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Ratational energies for the asymmetric rotor model for odd-mass nuclei
1 .D.2
] I
Nuclear Ph~tsics 3 6 (196"2) 6 6 6 - - 6 8 7 ; (~) North-Holland Publishing Co., .4 msterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ROTATIONAL ENERGIES FOR THE A S Y M M E T R I C ROTOR M O D E L FOR O D D - M A S S NUCLEI LUCY WU PERSON
Department of Chemistry and Lawrence Radiation Laboratory, University of California, Berkeley, California and J O H N O. R A S M U S S E N t
Institute for Theoretical Physics, University of Copenhagen, Denmark Received 3 M a y 1962 Abstract: W e s t u d y the effect o f ellipsoidal nuclear d e f o r m a t i o n in o d d - m a s s - n u c l e a r rotational b a n d structure, the magnetic m o m e n t , a n d electric q u a d r u p o l e reduced transition probabilities. Also, we s t u d y the relationship between the rotational b a n d s o f a n ellipsoidaUy deformed nucleus a n d the vibrational a n d rotational b a n d s o f a spheroidally d e f o r m e d nucleus with ),-vibrationrotation interaction in the limit o f ~' a p p r o a c h i n g 0 or ½~. E q u a t i o n s for the a s y m m e t r i c r o t o r m o t i o n are derived. By using T.D. N e w t o n ' s single-particle eigenvalues a n d eigenvectors, we then present numerical calculations s h o w i n g rotational spectra associated with an o d d nucleon in a n ellipsoidal well. The calculations for the N = 4 a n d N = 2 shells were done o n a n IBM 709 c o m p u t e r . N u m e r i c a l results are discussed in t e r m s o f the fl a n d ~' d e f o r m a t i o n parameters required to give the k n o w n spins o f the o d d - m a s s cesium isotopes Cs ~7 to Cs ~aT. T h e rotational energy spectrum, magnetic m o m e n t o f g r o u n d state, a n d various E2 transition probabilities are calculated for Cs ~3~ for several deformations, w i t h best energy s p e c t r u m fit at fl = 0.28, ~, = 38 deg. There are difficulties with the transition probabilities a n d magnetic m o m e n t .
1. Introduction The Davydov-Filippov model of an ellipsoidally deformed nucleus with three unequal principal axes has been applied extensively to even nuclei 1). For small values of the asymmetry parameter y the rotational spectra correspond closely to those of the symmetric rotor with y-vibrational excitations added. Perhaps the greatest utility of the model has been in the regions of nuclei outside the regions of definite spheroidal deformation, where the energy of the second excited 2 4- state may be only about twice the energy of the first excited state. We felt that it would be interesting to examine the model for odd-mass nuclei it. At a late stage in our calculation we learned of similar work 2) by Hecht and Satchler, who treat the rotational energies of asymmetric odd-mass nuclei with essentially similar results, except that their results are for the spectra of nuclei with A around 190. Watanabe and Blair have also treated the problem, with applications to d-shell t On sabbatical leave from the University o f California, Berkeley, California. tt Based on a Ph. D. thesis (by L.W.P.), University o f California, Berkeley, California (see ref. ~s)). 666
L. W. PERSON AND J. O. RASMUSSEN
667
nuclei 3). Filippov 4) has recently made calculations and general examinations of the problem of stability of the asymmetric nuclear shape. From his results it appears that noncylindrical shapes could possibly be of lowest energy in some cases, where one kind of nucleon has nearly completed a shell closure and the other kind is just beyond a closed shell. The region of the neutron-deficient cesium isotopes possibly satisfies these requirements. It is instructive to consider a trivial extension of the Davydov-Filippov model to odd-mass nuclei, namely, the case where the odd nucleon is in a pure J = ½ state, and hence completely uncoupled from the rotor. In such a case we have just the spectrum of an even nucleus, but with a ground state spin of ½ and all excited levels doubly degenerate, with spins differing by + ½ from the spin in the even nucleus. We see that in the rotational band there will be only one state with I = ½, two with I = 3, three with ~, and generally I + ½ states of a given spin L The sequence of ground-state spins of the odd-mass cesium isotopes qualitatively suggests a possible explanation fia terms of an ellipsoidal deformation which sets in as the neutron number departs sufficiently from 82. The measured spins for neutron numbers 72, 74, 76, 78, 80, 82 are, respectively, ½, ½, ~, {, }, {. The expected g; spherical shell model spin appears near the closed shell. Ellipsoidal deformation of the nuclear potential would tend to quench the orbital angular momentum of an odd particle so that, for sufficiently large deformation, spin ½ should lie lowest. The quantitative testing of the model involves considerable mathematical complication. Fortunately, nucleon eigenfunctions and eigenvalues for an eUipsoidal harmonicoscillator potential had been calculated and tabulated by T. D. Newton ~' 6) through the fourth oscillator shell. The Hamiltonian he used was more appropriate for N = 4. neutrons than for protons, and we are deeply indebted to him for recalculating, at our request, the fourth oscillator shell, using a proton parameter. With his eigenfunctions we have attempted to calculate rotational spectra of even parity for some odd-proton nuclei in the 50 to 82 shell. 2. Theory of Odd-Mass Nuclei with Fixed ElUpsoidal Deformation The Hamiltonian of the coupled system of nucleon and asymmetrical rotor core is as follows: = J~ap -~t-~ i n ,
-']- T R ,
(1)
where the terms on the right-hand side are as defined later. The rotational energy of the general rigid rotor may be expressed as 3
TR = i y ~ R 2~
-,
,
(2)
!¢=1
where R~ denotes the components of angular momentum along the principal axes. We assume, in accordance with the hydrodynamic model 7), that ~
= 4Bfl 2 sin 2 (?-x~zt),
(3)
668
L . W . PERSON AND J. O. RASMUSSEN
where fl and ~ are the usual parameters specifying a general ellipsoidal deformation, and B is the inertial parameter for quadrupole surface oscillations. The single-particle Hamiltonian°~d'p is given by ~ p ~-- Tp+
Vp(r)+C I. s+Dl 2,
(4)
where h2
= -½
V
(S)
Vp = ½M((0~ X~ + (02 X~ + (03 X32)•
(6)
Here M is the single-particle effective mass of a nucleon in the nucleus, X 1 , Xz, )(3 are the Cartesian coordinates of the nucleon in the body fixed system, hcol, h(0z, h(03 are the corresponding quantum energies along the three principal axes, satisfying the conservation of nuclear volume conditions (7)
(DI (02 (03 = ( 0 3 ,
while I is the infinitesimal pseudo-rotation operator, and C is a spin-orbit potential strength parameter. The V 2 operator in eq. (5) is evaluated in the body-fixed coordinate system. The D l 2 term in (4) is a correction which depresses high-angular-momentum orbitals, equivalent to the effect of a nuclear potential more square than pure harmonic, The term ~ i . t represents the interaction of the particle with the nuclear deformation. However, in Newton's 6) calculation for ellipsoidal nuclei the energy eigenvalues E s include both the single-particle energy and the interaction energy. We shall not consider nuclear shape vibrations but consider the shape fixed. In the case of non-axial nuclei, if the moments of inertia are sufficiently large, the particle motion will follow nearly adiabatically the rotations of the well, and the wave function will be a linear combination of Nilsson's wave function "s' 9) in the following way: ~,M = Ii'2I ~167z2 + 1'I]
~ btr3 ~ 13
J J3
~jj3(DI3M(Oi)~/NIJJ3.~_(__),
i-ZD_t3M(Oi)~stz_s3),,
(8)
where ~PNzJS3is as defined by Newton's 6 ) eq. (3); the summation o v e r J3 runs only over alternate half-integr:~', values between J and - J ; and 13 is summed between I a n d - / o n alternate values such that 13 --J3 = 2v. Here v = 0, _+ 1, ___2,..., ___N, which is due to the symmetry requirement of invariance with respect to a rotation through g about the 3 axis. By convention, b1½ and ~¢s~ are always taken with a positive sign. The ~'sJ3 are the mixing amplitudes which are the eigenvectors of Newton's table 6), and b i i 3 are the mixing amplitudes of the rotational states with different 13 values and the same L Therefore bH3 are the eigenvectors of the rotational spectra that we calculate. For axial nuclei (7 ~ 1N1r), I3 and J3 will be approximately constant, and their eigenvalues will be K a n d I2. The adiabatic-rotation assumption may not be
ROTATIONALENERGIES
(169
a good one for the cesium isotopes we later treat, since rotational energy spacings are not much smaller than the single-particle-level spacing. We have neglected the effect of the coriolis interaction in mixing different Newton eigenfunctions.
2.1. ROTATIONAL ENERGY 1N THE STRONG-COUPLING APPROXIMATION The rotational states may have energies much smaller than the single-particle energy and the phonon energy, so in eq. (1) we only have to consider the rotational energy term TR = ½ ~ R ~2 ~ - 1. I f we use the representation in which 12, 13, j2 and J3 are constants of motion, then we write TR in the following form given by BohrV): TR = TDRq- TOR-
(9)
Here TDR'is the diagonal term which is the same as the rotational-energy term in the nucleus with axial asymmetry, excluding the case of d 3 = ½:
TDR = (h24~1+ ~2h2) [ I ( I + l ) - I 2 + j ( J + l ) - J ] ] +
~h2( I 3 - J 3 ) z.
(10)
Further, TOR are the off-diagonal terms that vanish in case of axial symmetry, except for J3 = ½: ( hz To. = -
h2 11 S l +
) I2 J2
+
The nonvanishing matrix elements of eq. (11) will be governed by the following selection rule: The first term will connect states of AK = _+ 1 and A~2 = _ 1, and the second and the third terms will connect states of AK = 0 and A• = _+2, or AK = + 2 and A~2 = 0. Substitution of eq. (8) into the rotational-energy equation TR ku = Erot ~ gives a secular determinant of dimension I + ½ by I + ½ to be solved to determine the eigenvalues and the coefficients bm. All three terms in TDR contribute to the diagonal matrix element. O f the three terms in TOR, the second contributes a constant amount to the diagonal matrix element, and can - for our purposes - be ignored. The first and third terms provide the off-diagonal matrix elements f o r ~ 3 ,t3_1 and ~ l ~ , i~+2. Therefore the secular matrix is separable with only even values of 13 - I ~ coupled. The general formula for diagonal elements of rotational energy is ~F13,3 = h2 + 2~ 3-
~
+ ~
{I(I+1)-I2+ ~ Idss3i2[J(J+l)-J]]} ss3
./s3ZIdss3[Z(13-J3) z
(12)
670
L, w. PERSON AND J. O. RASMUSSEN
(h2 h2) E (-) '-' d~ds+½)
+-(-)&~.~(l+½) ~ 1 + ~
( hh2~ ) 4 ~~1
~ , ,'-~
~
d
d [(JWJ3)( J - J 3 + l ) ]
½ (12)
_(h2__ h%](i+½ ) E (-)'-%,,,v~_,,~+,,d.,~.~ \4~1
4~2]
dJ3J'S'3
× [(J -- d3)( J + J3 + 1)] ~. Due to the symmetry of the wave function, the other two matrix elements give the same numerical result, and so are not written out. The general formula for off-diagonal elements of ToR is J4g'~'13(13- 1) = -- 313 -(I'3 - 1 ) [ ( I + •3)(• -- 13 + 1)] ~ t(]12 __ h 2 ~ E ( × [\,4~,~1 4 ~ 2] JJ3J'J'3
'"~I-J(~Jd'VJ'3-(J3 +l)~dJ'3~JS3[(J-J3)(J-}-J3+l)]{ A
(13)
and
~'~'~-+~
(h2
-
8~1
]721
[(I-T-I3)(I-T-I3-1)(I+I3+l)(I+-I3+21] ½"
(14)
8,~J
Due to the symmetry of the wave function, the other two matrix elements give the same numerical result as eqs. (13) and (14), and so are not written out. 2.2. MAGNETIC MOMENT The m a g n e t i c - m o m e n t operator is t% -
e
2Mc
10)
[~_,(g, li+gss,)+gR, Rll+gR2R22+gR~R33].
