of nuclear electric quadrupole moments has played an important role in the development of both the shell model i) and the collective model z, 3). Throughout the years several studies of the systematics of quadrupole moments have been carried out 4-1 o). However, the only two investigations which were carried out since 1959 have been devoted to limited regions of the periodic table 9, 10). In all these papers the signs and the magnitudes of the quadrupole moments were usually related to the number of odd nucleons, to the numbers of protons and neutrons that occupy levels in specific subsheUs or to the number of nucleons outside major closed shells. Some of the conclusions that have been reached in these aforementioned analyses of the experimental data are the following: (i) The majority of the measured quadrupole moments have a positive sign. Quadrupole moments tend to be positive in the ground states of nuclei not in proximity to closed shells in either the protons or the neutrons. (ii) For odd-mass nuclei in which the odd particle is a single nucleon outside (or missing from) a major closed shell the quadrupole moments are always negative (positive). (iii) The quadrupole moments are usually larger, and frequently very much larger, than what one might expect on the basis of single-particle models. The quadrupole moments of some rare-earth nuclei are 30 times larger than the corresponding singleparticle values. The positive quadrupole moments often have appreciably larger magnitudes than those of the typical negative quadrupole moment. t Now at Department of Physics, Northeastern University, Boston Mass. tt Work supported in part by the U.S. Atomic Energy Commission. 321
322
Y.Y.
SHARON
The purpose of this paper is to point out that somewhat more quantitative empirical observations can be made if one considers the experimental quadrupole moment data from a new point of view. 2. Analysis of the data We have performed an empirical analysis of the signs of the ground state quadrupole moments of nuclei. We include all those moments whose magnitudes and signs have both been measured experimentally up to 1965. We must exclude from consideration
SIGNS
OF
MEASURED
QUADRUPOLE 0
MOMENTS
odd proton nuclei
[] odd neutron nuclei r'-, LIJ --I _J LL
1.0 ItO ll5
74E] I
~B (9~8 (9-n I []28 ,~7o (967
£,9
0.8
I '
!
b-<[ I
66(9@261
-
0.6
El 51
+~,5
(@16
(9,os
lot® ~,oz (9,oo
~L~,43
(999
~q4
94~7
(9,3
(934
ssE~
~61 G6o
I I r~ql2 I ', ,19E](~)121
(gsl
~. 97
E]93
(9s8 1 25 Z~37
(~) 105
(91o4
E~ 14
(96
I '
E{]~6
/~8
(945
(93~ I
0.4
(91°7 (9106
~4-," (946
[(9~z7
40 ~r~41E] [
(924
,~7
(~49
~_44
! i i 4z
@64 (965 /+16s (962 ~z~
T U) Z 0 n-I-Zb ILl Z
F] 48
@09 ®TZl
- 109112
io8
.--I .._1
/x d o u b l y o d d n u c l e i
I 75 ' 'r~!J
(~)59 E]57 Q58
(992
®u ~ ' z °
F]91
il6 i ~]118
89~[~
90
~ss
~s6 Q2t
LL
o 0.2
£z}
Z 0
@m
0
[ I
79r~ 7~
•2o
E]B, do 52~s°
(gs~ (953
~
0
I
0.2 FRACTION
0.4 OF
PROTON
(932
({)54
8o,~ra ~28.
]
r:19 u_
(98s
I
d 8 I
0.6 SHELL
I(~ 19
0.8 THAT
I
I.O IS
FILLED
Fig. 1. Signs of measured quadrupole moments in the ground states of nuclei. Each nucleus is denoted by a number. The key to this numbering appears in table 1. The abscissa for each nucleus is the fraction of the last major proton shell that is actually occupied by the protons in this nucleus. The ordinate for each nucleus is the fraction of the last major neutron shell is actually occupied by the neutrons in this nucleus.
