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Relative transition probabilities between rotational levels of non-axial nuclei
Nuclear Physics
12 (1969) liS-68;
@
North-Holland
Publishing
Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
RELATIVE
TRANSITION
ROTATIONAL
PROBABILITIES
LEVELS
A. S. DAVYDOV
OF NON-AXIAL
NUCLEI
and V. S. ROSTOVSKY
Moscow State University, Received
BETWEEN
1 January
Moscow 1959
Abstract: Energies and wave functions are calculated for the rotational states (J = 4, 6, 8) of non-axial nuclei. Relative reduced E2 transition probabilities for the rotational band levels of such nuclei are derived. Conditions under which the rotational states can be characterized by quantum numbers K are established. It is shown that if the nuclear shape differs from axial symmetry, the 1 : 3.3 : 7 : 12 interva1 rule observed in rotational bands of axial nuclei is violated. The theory is compared with the experiments! data.
1. Introduction Rotational levels in even nuclei were investigated by Davydov and Filippov 1) under the assumption that in a first approximation the equilibrium shape of the nucleus can be represented by a triaxial ellipsoid. Analytical expressions for the rotational level energies were derived for spins of 2, 3 and 6, and the transition probabilities between these levels were computed. It was demonstrated, in particular, that the theory yields a unique relation between the ratio of the energies of two spin 2 levels and the ratio of the reduced transition probabilities from the second spin 2 level to the first level (cascade transition) or directly to the ground state (direct transition), In the present paper some results of numerical calculations of the energies of rotational states with spins 4, 6, 8 are given as functions of the parameter y which specifies the deviation of the nuclear shape from axial symmetry “). The wave functions of these excited states and the transition probabilities between them are calculated. In section 4 we define the conditions under which rotational states can be described by approximate wave functions corresponding to states with a definite total momentum on axis 3 of the nucleus. In the same section approximate expressions are derived for reduced E2 probabilities between rotational nuclear states which only slightly differ from axial symmetry. The theory is compared with the experimental data in section 5. 2. Energy of Excited States Possessing
Spins of 4, 6 or 8
In the adiabatic approximation the rotational energy of a non-spherical even nucleus is defined by the Schroedinger equation: 58
RELATIVE
TRANSITION
(H-&)Y
69
PROBABILITIBS
= 0,
(2*1)
where E is measured in fi8(4B#9-1 units and the operator H is defined by
JA are projections of the total angular momentum operator on the axes of a coordinate system connected with the nucleus. The wave function Y which corresponds to a state with a total angular momentum J and satisfies the symmetry conditions established by Bohr *) can be written down as
where
In (2.4) the D&,r are functions of the Eulerian angles specifying the orientation of the principal nuclear axes in space. It can be shown that wave functions (2.3) form the basis of a completely symmetric representation of the group 92 (see ref.l)) whose elements are rotations through 180’ about each of the three principal nuclear axes. Inserting (2.3) in (2.1) and employing the values of the matrix elements of the rotational energy operator (2.2) on the wave functions (2.4), we obtain for each value of J a set of algebraic equations defining the values of the coefficients A, in the wave functions (1.3). For example, for J = 4 the Schroedinger equation (1.1) reduces to the set of equations [Ei(a+/?) -c]A,,+$dg(a-/?)A2 = 0, ~2/~(a-/?)Ao+[4(afB)+26--E]A,+~d~(a-_B)A, idY(a-_B)A,+[(a+@)+SS--e]A,
= 0,
(2.6)
= 0,
where a =
sir+(y--in),
p = sin-2(y+gz),
8 = sin+?.
The energies of the corresponding rotational states can be derived from the condition of solubility of such ,sets of equations. By solving each set (for corresponding values of e) one can determine the wave functions of these states. The wave functions (2.3) and the levels possessing spins 2, 3, 6 can be expressed in terms of parameter y by analytical formulas. The latter are given in ref.1). For the energies of spins 4, 6 and 8 levels algebraic equations of respectively third, fourth and fifth degree with respect to e are obtained. Thus, for example, the equation for the energy of spin 4 levels is
60
A. S. DAVYDOV
AND V. S. ROSTOVSKY
48 [27 + 26 sin2 3~1.5
g3 - - 90E2
sin2 3y
+
sin43y
640[27+7
-
sin23y]
o =.