(15)
I f we assume that the three collective g factors are equal to a value gR, then we m a y use Nilsson's 8) expression (20):
1 /~ ---- -[-(g~-- gl)(s " I ) + (g,-- g R ) ( J " I ) + gR I(I + 1)], I+1
(16)
where
13JJ3 x { b , , 3 b,_ (,, -1 )( -- )J - 4x~dJ, ~ J - (J, - 1)(½)
xE(I+la)(I-I3+l)(J+Ja)(J-J3+l)] ~
+ b,. b,_.~+,)(-)'-~C.~j_(~+,)(½) x [ ( I - 13)(1+ I3 + 1 ) ( J - Ja)(J + J3 + 1)]~'} -
(17)
ROTATIONAL ENERGIES
671
For the calculation of the matrix ( s . I) it is more convenient to work in the representation of (f2]NIA~), where Nilsson's expansion coefficient d~la Z is related to ~/ss3 in the following way s): "5~tA~r = E (I½Aff'[JJ3)d:~Js3. J
Then we have (s
I) •
=
~ btt3[~c~t(s3_~)~--.~i[l(s3+½)_~] 2 2 2 • ½I3 I31J3
+ ~ (-)'bmb,_(,~_l)d,(a~_~)~Ci_(a~_~)~" ½[(I+I3)(I-I3+1)] ~
(18)
13|J3
+ ~ (-)lb,,3b,-(,3+x)dua3+~)-½~c,-(a3+~)-~" ½[(I-I3)(I+I3+1)] ½. I31J3
2.3. ELECTR1C-QUADRUPOLE REDUCED TRANSITION PROBABILITIES The transition probability for electric-quadrupole radiation is given by 11)
T(E2;Ii-~If)= 4~ ( he E ) 5 B(E2;I, --* If),
(19)
where B(E2; Ii ~ If) is the reduced transition probability between rotational states I i and If. It can be written as 5 B(E2; Ii -~ If) - 16~(2Ii + 1) ~ f (fIQ2u[i)2"
(20)
By using the model of a nucleus as an incompressible classical liquid drop with a uniform charge distribution, the nuclear quadrupole-moment operator Qz~, can be written as 12)
Q2~, = eQo[D2o cos ?+(D22+D2_2) sin 7 (2)-~],
(21)
where Qo is the intrinsic quadrupole moment of an axial nucleus and is related to the deformation parameter/3 by Qo = 3ZR2fl(5~z) -~. After simple algebraic manipulation, and assuming that there is no change in the internal state of the nucleus in the transition, the reduced transition probability can be expressed in terms of the average value of fl and ? as 12) B(E2; I~ ~ If) --- 5 e2Q2] ~ bl~r3bitra(COS y(ii2130[ifi,a) 16g I~r~ + [sin ? (2)-~](Ii 213 ± 2llflj, )[2. (22~ 2.4. E L E C T R I C - Q U A D R U P O L E
MOMENT
The formula for the spectroscopic electric quadrupole moment can readily be derived by a specialization of the formulas of subsect. 2.3; that is Qsp,c = -1 (Q2o)n,=1, e
(23)
672
L.W.
PERSON AND
J. O. R A S M U S S E N
where Q2o = Qo{D2o cos 7 +(D22 +Do2_ 2)[sin y (2)-½]}.
(24)
Substituting eq. (24) into eq. (23), we get Qspec = Qo(I2IOI II) ~ bu~bu'3[ cos y(12130[II3)~,'3t3 131"3
+ [sin ~, (2)-~]((I213 21113 + 2)6r3h + 2 + (I213 - 21113 - 2)6r~13- 2)] -
Qo (1+1)(21+3)
{cos ~, ~ bu31313 ~ - I(I + 1)] ,3
+ sin ~, (k)~ E b,t3 b,,,3([(1 + 13 + 1)(I + I3 + 2)(• - 13 - 1)(I - I3)]~6v3,3 +2 (25) I31'3
+ [ ( I - 13 + 1)(I-- 13 + 2)(• + 13 -- 1)(I + 13)]*(~I,313 _ 2)}. 3. Numerical Studies and Comparison with Data We have used Newton's eigenfunctions and eigenvectors 5,6) for our calculations of rotational spectra; other nuclear properties were calculated by using the formulas of sect. 2. 3.1. ROTATIONAL SPECTRA In the low-lying rotational-energy calculation, we assumed that the odd-nucleon state of motion in the nuclear well is not changed for different states of rotational motion. We also neglected possible vibrational effects and the collective rotationvibration interaction. The rotational energy ER and its associated eigenvector bH3, which are obtained by diagonalJzation of an (I+½) by ( I + ½) matrix for a state with nuclear spin/, were calculated by means of an IBM 709 digital computer 13). Both the calculated values of the rotational energy ER and the mixing coefficient bu3 fore ach value of the single-particle energy E S = E s - ( N + ~ } ) are tabulated in Appendix B of ref. 13) from 0 to ½n in steps of ~¢~z. By symmetry, the states with deformation parameters fl and y, and the states with deformation parameters - f l and ½~-~ are equivalent. Curves fitted through the calculated values of fl and ), were used for interpolation. Curves showing the total energy E = Es+ER for the lowest single-particle state of the N = 2 shell, 3 = 0.2, B/h 2 = 77.25 MeV-*, are plotted as a function of 7 in fig. 1 for the first rotational band, in fig. 2 for the second rotational band, and in fig. 3 for the third rotational band. For comparison, the rotational energy of even nuclei 14) with the same values of 3 and B are plotted as a function of ~ in fig. 4. From these four figures we can clearly see the analogy between the rotational spectra of an odd nucleon in an ellipsoidally deformed nucleus and the Davydov-Filippov rotational energy spectra of even nuclei 1, xs). This is expressed schematically in the correlation diagram in fig. 5.
l
I
-~--
l
I
i
I
I
I
I
I
I
I
E
I
I 45
I
I
62 60
s9
•
E°(11/2+)
z~
Eo(7/2
(:P f
61
.
E,(9/z+)
~
60
o
E(3/z+)
!I
•
El(5/24")
:I
E(i/a+)
il
58
57~
,& EZ ( 1 1 / 2 ~.) z~ E z l E2 0 Ez lIE z
+)
( 7 1 2 +) ( 9 / 2 ".-) {3/2 *) (512")
59
,1 ,f ,I
i
ta
//
I I /
,;'
54~ r I
t
,i ,J
53-
!
.9
52~
;
5,~-
~-~~
--~,~ _
_ _-o~
5~--.50
.e~--~..o , /
.--~'~-~"~e
o
-- ""
---"cr
._.~.~,
15
~
~ ~-~
z~
-
..l"-'~
30 7
_--~_
"--'~
."
45
60
I
I
I 30
T--
I
I
I
I0
A E3(7/2"+)
59_!
uE3(5/2+
60
I
(6+) (,5+) (4+1 {6+)
IE, (3*)
) 9
~E I (4+) eE 2 (2+)
60
OEm (2+)
8
~
s7 Lid
I
• E2 o£ a A£ z ~.E I
I1
eE3(9/2+) 61
I ( de{] )
Fig. 2. Energy d i a g r a m o f the second rotational b a n d for the lowest single-particle state o f the N = 2 shell as a function o f 7.
• E 3 (11 / 2 + )
62
I 15
7
Fig. l. Energy d i a g r a m o f the first rotational b a n d for the lowest single-particle state o f the N = 2 shell as a function o f 7r
I
0
( deg )
7 6
56
'\\ ~
4': 55
•
\ 54
zx
,
'\
/
"e
.
/
/ O
o....._..~
\ram ~
',\\
-
\
\
5.3
I 0
15
"T
7"
I
30
(
deq
415
..... 60
)
Fig. 3. Energy d i a g r a m o f the third rotational b a n d for the lowest single-particle state o f the N = 2 shell as a function o f 7.
OI 0
o-.---- a ..... I I 15
7
°-- "'--° I I :30
I
( deg )
Fig. 4. Rotational-energy d i a g r a m o f an even nucleus ER (in MeV) as a function o f 9).
674
L.
W.
PERSON
AND
.L
O.
RASMUSSEN
As discussed in the Introduction, in the limiting case of a pure J = ½ odd nucleon, the nucleon motion is completely decoupled from the collective surface motion. Also, one gets a system of levels with the same spacing as in an even nucleus but with a doubly degenerate state of spin I_+ ½in place of each level of spin I i n the correspondEnergy levels of on even osymmetrnc rotor
Energy levels of on odd nucleon
I ,Tr
I,TT -
-
Energy levels in the limits of o pure S 1 / 2 odd nucleon
5+-~-
.
.
4+
__
.
.
.
11/24-
I ,7 . . . . .
11/2+
- - 9 / 2 + -
:5+=
-
-
-
-
7/2
..... -
-
-
.
.
.... +-
.
.
.
.
. -
-
-
4+ 2+
7/2 .
.
.
. . . . .
0+
712+- - 5 1 2 + -
-
I
/
.
.
.
-
_
4+
7 / 2 4. 5/2 +
-
.
-
_
512+ 3/2 +
13/2+ 11124-
9/2 +
- -~......
_ _ 5 1
3/2+
2
. -
1312+ . . . . . 1112+----
--9/2+--__ -
.
5•2+ 312+---
-
~ -
2+
9/2
5/2+-
2+~--_
6+
- - 9 /
9/2+ 7/2+
+
-
7/~
+ +
3/~+
-
I/
Fire 5. Relation o f spectrum between an even asymmetric r o t o r and an odd-mass nucleus o f el[ipsoidal shape. i
i
i
J
i
,
,
65
63, 62 >
6o
• A • o • [3
E ( I I I 2 +) E(7/2+) E ( 9 / 2 ÷) E (3/2") E (5/2*) E{I/2*)
~ 60 ¢ w
59 #"
""~.... -
58' 56 55 ~ , . --0---.[3. " "--G----G----~---t:~--'~---~" 0
15
30 )" (deg)
45
60
Fig. 6. Energy d i a ~ ' a m o f a nearly pure s t state as a function o f 7, for the N = 2 shell, with/3 = 0.1 and B/t~== 77.25 MeV -1.
675
ROTATIONAL ENERGIES
ing even nucleus, except that a spin 0 goes to a spin ½ level. T h e existence o f the nearly degenerate states is clearly seen in fig. 6, where the energy is p l o t t e d a g a i n s t 7 for a single-particle state o f a l m o s t p u r e s t. 3.2. NUCLEAR PROPERTIES OF Cs TM O f the light cesium isotopes with g r o u n d - s t a t e spin less t h a n ~-,7 o n l y Cs T M has e n o u g h k n o w n even-partly excited states for significant c o m p a r i s o n . T h e r o t a t i o n a l energy s p e c t r u m is calculated for B f l 2 / h 2 = 13.85 M e V -1 a n d hco = 8.07 MeV. W e chose the N e w t o n p r o t o n state no. 27, c o r r e s p o n d i n g to o u r fig. 13; such choice presumes one p r o t o n p a i r in an h ~ orbital. T h e results are t a b u l a t e d in table 1 together with theoretical spins a n d the experim e n t a l energy values x6, 17). I n t e r p o l a t e d energy values for o p t i m u m p a r a m e t e r s fl = 0.28 a n d ? = 38 ° are given, a l o n g with the energy levels calculated for neighb o u r i n g values o f d e f o r m a t i o n parameters. T h e ~ value used in table 1 is r a t h e r TABLE 1 Energy levels, experimental and theoretical energy values for Cs13x Level label
J I H G F E D C B A
Ee~p b)
Eth~oc)
Spin~the°~) (keV)
½+' j-k-' ~+ ' ~q-' ~ + ~+ ½-t½q~ q~+
1039 620 -373 -216 133 124 -0
(keV)
0.28, 38°
0.3, 30°
0.3, 37.5°
0.3, 45 °
0.2, 37.5°
1022 644 448 380 264 224 136 122 40 0
1682.7 1404.4 722.9 899.7 531.7 780.6 529.2 717.8 330.8 383.5
1106.7 650.7 462.6 452.1 260.2 276.3 196.7 183.8 0 2.61
1873.7 1714.7 684.9 458.8 419.1 377.3 183.9 341.9 304.1 97.4
1224.0 1023.1 822.4 575.2 731.2 466.9 478.9 432.6 606.4 373.9
• ) Underlined spins have been experimentally determined or inferred. b) See ref. xs) for compilation of various determinations of energies. e) Interpolated energy values of this column are mostly subject to an uncertainty of about 8 keV. All values are for Newton state no. 27, with Bfl2/h 2 = 13.85. close to the value required to fit levels o f the n e i g h b o u r i n g even nucleus X e 13° T h e ratios o f r o t a t i o n a l energy o f Xe 13° are 18) E 1 ( 4 + ) - 2.25, E1(2+)
E 1 ( 5 + ) - 4.43, E1(2+)
E l ( 6 + ) -- 3.66, E1(2+)
w h i c h c o r r e s p o n d s to a d e f o r m a t i o n ~ = 35 ° o r 25 ° in D a y a n d M a l l m a n n ' s table 19). T h e decay scheme o f Ba T M in fig. 7 is t a k e n f r o m the recent study b y B o d e n s t e d t et al. 2o). T h e y p r o p o s e a n d j u s t i f y spins o f ½ a n d ~ for excited states at 124 a n d 133 keV, respectively. T h e l o c a t i o n s a n d spins o f these states, together with the spin o f the g r o u n d state, essentially fix the p a r a m e t e r s / ~ a n d 7 in our calculation. T h e energy
676
L.W.