QUADRUPOLE MOMENTS
323
doubly even nuclei as well as odd-mass or doubly odd nuclei whose ground state spins are less than 1. The vast majority of data that we use is taken from the compilations of nuclear moments by Lindgren 11), and Fuller and Cohen az). Usually whenever one of these compilations reports a definite sign for a quadrupole moment we accept that sign as valid t,. In very few cases we supplement the information in these compilations on the basis of more up-to-date research papers or private communications (see table 1). It should be noted that our analysis covers the moments of 121 nuclei, i.e. that our investigation is based on many more data than were available to the authors of previous studies of quadrupole moments throughout the periodic table 4- s). We base our analysis on the j-j shell model. In this model the magic numbers, which correspond to closed shells in either protons or neutrons, are 2, 8, 20, 28, 50, 82, 126, 184 and so on. Successive shells of identical nucleons thus contain up to 2, 6, 12, 8, 22, 32, 44 and 58 nucleons, respectively. In fig. l each nucleus is represented as a point on a two-dimensional grid, i.e., by a pair of Cartesian coordinates. The abscissa for each nucleus is the fraction of the last proton shell that is actually occupied by the protons in this nucleus. Similarly, for each nucleus in fig. 1, the ordinate is the fraction of the last neutron shell that is actually occupied by the neutrons in this nucleus. A closed shell nucleus is considered to have its last shell totally unoccupied, so that in fig. 1 the abscissas and ordinates are always less than one and greater than or equal to zero tit. To illustrate our notation, let us recall that 12 identical particles can be placed in the 2s-ld shell. Thus 160 8 is denoted by (0, 0), 1709 by (0,~3-), 197F8 by (x~-,0), ISF 9 by (T12,TI~-), 20°Nelo by (-~,-~), 377C120 by (¼, 0) and 40 11 ~). In fig. 1 we place a plus sign at the coordinates of a nucleus with a 19Kzl by (~y, positive quadrupole moment and a minus sign at the coordinates of a nucleus with a negative quadrupole moment. Odd-proton nuclei (i.e. nuclei with an odd number of protons and an even number of neutrons) are denoted by circles, odd-neutron nuclei by squares and doubly odd nuclei by triangles. Each of the 121 nuclei in fig. 1 is denoted by a number. The key to this numbering appears in table 1. 3. Nuclei with the last proton and neutron shells both more than one-fifth filled
On the basis of fig. 1 the following observations can be made: (i) With very few exceptions, whenever the major proton and the major neutron shells are both ~ or more filled, then the quadrupole moments are positive. Out of over 60 cases there are only four exceptions to this rule. Three of these exceptions involve nuclei with 17 odd nucleons (16517, 33 17 17 35Clls and 36Cl19). These three exceptions could be accounted for if we were to assume that 16 identical nucleons form a semi-closed shell. The assumption that the ld~ single-particle level lies considerably higher than the ld~ t The author is grateful to Dr. Lindgren for a preprint of his table and for several private communications. Typographical errors which appear in ref. n) are usually corrected in ref. 4~). tt Thus, for example, we do not include the moments of l~]Pmo0(ref.44)) or l~prs3(ref. 45)). ttt The reader note that our grid should really be a closed surface rather than a square in flat space. The horizontal (or vertical) lines denoting 0 and 1 should coincide.
324
Y.Y. SHARON
TABLE 1 Nuclei wl~ose quadrupole moments are included in fig. 1. A minus sign appears before the n u m b e r designating a nucleus with a negative m o m e n t 1
~H 1
25
"6~c 21 ~ 25
49
85 37Rb48
73
137~ 55t~s82
97 b) 177bl f 72~--105
--2
6Li 3
--26 b)
47 215C26
50
87 37Rb50
74
135 56B479
98 b) 179i..1f 72---107
--3
37Li4
27 d)
,$7 • 22T125
51
87 385r49
75
137R^ 56-~t81
99
181 73Ta108
9Be 5
28 a)
4922T127
_ 5 2 a)
90 39Y51
76
1391n 57--82
100
185R^ 75-~Cl 10
51 23V28
--53
93 --77 41Nb52
141p_ 59-'82
101
187R 75 e112
4 ~) 5
l°B 5
29
6
lIB 6
--30
7 e)
1~C 5
8
174N7
--9
53 24Cr29
54
99 43Tc56
--78 b) 143Nd 60- "-83
102
189~ 7 6 u s i 13
31
55 25Mn30
55 d) l°5pcl 46--59
--79 b) 145Ncl 60"~85
103
191177~ti14
32
59 27 Co32
56
107Cd 48---59
80 e) 1 4 8 p ~ 61 ~ul87
104
1931" 77-'116
170 9
- - 3 3 13) 63 29Cu34
57
109Ccl --81 a) 147g I_ 48---61 62-11185
105
197A" 79" ~Ul 18
10
121Nell
--34 b) 6295Cu36
58
1°9I4.9-t160
82 a) 1 62~u187 49~_
t06
2011480--~121
11
23Na12
--35
65 30Zn35
59
1111~ 49-1162
83 b) 1 5 3 ~ _ 62~n191
107 f) 203i_1_ 80-~123
12
225Mg13
36
67 3021137
60
113149-1'64
84
151w63~u88
--108
2°31q: 83~'120
13
27 13Al14
37
67 31Ga36
61
115I49-~66
85
1531~ " 63~u90
--109
204~. 831~1121
--14
33 16517
38
69 31Ga38
--62
12 l,~h 51--70
86
155G,t 64~91
--110
2°6ri: 83-'123
15
365519
39
71 31Ga40
63
122~h 51~71
87
157Gcl 64~93
--111
2°9R: 83-t126
--16
135C118
40
72 31Ga41
--64
123 515672
88 a) 156Th 65~91
112
205p^ 84-u121
--17
376C119
--41 b)
73Ge41
--65
125I 53-72
89 a) 160Th 65--~95
113
207084--ut23
--18
37C120
42
75 33As42
--66
127I 53-74
90 a) 161r~. 66~Y95
114
227A~ 89--~138
19
39 19K20
43
75 345e41
--67
1291 53"76
91 a) 16315. 66~Y97
_ 1 1 5 a) 229Th 90--'139
--20
40 19K21
44
79 345245
--68
131[ 53-78
92
I6514~ 67"-t)98
_ 1 1 6 b) 233p. 91-~142
1491K22
45
79 35Br44 81 35Br46
--69 b) 1331 53-80 --70 13t 54Xe77
93
173Vh 70---103 1751" 71~Ulo4
117 b) 23311 92--141 118 b) 23511 92~143
21 a)
_ 2 2 b) 43 215c22 23 b) --24
46
94
44 215c23
47
83 36Kr47
--71
133 55Cs78
95
17671LU105
119 ~) 2 4 1 p . 94--u147
45 218C24
48
85 36Kr49
72
135~ 55~'~$80
96
1771" 7 1 - u l 06
120
M o s t of the data appear in b o t h refs. 11,12). a) Data are from ref. 11) only. b) Data are from ref. 12) only. e) Ref. ~1). d) Ref. 32). e) Ref. 33). r) Ref. 34). The author is grateful to Dr. Redi for this information.