sin4 3y
(2.6)
The explicit form of the equations for the energy of spin 6 and 8 levels has been presented in the previous paper of the authors 13). Results of numerical solution of these equations for several values of y are given in table 1. The number in parenthesis after E refers to the spin of the excited state and the subscript denotes the level number. TABLE 1 Dependence of rotationalenergy (inP/4Boa units)of even nucleus on parameter y Y0
0
G(4) s*(4) &a(4) ~(6) &a(6) %ts)
5
13.33 03 00 23 00 48
10
15
13.64 14.12 274.1 77.73 1056 268.1 28.42 29.51 270.6 93.51 48.60 50.26
3. Quadrupole
20
16.00 42.48 122.5 30.65 60.46 50.87
Electric
15.81 32.26 71.92 30.78 54.52 49.60
22.5
25
27.6
30
16.01 30.92 68.51 30.51 55.74 48.94
16.06 31.21 49.20 30.33 57.33 48.38
16.02 32.69 42.85 30.06 59.71 48.10
16 34 40 30 60 48
Transition
in the Rotational
Probabilities
Band
The reduced probability for electric quadrupole transitions between two states described by the functions PJmi and !PJ#,#r can be described by the expression 6 B(E2; i + f) = Z I W4C72pl.T~ij12~ 16~“(2J+ 1) pm’m where (3.13 8,,
=
(Ds~0s y + 3
eQo
(Dz2+D:,
-,)),
Q.
=‘3.
The wave functions for states with spins of 2 and 3 have been computed in ref.1). The coefficients A, for wave functions of states of spin 4 %2+%-2 Y4mi =
42
DL+D:,_,
+A4i
42
1
(3.2)
can be evaluated by solving the set of equations (2.5) for each of the roots of the equation (2.6) which defines the energy values of these levels. Values of coefficients A, for two spin 4 levels are presented in table 2. The values of the coefficients of the wave functions
for spin 6 states are also given in table 2.
RELATIVE TRANSITION PROBABILITIES
61
TABLE 2 Coefficients definingwave functions(3.2)and 3.3)of stateswith spinsof 4and 6 10 A 01
1
A a1 A., A 0s A ,*
0 0 0
A 4, &u 41 B.1 &I
0 1 0 0 0
1
1 0.003 IO-' -0.003 1 6x10-' 1 0.0076 10-L 10-1
0.000 0.030 104 -0.030 0.000 0.004 0.008 0.066 6X10_ lo-'
16
20
22.6
26
27.6
0.003 0.114 0.001 -0.114 0.003 0.015 0.973 0.232 8X10_" 6x10-6
0.066 0.296 0.010 -0.296 0.984 0.043 0.878 0.416 0.043 9x10.
0.910 0.414 0.022 -0.416 0.907 0.014 0.817 0.670 0.081 2.8x10-*
0.862 0.622 0.043 -0.523 0.842 0.128 0.766 0.633 0.113 6.1x10-*
0.792 0.606 0.076 -0.602 0.764 0.267 0.714 0.674 0.180 0.016
30
I
0.739 0.661 0.126 -0.669 0.500 0.661 0.672 0.606 0.264 0.031
By using (3.1) and the wave functions of states of spin 4 and 2 one can compute the reduced transition probabilities between these states. Thus, the reduced probabilities (in eaQoz/16nunits) for the quadrupole transition between spin 4 and spin 2 levels can be expressed by the formula b(E2; 4i + 2f) = &{cos
Y(f-%~*S~~&~,)
+sin r(2/1~A,,~i+A,ibi+1/35;4,,bi))~,
(3.4)
where ai, b, are coefficients specifying the wave functions of states possessing a spin of 2 (see ref. l)). The reduced probabilities for transitions between levels with a spin of 4 are given by the expression b(E2; Pi + 4f) = A{2 cos yC7A,IA,I-6A,,A,-2A,,A,,]
(3.5)
+ sin YC3~(A,IA~+~o*~*I)+~/21(A,IA,*+A,,A,*)l}~. Finally, the expressions for the reduced quadrupoIe electric transitions between levels with spins 4, 3 and 6, 4 are b(E2; 4i -+ 3) = +(21/i cos y - Asi+.& b(E2; 6i + 4f) = &
cos y[3d%$1~m+4~
+ sin y V5&A,,+
B,iA,f +1/=&i GO
A,+*B,,
~[&4,,-d?z/jA,,]}a,
B,,&+~B,~A,~I A,f+d49.6B,,A,I
(3.0) (3.‘) a II
.