PERSON A N D J. O. RASMUSSEN
match with four additional higher levels test the model, and the agreement is good. The theoretical spins are not inconsistent with the observation of all the gamma transitions in the experimental decay scheme shown in fig. 7, except for one transition; namely, there cannot be a transition JC because of a spin change of three by our assignments. We suggest that there is enough uncertainty in the experimental energies so that the reported 917 keV transition could fit as transition JD. The experimental literature is somewhat contradictory with respect to multipolarity assignments. It is clear that our spin assignments could be tested by multipolarity assignments. Transitions CA, GC, ID, and JE must be pure E2, since ]All = 2. A search for the unobservBo
131
/
/ 1039
/ (rl/2: II5d )
I
1039
824
703 - - 620
I 908
216
487
i
I
I I
416
250
5OO
i
--T 240
....
373
1
216
124
I
~'~/2: ( 1 3 . 3 : 0 5 )
nsec
133
1
O--
/ "el/z = ( 3 . 7 7 *- 0 . 0 5 ) nsec
o
13 Fig.
7.
D e c a y s c h e m e o f B a xSt.
ed higher spin levels predicted here could be valuable. The experimental level at 703 keV does not seem to have a counterpart among the theoretical values. We are also somewhat concerned about the difficulty of populating a 7 + state at 1039 keV, presuming Ba 131 to have a ½+ or 3 + ground state. The reason for not seeing the two z9 + states and higher spin states could be that the high spin itself and the predominance of high-K components in these states inhibit population by fl decay from Ba 131 or by ~, transition from higher energy states. Before presenting the calculations of the ground-state magnetic moment and some relative transition probabilities, we list the eigenfunctions of the states involved, in table 2 the fifteen d jl ~ Newton coefficients used, and in table 3 the three btK values and the four b~K values." We calculated the magnetic moment of the ground state of Cs lal by setting collec -
ROTATIONAL ENERGIES
677
TABLE 2 N e w t o n ' s particle-state-eigenvector c o m p o n e n t s for N = 4, fl = + 0 . 3 a n d ~ ~ 37.5 ° (eigenvectors expressed as fl = --0.3 a n d ~ = 22.5 °)
1
J
Ja
4 4 4 4 4 2 4 4 2 2 4 4 2 2 0
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ½ ~ ~ ½
d.lJ 3
~ --~ --5 ~ ~ ~ --~ --6 --~ --~ ½ ½ ½ ½ ½
0.04129 0.14200 0.01109 --0.28351 0.00604 --0.14104 --0.51944 --0.01793 --0.24832 --0.05451 0.61698 0.02807 0.35165 0.04707 0.18867
TABLE 3 Calculated rotation-eigenvector c o m p o n e n t s I = ~ (ground state)
K ½ --]
I = ]~
b[K
K
--0.87287 0.47423 0.11491
b]K
½ --~ ~ --½
--0.44320 --0.78613 0.42070 0.09263
tive ORvalues equal to Z/A, 9~ = 1 and first setting 9s for the odd proton equal to the "quenched" value 4.0 found generally applicable by Chiao and Rasmussen 2~). A second calculation sometimes follows in parentheses, using free-space gs = 5.585. We feel that the quenched values are more appropriate, since they take account of a general tendency by the odd nucleon toward polarizing the spins of neighbouring nucleons in such a way as to reduce the intrinsic-spin contribution to the magnetic moment. The experimental magnetic moment of Cs T M is +3.48 n.m. 22). Table 4 gives TABLE 4 Theoretical magnetic m o m e n t ), = 30 ° fl = 0.2 fl = 0.3
2.35 2.88
2, = 37.5 ° 2.66 2.86
), = 45 ° 2.79 2.74
~, = 38 ° 2.70 2.85
our theoretical moment at several values of fl and y near the optimum (0.28, 38 °) in units of the nuclear magneton. The interpolated moment for fl = 0.28, y = 38 °
6'78
L. W . PERSON AND J. O. RASMUSSEN
is 2.82 (unquenched value is 3.14). For the purpose o f comparison, the spherical shell model with quenched g factor gives # = 4.0 for d t (unquenched, 4.793) and # = 1.68 for (gi÷)j-_-3t (unquenched, 1.225). Nilsson's model 8, 9) for a pure gt proton with projection f2 = ~ gives # = 1.49 (unquenched, 1.18). The agreement is not good for all the models and the experimental magnetic m o m e n t lies in between that of asymmetric-rotor and single-particle models. It is interesting to note that the magnetic moments of the spin ~ nuclei 531125 and 531127(__+2.6 nm and + 2.7935 nm, respectively) are close to the values of table 4, suggesting the possible applicability of this ellipsoidal model to these nuclei. We have used our formulas of subsect. 2.3 to calculate reduced E2 transition probabilities for a number of transitions. In most cases the experimental data are 200
,
,
,
,oo ~._~,o~O~ (c,) ÷
- - ~
~o
~~ /
.
.
.
.
_
/i \ \Experimentol
I
•
Ioea
B,p
~,,
",1
5
2
0.8
I 15
I 30 Z
45
~
I
60
(deg)
Fig. 8. C o m p a r i s o n o f the experimental B(E2½+ ~ t + ) value with the theoretical values as a function o f 7, with fl = 0.3.
at present not sufficient to test the calculations carefully, since M1-E2 mixing ratios are not well
ROTATIONAL
ENERGIES
679
experimental value by about a factor of 1.5. In retrospect it would have been more realistic to use an inertial parameter B considerably larger than the irrotational value. Since we used such a small B, we were forced to a deformation value of fl = 0.28 in order to fit the energy spectrum. Certainly the experience from the spheroidal nuclear region would lead us to believe that a fl value about half of this is more realistic for Cs 131. Such a modification of our calculations away from the hydrodynamic moments of inertia would lower the predicted B(E2) value, since they vary roughly as f12. With a lower fl the strong gt admixture of table 2 would be diminished. We find a serious discrepancy in trying to compare experimental and theoretical transition rates for the 133-keV transition presumed to be ~-+ to ~ + and predominantly M1. Its lifetime is 13.3 x 10 -9 sec (ref. 20)) and, allowing for a conversion 100
÷
ea
I
50~
I
4
20 D(e%m'• ;
k¢)
"-t,(M uJ Q3 e4
I
--]
', V ,'
/
%% / ~ J
t
I0
I
Bsp
Experiment j_j_. value r l / 2 1
5 -- ~
2 I
0
I
15
I
30
I
45
I
60
/
7 (deg) Fig. 9. C o m p a r i s o n o f the experimental B ( E 2 } + ~ ~q-) values with the theoretical values as a function o f y, with fl = 0.3.
coefficient of ~ 0.5, this given an upper limit B(E2) < 7.0 x 10 -50 e 2 • cm 4. Our theoretical calculation (fig. 9) for B(E2) is 3.4 times this limit. I f future experimental work firmly establishes the position and spin associated with the 13.3 x 10 -9 sec state according to the proposal of Bodenstedt et al. 2o), then it will be clear that we should not have fitted that state as the 7 + number of the ground rotational family. It may be that the 13.3 nsec state belongs to a different intrinsic Newton proton state than the ground state. We have also calculated the relative B(E2) values for the three transitions each from the first two excited states of spin ½ (i.e., at 216 keV and 620 keV). The ratios to the ground transition are shown in figs. 10 and 11. Points at 0 ° and 60 ° are calculated in the spheroidal limits through squares of Clebsch-Gordan coefficients with K = ½ for I equal to ½, i , ~ on the prolate side; and K = ½ for Ii; K = ~r for I i, I i on the oblate side. Experimental data on M1-E2 mixing ratios are presently insufficient to test any of these calculations. They are presented here in part to illustrate the drastic differences between ½+ and ½' + states, whereby there appears clearly an approximate
680
L.W. IO0
PERSON t
1
1
AND 1
1
J. O. R A S M U S S E N 1
1
t
l
80 60 40
8(E2 3/2+-~ I/2÷)
io B
6A 4
0 o C~ L~J a3
2 lO
-
\
-
0.8 06
\ B{E2 3 / 2 + ~ 7 / 2 + )
8(E2 3/2÷-~ 512~-)
\ L o,~ 7,5
225 7" ( d e g )
Fig. 10. T h e o r e t i c a l ratios B ( E 2 ] + -+ ½ + ) / ( B ( E 2 ~ + -+ ~ ] + ) a n d B(E2]-+- -~ ~ - k ) / B ( E 2 ~ + as a f u n c t i o n o f 7, w i t h fl = 0.3. I
I
I
I
I
I
-~ ~ - k )
I
7
~ /'
6
o
5
4 oJ Ld
3
-,, B(E2 3/2"+'-,i / 2 ~ . ) / /
iI
o
2
I
/
\
k,%(E2 3/2'+-~ 7/2+)\\
i
B(E2 3/2'+--~5/2+}
/
~L___.~
/ /
c~c ~
_1
I 7.5
I I I I I 22.5 30 3z5 45 52.5 )" ( d e g ) Fig. I 1. T h e l o g a r i t h m i c theoretical ratios B(E2~-'+ -+ ½ + ) / B ( E 2 ~ ' + --* ~ + ) a n d B ( E 2 ~ t ' + -+ 7 z + ) / B(E2~-'÷ -+ ~ + ) as a f u n c t i o n o f 7, w i t h fl = 0.3. 0
I I5
ROTATIONAL ENERGIES
681
selection rule from the limiting model of y vibrations of a spheroid (see Appendix). In this latter model the ~r + and ½ + states have no phonons of gamma vibration, the ½+ and ½+ have one phonon, and the ½+ ' has two phonons. This can be seen from the top curve of fig. I 1. 3.3. G R O U N D
STATE SPIN
We have determined the lowest state for a variety of fl and 7 values for nucleon numbers of 51, 53, 55 and 57; and have plotted the deformation regions where a given spin is lowest 22, 24, 25). This map is in polar coordinates, where fl is the radial coordinate and 7 is the angular coordinate. All shapes of quadrupole deformation y (deg) Z'J) 20
~0/ ~, ~, 5 o
T (deg) tO
' i/z.
~
~,0 ~0
£ o.3
0
oo
o.3
-Q/
"~
Fig. 12. Plot of the ground-state spin of the nucleus with 51 protons, as a function offl, y; with BflZ/h 2 = 9.41 MeV -1 and hoJ o = 8.16 MeV.
°1
Fig. 13. Plot o f the ground-state spin of the nucleus with 55 protons, as a function o f f l , y; with Bfl2/~ 2 = 9.41 MeV -1 and ¥~to0 = 8.16 MeV. A pair of p r o t o n s are assumed in the h~k orbital. If one assumes no p r o t o n s to be in the 5th oscillator shell, the above diagram applies to atomic n u m b e r 53.
7"(deg)
f (deg)
~oz9
2,0 !0
~P
o
FO.I
Fig. 14. Plot of the ground-state spin o f the nucleus with 5 5 + p protons, as a function of fl, y; B~2/h 2 = 9.41 MeV -1 and //co0 = 8.16 MeV. The quantity p is the n u m b e r of p r o t o n pairs in the fifth oscillator shell.
~,0
50 ,
~
20
I0
,
,
0
'0./
Fig. 15. Plot o f the ground-state spin of the nucleus with 5 7 + p protons, as a function o f fl, y; with Bfl2/li 2 = 9.41 MeV -1 and ~to 0 = 8.16 MeV. The quantity p is the n u m b e r o f p r o t o n pairs in the fifth oscillator shell.