2"1A~ 95 "-lu 146 121 b) 2 4 3 A ~ 95-~1u148
QUADRUPOLE MOMENTS
325
and 2s~ levels is not in disagreement with the usual interpretation of the spectrum of 1~O 9 . Furthermore, Talmi and Unna 13) were able to account for the binding energies of nuclei at the end of the 2s-ld shell by treating the ld~ subshell as a distinct entity t The fourth apparent exception to rule (i) 233n 911"a142, is really an exception that proves 31n " i 233n the rule. The ground states of the odd isotopes of Pa, rt 2 91t'a140 ana 91ra142), have angular momenta of 3 and are the only nuclear ground states that are usually interpreted as the 3 states of rotational bands with K = ½ (refs. 15, 26)). For such a state the intrinsic quadrupole moment has an opposite sign from the measured one tt Thus the measured negative quadrupole moment indicates that 233r,, 91r'a142 has a prolate deformation ttt In the region of fig. 1 to which rule (i) does not apply we find approximately equal numbers of positive and negative quadrupole moments (out of 60 values of Q, 27 are positive and 33 are negative). On the other hand, out of the 61 nuclei to which this empirical rule is germane, not fewer than 57 possess positivequadrupole moments*. This result is especially impressive in view of the fact that our analysis involves nuclei from all the parts of the periodic table with atomic weights that range from 2 to 243. The nuclei which we consider - doubly odd, odd-neutron and odd-proton - have ground state spins which range from I to 7. Some of these nuclei are slightly deformed and others are strongly deformed. Yet, in spite of this great diversity, the specific details of the shell-model configurations for the ground-state wave functions do not seem to matter greatly in the region that is covered** by rule (i). Also, with the aforenoted exception of the ld~ subshell, one can apparently ignore the details of the subshells in the major shell structure when one predicts the signs of the nuclear quadrupole moments. Moszkowski and Townes 6) have shown that some positive quadrupole moments (as well as ground state spins and other properties) could be t However, the usual interpretation of some of the experimental data in the 2 s - l d shell seems to suggest a semi-closed shell at 14 identical nucleons rather than 16 (see refs. 1~.,9)). The existence of a semi-closed shell at 14, but not at 16, nucleons is also strongly suggested by the large binding energy of ~sSi~. All three exceptions with 17 odd nucleons have ground states with angular m o m e n t a 3 It is unlikely that we are dealing with K ½ bands whose lowest state has an angular m o m e n t u m of 2. However, if this were the case, then the negative quadrupole moments would be indicative of prolate equilibrium shapes. ** Consider a rotational band that is characterized by the quantum number K. The measured quadrupole moment Q in the state with angular m o m e n t u m I is related to the intrinsic quadrupole m o m e n t Qo by the equation Q = [(3K2-1(Iq-1))/(l+l)(21+3)]Qo. W e note in passing that this formula helps to account for the negative quadrupole moments that have been measured for the 2 + (first-excited) states of some even-even nuclei c o m m o n l y believed to be prolately deformed. *tt For 2~lPa~40, where the ground state is again characterized by K = ~ and I = ~ (ref. 17)), the sign of Q has not been measured. We expect it to be negative. Dr. R. Marrus has called the author's attention to another nucleus, 2~Am147, in which the measured quadrupole moment in the ground state (I = 1, K = 0) is negative but the actual deformation is probably prolate (ref. ~8)). This doubly odd nucleus does not appear in table 1 of this paper because it is not included in ref. 11) and because its Q is reported with a ± sign in ref. ~ ) . ++ A b o u t 20 of these lie in the rare-earth region. The linear trend of points 8 2 - - 1 0 7 , 41-49 and 114-121 correspond to portions of the stability line of neutrons with protons. ~:,t These details should be considered more carefully when one studies also the magnitudes of the qnadrupole moments and not only their signs.