For the sake of brevity we shah call the energy levels 0, 21, 41, 61, 81 levels of the “ground rotational band”. They are depicted in fig. 1 by solid lines. For y + 0 the levels of the ground rotational state go over to levels of an axially-symmetric nucleus. All other energy levels (depicted in fig. 1 by dashed lines) tend to infinity for y + 0. These energy levels will be called “anomalous” levels.
62
A. S. DAVYDOV
AND
V. S. ROSTOVSKY
The reduced electric quadrupole transition probabilities between different rotational states of even nuclei can be calculated by using the wave function coefficients given in table 2 and ref.l). The values of some of the reduced probabilities are given in table 3. In the present paper reduced probabilities for all transitions are expressed in e2Q02/16z units.
5
10
15
20
25
30
a"Fig. 1 TABLE
3
Reduced probabilities (in e*$&,*/Mz units) for electric quadrupole transitions between some rotational states of an even nucleus 0
Y0 b(E2; b(E2; b(E2; b(E2; b(E2; b(E2; b(E2; b(E2; b(E2;
3 41 41 42 42 42 42 61 61
+ + + -+ -+ -+ -+ --f -+
41) 21) 22) 21) 22) 3) 41) 41) 42)
0 1.429 0 0 0.596 1.333 0 1.673 0
5 0.0060 1.418 3.6 x lo-’ 4.1 x 10-a 0.591 1.323 0.0138 1.663 10-a
10 0.034 1.396 0.0023 0.011 0.576 1.282 0.0024 1.647 7.7 x lo-
T -
16
T -
20
22.5
T-
26
27.5
T -
*1 1I 1 0.130 1.377 0.010 0.008 0.643 1.172 0.167 1.662 0.036
0.406 1.372 0.033 4 x lo-’ 0.481 0.978 0.313 1.623 0.062
0.619 1.366 0.044 0.009 0.447 0.680 0.339 1.671 0.033
0.821 1.366 0.039 0.021 0.435 0.448 0.311 1.703 0.011
0.955 1.378 0.016 0.018 0.484 0.210 0.271 1.726 0.0023
0 0 0.596 0 0.273 1.731 0
i
From the results of ref. 1) and the data in table 3 it can be concluded that the reduced electric quadrupole transition probabilities between various rotational states of even nuclei can be divided into three types: a) Transitions for which the reduced probabilities are of the order of
RELATIVE
TRANSITION
63
PROBABILITIES
unity. Of this type are cascade transitions between levels of the ground rotational band and cascade transitions between “anomalous” rotational levels; examples are the transitions 3 + 22; 42 + 3; 42 + 22. b) Transitions between levels of the ground rotational band and anomalous rotational levels of different spin. Examples are the transitions 3 + 21; 41 + 22, 42 --f 2X; 61 --z 42. The reduced probabilities for such transitions are equal to zero for y = 0 or 30” and are very small for other values of y. c) Transitions between levels of identical spin. Examples are 22 --+ 21, 42 --f 41. The reduced probabilities for such transitions are equal to zero for y = Oand th en significantly increase with increase of y, attaining a maximum (of the order of unity) for y = 30’. The 3 + 41 transition also belongs to this type of transition. 4. Quantum Number K and K Forbiddenness The wave functions (2.3) for rotational states of even nuclei are represented by linear combinations t of the functions (2.4) corresponding to states with a definite value of the total nuclear angular momentum (quantum number K) projection on axis 3 in a coordinate system fixed to the nucleus. The magnitude of the coefficients determining the contribution of various terms to such a linear combination appreciably depends on the parameter y which characterizes the deviation of the shape of the nucleus from axial symmetry. It can be seen from table 2 and ref. 1) that for y < 15 the wave functions of the nuclear rotational states can be approximated by expressions which contain only one value of K. Thus, the wave functions of spin 2 states can be replaced by the approximate functions ypl = 120) = y& = 122) =
5 4 g--g %I~
() (
&
(4.1)
i(qL+x,-P). 1
The wave functions of spin 4 states can be approximated
by the expressions
etc. t Only the function for a rotational state with a spin of 3 corresponds to a definite value of K = 2 (see ref. I)).
64
A.