~ 8 "2
L. W .
P E R S O N A N D J. O. RASMUSSEN
are of course represented within the 60 ° sector. Fig. 12 represents the 51st level of the proton, fig. 13 the 53rd level, fig. 14 the 55th level, and fig. 15 the 57th level, if one assumes no protons in the 5th oscillator shell (h~). For the larger fl values one expects the occupation of h~ to begin even before atomic number 55. The lowest spin state along 7 = 0 in these four figures should correspond with the lowest Nilsson state for that proton number from the diagram of fig. 3 of Mottelson and Nilsson 9), except that the h~k levels are missing. One sees from all figures except fig. 12 that our intuition about the stabilization of spin ½ states in asymmetric wells is borne out by the calculations. Note that at the Cs laa reference point (fl = 0.28, y = 38 °) on fig. 13 the ~ + spin is lowest; for the Cs 127 and Cs ~29 only a slight shift toward a smaller y value brings spin ½ lowest. Some experimental knowledge of excited states of Cs 127 and Cs 129 would be most desirable, to test between predictions of the spheroidal and the asymmetric nuclear models. 3.4. MAGNETIC MOMENTS OF SPIN ½ NUCLEI There is one region in which the approximate calculation of magnetic moments for spin ½ nuclei is especially simple. This is the region of neutron number 65 to 81, where the g~ and d~ orbitals are presumably filled, and even-partiy states will generally consist of configurations involving the close-lying d~ and s~ orbitals. An examination of the Newton eigenfunctions of the top two states in the fourth oscillator shell for y < 30 ° shows that these states consists mainly of s~-d~ admixture in comparable amounts. Likewise, Nilsson's highest f2 = ½ state number 51 in his two sets of calculations shows the same admixture s). The relative mixing ratios do not depend strongly on deformation. I f a Nilsson-type calculation is made with only degenerate s½ and d~ orbitals considered, and if the radial matrix elements of r E between 3s and 2d and between 2d and itself are considered equal (Nilsson's eqs. (1 la) and (1 lb) have them differing by only 4 ~o)8), then the eigenfunction of the top t2 = ½ state for prolate (next to the top for oblate) has as
=
as
=
independent of deformation. The predicted moment is two-thirds of the simple st nucleon value plus one-third of the moment resulting from a d~ nucleon coupled to a core angular m o m e n t u m of 2 to a resultant ½. The magnetic moment of the d~ plus phonon coupled to ½ is /~ = g R - - 9 s . I f we use the quenched 9s factor of - 2 . 4 for an odd neutron and a 0R factor for collective motion of about 0.4, the magnetic moment of a pure s½ neutron state is about - 1.2, and the moment of the d~ plus phonon is about zero. Thus, the nucleus with an odd neutron in this state should have a magnetic moment of about - 0 . 8 n.m. Likewise, the second from the top ~ = ½ state in the prolate spheroid (top on
ROTATIONAL ENERGIES
683
oblate) is one-third of sl character and two-thirds of dj plus phonon. An odd neutron in this state should give rise to a magnetic moment of - 0 . 4 n.m. Intuitively, we would expect that models with a stable nonaxial deformation or with quadrupole oscillations about a spherical equilibrium shape might give rise to intermediate theoretical values for the odd neutron in either state. TABLE 5 Magnetic m o m e n t values o f certain spin ½ nuclei
Isotope
Experimental /t (n.m.)
,aCd~]1 ,aCd~[S ~oSn~9 5~Te~3
--0.59 --0.62 -- 0.73
5~Te~ 5
-- 0.88
-
54Xe~9
-
1.04
-
0.77
Table 5 lists nuclear moments of those spin ½ nuclei to which the model might possibly apply. The tin isotope in is obvious disagreement, but the other isotopes are consistent with some sort of quadrupole coupling-mixed s-d model, as discussed above. A reduction in the collective gR factor for tin, corresponding to the proton closed shell, could explain its deviation in terms of the coupling scheme here proposed. The magnetic moment of spin ½ nuclei does not give much help in deciding between the nonaxial model and other quadrupole deformation models. We shall examine Newton's 6) eigenfunctions for the three highest states in the fourth oscillator shell in order to be able to understand more quantitatively the predictions of his model. If much dt- ~ admixture is involved in the wave function, there will be serious shifts in the calculated moment away from the simple formula. At fl = 0.1 and 7 = 30 °, for example, the Newton wave function with the most s½ character is actually 28 % sl, 55 ~ d~, 8.3 % dt, and 9 % other. It has the second from the highest eigenvalue. We calculate a magnetic moment of about - 0 . 8 n.m., but here the d~-d t crossterm (calculated with eq. (A.I[.5) of ref. z6)) contributes almost half the value. Therefore, the basis of the previously derived simple approximate formula (/~ = - 1.2 x fraction %), based on s½-d~ admixture, is clearly violated, and may be trusted only at deformation parameters well below fl = 0.I for asymmetric shapes. For prolate (7 = 0) symmetric shapes the validity goes to somewhat larger ft. That is, for 7 = 0, fl = 0.1 we have 62 % st, 3 1 % d~, and 3.8 % d~; and the d~-d I cross term contributes only - 0 . 1 7 n.m shift to the magnetic moment. 4.
Conclusions
We have attempted serious and detailed testing of the fixed asymmetric-rotor model for odd-mass nuclei. Certainly the large E2 transition probability in odd-mass nuclei somewhat removed from closed shells yet not within the regions of spheroidal
684
L. W. PERSON AND J. O. RASMUSSEN
nuclei are strong indications of collective motion. The availability of nuclear eigenfunctions for an asymmetric well as calculated by Newton 6) make various calculations feasible. Our attempt to fit all the well-established levels of Cs 13a as a single rotational band gives encouragement but is inconclusive. Better experimental information is much needed. Also, our difficulties with the E2 transition probability of the 133-keV transition suggests that the theoretical calculations need to be repeated to include perhaps two or more Newton intrinsic states with rotation-particle coupling, a coupling we have completely ignored for the sake of simplifying calculations. The magnetic moments of spin ½ nuclei ranging from 65 to 81 neutrons agree with predictions of a model involving coupling to a core angular momentum, but do not distinguish to any degree the core shape or magnitude of deformation. Successful results with this model do not necessarily prove that the nucleus literally has fixed asymmetric deformation. We show in the Appendix a spheroid with Y vibrations gives quite similar results at small ~ values. The asymmetric rotor model does provide a prescription for calculations and a point of attack on nuclei in regions not yet very amenable to theoretical interpretation. Appendix RELATIONSHIP
OF A SYMMETRIC R O T O R P L U S ~' V I B R A T I O N ASYMMETRIC ROTOR
WITH
A SLIGHTLY
It is evident that at 7 = 0 and 7 = ½n, the first rotational band has the same structure as that of a spheroid; and the energy values of the higher rotational band become infinite, with hydrodynamic moments of inertia. However, in the cases of = l ~ n or {4rc the computed spectra of the ellipsoidally deformed nuclei (see Appendix B of ref. 13)) show the general features of the rotational-7-vibrational spectra of a spheroid. In this section we shall see that in these limits it is possible to reproduce the properties of the ground, second, and third rotational bands of an ellipsoidal nucleus by considering the y-vibration-rotation interaction between rotational bands, and the one- and two-phonon ~-vibrational bands of a spheroidal nucleus. Since it is impossible to write a simple expression for ellipsoidal nuclei in the general case, one specific example will be explained in detail; the other cases are quite analogous. The following illustration is for an intrinsic state with projection 12 = ~}: I .~
~z -~+
K ~ }
n 2 2
:~ .~
+ +
½ ½
l J
+ +
~ ~
o o
ROTATIONAL ENERGIES
685
The Hamiltonian oYF' for 7-vibration-rotation interaction may be derived from the last term of eq. (11) when ~ is small 27)
•~ ' =
h2(I2-IZ)Y
(A.1)
In the y-asymmetrical model the energies of the bands are determined by the fl and values. In axially symmetric nuclei the energies of the rotational and the vibrational bands depend upon the phonon energy and the moment of inertia. If the adiabatic approximation holds and if the y-vibration phonon energy [hco = (Cr/flr) ~] and the moment of inertia (5 = 3Brfl 2) are the same for all the bands, the energy of the nth highest energy state with spin I is expressed as h2
E,(I) = ~ {[I(I+l)-K2]-(-)'-s(J+½)(I+½)6r~t6~½}+Qhto,
(A.2)
where
for, for 1 = ½, ] for the first three bands. Since the mixing amplitudes and the B(E2) values for different bands are proportional to the term (By Crfl4) j, which is approximately equal to (½hog~), we define an energydifference ratio P _ E2(~)- E,(~)
(A.3)
'
which is related to the (B~Crfl4)~ in a linear way, and we shall express the mixing amplitudes and the B(E2) ratio in terms of p by second-order perturbation theory. For comparison, the values of fl and the values of the mixing amplitudes a(~, ~; ~}, ½),
a(~, 3; ~:, ½),
and the ratio B[E2~(2) -, ½(1)] B[E2~(2) ~ ~2(1)] are given in table 6. The notation a(I, K; I, K') signifies the mixing amplitude of the TABLE 6 C o m p a r i s o n o f mixing a m p l i t u d e and B(E2) ratio for ellipsoidal a n d spheroidal models a(~, ~-; t , ½)
0.1 0.2 0.3
a(], ~; ~,, ½)
B[E2~(2) -~ ½(1)] B[E2~(2) --~ ~(1)1
spheroidal
ellipsoidal
spheroidal
ellipsoidal
spheroidal
etlipsoidal
0.00278 0.00285 0.00318
0.00276 0.00226 0.00304
--0.00238 --0.00250 --0.00273
--0.00049 -- 0.00046 --0.00049
2.93 × 10 a 2.50 × 10 a 1.87 × l09
1.59 × 107 4.82 × 10 ~ 1.75 × l0 T
686
L.W.
PERSON A N D J. O. RASMUSSEN
a n g u l a r m o m e n t u m projection K ' in the state o f spin I a n d p r e d o m i n a n t c o m p o n e n t K. The p values for the spheroidal model are calculated using the results i n A p p e n d i x B o f ref. 13) for ~ = 52.5 ° . F r o m table 6 we see that b o t h models yield similar results, a n d that the B(E2) ratio shows the effect of the f o r b i d d e n t r a n s i t i o n o f An = 2, a l t h o u g h the m a g n i t u d e s are n o t equal. W e believe that as 7 goes to smaller values the two models will a p p r o a c h identical predictions. Note that in the B(E2) ratio calculation the d e n o m i n a t o r c o n t a i n s two terms o f opposite sign a n d nearly equal values, which will cause the B(E2) ratio to be very sensitive to the a values. W e wish to express again our gratitude to Dr. T. D. N e w t o n for p e r f o r m i n g a special set of N = 4 e i g e n f u n c t i o n calculations at our request. O n e o f us (J. O. R.) wishes to acknowledge fellowship s u p p o r t of the N a t i o n a l Science F o u n d a t i o n , a n d the hospitality of the Institute for Theoretical Physics i n C o p e n h a g e n d u r i n g the writing of this paper. The calculational work was d o n e u n d e r the auspices of the U.S. A t o m i c Energy C o m m i s s i o n .
References 1) A. S. Davydov and G. F. Filippov, Nuclear Physics 8 (1957) 237 2) K. T. Hecht and G. R. Satchler, Nuclear Physics 32 (1962) 286; K. T. Hecht, Bull. Amer. Phys. Soc. 6 (1961) 78 3) S. Watanabe, Ph. D. Thesis, University of Washington (Seattle, 1961) 4) G. F. Filippov, Soviet Physics (JETP) 11 (1960) 949 5) T. D. Newton, Can. J. Phys. 38 (1960) 700 6) T. D. Newton, Atomic Energy of Canada Limited, Chalk River, Ontario, Report, CRT-886, January 1960, (unpublished) 7) Aage Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1952) 8) Sven G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) 9) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skrifter Dan. Vid. Selsk. 1, No. 8 (1958) 10) R. J. Blin-Stoyle, Theories of nuclear moments (Oxford University Press, London, 1957) 11) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (John Wiley and Sons, Inc., New York, 1952), p. 595 12) A. S. Davydov and G. F. Filippov, Soviet Physics (JETP) 8 (1960) 305 13) L. W. Person, Ph.D. Thesis, Lawrence Radiation Laboratory Report UCRL-9753, July 1961 (unpublished) 14) B. R. Moore and W. White, Can. J. Phys. 38 (1960) 1149 15) A. S. Davydov and V. S. Rostovsky, Nuclear Physics 12 (1959) 58 16) W. C. Beggs, B. L. Robinson and R. W. Fink, Phys. Rev. 101 (1956) 149 17) J. M. Cook, J. M. LeBlanc, W. H. Nester and M. K. Brice, Phys. Rev. 91 (1953)76 18) B. S. D~helepov and L. K. Peker, Decay schemes of radioactive nuclei (Academy of Sciences of the USSR Press, Moscow, 1958) 19) P. P. Day, E. D. Klema and C. A. Mallmann, Argonne National Laboratory Report ANL-6220, November 1960 (unpublished) 20) E. Bodenstedt, H. J. K6rner, C. Giinter, D. Honestadt and J. Radeloff, Nuclear Physics 20 (1960) 557 21) J. O. Rasmussen and L. W. Chiao, in Proc. Intern. Conf. on Nuclear Structure, Kingston, Canada (University of Toronto Press, Toronto, 1960) p. 646
ROTATIONAL ENERGIES
687
22) E. Bellamy and K. Smith, Phil. Mag. 44 (1953) 33 23) L. A. Sliv and I. M. Band, Coetticients of internal conversion of gamma radiation (Academy of Sciences of the USSR Press, Moscow, 1958) 24) J. E. Mack, Revs. Mod. Phys. 22 (1950) 64 25) W. A. Nierenberg, H. A. Shugart, H. B. Silsbee and R. J. Sunderland, Phys. Rev. 104 (1956) 1380 26) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) 27) T. Tamura and F. Udagawa, Nuclear Physics 16 (1960) 460
] I
Nuclear Ph~tsics 3 6 (196"2) 6 6 6 - - 6 8 7 ; (~) North-Holland Publishing Co., .4 msterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ROTATIONAL ENERGIES FOR THE A S Y M M E T R I C ROTOR M O D E L FOR O D D - M A S S NUCLEI LUCY WU PERSON
Department of Chemistry and Lawrence Radiation Laboratory, University of California, Berkeley, California and J O H N O. R A S M U S S E N t
Institute for Theoretical Physics, University of Copenhagen, Denmark Received 3 M a y 1962 Abstract: W e s t u d y the effect o f ellipsoidal nuclear d e f o r m a t i o n in o d d - m a s s - n u c l e a r rotational b a n d structure, the magnetic m o m e n t , a n d electric q u a d r u p o l e reduced transition probabilities. Also, we s t u d y the relationship between the rotational b a n d s o f a n ellipsoidaUy deformed nucleus a n d the vibrational a n d rotational b a n d s o f a spheroidally d e f o r m e d nucleus with ),-vibrationrotation interaction in the limit o f ~' a p p r o a c h i n g 0 or ½~. E q u a t i o n s for the a s y m m e t r i c r o t o r m o t i o n are derived. By using T.D. N e w t o n ' s single-particle eigenvalues a n d eigenvectors, we then present numerical calculations s h o w i n g rotational spectra associated with an o d d nucleon in a n ellipsoidal well. The calculations for the N = 4 a n d N = 2 shells were done o n a n IBM 709 c o m p u t e r . N u m e r i c a l results are discussed in t e r m s o f the fl a n d ~' d e f o r m a t i o n parameters required to give the k n o w n spins o f the o d d - m a s s cesium isotopes Cs ~7 to Cs ~aT. T h e rotational energy spectrum, magnetic m o m e n t o f g r o u n d state, a n d various E2 transition probabilities are calculated for Cs ~3~ for several deformations, w i t h best energy s p e c t r u m fit at fl = 0.28, ~, = 38 deg. There are difficulties with the transition probabilities a n d magnetic m o m e n t .