326
Y.Y. SHARON
explained on the basis of a subshell structure in which subshells were not completely filled t. It seems to us, however, that the success of rule (i) in about 95 ~ of the relevant cases indicates the presence of a deformed core. Consequently, the subshell structure plays only a minor role in so far as the signs of the quadrupole mements are concerned. At one time two experimental groups ~9,2o) reported a negative value for the quadrupole moment of 237 93N P144, a nucleus whose coordinates in our scheme would be (0.25, 0.31). This would have been a significant exception ,t to rule (i). However, further experiments have now been carried out 2 3), and the best present information indicates strongly that the quadrupole moment of 237Np is indeed positive t tt
4. "One-particle" and "one-hole" rules Let us next consider an odd-proton nucleus which consists of closed major shells of nucleons plus an additional even number of neutrons and one proton. Let us also make the simplifying assumption that the neutrons couple so that their net angular momentum is zero. It has been shown long ago that the quadrupole moment of such a nucleus should be negative if this moment is due only to the effect of the last proton. The above result follows from the fact that the expectation value of the quadrupole moment operator has to be evaluated in the state with M = I, i.e. with the particle moving in an equatorial orbit. By an analogous argument, we also expect to find a negative quadrupole moment for an odd-neutron nucleus with one valence neutron outside a closed major neutron shell. This latter result follows from the argument presented above if we assume that the neutron polarizes the proton charge distribution or possesses a positive effective electric charge. From shell model considerations it can similarly be shown (see, for example, ref. t)) that the quadrupole moment should be positive for any odd-neutron nucleus where one neutron is missing from a major shell, and also for any odd-proton nucleus where one proton is missing from a major shell. These results are reasonable in view of our previous arguments and in view of the spherical charge distribution which characterizes a closed shell. The argument runs as follows. We consider a closed shell as consisting of one particle plus the "rest of the shell". The quadrupole moment of a closed shell is equal to zero. Since the quadrupole moment of the rest of the shell is evaluated in the t T h e point m a d e in ref. 6) is that prolate configurations are energetically favoured. Moszkowski a n d T o w n e s 6) suggest that this effect is large e n o u g h to prevail over the extra stability that is associated with closed subshells. In this way they explain the positive q u a d r u p o l e m o m e n t s o f the o d d - p r o t o n nuclei beyond Cu a n d also the g r o u n d state spins o f doubly o d d nuclei with N = Z. Moszkowski a n d Townes also a c c o u n t for s o m e observed g r o u n d state spins that are in disagreement with the predictions o f the simple shell m o d e l o f M a y e r a n d Jensen. ~t It would also be an exception to the systematics o f the T o w n e s - F o l e y - L o w plot (see ref. 5)) a n d also to the results o f Hill a n d Wheeler 31) on alpha decay. F u r t h e r m o r e , since the g r o u n d state spin o f ~37Np is ~+, we should expect to find a prolate d e f o r m a t i o n o n the basis o f the Nilsson level
diagram for this region (see refs. 15,22)). ttt A private communication from Dr. J. W. T. Dabbs, to whom the author is grateful for information on 2~Np144.
QUADRUPOLE MOMENTS
327
state with M = / , the particle must be in the state with M = - L It can be shown that the particle again has a negative quadrupole moment, of the same magnitude that it would have in the state with M = L Consequently, the rest of the shell must have a positive moment of the same magnitude. Moszkowski and Townes 6) applied these predictions of the shell model (which we shall designate as the "one-particle" and "one-hole" rules) to the limited amount of data that was available in 1954. They found these rules to be verified in every case that was relevant. With the present, more extensive data, we still obtain perfect agreement with these predictions for the signs of the pertinent quadrupole moments. There are 13 one-particle nuclei for which quadrupole moments have been measured; ~Li4 , ,~09, ~Sc22 ' 45. 47 53 63 65 12,q.~, 123q~, 21Sc24, 21SCz6, 2~Cr29, 29Cu34, 29Cu36, 51~70, 51~v72, a433xTA60~,,.,83,203n-83~112° and 209n-83161a26. In every case the quadrupole moment is negative. It is interesting to note that none of these 13 nuclei lies in that region of fig. 1 where rule (i) is applicable; thus there is never any conflict between rule (i) and the oneparticle rule. There are 12 other cases which represent one-hole nuclei and in all 35 39 41 of these instances we find positive quadrupole moments: 16819, 19K20, 19K22, 49 • 59 85 87 109r lllr ll3[ l15r and 1 3 7 ~ 22T127, 27C032, 36Kr49, 38Sr49, 491n60, 491n62, 49 n64, 491n66 56tJa81 • Nine of these one-hole nuclei fall in the region of fig. 1 to which rule (i) applies. In these cases the positive quadrupole moments can be explained either on the basis of rule (i) or on the basis of the one-hole rule. Rule (i), however, could not have been used to predict the positive quadrupole moments of 137n56tsa81, 27Co3259 and 3919K2o. The one-particle rule explains why, with the hypothetical semi-closed shell at 16 identical nucleons, the three nuclei 35 17Cl18, 37 17C120 and 33 ~6S17 could be expected to have negative quadrupole moments. This explanation would have to assume that the one-particle rule takes precedence over our empirical rule (i) in the very unusual cases where both of these rules apply. We have seen that the one-particle and one-hole rules account correctly for the signs ot the quadrupole moments of all 25 nuclei to which these rules can be applied. At first glance this success would seem to suggest that the simple spherical shell picture (with a positive effective charge for the odd neutron) is fully able to account for the quadrupole moments of one-particle and one-hole nuclei. However, as was first noted by Townes, Foley and Low 5), the more detailed shell-model predictions often fail even in these simple cases. To show this inadequacy of the simple shell picture we present in table 2 the calculated and the measured magnitudes of the quadrupole moments O in the ground states of one-particle and one-hole nuclei. Most of the experimental data are taken from ref. 11). The calculated values have been found by applying the single-particle formula Q _ _T_2J- 1 s Ag;s-v2 e 2 j + l 3--- - 0 x 1 0 - 2 b both to odd-proton nuclei and to odd-neutron nuclei (see for example ref. ~0)). Here j is the angular momentum of the ground state, A the atomic weight and the
328
Y.