S. DAVYDOV
AND
V. S. ROSTOVSKY
In those cases (y < 16”) when the rotational motion of the nucleus can be approximately described by wave functions of the (4.1) (4.2) type the rotational state of an even nucleus can be characterized by two quantum numbers, J and K. In this approximation values K = 0 correspond to levels of the “ground rotational band”. Anomalous rotational levels can then be divided into a set of levels with K = 2, 4, 6, . . . The non-vanishing reduced probabilities (in our e2Q,/l&z units} for electric quadrupole transitions between states described by the approximate functions IJK) have the form b,(EZ; JK + J’K)
= 5 cos”y(2JOK/J’K)“,
b,(E2; JK + J’K+2)
= $(l+s,,)
sin2y(2J2K/ J’K+2)2,
b,(E2; JK + J’K-2)
= ~(l+&,)
sin2r(2J-2KJ
J’K-2)2.
The rules given at the end of the preceding section reduce in the approximation under consideration to the condition of validity of the equality AK=0
(44
for the most probable transitions. Those transitions which occur when conditions (4.3) are not satisfied are usually called K-forbidden transitions. Since quantum number K is only approximate, rule (3.3) can be important only for nuclei with y < 15’. If y > MO, the exact functions (2.3) should be employed. The results obtained in section 3 permit one to estimate the error which arises when the approximate functions )JK) are substituted for the exact functions (1.3). It should be mentioned that in analyzing the data on the relative transition intensities a number of authors 3--6)were compelled to assign a quantum number K = 2 to some of the levels possessing spins 3 and 2. Such excited states were ascribed by the authors to the so-called y-vibrations, the frequency of the latter being chosen arbitrarily. However, if one views such states as rotational states of a non-axial nucleus a simple explanation of the spacing of the levels and of the relative transition probabilities between them can be proposed. In this case, only a single parameter y which is uniquely defined by the ratio of the energies of two spin 2 levels is involved in the theory (see the following section). 5. Comparison
with Experiment
In order to facilitate comparison of the results with the experimental data, plots of the dependence of parameter y on the ratios of the rotational level energies for various spins are shown in fig. 1. The parameter y can be uniquely determined from the ratio of the energies of two spin 2 levels with the aid of the formula
RELATIVE
E,(2) p= El (2)
TRANSITION
PROBABILITIES
l+dl--_%sinB3y. l-dl-i
sin*3y
66
w
After determining in this way the value of y, one can use fig. 1 to find the position of the rotational levels of even nuclei. It can be seen from fig. 1 that as a result of deviation of the shape of the nucleus from axial symmetry the interval rule 1 : 3.3 : 7 : 12 which is observed in the rotational band of axially symmetric nuclei is violated in the “ground rotational band”. Thus, for example, for y = 30” the levels of the “ground rotational band” should satisfy the interval rule 1 : 2.67 : 6 : 8. In fig. 1 the circles denote the experimental values of the ratio of the excitation energies of Oslso, OS”~~,Dy18*, UBa, Puss8 7), Um “), 0~186,0~18814) and Erl66, Er168l4) nuclei to the energies of the first excited level of these nuclei. The figure shows that the theory yields spin sequences and energy ratios which are in good agreement with the experimental data. The slight difference between the experimental points and theoretical values can be removed by introducing a correction factor
which accounts for the relation between rotation and internal excitation of the nucleus. The magnitude of this correotion factor may serve as a criterion of validity of the adiabatic approximation. It should be noted that deviations of the experimental energy ratios from the interval rule 1 : 3.3 : 7 : 12 was wholly ascribed to violation of adiabaticity when the rotational spectra were analyzed from the viewpoint of axial symmetry of the nucleus. As a consequence the significance of the correction term (6.2) was overestimated. As a matter of fact deviation of axial nuclei from the interval rule is largely due to violation of axial symmetry of the nucleus. In this respect the experimental data on the 2+, 4+, 6+ and 8+ levels of the Osfe* nucleus are especially interesting. The positions of these levels have been determined by ScharffGoldhaber, Alburger, Harbottle and McKeown ‘) and Aten, de Feyfer, Sterk and Wapstra 10) who studied cascade y transitions in the decay of the 10 min 0~190isomer with a spin lo-. The aforementioned authors 9110)note that the experimental spin sequence satisfactorily agrees with the spin sequence of the levels in the HP** nucleus which possesses a pronounced rotational spectrum. However, the observed energy ratios were found to be appreciably different from the theoretical values for axially symmetric nuclei. It is also noted in the papers cited above that even the introduction of the correction (6.2) does not petit one to attain the experimental value 1 : 2.93 : 6.62 : 8.93. Apart from a change in the interval rule for the levels of the ground rotational band, violation of axial symmetry of the nucleus leads to the appear-
A.