1. Introduction The Davydov-Filippov model of an ellipsoidally deformed nucleus with three unequal principal axes has been applied extensively to even nuclei 1). For small values of the asymmetry parameter y the rotational spectra correspond closely to those of the symmetric rotor with y-vibrational excitations added. Perhaps the greatest utility of the model has been in the regions of nuclei outside the regions of definite spheroidal deformation, where the energy of the second excited 2 4- state may be only about twice the energy of the first excited state. We felt that it would be interesting to examine the model for odd-mass nuclei it. At a late stage in our calculation we learned of similar work 2) by Hecht and Satchler, who treat the rotational energies of asymmetric odd-mass nuclei with essentially similar results, except that their results are for the spectra of nuclei with A around 190. Watanabe and Blair have also treated the problem, with applications to d-shell t On sabbatical leave from the University o f California, Berkeley, California. tt Based on a Ph. D. thesis (by L.W.P.), University o f California, Berkeley, California (see ref. ~s)). 666
L. W. PERSON AND J. O. RASMUSSEN
667
nuclei 3). Filippov 4) has recently made calculations and general examinations of the problem of stability of the asymmetric nuclear shape. From his results it appears that noncylindrical shapes could possibly be of lowest energy in some cases, where one kind of nucleon has nearly completed a shell closure and the other kind is just beyond a closed shell. The region of the neutron-deficient cesium isotopes possibly satisfies these requirements. It is instructive to consider a trivial extension of the Davydov-Filippov model to odd-mass nuclei, namely, the case where the odd nucleon is in a pure J = ½ state, and hence completely uncoupled from the rotor. In such a case we have just the spectrum of an even nucleus, but with a ground state spin of ½ and all excited levels doubly degenerate, with spins differing by + ½ from the spin in the even nucleus. We see that in the rotational band there will be only one state with I = ½, two with I = 3, three with ~, and generally I + ½ states of a given spin L The sequence of ground-state spins of the odd-mass cesium isotopes qualitatively suggests a possible explanation fia terms of an ellipsoidal deformation which sets in as the neutron number departs sufficiently from 82. The measured spins for neutron numbers 72, 74, 76, 78, 80, 82 are, respectively, ½, ½, ~, {, }, {. The expected g; spherical shell model spin appears near the closed shell. Ellipsoidal deformation of the nuclear potential would tend to quench the orbital angular momentum of an odd particle so that, for sufficiently large deformation, spin ½ should lie lowest. The quantitative testing of the model involves considerable mathematical complication. Fortunately, nucleon eigenfunctions and eigenvalues for an eUipsoidal harmonicoscillator potential had been calculated and tabulated by T. D. Newton ~' 6) through the fourth oscillator shell. The Hamiltonian he used was more appropriate for N = 4. neutrons than for protons, and we are deeply indebted to him for recalculating, at our request, the fourth oscillator shell, using a proton parameter. With his eigenfunctions we have attempted to calculate rotational spectra of even parity for some odd-proton nuclei in the 50 to 82 shell. 2. Theory of Odd-Mass Nuclei with Fixed ElUpsoidal Deformation The Hamiltonian of the coupled system of nucleon and asymmetrical rotor core is as follows: = J~ap -~t-~ i n ,
-']- T R ,
(1)
where the terms on the right-hand side are as defined later. The rotational energy of the general rigid rotor may be expressed as 3
TR = i y ~ R 2~
-,
,
(2)
!¢=1
where R~ denotes the components of angular momentum along the principal axes. We assume, in accordance with the hydrodynamic model 7), that ~
= 4Bfl 2 sin 2 (?-x~zt),
(3)
668
L . W . PERSON AND J. O. RASMUSSEN
where fl and ~ are the usual parameters specifying a general ellipsoidal deformation, and B is the inertial parameter for quadrupole surface oscillations. The single-particle Hamiltonian°~d'p is given by ~ p ~-- Tp+
Vp(r)+C I. s+Dl 2,
(4)
where h2
= -½
V
(S)
Vp = ½M((0~ X~ + (02 X~ + (03 X32)•
(6)
Here M is the single-particle effective mass of a nucleon in the nucleus, X 1 , Xz, )(3 are the Cartesian coordinates of the nucleon in the body fixed system, hcol, h(0z, h(03 are the corresponding quantum energies along the three principal axes, satisfying the conservation of nuclear volume conditions (7)
(DI (02 (03 = ( 0 3 ,
while I is the infinitesimal pseudo-rotation operator, and C is a spin-orbit potential strength parameter. The V 2 operator in eq. (5) is evaluated in the body-fixed coordinate system. The D l 2 term in (4) is a correction which depresses high-angular-momentum orbitals, equivalent to the effect of a nuclear potential more square than pure harmonic, The term ~ i . t represents the interaction of the particle with the nuclear deformation. However, in Newton's 6) calculation for ellipsoidal nuclei the energy eigenvalues E s include both the single-particle energy and the interaction energy. We shall not consider nuclear shape vibrations but consider the shape fixed. In the case of non-axial nuclei, if the moments of inertia are sufficiently large, the particle motion will follow nearly adiabatically the rotations of the well, and the wave function will be a linear combination of Nilsson's wave function "s' 9) in the following way: ~,M = Ii'2I ~167z2 + 1'I]
~ btr3 ~ 13
J J3
~jj3(DI3M(Oi)~/NIJJ3.~_(__),
i-ZD_t3M(Oi)~stz_s3),,
(8)
where ~PNzJS3is as defined by Newton's 6 ) eq. (3); the summation o v e r J3 runs only over alternate half-integr:~', values between J and - J ; and 13 is summed between I a n d - / o n alternate values such that 13 --J3 = 2v. Here v = 0, _+ 1, ___2,..., ___N, which is due to the symmetry requirement of invariance with respect to a rotation through g about the 3 axis. By convention, b1½ and ~¢s~ are always taken with a positive sign. The ~'sJ3 are the mixing amplitudes which are the eigenvectors of Newton's table 6), and b i i 3 are the mixing amplitudes of the rotational states with different 13 values and the same L Therefore bH3 are the eigenvectors of the rotational spectra that we calculate. For axial nuclei (7 ~ 1N1r), I3 and J3 will be approximately constant, and their eigenvalues will be K a n d I2. The adiabatic-rotation assumption may not be
ROTATIONALENERGIES
(169
a good one for the cesium isotopes we later treat, since rotational energy spacings are not much smaller than the single-particle-level spacing. We have neglected the effect of the coriolis interaction in mixing different Newton eigenfunctions.
2.1. ROTATIONAL ENERGY 1N THE STRONG-COUPLING APPROXIMATION The rotational states may have energies much smaller than the single-particle energy and the phonon energy, so in eq. (1) we only have to consider the rotational energy term TR = ½ ~ R ~2 ~ - 1. I f we use the representation in which 12, 13, j2 and J3 are constants of motion, then we write TR in the following form given by BohrV): TR = TDRq- TOR-
(9)
Here TDR'is the diagonal term which is the same as the rotational-energy term in the nucleus with axial asymmetry, excluding the case of d 3 = ½:
TDR = (h24~1+ ~2h2) [ I ( I + l ) - I 2 + j ( J + l ) - J ] ] +
~h2( I 3 - J 3 ) z.
(10)
Further, TOR are the off-diagonal terms that vanish in case of axial symmetry, except for J3 = ½: ( hz To. = -
h2 11 S l +
) I2 J2
+
The nonvanishing matrix elements of eq. (11) will be governed by the following selection rule: The first term will connect states of AK = _+ 1 and A~2 = _ 1, and the second and the third terms will connect states of AK = 0 and A• = _+2, or AK = + 2 and A~2 = 0. Substitution of eq. (8) into the rotational-energy equation TR ku = Erot ~ gives a secular determinant of dimension I + ½ by I + ½ to be solved to determine the eigenvalues and the coefficients bm. All three terms in TDR contribute to the diagonal matrix element. O f the three terms in TOR, the second contributes a constant amount to the diagonal matrix element, and can - for our purposes - be ignored. The first and third terms provide the off-diagonal matrix elements f o r ~ 3 ,t3_1 and ~ l ~ , i~+2. Therefore the secular matrix is separable with only even values of 13 - I ~ coupled. The general formula for diagonal elements of rotational energy is ~F13,3 = h2 + 2~ 3-
~
+ ~
{I(I+1)-I2+ ~ Idss3i2[J(J+l)-J]]} ss3
./s3ZIdss3[Z(13-J3) z
(12)
670
L, w. PERSON AND J. O. RASMUSSEN
(h2 h2) E (-) '-' d~ds+½)
+-(-)&~.~(l+½) ~ 1 + ~
( hh2~ ) 4 ~~1
~ , ,'-~
~
d
d [(JWJ3)( J - J 3 + l ) ]
½ (12)
_(h2__ h%](i+½ ) E (-)'-%,,,v~_,,~+,,d.,~.~ \4~1
4~2]
dJ3J'S'3
× [(J -- d3)( J + J3 + 1)] ~. Due to the symmetry of the wave function, the other two matrix elements give the same numerical result, and so are not written out. The general formula for off-diagonal elements of ToR is J4g'~'13(13- 1) = -- 313 -(I'3 - 1 ) [ ( I + •3)(• -- 13 + 1)] ~ t(]12 __ h 2 ~ E ( × [\,4~,~1 4 ~ 2] JJ3J'J'3
'"~I-J(~Jd'VJ'3-(J3 +l)~dJ'3~JS3[(J-J3)(J-}-J3+l)]{ A
(13)
and
~'~'~-+~
(h2
-
8~1
]721
[(I-T-I3)(I-T-I3-1)(I+I3+l)(I+-I3+21] ½"
(14)
8,~J
Due to the symmetry of the wave function, the other two matrix elements give the same numerical result as eqs. (13) and (14), and so are not written out. 2.2. MAGNETIC MOMENT The m a g n e t i c - m o m e n t operator is t% -
e
2Mc
10)
[~_,(g, li+gss,)+gR, Rll+gR2R22+gR~R33].