Y.
S H A R O N
TABLE 2 Quadrupole moments of one-particle or one-hole nuclei Nucleus
J
ro
Q calculated (b. e) (single-particle value)
Q measured (b. e)
Q calculated Q measured
A. Odd-proton one-particle nuclei 7Li 4
~--
1.91
--
,*3 21Sc22 45 21Sc2¢ 47 218c26
~--
1.30
--0.093
--0.05 ~)
1.86
~--7_
1.30 1.30
--0.096
--0.22
0.44
--0.099
--0.22 ~)
0.45
23Cu34
~--
1.29
--0.079
--0.18 ~)
0.44
265Cu36
~--
1.29
--0.083
--0.19
0.44
121Sh 51~70 123Rh 51-~72 20~Bi120
~+
1.20
--0.141
--0.29 b)
0.47
~+
1.20
--0.160
--0.37 ~)
0.43
,~--
1.18
--0.231
--0.64
0.36
1.18
--0.235
--0.37 e)
0.64
B. Odd-proton one-hole nuclei 39 19K20 ,~+ 1.32 41 19K22 3+ 1.32
+0.060
+0.09
0.67
209Bi126
~9Co32
.~--
~--
0.040
-- 0.045
0.89
40.062
+0.11
0.56
1.27
+0.110
+0.404
0.27
+0.155
+1.20
0.129
1O991n60
~-+
1.19
111149-n62 113r49-"64 llsl49~tJ66
,~+
1.19
+0.157
+1.18
0.133
~9 +
1.19
+0.159
+1.14
0.139
-~-+
1.19
+0.161
+1.16
0.139
C. Odd-neutron one-particle nuclei
1~0 9 53 24Cr29 143Nd 60~ "-83
--
0.048
-- 0.027
1.78
1.25
--
0.066
-- 0.03
2.20
1.20
--0.177
--0.48 a)
0.37
+0.054
1.00
~+
1.35
~-~--
D. Odd-neutron one-hole nuclei 36SS~9
{+
1.30
+0.054
22T12749 •
~--
1.25
+0.094
+0.24 a)
0.39
8~Kr49
~+
1.20
+0.134
+0.45
0.30
87Sr49
2 ,
1.20
+0.136
40.36
0.38
137R~ 50-,~81
~+
1.20
+0.115
+0.28
0.41
The data for this column, unless otherwise indicated, are taken directly from reLn). The exceptions are: a) Data from ref. 35). e) Average of several reported values. b) Data from ref. 12). a) One of several reported values.
QUADRUPOLE MOMENTS
329
e - p a r t i c l e nuclei. The value of r 0 the constant of proportionality -T- sign refers to o none-ho~e in the law relating the nuclear radius (in fm) and A s, is taken from Hofstadter's tables 24) or extrapolated from them. The quantity e designates the effective charge of the odd particle. In table 2 a value of unity was always arbitrarily assumed for the effective charge e in all the cases (for odd-neutron nuclei as well as for odd-proton nuclei). Admittedly there is a considerable amount of uncertainty that is associated with the experimental measurements of quadrupole moments and also with the interpretation of, and the shielding corrections to, such measurements. Just the same, however, it is clear from table 2 that the one-particle model, which had such outstanding success in accounting for the signs of the quadrupole moments of oneparticle and one-hole nuclei, cannot account for the magnitudes of these moments. The marked change that we find in the quadrupole moments of the several odd-mass isotopes of Sc (or of Bi) and the difference in the ground state spins of 1211Sb70 and ~23~h all show that in some cases the even group of nucleons cannot be ignored. We also see that (except in the cases of ~s709 , 24Cr29 53 and 43 21Scz2 ) the one-particle model predicts moments that are much too small. We cannot account for the experimental data without invoking effective proton and neutron charges which vary from case to case *. Thus, even in these simple cases, core deformation details must be taken into account.
5. An alternative to the one-particle and one-hole rules
Since the one-particle and the one-hole rules should account for the magnitudes of the relevant quadrupole moments but fail to do so, we would like to propose a possible rule to supplement them. This rule, which we shall designate as rule (ii), is strictly an empirical one. It is similar to rule (i) in its emphasis on the fraction of the last shell that is filled, rather than on the number of nucleons outside closed shells. Rule (ii) makes reference only to the signs of quadrupole moments and has nothing to say about their magnitudes. Rule (ii) can be stated as follows: ~Or odd neutron odd-proton nuclei whose last major neutron shell is ~ or less filled (~ or more filled) proton
protons the quadrupole moments are always negative (positive). The number of even neut ....