06
S. DAVYDOV
AND
V.
S. ROSTOVSKY
ante of new rotational (“anomalous”) levels. Such are, for example, the second levels with spins of 2 and 4 in the Oslo0 nucleus. These levels are not noticeable in decay of the 10 minute OsIQOisomer but exhibit themselves in K-capture decay of Ir lQo. The peculiarities of the spectrum of the OPO nucleus are related to the large value of y = 21.4’ which corresponds to the experimental energy ratio &(2)/J&(2) = 3.15. With the value y = 21.4 the reduced probabilities and relative electric quadrupole transition probabilities can be calculated for the Oslo0 nucleus. The values are given in table 4; TABLE
4
ReIative transition probabilities between various levels in the Osl*O nucleus
E2 transitions
21+ 41 + 22 + 22-+ 22 4 3 + 3+ 3 --f 61 -+ 42 + 42 + 42 + 42+ 42 +
0 21 21 0 41 21 22 41 41 22 41 21 3 61
Transition energy &eV) 186 360 400 586 40 604 204 244 600 640
580 940 336 80
Reduced probabilities in ePQ,a/16n units 0.934 1.33 0.467 0.066 0.071 0.18 1.60 0.516 1.6 0.461 0.319 0.007 0.813 0.06
Relative probability
1 39.3 21.5 21.9 3.5 x 10-b 69 2.9 2.14 251 100 100 24.6 16.5 9 x LO-’
transitions referring to a theoretical level of spin 3 (which so far has not been observed experimentally) are also included. Using OP” as an example we have demonstrated that the theory permits one to calculate the relative electric quadrupole transition probabilities between all rotational states. The values for the relative transition intensities thus obtained yield a qualitative explanation of the observed decay schemes of the excited states of the Osreo nucleus. For example, decay of the isomer state (spin 10-J of the OP” nucleus occurs only as a result of a series of cascade transitions via rotational levels of the ground rotational band without noticeable excitation of anomalous rotational levels. The 4+ and 2+ anomalous rotational levels are excited in K capture of the Irlao nucleus. From table 4 it follows that by emitting a E2 y quantum a nucleus can go over from an excited state corresponding to the 4+ anomalous rotational level to either a rotational state of spin 4 in the ground rotational band with subsequent cascade emission of two y quanta or (with about the same probability) to a state 2+ corresponding to the anomalous rotational band. From an anoma-
I&LATIVE
~mNsmoR
67
PRCIEUAYUL~T~E~
lous spin 2 level the transition can occur either to a state of spin 2 of the ground rotational band or (with about equal probability) directly to the ground state. According to the theory the OslSOnucleus should possess an excited state (about 790 keV) with a spin of 3. However, as can be seen from the table, excitation of this rotational state is not probable inasmuch as transition to other rotational states are much more probable from the (spin 4 and 6) levels located above it. This may explain why this rotational state hitherto has not yet been detected. A comparison between the theoretical values of the transition probabilities and experimental values has been presented in ref. 1). Since this paper was published, new experimental data on the ratio of the cascade (22 + 21) and direct (22 + 0) reduced probabilities have appeared. The experimental data of McGowan 11)and Nathan 4) and the theoretical values obtained on basis of the formula b(E2; 22 --f 21) b(E2; 22 + 0)
=
20 sin2 3y
7{9-8 sin*3y---3-2
sins3yld9-8
sin*3y}
are given in table 5. Ratio of reduced quadrupole
Nuclei
transition probabilities for various nuclei
E,(2) E,(2)
Y0
16.4 8.11 8.02 4.08 2.98 2.37 2.23 6.0
10.3 14.0 14.1 19.3 21.4 26.4 20.3 16.2
b(E2; 22 + 21)
b(E2; 22 -+ 21)
b(E2; 22 + 0)
b(E2; 21 3 0)
them
*
1.7 2.0 3.0 4.5 7.1 21 40 2.7
exp 1.8 1.94 2.30 2.7 9.7 9.1 30 2.38
them
exp
0.062 -
0.014 -
0.18 0.32 0.60 0.90 1.29 0.16
0.12 0.19 0.62 0.68 2.8 0.12
The parameter y was computed from the ratio (6.1). The experimental data lr) and theoretical values of b(E2; 22 -+ 21)
20 sins3y
b(E2; 21+ 0) = 7{[9--8 sin*3y+ 13-2 sins(3y)]2/9-8
sin*3y}
for the ratio of the probabilities of transitions (22 -+ 21) and (21+ 0) are given in the same table. It can be concluded from tabIe 6 that the theory yields a unique relation between the indicated reduced transition probability ratios and ratio of the energies of both spin 2 levels.