(15)
I f we assume that the three collective g factors are equal to a value gR, then we m a y use Nilsson's 8) expression (20):
1 /~ ---- -[-(g~-- gl)(s " I ) + (g,-- g R ) ( J " I ) + gR I(I + 1)], I+1
(16)
where
13JJ3 x { b , , 3 b,_ (,, -1 )( -- )J - 4x~dJ, ~ J - (J, - 1)(½)
xE(I+la)(I-I3+l)(J+Ja)(J-J3+l)] ~
+ b,. b,_.~+,)(-)'-~C.~j_(~+,)(½) x [ ( I - 13)(1+ I3 + 1 ) ( J - Ja)(J + J3 + 1)]~'} -
(17)
ROTATIONAL ENERGIES
671
For the calculation of the matrix ( s . I) it is more convenient to work in the representation of (f2]NIA~), where Nilsson's expansion coefficient d~la Z is related to ~/ss3 in the following way s): "5~tA~r = E (I½Aff'[JJ3)d:~Js3. J
Then we have (s
I) •
=
~ btt3[~c~t(s3_~)~--.~i[l(s3+½)_~] 2 2 2 • ½I3 I31J3
+ ~ (-)'bmb,_(,~_l)d,(a~_~)~Ci_(a~_~)~" ½[(I+I3)(I-I3+1)] ~
(18)
13|J3
+ ~ (-)lb,,3b,-(,3+x)dua3+~)-½~c,-(a3+~)-~" ½[(I-I3)(I+I3+1)] ½. I31J3
2.3. ELECTR1C-QUADRUPOLE REDUCED TRANSITION PROBABILITIES The transition probability for electric-quadrupole radiation is given by 11)
T(E2;Ii-~If)= 4~ ( he E ) 5 B(E2;I, --* If),
(19)
where B(E2; Ii ~ If) is the reduced transition probability between rotational states I i and If. It can be written as 5 B(E2; Ii -~ If) - 16~(2Ii + 1) ~ f (fIQ2u[i)2"
(20)
By using the model of a nucleus as an incompressible classical liquid drop with a uniform charge distribution, the nuclear quadrupole-moment operator Qz~, can be written as 12)
Q2~, = eQo[D2o cos ?+(D22+D2_2) sin 7 (2)-~],
(21)
where Qo is the intrinsic quadrupole moment of an axial nucleus and is related to the deformation parameter/3 by Qo = 3ZR2fl(5~z) -~. After simple algebraic manipulation, and assuming that there is no change in the internal state of the nucleus in the transition, the reduced transition probability can be expressed in terms of the average value of fl and ? as 12) B(E2; I~ ~ If) --- 5 e2Q2] ~ bl~r3bitra(COS y(ii2130[ifi,a) 16g I~r~ + [sin ? (2)-~](Ii 213 ± 2llflj, )[2. (22~ 2.4. E L E C T R I C - Q U A D R U P O L E
MOMENT
The formula for the spectroscopic electric quadrupole moment can readily be derived by a specialization of the formulas of subsect. 2.3; that is Qsp,c = -1 (Q2o)n,=1, e
(23)
672
L.W.
PERSON AND
J. O. R A S M U S S E N
where Q2o = Qo{D2o cos 7 +(D22 +Do2_ 2)[sin y (2)-½]}.
(24)
Substituting eq. (24) into eq. (23), we get Qspec = Qo(I2IOI II) ~ bu~bu'3[ cos y(12130[II3)~,'3t3 131"3
+ [sin ~, (2)-~]((I213 21113 + 2)6r3h + 2 + (I213 - 21113 - 2)6r~13- 2)] -
Qo (1+1)(21+3)
{cos ~, ~ bu31313 ~ - I(I + 1)] ,3
+ sin ~, (k)~ E b,t3 b,,,3([(1 + 13 + 1)(I + I3 + 2)(• - 13 - 1)(I - I3)]~6v3,3 +2 (25) I31'3
+ [ ( I - 13 + 1)(I-- 13 + 2)(• + 13 -- 1)(I + 13)]*(~I,313 _ 2)}. 3. Numerical Studies and Comparison with Data We have used Newton's eigenfunctions and eigenvectors 5,6) for our calculations of rotational spectra; other nuclear properties were calculated by using the formulas of sect. 2. 3.1. ROTATIONAL SPECTRA In the low-lying rotational-energy calculation, we assumed that the odd-nucleon state of motion in the nuclear well is not changed for different states of rotational motion. We also neglected possible vibrational effects and the collective rotationvibration interaction. The rotational energy ER and its associated eigenvector bH3, which are obtained by diagonalJzation of an (I+½) by ( I + ½) matrix for a state with nuclear spin/, were calculated by means of an IBM 709 digital computer 13). Both the calculated values of the rotational energy ER and the mixing coefficient bu3 fore ach value of the single-particle energy E S = E s - ( N + ~ } ) are tabulated in Appendix B of ref. 13) from 0 to ½n in steps of ~¢~z. By symmetry, the states with deformation parameters fl and y, and the states with deformation parameters - f l and ½~-~ are equivalent. Curves fitted through the calculated values of fl and ), were used for interpolation. Curves showing the total energy E = Es+ER for the lowest single-particle state of the N = 2 shell, 3 = 0.2, B/h 2 = 77.25 MeV-*, are plotted as a function of 7 in fig. 1 for the first rotational band, in fig. 2 for the second rotational band, and in fig. 3 for the third rotational band. For comparison, the rotational energy of even nuclei 14) with the same values of 3 and B are plotted as a function of ~ in fig. 4. From these four figures we can clearly see the analogy between the rotational spectra of an odd nucleon in an ellipsoidally deformed nucleus and the Davydov-Filippov rotational energy spectra of even nuclei 1, xs). This is expressed schematically in the correlation diagram in fig. 5.
l
I
-~--
l
I
i
I
I
I
I
I
I
I
E
I
I 45
I
I
62 60
s9
•
E°(11/2+)
z~
Eo(7/2
(:P f
61
.
E,(9/z+)
~
60
o
E(3/z+)
!I
•
El(5/24")
:I
E(i/a+)
il
58
57~
,& EZ ( 1 1 / 2 ~.) z~ E z l E2 0 Ez lIE z
+)
( 7 1 2 +) ( 9 / 2 ".-) {3/2 *) (512")
59
,1 ,f ,I
i
ta
//
I I /
,;'
54~ r I
t
,i ,J
53-
!
.9
52~
;
5,~-
~-~~
--~,~ _
_ _-o~
5~--.50
.e~--~..o , /
.--~'~-~"~e
o
-- ""
---"cr
._.~.~,
15
~
~ ~-~
z~
-
..l"-'~
30 7
_--~_
"--'~
."
45
60
I
I
I 30
T--
I
I
I
I0
A E3(7/2"+)
59_!
uE3(5/2+
60
I
(6+) (,5+) (4+1 {6+)
IE, (3*)
) 9
~E I (4+) eE 2 (2+)
60
OEm (2+)
8
~
s7 Lid
I
• E2 o£ a A£ z ~.E I
I1
eE3(9/2+) 61
I ( de{] )
Fig. 2. Energy d i a g r a m o f the second rotational b a n d for the lowest single-particle state o f the N = 2 shell as a function o f 7.
• E 3 (11 / 2 + )
62
I 15
7
Fig. l. Energy d i a g r a m o f the first rotational b a n d for the lowest single-particle state o f the N = 2 shell as a function o f 7r
I
0
( deg )
7 6
56
'\\ ~
4': 55
•
\ 54
zx
,
'\
/
"e
.
/
/ O
o....._..~
\ram ~
',\\
-
\
\
5.3
I 0
15
"T
7"
I
30
(
deq
415
..... 60
)
Fig. 3. Energy d i a g r a m o f the third rotational b a n d for the lowest single-particle state o f the N = 2 shell as a function o f 7.
OI 0
o-.---- a ..... I I 15
7
°-- "'--° I I :30
I
( deg )
Fig. 4. Rotational-energy d i a g r a m o f an even nucleus ER (in MeV) as a function o f 9).
674
L.
W.
PERSON
AND
.L
O.
RASMUSSEN
As discussed in the Introduction, in the limiting case of a pure J = ½ odd nucleon, the nucleon motion is completely decoupled from the collective surface motion. Also, one gets a system of levels with the same spacing as in an even nucleus but with a doubly degenerate state of spin I_+ ½in place of each level of spin I i n the correspondEnergy levels of on even osymmetrnc rotor
Energy levels of on odd nucleon
I ,Tr
I,TT -
-
Energy levels in the limits of o pure S 1 / 2 odd nucleon
5+-~-
.
.
4+
__
.
.
.
11/24-
I ,7 . . . . .
11/2+
- - 9 / 2 + -
:5+=
-
-
-
-
7/2
..... -
-
-
.
.
.... +-
.
.
.
.
. -
-
-
4+ 2+
7/2 .
.
.
. . . . .
0+
712+- - 5 1 2 + -
-
I
/
.
.
.
-
_
4+
7 / 2 4. 5/2 +
-
.
-
_
512+ 3/2 +
13/2+ 11124-
9/2 +
- -~......
_ _ 5 1
3/2+
2
. -
1312+ . . . . . 1112+----
--9/2+--__ -
.
5•2+ 312+---
-
~ -
2+
9/2
5/2+-
2+~--_
6+
- - 9 /
9/2+ 7/2+
+
-
7/~
+ +
3/~+
-
I/
Fire 5. Relation o f spectrum between an even asymmetric r o t o r and an odd-mass nucleus o f el[ipsoidal shape. i
i
i
J
i
,
,
65
63, 62 >
6o
• A • o • [3
E ( I I I 2 +) E(7/2+) E ( 9 / 2 ÷) E (3/2") E (5/2*) E{I/2*)
~ 60 ¢ w
59 #"
""~.... -
58' 56 55 ~ , . --0---.[3. " "--G----G----~---t:~--'~---~" 0
15
30 )" (deg)
45
60
Fig. 6. Energy d i a ~ ' a m o f a nearly pure s t state as a function o f 7, for the N = 2 shell, with/3 = 0.1 and B/t~== 77.25 MeV -1.
675
ROTATIONAL ENERGIES
ing even nucleus, except that a spin 0 goes to a spin ½ level. T h e existence o f the nearly degenerate states is clearly seen in fig. 6, where the energy is p l o t t e d a g a i n s t 7 for a single-particle state o f a l m o s t p u r e s t. 3.2. NUCLEAR PROPERTIES OF Cs TM O f the light cesium isotopes with g r o u n d - s t a t e spin less t h a n ~-,7 o n l y Cs T M has e n o u g h k n o w n even-partly excited states for significant c o m p a r i s o n . T h e r o t a t i o n a l energy s p e c t r u m is calculated for B f l 2 / h 2 = 13.85 M e V -1 a n d hco = 8.07 MeV. W e chose the N e w t o n p r o t o n state no. 27, c o r r e s p o n d i n g to o u r fig. 13; such choice presumes one p r o t o n p a i r in an h ~ orbital. T h e results are t a b u l a t e d in table 1 together with theoretical spins a n d the experim e n t a l energy values x6, 17). I n t e r p o l a t e d energy values for o p t i m u m p a r a m e t e r s fl = 0.28 a n d ? = 38 ° are given, a l o n g with the energy levels calculated for neighb o u r i n g values o f d e f o r m a t i o n parameters. T h e ~ value used in table 1 is r a t h e r TABLE 1 Energy levels, experimental and theoretical energy values for Cs13x Level label
J I H G F E D C B A
Ee~p b)
Eth~oc)
Spin~the°~) (keV)
½+' j-k-' ~+ ' ~q-' ~ + ~+ ½-t½q~ q~+
1039 620 -373 -216 133 124 -0
(keV)
0.28, 38°
0.3, 30°
0.3, 37.5°
0.3, 45 °
0.2, 37.5°
1022 644 448 380 264 224 136 122 40 0
1682.7 1404.4 722.9 899.7 531.7 780.6 529.2 717.8 330.8 383.5
1106.7 650.7 462.6 452.1 260.2 276.3 196.7 183.8 0 2.61
1873.7 1714.7 684.9 458.8 419.1 377.3 183.9 341.9 304.1 97.4
1224.0 1023.1 822.4 575.2 731.2 466.9 478.9 432.6 606.4 373.9
• ) Underlined spins have been experimentally determined or inferred. b) See ref. xs) for compilation of various determinations of energies. e) Interpolated energy values of this column are mostly subject to an uncertainty of about 8 keV. All values are for Newton state no. 27, with Bfl2/h 2 = 13.85. close to the value required to fit levels o f the n e i g h b o u r i n g even nucleus X e 13° T h e ratios o f r o t a t i o n a l energy o f Xe 13° are 18) E 1 ( 4 + ) - 2.25, E1(2+)
E 1 ( 5 + ) - 4.43, E1(2+)
E l ( 6 + ) -- 3.66, E1(2+)
w h i c h c o r r e s p o n d s to a d e f o r m a t i o n ~ = 35 ° o r 25 ° in D a y a n d M a l l m a n n ' s table 19). T h e decay scheme o f Ba T M in fig. 7 is t a k e n f r o m the recent study b y B o d e n s t e d t et al. 2o). T h e y p r o p o s e a n d j u s t i f y spins o f ½ a n d ~ for excited states at 124 a n d 133 keV, respectively. T h e l o c a t i o n s a n d spins o f these states, together with the spin o f the g r o u n d state, essentially fix the p a r a m e t e r s / ~ a n d 7 in our calculation. T h e energy
676
L.W.