[or
the fraction of the last neutron proton shell that is actually filled] does not matter. This rule is verified by the signs of the quadrupole moments of 24 out of the 25 one-particle and one-hole nuclei; it does not apply to the twenty-fifth case (~Li4). In addition this rule is also verified by the negative quadrupole moments of the five odd-mass iodine isotopes, of 14S6oNd85and of ~7Sm85 (all with three valence protons or neutrons). Rule (ii) also predicts correctly the positive quadrupole moments of seven 5n nuclei with three or five neutron or proton holes each. In three of these cases t/ 1 3561Ja79, 205~ 207r~ ~ 83 s4vo12x and s4vo123), rule fi) does not apply; in the other four cases (36K47, t We note, however, that in table 2 the values in the last column are constant over a part (Sc-Sb) of section A. The values are also approximately constant over a part (Ti-Ba) of section B.
330 197--
Y.Y. 201T'r
SHARON
203
79,~ulls, sortg121 and soHgla3) rule (i) is relevant. Rules (ii) and (i) are never in conflict, and when both apply (for example in the cases of nine one-hole nuclei) both predict positive quadrupole moments. We saw that rule (ii) makes definite predictions for the signs of the moments of some nuclei with three or five identical odd valence nucleons (or holes). In other cases, where there are three or five valence particles (holes) but where rule (ii) does not apply, the quadrupole moments may have other signs. Thus 23Na~2, ~Zo~Net t and the odd 31Ga isotopes, all have three identical odd valence particles and yet all possess positive quadrupole moments. Rule (ii) predicts correctly the signs of the quadrupole moments of 39 nuclei (including 12 cases in which it overlaps with rule (i)). The positive quadrupole moment of 14962Sm87 (with the last neutron shell O. 113 filled) is the only exception that we could find to this rule *. 6. O t h e r c a s e s
Quadrupole moments have been measured for ten doubly odd nuclei which do not fall into the region of fig. 1 where both coordinates are larger than one-fifth. The ten quadrupole moments, which are covered neither by any of our rules nor by the oneparticle and the one-hole rules, seem to depend strongly on the details of the coupling schemes *t. The moments of these doubly odd nuclei do not fall into a general pattern that is easily identified. Moszkowski and Townes 6) have shown that the ground state spins of doubly odd nuclei with N = Z are usually such as to avoid negative quadrupole moments. It would be interesting to study the quadrupole moments of doubly odd nuclei more carefully in order to account for the relationships between the moments of odd-mass and doubly odd isotopes of the same element. For 3Li, 5B, 17C1, 3tGa, 71Lu or 83Bi all the isotopes of any one element have quadrupole moments of one sign. However, for 19K, 21Sc or 51Sb, the doubly odd isotopes have quadrupole moments of the opposite signs from the odd ones. Doubly odd nuclei with one proton or one neutron outside (or missing from) major closed shells should be especially interesting. Quadrupole moments have been measured for nine singly magic nuclei. The negative signs for the quadrupole moments of ~s709 and of 209-.s3t~1126follow from the fact that in each of these cases there is only one nucleon outside doubly magic closed shells. If we assume that 16 identical nucleons form a closed shell, then we can likewise explain why the singly magic nucleus 37VC120has a negative quadrupole moment. Of the remaining six singly magic nuclei, all of which are magic in the neutrons and lie at the bottom of fig. 1, five nuclei possess quadrupole moments that are positive. The sixth case is that of 14L~ 591~r82. This nucleus has a negative quadrupole moment which t The recent work by Woodgate 36) seems to provide further evidence that the moment of X~SmsT is indeed positive. ** In ref. 7), Townes found this to be true for all doubly odd nuclei. We, however, can account for the signs of the quadrupole moments of those odd-odd nuclei to which rule (i) applies (with ~C119 being the only possible exception).
QUADRUPOLE MOMENTS
331
is apparently due to subshell details [the probable proton configuration for its ground state being (g~)8 d~_]. Finally, there are about eleven odd-proton nuclei* and seven odd-neutron** nuclei which lie in those regions of fig. 1 to which no rule applies. For these 18 nuclei subshell 93 details can be important. (For example both 73 32Ge41 and ~1Nb52 have negative quadrupole moments; this is probably due to the existence of a semi-closed shell of 40 identical nucleons. We noted earlier, however, that in those cases[such as 37~Se41] where rule (i) applies the effect of the semi-closed shell of 40 nucleons is not evident.) The moments of the odd-proton nuclei in this group tend strongly to be positive, while the moments of the odd-neutron nuclei show no such tendency. In some of the 18 cases the plot of Townes (see ref. 7), p. 450) can be very useful; often, however, these cases lie in that portion of Townes' plot where there is a transition from negative to positive quadrupole moments and no definite information can be obtained.