68
A,
S.
DAVYDOV
AND
V.
53.
ROSTOVSKY
Nathan and Waggoner “) have presented experimental values for the ratio of the intensities of the E2 transitions 3 + 41 and 3 + 21 in the Smls2 nucleus which imply that the experimental ratio of the reduced probabilities b(E2; 3 + 41) = 1.88.
i b(E2; 3 + 21) Iexp
From the ratio E2(2)/E,(2) = 8.9 for the Smfb2 nucleus it follows that y = 13.5’ and the theory thus yields b(E2; 3 + 41) b(E2; 3 +
21) theor
= 1.37,
which is in good agreement with the experimental ratio. References 1) A. S. Davydov and G. F. Filippov JETP 35 (1968) 440; (lQ58) 703; Nuclear Physics 8 (1968) 237 2) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1962); A. Bohr and B. Mottelson, Mat, Fys. Meld. Dan. Vid. Selsk. 27, No. 16 (1953) 3) 0. Nathan and M. Waggoner, Nuclear Physics 2 (1966/67) 548 4) 0. Nathan, Nuclear Physics 4 (1957) 126 5) J. Juliano and F. Stephens, Phys. Rev. 108 (1958) 341 6) G. D. Hickman and M. Wiedenbeck, Phys. Rev. 111 (1958) 539 7) B. S. DBelepov and L. K. Peker, Decay Schemes of Radioactive Nuclei, Acad. Sci. USSR (1968) 8) D. Strominger, J, M. Hollander and G. T. Seaborg, Revs. Mod. Phys. 30 (1968) 686 9) G. Scharff-Goldhaber, D. Alburger, G. Harbottle and M. McKeown, Bull. Amer. Phys. Sot. 2 (1957) 26; preprint (1958) 10) A. Aten, G. de Feyfer, M. Sterk and A. Wapstra, Physica 21 (1965) 740, 990 11) F. McGowan, Report at International Conference on Nuclear Physics, Paris (July, 1968); F. McGowan and Stelson, Bull. Am. Phys. Sot. 3 (1968) 228 12) R. M. Diamond and J. M. Hollander, Nuclear Physics 8 (1958) 143 13) A. S. Davydov and W. S. Rostovsky, JETP 36 (1959) no. 6 14) K. P. Jacob, J. Mihelich, B. Harmatz and T, Handley, Bull. Am. Phys. Sot. 3 (1968) 558
12 (1969) liS-68;
@
North-Holland
Publishing
Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
RELATIVE
TRANSITION
ROTATIONAL
PROBABILITIES
LEVELS
A. S. DAVYDOV
OF NON-AXIAL
NUCLEI
and V. S. ROSTOVSKY
Moscow State University, Received
BETWEEN
1 January
Moscow 1959
Abstract: Energies and wave functions are calculated for the rotational states (J = 4, 6, 8) of non-axial nuclei. Relative reduced E2 transition probabilities for the rotational band levels of such nuclei are derived. Conditions under which the rotational states can be characterized by quantum numbers K are established. It is shown that if the nuclear shape differs from axial symmetry, the 1 : 3.3 : 7 : 12 interva1 rule observed in rotational bands of axial nuclei is violated. The theory is compared with the experiments! data.