PERSON A N D J. O. RASMUSSEN
match with four additional higher levels test the model, and the agreement is good. The theoretical spins are not inconsistent with the observation of all the gamma transitions in the experimental decay scheme shown in fig. 7, except for one transition; namely, there cannot be a transition JC because of a spin change of three by our assignments. We suggest that there is enough uncertainty in the experimental energies so that the reported 917 keV transition could fit as transition JD. The experimental literature is somewhat contradictory with respect to multipolarity assignments. It is clear that our spin assignments could be tested by multipolarity assignments. Transitions CA, GC, ID, and JE must be pure E2, since ]All = 2. A search for the unobservBo
131
/
/ 1039
/ (rl/2: II5d )
I
1039
824
703 - - 620
I 908
216
487
i
I
I I
416
250
5OO
i
--T 240
....
373
1
216
124
I
~'~/2: ( 1 3 . 3 : 0 5 )
nsec
133
1
O--
/ "el/z = ( 3 . 7 7 *- 0 . 0 5 ) nsec
o
13 Fig.
7.
D e c a y s c h e m e o f B a xSt.
ed higher spin levels predicted here could be valuable. The experimental level at 703 keV does not seem to have a counterpart among the theoretical values. We are also somewhat concerned about the difficulty of populating a 7 + state at 1039 keV, presuming Ba 131 to have a ½+ or 3 + ground state. The reason for not seeing the two z9 + states and higher spin states could be that the high spin itself and the predominance of high-K components in these states inhibit population by fl decay from Ba 131 or by ~, transition from higher energy states. Before presenting the calculations of the ground-state magnetic moment and some relative transition probabilities, we list the eigenfunctions of the states involved, in table 2 the fifteen d jl ~ Newton coefficients used, and in table 3 the three btK values and the four b~K values." We calculated the magnetic moment of the ground state of Cs lal by setting collec -
ROTATIONAL ENERGIES
677
TABLE 2 N e w t o n ' s particle-state-eigenvector c o m p o n e n t s for N = 4, fl = + 0 . 3 a n d ~ ~ 37.5 ° (eigenvectors expressed as fl = --0.3 a n d ~ = 22.5 °)
1
J
Ja
4 4 4 4 4 2 4 4 2 2 4 4 2 2 0
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ½ ~ ~ ½
d.lJ 3
~ --~ --5 ~ ~ ~ --~ --6 --~ --~ ½ ½ ½ ½ ½
0.04129 0.14200 0.01109 --0.28351 0.00604 --0.14104 --0.51944 --0.01793 --0.24832 --0.05451 0.61698 0.02807 0.35165 0.04707 0.18867
TABLE 3 Calculated rotation-eigenvector c o m p o n e n t s I = ~ (ground state)
K ½ --]
I = ]~
b[K
K
--0.87287 0.47423 0.11491
b]K
½ --~ ~ --½
--0.44320 --0.78613 0.42070 0.09263
tive ORvalues equal to Z/A, 9~ = 1 and first setting 9s for the odd proton equal to the "quenched" value 4.0 found generally applicable by Chiao and Rasmussen 2~). A second calculation sometimes follows in parentheses, using free-space gs = 5.585. We feel that the quenched values are more appropriate, since they take account of a general tendency by the odd nucleon toward polarizing the spins of neighbouring nucleons in such a way as to reduce the intrinsic-spin contribution to the magnetic moment. The experimental magnetic moment of Cs T M is +3.48 n.m. 22). Table 4 gives TABLE 4 Theoretical magnetic m o m e n t ), = 30 ° fl = 0.2 fl = 0.3
2.35 2.88
2, = 37.5 ° 2.66 2.86
), = 45 ° 2.79 2.74
~, = 38 ° 2.70 2.85
our theoretical moment at several values of fl and y near the optimum (0.28, 38 °) in units of the nuclear magneton. The interpolated moment for fl = 0.28, y = 38 °
6'78
L. W . PERSON AND J. O. RASMUSSEN
is 2.82 (unquenched value is 3.14). For the purpose o f comparison, the spherical shell model with quenched g factor gives # = 4.0 for d t (unquenched, 4.793) and # = 1.68 for (gi÷)j-_-3t (unquenched, 1.225). Nilsson's model 8, 9) for a pure gt proton with projection f2 = ~ gives # = 1.49 (unquenched, 1.18). The agreement is not good for all the models and the experimental magnetic m o m e n t lies in between that of asymmetric-rotor and single-particle models. It is interesting to note that the magnetic moments of the spin ~ nuclei 531125 and 531127(__+2.6 nm and + 2.7935 nm, respectively) are close to the values of table 4, suggesting the possible applicability of this ellipsoidal model to these nuclei. We have used our formulas of subsect. 2.3 to calculate reduced E2 transition probabilities for a number of transitions. In most cases the experimental data are 200
,
,
,
,oo ~._~,o~O~ (c,) ÷
- - ~
~o
~~ /
.
.
.
.
_
/i \ \Experimentol
I
•
Ioea
B,p
~,,
",1
5
2
0.8
I 15
I 30 Z
45
~
I
60
(deg)
Fig. 8. C o m p a r i s o n o f the experimental B(E2½+ ~ t + ) value with the theoretical values as a function o f 7, with fl = 0.3.
at present not sufficient to test the calculations carefully, since M1-E2 mixing ratios are not well
ROTATIONAL
ENERGIES
679
experimental value by about a factor of 1.5. In retrospect it would have been more realistic to use an inertial parameter B considerably larger than the irrotational value. Since we used such a small B, we were forced to a deformation value of fl = 0.28 in order to fit the energy spectrum. Certainly the experience from the spheroidal nuclear region would lead us to believe that a fl value about half of this is more realistic for Cs 131. Such a modification of our calculations away from the hydrodynamic moments of inertia would lower the predicted B(E2) value, since they vary roughly as f12. With a lower fl the strong gt admixture of table 2 would be diminished. We find a serious discrepancy in trying to compare experimental and theoretical transition rates for the 133-keV transition presumed to be ~-+ to ~ + and predominantly M1. Its lifetime is 13.3 x 10 -9 sec (ref. 20)) and, allowing for a conversion 100
÷
ea
I
50~
I
4
20 D(e%m'• ;
k¢)
"-t,(M uJ Q3 e4
I
--]
', V ,'
/
%% / ~ J
t
I0
I
Bsp
Experiment j_j_. value r l / 2 1
5 -- ~
2 I
0
I
15
I
30
I
45
I
60
/
7 (deg) Fig. 9. C o m p a r i s o n o f the experimental B ( E 2 } + ~ ~q-) values with the theoretical values as a function o f y, with fl = 0.3.
coefficient of ~ 0.5, this given an upper limit B(E2) < 7.0 x 10 -50 e 2 • cm 4. Our theoretical calculation (fig. 9) for B(E2) is 3.4 times this limit. I f future experimental work firmly establishes the position and spin associated with the 13.3 x 10 -9 sec state according to the proposal of Bodenstedt et al. 2o), then it will be clear that we should not have fitted that state as the 7 + number of the ground rotational family. It may be that the 13.3 nsec state belongs to a different intrinsic Newton proton state than the ground state. We have also calculated the relative B(E2) values for the three transitions each from the first two excited states of spin ½ (i.e., at 216 keV and 620 keV). The ratios to the ground transition are shown in figs. 10 and 11. Points at 0 ° and 60 ° are calculated in the spheroidal limits through squares of Clebsch-Gordan coefficients with K = ½ for I equal to ½, i , ~ on the prolate side; and K = ½ for Ii; K = ~r for I i, I i on the oblate side. Experimental data on M1-E2 mixing ratios are presently insufficient to test any of these calculations. They are presented here in part to illustrate the drastic differences between ½+ and ½' + states, whereby there appears clearly an approximate
680
L.W. IO0
PERSON t
1
1
AND 1
1
J. O. R A S M U S S E N 1
1
t
l
80 60 40
8(E2 3/2+-~ I/2÷)
io B
6A 4
0 o C~ L~J a3
2 lO
-
\
-
0.8 06
\ B{E2 3 / 2 + ~ 7 / 2 + )
8(E2 3/2÷-~ 512~-)
\ L o,~ 7,5
225 7" ( d e g )
Fig. 10. T h e o r e t i c a l ratios B ( E 2 ] + -+ ½ + ) / ( B ( E 2 ~ + -+ ~ ] + ) a n d B(E2]-+- -~ ~ - k ) / B ( E 2 ~ + as a f u n c t i o n o f 7, w i t h fl = 0.3. I
I
I
I
I
I
-~ ~ - k )
I
7
~ /'
6
o
5
4 oJ Ld
3
-,, B(E2 3/2"+'-,i / 2 ~ . ) / /
iI
o
2
I
/
\
k,%(E2 3/2'+-~ 7/2+)\\
i
B(E2 3/2'+--~5/2+}
/
~L___.~
/ /
c~c ~
_1
I 7.5
I I I I I 22.5 30 3z5 45 52.5 )" ( d e g ) Fig. I 1. T h e l o g a r i t h m i c theoretical ratios B(E2~-'+ -+ ½ + ) / B ( E 2 ~ ' + --* ~ + ) a n d B ( E 2 ~ t ' + -+ 7 z + ) / B(E2~-'÷ -+ ~ + ) as a f u n c t i o n o f 7, w i t h fl = 0.3. 0
I I5
ROTATIONAL ENERGIES
681
selection rule from the limiting model of y vibrations of a spheroid (see Appendix). In this latter model the ~r + and ½ + states have no phonons of gamma vibration, the ½+ and ½+ have one phonon, and the ½+ ' has two phonons. This can be seen from the top curve of fig. I 1. 3.3. G R O U N D
STATE SPIN
We have determined the lowest state for a variety of fl and 7 values for nucleon numbers of 51, 53, 55 and 57; and have plotted the deformation regions where a given spin is lowest 22, 24, 25). This map is in polar coordinates, where fl is the radial coordinate and 7 is the angular coordinate. All shapes of quadrupole deformation y (deg) Z'J) 20
~0/ ~, ~, 5 o
T (deg) tO
' i/z.
~
~,0 ~0
£ o.3
0
oo
o.3
-Q/
"~
Fig. 12. Plot of the ground-state spin of the nucleus with 51 protons, as a function offl, y; with BflZ/h 2 = 9.41 MeV -1 and hoJ o = 8.16 MeV.
°1
Fig. 13. Plot o f the ground-state spin of the nucleus with 55 protons, as a function o f f l , y; with Bfl2/~ 2 = 9.41 MeV -1 and ¥~to0 = 8.16 MeV. A pair of p r o t o n s are assumed in the h~k orbital. If one assumes no p r o t o n s to be in the 5th oscillator shell, the above diagram applies to atomic n u m b e r 53.
7"(deg)
f (deg)
~oz9
2,0 !0
~P
o
FO.I
Fig. 14. Plot of the ground-state spin o f the nucleus with 5 5 + p protons, as a function of fl, y; B~2/h 2 = 9.41 MeV -1 and //co0 = 8.16 MeV. The quantity p is the n u m b e r of p r o t o n pairs in the fifth oscillator shell.
~,0
50 ,
~
20
I0
,
,
0
'0./
Fig. 15. Plot o f the ground-state spin of the nucleus with 5 7 + p protons, as a function o f fl, y; with Bfl2/li 2 = 9.41 MeV -1 and ~to 0 = 8.16 MeV. The quantity p is the n u m b e r o f p r o t o n pairs in the fifth oscillator shell.