7. Concluding remarks We note that out of the 121 quadrupole moments that we consider, 84 (or 69.4 %) are positive and 37 are negative. These totals can be broken down into subcategories as follows: In the group of 17 doubly odd nuclei, 11 nuclei (64.7 %) have positive quadrupole moments and six have negative moments. In the group of 61 oddproton nuclei, 40 nuclei (65.6 %) have positive moments and 21 have negative moments. Finally, from among the 43 odd-neutron nuclei whose moments we analyse, 33 nuclei (76.7 %) have positive moments and 10 have negative moments. We do not believe that much significance should be attached to the somewhat larger percentage of positive moments for odd-neutron nuclei. This result occurs partly because in the region of fig. 1 to which rule (i) applies we find 27 of our 43 odd-neutron cases, but only seven of the 17 doubly odd cases and 27 of the 61 odd-proton cases. It is also due to the fact that of the 13 one-particle nuclei for which quadrupole moments (negative in every case) have been measured, 10 are odd-proton nuclei and only 3 are odd-neutron. Throughout our work we have found a surprisingly complete symmetry between the general effects that protons and neutrons have on the signs of the quadrupole moments. It had already been noted in earlier works 7,8, lo) that the quadrupole moments of odd-neutron and of odd-proton nuclei are usually of about the same order of magnitude. For odd-neutron or odd-proton nuclei we do not find any empirical correlations between the spins of the ground states and the signs of the measured quadrupole moments. The success of rule (i) suggests a possible connection between the proton-neutron force and the preponderance of positive nuclear quadrupole moments that is observed in nature. It has been noted that the positive quadrupole moments often have appreciably larger 5 t ~sMn3o ' alGaa5 ,8, atGaa8 , 8 8 8tGa4o , - / 1 4iNb52 ,98 43Tc56 ,99 1~53Cs78,I3~Cs80,161Eu88,I~aaEuso,2e'A89clas.
332
Y.Y. SHARON
magnitudes than those of the typical negative quadrupole moment 7). F r o m the diagram on page 450 of Townes' Handbuch article 7) it is clear that the average positive quadrupole moment is at least three times larger than the average negative quadrupole moment. However, it is precisely when rule (i) is applicable that we encounter very large positive quadrupole moments, moments that often are an order of magnitude larger than the single-particle estimates. This fact is illustrated when we consider the quantity [Ore. . . . . . a/A~] for the 60 quadrupole moments (27 positive, 33 negative) to which rule (i) does not apply. The average value of this quantity for the 27 positive quadrupole moments is 0.0163 b; the average value for the 33 negative quadrupole moments is -0.0139 b, i.e. just 15 ~o less in absolute value. It must be emphasized that our analysis has dealt only with the signs of quadrupole moments. It is possible, however, to extend our empirical analysis (in terms of the filled fractions of the major shells in both protons and neutrons) to include also the systematics of the magnitudes of the quadrupole moments. These magnitudes have been considered, from several other points of view, in the previous studies of quadrupole moments 4-1o). In these studies most authors found it fruitful to follow the approach of Townes 7) by concentrating on the "inferred intrinsic quadrupole moments" (see ref. 7), p. 449) rather than concentrating on the measured moments. One is then able to include in the analysis also the inferred magnitudes of the intrinsic quadrupole moments of nuclei with ground state spins of less than unity t The magnitudes of quadrupole moments increase rapidly with distance from closed shells. Consequently, the effects of the shell structure become very obvious in plots such as those that appear on p. 450 ofref. 7). The general picture that has emerged in this paper has been well known for a long time. However, our new parametrization makes possible observations that are more general and quantitative. We have already seen how our two simple empirical rules have enabled us to account for a large majority of the signs of the measured quadrupole moments. On the basis of our study we can make predictions of the signs of many quadrupole moments that have not yet been measured and whose signs sometimes could not have been predicted on the basis of the previous studies of moment systematics 4-8). We can also make an informed guess as to the signs of quadrupole moments whose magnitudes have been measured but whose signs have not been definitely established. To illustrate with several such cases from refs. 11, 12, 3o), '76 76 80 82 138T_ we predict positive quadrupole moments for 33As43,35Br41,35Br45,35Br47, 57La81, 45 • 167r~ 166~__ 170 22"1"123, 68Jc-~r99, 69111197and 69Tmlol. Our analysis can also be usefulin a case where two reported measurements of a given quadrupole moment disagree in both magnitude and sign. Thus, in ref. 11), we would have selected Q = - 0 . 2 0 b rather than Q = +0.16 b for 29Cu34, 63 Q = - 0 . 1 9 rather than Q = +0.15 for 65 29Cu36, Q = - 0 . 4 8 2 rather than Q = +0.02 for 143~A 6 0 1 " ~ 8 3 and Q = - 0 . 2 5 5 rather than Q = +0.01 * We believe that rules (i) and (ii) may also apply to the signs of the intrinsic quadrupole moments of all nuclei, with A > 40 including doubly even ones.