1. Introduction Rotational levels in even nuclei were investigated by Davydov and Filippov 1) under the assumption that in a first approximation the equilibrium shape of the nucleus can be represented by a triaxial ellipsoid. Analytical expressions for the rotational level energies were derived for spins of 2, 3 and 6, and the transition probabilities between these levels were computed. It was demonstrated, in particular, that the theory yields a unique relation between the ratio of the energies of two spin 2 levels and the ratio of the reduced transition probabilities from the second spin 2 level to the first level (cascade transition) or directly to the ground state (direct transition), In the present paper some results of numerical calculations of the energies of rotational states with spins 4, 6, 8 are given as functions of the parameter y which specifies the deviation of the nuclear shape from axial symmetry “). The wave functions of these excited states and the transition probabilities between them are calculated. In section 4 we define the conditions under which rotational states can be described by approximate wave functions corresponding to states with a definite total momentum on axis 3 of the nucleus. In the same section approximate expressions are derived for reduced E2 probabilities between rotational nuclear states which only slightly differ from axial symmetry. The theory is compared with the experimental data in section 5. 2. Energy of Excited States Possessing
Spins of 4, 6 or 8
In the adiabatic approximation the rotational energy of a non-spherical even nucleus is defined by the Schroedinger equation: 58
RELATIVE
TRANSITION
(H-&)Y
69
PROBABILITIBS
= 0,
(2*1)
where E is measured in fi8(4B#9-1 units and the operator H is defined by
JA are projections of the total angular momentum operator on the axes of a coordinate system connected with the nucleus. The wave function Y which corresponds to a state with a total angular momentum J and satisfies the symmetry conditions established by Bohr *) can be written down as
where
In (2.4) the D&,r are functions of the Eulerian angles specifying the orientation of the principal nuclear axes in space. It can be shown that wave functions (2.3) form the basis of a completely symmetric representation of the group 92 (see ref.l)) whose elements are rotations through 180’ about each of the three principal nuclear axes. Inserting (2.3) in (2.1) and employing the values of the matrix elements of the rotational energy operator (2.2) on the wave functions (2.4), we obtain for each value of J a set of algebraic equations defining the values of the coefficients A, in the wave functions (1.3). For example, for J = 4 the Schroedinger equation (1.1) reduces to the set of equations [Ei(a+/?) -c]A,,+$dg(a-/?)A2 = 0, ~2/~(a-/?)Ao+[4(afB)+26--E]A,+~d~(a-_B)A, idY(a-_B)A,+[(a+@)+SS--e]A,
= 0,
(2.6)
= 0,
where a =
sir+(y--in),
p = sin-2(y+gz),
8 = sin+?.
The energies of the corresponding rotational states can be derived from the condition of solubility of such ,sets of equations. By solving each set (for corresponding values of e) one can determine the wave functions of these states. The wave functions (2.3) and the levels possessing spins 2, 3, 6 can be expressed in terms of parameter y by analytical formulas. The latter are given in ref.1). For the energies of spins 4, 6 and 8 levels algebraic equations of respectively third, fourth and fifth degree with respect to e are obtained. Thus, for example, the equation for the energy of spin 4 levels is
60
A. S. DAVYDOV
AND V. S. ROSTOVSKY
48 [27 + 26 sin2 3~1.5
g3 - - 90E2
sin2 3y
+
sin43y
640[27+7
-
sin23y]
o =.
sin4 3y
(2.6)
The explicit form of the equations for the energy of spin 6 and 8 levels has been presented in the previous paper of the authors 13). Results of numerical solution of these equations for several values of y are given in table 1. The number in parenthesis after E refers to the spin of the excited state and the subscript denotes the level number. TABLE 1 Dependence of rotationalenergy (inP/4Boa units)of even nucleus on parameter y Y0
0
G(4) s*(4) &a(4) ~(6) &a(6) %ts)
5
13.33 03 00 23 00 48
10
15
13.64 14.12 274.1 77.73 1056 268.1 28.42 29.51 270.6 93.51 48.60 50.26
3. Quadrupole
20
16.00 42.48 122.5 30.65 60.46 50.87
Electric
15.81 32.26 71.92 30.78 54.52 49.60
22.5
25
27.6
30
16.01 30.92 68.51 30.51 55.74 48.94
16.06 31.21 49.20 30.33 57.33 48.38
16.02 32.69 42.85 30.06 59.71 48.10
16 34 40 30 60 48
Transition
in the Rotational
Probabilities
Band
The reduced probability for electric quadrupole transitions between two states described by the functions PJmi and !PJ#,#r can be described by the expression 6 B(E2; i + f) = Z I W4C72pl.T~ij12~ 16~“(2J+ 1) pm’m where (3.13 8,,
=
(Ds~0s y + 3
eQo
(Dz2+D:,
-,)),
Q.
=‘3.