~ 8 "2
L. W .
P E R S O N A N D J. O. RASMUSSEN
are of course represented within the 60 ° sector. Fig. 12 represents the 51st level of the proton, fig. 13 the 53rd level, fig. 14 the 55th level, and fig. 15 the 57th level, if one assumes no protons in the 5th oscillator shell (h~). For the larger fl values one expects the occupation of h~ to begin even before atomic number 55. The lowest spin state along 7 = 0 in these four figures should correspond with the lowest Nilsson state for that proton number from the diagram of fig. 3 of Mottelson and Nilsson 9), except that the h~k levels are missing. One sees from all figures except fig. 12 that our intuition about the stabilization of spin ½ states in asymmetric wells is borne out by the calculations. Note that at the Cs laa reference point (fl = 0.28, y = 38 °) on fig. 13 the ~ + spin is lowest; for the Cs 127 and Cs ~29 only a slight shift toward a smaller y value brings spin ½ lowest. Some experimental knowledge of excited states of Cs 127 and Cs 129 would be most desirable, to test between predictions of the spheroidal and the asymmetric nuclear models. 3.4. MAGNETIC MOMENTS OF SPIN ½ NUCLEI There is one region in which the approximate calculation of magnetic moments for spin ½ nuclei is especially simple. This is the region of neutron number 65 to 81, where the g~ and d~ orbitals are presumably filled, and even-partiy states will generally consist of configurations involving the close-lying d~ and s~ orbitals. An examination of the Newton eigenfunctions of the top two states in the fourth oscillator shell for y < 30 ° shows that these states consists mainly of s~-d~ admixture in comparable amounts. Likewise, Nilsson's highest f2 = ½ state number 51 in his two sets of calculations shows the same admixture s). The relative mixing ratios do not depend strongly on deformation. I f a Nilsson-type calculation is made with only degenerate s½ and d~ orbitals considered, and if the radial matrix elements of r E between 3s and 2d and between 2d and itself are considered equal (Nilsson's eqs. (1 la) and (1 lb) have them differing by only 4 ~o)8), then the eigenfunction of the top t2 = ½ state for prolate (next to the top for oblate) has as
=
as
=
independent of deformation. The predicted moment is two-thirds of the simple st nucleon value plus one-third of the moment resulting from a d~ nucleon coupled to a core angular m o m e n t u m of 2 to a resultant ½. The magnetic moment of the d~ plus phonon coupled to ½ is /~ = g R - - 9 s . I f we use the quenched 9s factor of - 2 . 4 for an odd neutron and a 0R factor for collective motion of about 0.4, the magnetic moment of a pure s½ neutron state is about - 1.2, and the moment of the d~ plus phonon is about zero. Thus, the nucleus with an odd neutron in this state should have a magnetic moment of about - 0 . 8 n.m. Likewise, the second from the top ~ = ½ state in the prolate spheroid (top on
ROTATIONAL ENERGIES
683
oblate) is one-third of sl character and two-thirds of dj plus phonon. An odd neutron in this state should give rise to a magnetic moment of - 0 . 4 n.m. Intuitively, we would expect that models with a stable nonaxial deformation or with quadrupole oscillations about a spherical equilibrium shape might give rise to intermediate theoretical values for the odd neutron in either state. TABLE 5 Magnetic m o m e n t values o f certain spin ½ nuclei
Isotope
Experimental /t (n.m.)
,aCd~]1 ,aCd~[S ~oSn~9 5~Te~3
--0.59 --0.62 -- 0.73
5~Te~ 5
-- 0.88
-
54Xe~9
-
1.04
-
0.77
Table 5 lists nuclear moments of those spin ½ nuclei to which the model might possibly apply. The tin isotope in is obvious disagreement, but the other isotopes are consistent with some sort of quadrupole coupling-mixed s-d model, as discussed above. A reduction in the collective gR factor for tin, corresponding to the proton closed shell, could explain its deviation in terms of the coupling scheme here proposed. The magnetic moment of spin ½ nuclei does not give much help in deciding between the nonaxial model and other quadrupole deformation models. We shall examine Newton's 6) eigenfunctions for the three highest states in the fourth oscillator shell in order to be able to understand more quantitatively the predictions of his model. If much dt- ~ admixture is involved in the wave function, there will be serious shifts in the calculated moment away from the simple formula. At fl = 0.1 and 7 = 30 °, for example, the Newton wave function with the most s½ character is actually 28 % sl, 55 ~ d~, 8.3 % dt, and 9 % other. It has the second from the highest eigenvalue. We calculate a magnetic moment of about - 0 . 8 n.m., but here the d~-d t crossterm (calculated with eq. (A.I[.5) of ref. z6)) contributes almost half the value. Therefore, the basis of the previously derived simple approximate formula (/~ = - 1.2 x fraction %), based on s½-d~ admixture, is clearly violated, and may be trusted only at deformation parameters well below fl = 0.I for asymmetric shapes. For prolate (7 = 0) symmetric shapes the validity goes to somewhat larger ft. That is, for 7 = 0, fl = 0.1 we have 62 % st, 3 1 % d~, and 3.8 % d~; and the d~-d I cross term contributes only - 0 . 1 7 n.m shift to the magnetic moment. 4.
Conclusions
We have attempted serious and detailed testing of the fixed asymmetric-rotor model for odd-mass nuclei. Certainly the large E2 transition probability in odd-mass nuclei somewhat removed from closed shells yet not within the regions of spheroidal
684
L. W. PERSON AND J. O. RASMUSSEN
nuclei are strong indications of collective motion. The availability of nuclear eigenfunctions for an asymmetric well as calculated by Newton 6) make various calculations feasible. Our attempt to fit all the well-established levels of Cs 13a as a single rotational band gives encouragement but is inconclusive. Better experimental information is much needed. Also, our difficulties with the E2 transition probability of the 133-keV transition suggests that the theoretical calculations need to be repeated to include perhaps two or more Newton intrinsic states with rotation-particle coupling, a coupling we have completely ignored for the sake of simplifying calculations. The magnetic moments of spin ½ nuclei ranging from 65 to 81 neutrons agree with predictions of a model involving coupling to a core angular momentum, but do not distinguish to any degree the core shape or magnitude of deformation. Successful results with this model do not necessarily prove that the nucleus literally has fixed asymmetric deformation. We show in the Appendix a spheroid with Y vibrations gives quite similar results at small ~ values. The asymmetric rotor model does provide a prescription for calculations and a point of attack on nuclei in regions not yet very amenable to theoretical interpretation. Appendix RELATIONSHIP
OF A SYMMETRIC R O T O R P L U S ~' V I B R A T I O N ASYMMETRIC ROTOR
WITH
A SLIGHTLY
It is evident that at 7 = 0 and 7 = ½n, the first rotational band has the same structure as that of a spheroid; and the energy values of the higher rotational band become infinite, with hydrodynamic moments of inertia. However, in the cases of = l ~ n or {4rc the computed spectra of the ellipsoidally deformed nuclei (see Appendix B of ref. 13)) show the general features of the rotational-7-vibrational spectra of a spheroid. In this section we shall see that in these limits it is possible to reproduce the properties of the ground, second, and third rotational bands of an ellipsoidal nucleus by considering the y-vibration-rotation interaction between rotational bands, and the one- and two-phonon ~-vibrational bands of a spheroidal nucleus. Since it is impossible to write a simple expression for ellipsoidal nuclei in the general case, one specific example will be explained in detail; the other cases are quite analogous. The following illustration is for an intrinsic state with projection 12 = ~}: I .~
~z -~+
K ~ }
n 2 2
:~ .~
+ +
½ ½
l J
+ +
~ ~
o o
ROTATIONAL ENERGIES
685
The Hamiltonian oYF' for 7-vibration-rotation interaction may be derived from the last term of eq. (11) when ~ is small 27)
•~ ' =
h2(I2-IZ)Y
(A.1)
In the y-asymmetrical model the energies of the bands are determined by the fl and values. In axially symmetric nuclei the energies of the rotational and the vibrational bands depend upon the phonon energy and the moment of inertia. If the adiabatic approximation holds and if the y-vibration phonon energy [hco = (Cr/flr) ~] and the moment of inertia (5 = 3Brfl 2) are the same for all the bands, the energy of the nth highest energy state with spin I is expressed as h2
E,(I) = ~ {[I(I+l)-K2]-(-)'-s(J+½)(I+½)6r~t6~½}+Qhto,
(A.2)
where
for, for 1 = ½, ] for the first three bands. Since the mixing amplitudes and the B(E2) values for different bands are proportional to the term (By Crfl4) j, which is approximately equal to (½hog~), we define an energydifference ratio P _ E2(~)- E,(~)
(A.3)
'
which is related to the (B~Crfl4)~ in a linear way, and we shall express the mixing amplitudes and the B(E2) ratio in terms of p by second-order perturbation theory. For comparison, the values of fl and the values of the mixing amplitudes a(~, ~; ~}, ½),
a(~, 3; ~:, ½),
and the ratio B[E2~(2) -, ½(1)] B[E2~(2) ~ ~2(1)] are given in table 6. The notation a(I, K; I, K') signifies the mixing amplitude of the TABLE 6 C o m p a r i s o n o f mixing a m p l i t u d e and B(E2) ratio for ellipsoidal a n d spheroidal models a(~, ~-; t , ½)
0.1 0.2 0.3
a(], ~; ~,, ½)
B[E2~(2) -~ ½(1)] B[E2~(2) --~ ~(1)1
spheroidal
ellipsoidal
spheroidal
ellipsoidal
spheroidal
etlipsoidal
0.00278 0.00285 0.00318
0.00276 0.00226 0.00304
--0.00238 --0.00250 --0.00273
--0.00049 -- 0.00046 --0.00049
2.93 × 10 a 2.50 × 10 a 1.87 × l09
1.59 × 107 4.82 × 10 ~ 1.75 × l0 T
686
L.W.
PERSON A N D J. O. RASMUSSEN
a n g u l a r m o m e n t u m projection K ' in the state o f spin I a n d p r e d o m i n a n t c o m p o n e n t K. The p values for the spheroidal model are calculated using the results i n A p p e n d i x B o f ref. 13) for ~ = 52.5 ° . F r o m table 6 we see that b o t h models yield similar results, a n d that the B(E2) ratio shows the effect of the f o r b i d d e n t r a n s i t i o n o f An = 2, a l t h o u g h the m a g n i t u d e s are n o t equal. W e believe that as 7 goes to smaller values the two models will a p p r o a c h identical predictions. Note that in the B(E2) ratio calculation the d e n o m i n a t o r c o n t a i n s two terms o f opposite sign a n d nearly equal values, which will cause the B(E2) ratio to be very sensitive to the a values. W e wish to express again our gratitude to Dr. T. D. N e w t o n for p e r f o r m i n g a special set of N = 4 e i g e n f u n c t i o n calculations at our request. O n e o f us (J. O. R.) wishes to acknowledge fellowship s u p p o r t of the N a t i o n a l Science F o u n d a t i o n , a n d the hospitality of the Institute for Theoretical Physics i n C o p e n h a g e n d u r i n g the writing of this paper. The calculational work was d o n e u n d e r the auspices of the U.S. A t o m i c Energy C o m m i s s i o n .
References 1) A. S. Davydov and G. F. Filippov, Nuclear Physics 8 (1957) 237 2) K. T. Hecht and G. R. Satchler, Nuclear Physics 32 (1962) 286; K. T. Hecht, Bull. Amer. Phys. Soc. 6 (1961) 78 3) S. Watanabe, Ph. D. Thesis, University of Washington (Seattle, 1961) 4) G. F. Filippov, Soviet Physics (JETP) 11 (1960) 949 5) T. D. Newton, Can. J. Phys. 38 (1960) 700 6) T. D. Newton, Atomic Energy of Canada Limited, Chalk River, Ontario, Report, CRT-886, January 1960, (unpublished) 7) Aage Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1952) 8) Sven G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) 9) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skrifter Dan. Vid. Selsk. 1, No. 8 (1958) 10) R. J. Blin-Stoyle, Theories of nuclear moments (Oxford University Press, London, 1957) 11) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (John Wiley and Sons, Inc., New York, 1952), p. 595 12) A. S. Davydov and G. F. Filippov, Soviet Physics (JETP) 8 (1960) 305 13) L. W. Person, Ph.D. Thesis, Lawrence Radiation Laboratory Report UCRL-9753, July 1961 (unpublished) 14) B. R. Moore and W. White, Can. J. Phys. 38 (1960) 1149 15) A. S. Davydov and V. S. Rostovsky, Nuclear Physics 12 (1959) 58 16) W. C. Beggs, B. L. Robinson and R. W. Fink, Phys. Rev. 101 (1956) 149 17) J. M. Cook, J. M. LeBlanc, W. H. Nester and M. K. Brice, Phys. Rev. 91 (1953)76 18) B. S. D~helepov and L. K. Peker, Decay schemes of radioactive nuclei (Academy of Sciences of the USSR Press, Moscow, 1958) 19) P. P. Day, E. D. Klema and C. A. Mallmann, Argonne National Laboratory Report ANL-6220, November 1960 (unpublished) 20) E. Bodenstedt, H. J. K6rner, C. Giinter, D. Honestadt and J. Radeloff, Nuclear Physics 20 (1960) 557 21) J. O. Rasmussen and L. W. Chiao, in Proc. Intern. Conf. on Nuclear Structure, Kingston, Canada (University of Toronto Press, Toronto, 1960) p. 646
ROTATIONAL ENERGIES
687
22) E. Bellamy and K. Smith, Phil. Mag. 44 (1953) 33 23) L. A. Sliv and I. M. Band, Coetticients of internal conversion of gamma radiation (Academy of Sciences of the USSR Press, Moscow, 1958) 24) J. E. Mack, Revs. Mod. Phys. 22 (1950) 64 25) W. A. Nierenberg, H. A. Shugart, H. B. Silsbee and R. J. Sunderland, Phys. Rev. 104 (1956) 1380 26) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) 27) T. Tamura and F. Udagawa, Nuclear Physics 16 (1960) 460