QUADRUPOLE MOMENTS
333
for 14s~,t601,,~85.These selections are identical with those that were made, probably on other grounds *, by the authors of the compilation in ref. 12). 8. Possible explanations for the validity of our rules
The simple spherical shell model of Mayer and Jensen 1) predicts that there should be equal numbers of positive and negative quadrupole moments. This result also follows from calculations which minimize the total single-particle energies of particles which are placed in the orbitals of a deformed harmonic oscillator well 2s). The singleparticle spin-orbit force may change the order of levels in an oscillator well but its average effect should favour neither oblate nor prolate deformations 26). Many explanations have been advanced for the preponderance of positive quadrupole moments. Among the possible causes thus cited have been the effects of: (i) surface tension plus Coulomb repulsion, in conjunction with large core deformations 6). In a calculation along these lines Moszkowski and Townes 6) found that the quadrupole moments should be positive whenever a shell of identical nucleons is more than one-third full. However, in a slightly different calculation, Gottfried 27) showed that these effects are not a major reason for the preponderance of prolate shapes. His results, which are valid only for an oscillator well, are also valuable because they suggest that for our purposes it is really not necessary to consider shapes that are axially asymmetric. (ii) tensor Jbrces. It is commonly accepted that tensor forces are responsible for the positive quadrupole moment of the deuteron. There is some reason to suppose that these forces favour elongated shapes also in heavier nuclei tt. However, further work should be done using two-body tensor forces which have opposite signs in states with T = 0 and with T = 1. (iii) the deviations of the real single-particle well from its assumed harmonic-oscillator shape in which there is a heavy degeneracy of energy levels 25' 28, 26). The observed preponderance of positive quadrupole moments has beenlinked to the tendency of a square well to lower the energy of single-particle states with high orbital angular momenta 29) (or with low values of Iz (ref. 28)). This effect may turn a delicate balance in favour of prolate shapes. Further investigations into this effect were carried out by Lemmer and Weisskopf 26), who started by considering a deformed harmonic oscillator potential well. They found that prolate equilibrium shapes are energetically favoured in the first half of a shell and that oblate equilibrium shapes are favoured in the second half. This result is in agreement with the result that was obtained in ref. 25) but is in contradiction to the trend of the experimental data. Lemmer and Weisskopf showed, however, that the differences between the energies of the prolate and the oblate shapes are smaller at the end of a shell [when the oblate ? Negative values for the quadrupole moments of the Nd isotopes were indeed obtained in recent experiments (see ref. 42)). ?? See ref. 6), p. 306 where this point is discussed and further references are given.
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shapes are favoured] and larger at the beginning of a shell [when the prolate shapes are favoured]. They concluded that a perturbation of the oscillator well (such as the 12 term which is used in the Nilsson model to simulate square well effects, or a r 4 term to make the well steeper) is capable of accounting for the observed preponderance of prolate nuclear shapes. It would be interesting to extend these calculations to manyparticle states which are eigenstates of the total angular momentum. Our empirical observations seem to suggest that despite the complexity of the problem it may be possible to construct a crude model which would explain quadrupole moments of nuclei over the entire periodic table. Such a model should take into account the fractions of both the last major proton and the last major neutron shells that are occupied in the case of each nucleus. It is not unreasonable to expect that the mechanism which favours prolate shapes would predominate when certain fractions of shells are filled rather than when there are a certain number of particles outside closed shells. This is because for heavier nuclei, where the closed-shell cores are larger, more valence particles would be required to achieve corresponding core deformation effects. It is worthwhile to recall here the results that were obtained by Marshalek, Person and Sheline 9). Unlike the calculations cited earlier in this section, these authors explicitly considered both the protons and the neutrons in each nucleus. Marshalek, Person and Sheline 9) carried out calculations of equilibrium deformations in several regions of the periodic table. In their calculations they considered several possible Nilsson configurations for the ground state of each doubly even nucleus in the region of interest. Subsequently, in each case, they minimized the total single particle energies, for a given number of neutrons and a given number of protons, as functions of the prolate deformation. These authors confined themselves to elongated (and spherical) shapes and did not consider oblate deformations; the deformation contours that they calculated reproduce the observed systematics of the intrinsic positive quadrupole moments in the rare-earth and transuranic nuclei. However, the results of another calculation within the same general framework are in strong disagreement with the empirical rules that we deduced here. Kumar and Baranger 3s) carried out similar calculations for neutron deficient isotopes in the vicinity of Ba. In their work they used Nilsson wave functions and pairing plus quadrupole forces, taking into account the possibility of oblate as well as prolate deformations. Kumar and Baranger predict that oblate deformations should prevail throughout most of the region where both N and Z are between 50 and 82, including isotopes with Z > 56. This prediction has not been proved or disproved experimentally, but its validity seems somewhat unlikely in view of our analysis of the nuclear quadrupole moment data. If we accept the results of Kumar and Baranger then we must conclude that the simple Nilsson picture will not be able to explain the general features of rules (i) and (ii) that were devoloped in this paper. These rules are of course purely empirical, but they do seem to have wide validity.
Note added in proof." In the last two years the signs of a few additional ground-state
QUADRUPOLE MOMENTS
335
quadrupole moments have been determined. Positive moments have been found for 147~61t,m86 (ref.40)), 152~6312,1389(ref.42)) a n d 159,vr,65xv94 (ref.41)). N e g a t i v e m o m e n t s h a v e b e e n f o u n d f o r ~17515b66 (ref.37)) a n d 137r53184 (ref.39)). W e also n o t e t h a t t h e p o s i t i v e q u a d r u p o l e m o m e n t o f 16~Er99 (refs. 11, 12)) was i n a d v e r t e n t l y o m i t t e d f r o m t a b l e 1 a n d fig. 1 o f o u r p a p e r . O f t h e s e n e w results the n e g a t i v e m o m e n t s are a c c o u n t e d f o r by r u l e (ii) a n d t h e p o s i t i v e m o m e n t s by r u l e (i), e x c e p t t h a t n e i t h e r rule applies to 9s 14~7~61t,m86. F o r t h e m o m e n t o f a2M053, w h o s e m a g n i t u d e has b e e n r e c e n t l y m e a s u r e d (ref.38)), we p r e d i c t a n e g a t i v e sign.
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