The wave functions for states with spins of 2 and 3 have been computed in ref.1). The coefficients A, for wave functions of states of spin 4 %2+%-2 Y4mi =
42
DL+D:,_,
+A4i
42
1
(3.2)
can be evaluated by solving the set of equations (2.5) for each of the roots of the equation (2.6) which defines the energy values of these levels. Values of coefficients A, for two spin 4 levels are presented in table 2. The values of the coefficients of the wave functions
for spin 6 states are also given in table 2.
RELATIVE TRANSITION PROBABILITIES
61
TABLE 2 Coefficients definingwave functions(3.2)and 3.3)of stateswith spinsof 4and 6 10 A 01
1
A a1 A., A 0s A ,*
0 0 0
A 4, &u 41 B.1 &I
0 1 0 0 0
1
1 0.003 IO-' -0.003 1 6x10-' 1 0.0076 10-L 10-1
0.000 0.030 104 -0.030 0.000 0.004 0.008 0.066 6X10_ lo-'
16
20
22.6
26
27.6
0.003 0.114 0.001 -0.114 0.003 0.015 0.973 0.232 8X10_" 6x10-6
0.066 0.296 0.010 -0.296 0.984 0.043 0.878 0.416 0.043 9x10.
0.910 0.414 0.022 -0.416 0.907 0.014 0.817 0.670 0.081 2.8x10-*
0.862 0.622 0.043 -0.523 0.842 0.128 0.766 0.633 0.113 6.1x10-*
0.792 0.606 0.076 -0.602 0.764 0.267 0.714 0.674 0.180 0.016
30
I
0.739 0.661 0.126 -0.669 0.500 0.661 0.672 0.606 0.264 0.031
By using (3.1) and the wave functions of states of spin 4 and 2 one can compute the reduced transition probabilities between these states. Thus, the reduced probabilities (in eaQoz/16nunits) for the quadrupole transition between spin 4 and spin 2 levels can be expressed by the formula b(E2; 4i + 2f) = &{cos
Y(f-%~*S~~&~,)
+sin r(2/1~A,,~i+A,ibi+1/35;4,,bi))~,
(3.4)
where ai, b, are coefficients specifying the wave functions of states possessing a spin of 2 (see ref. l)). The reduced probabilities for transitions between levels with a spin of 4 are given by the expression b(E2; Pi + 4f) = A{2 cos yC7A,IA,I-6A,,A,-2A,,A,,]
(3.5)
+ sin YC3~(A,IA~+~o*~*I)+~/21(A,IA,*+A,,A,*)l}~. Finally, the expressions for the reduced quadrupoIe electric transitions between levels with spins 4, 3 and 6, 4 are b(E2; 4i -+ 3) = +(21/i cos y - Asi+.& b(E2; 6i + 4f) = &
cos y[3d%$1~m+4~
+ sin y V5&A,,+
B,iA,f +1/=&i GO
A,+*B,,
~[&4,,-d?z/jA,,]}a,
B,,&+~B,~A,~I A,f+d49.6B,,A,I
(3.0) (3.‘) a II
.
For the sake of brevity we shah call the energy levels 0, 21, 41, 61, 81 levels of the “ground rotational band”. They are depicted in fig. 1 by solid lines. For y + 0 the levels of the ground rotational state go over to levels of an axially-symmetric nucleus. All other energy levels (depicted in fig. 1 by dashed lines) tend to infinity for y + 0. These energy levels will be called “anomalous” levels.
62
A. S. DAVYDOV
AND
V. S. ROSTOVSKY
The reduced electric quadrupole transition probabilities between different rotational states of even nuclei can be calculated by using the wave function coefficients given in table 2 and ref.l). The values of some of the reduced probabilities are given in table 3. In the present paper reduced probabilities for all transitions are expressed in e2Q02/16z units.
5
10
15
20
25
30
a"Fig. 1 TABLE
3
Reduced probabilities (in e*$&,*/Mz units) for electric quadrupole transitions between some rotational states of an even nucleus 0
Y0 b(E2; b(E2; b(E2; b(E2; b(E2; b(E2; b(E2; b(E2; b(E2;
3 41 41 42 42 42 42 61 61
+ + + -+ -+ -+ -+ --f -+
41) 21) 22) 21) 22) 3) 41) 41) 42)
0 1.429 0 0 0.596 1.333 0 1.673 0
5 0.0060 1.418 3.6 x lo-’ 4.1 x 10-a 0.591 1.323 0.0138 1.663 10-a
10 0.034 1.396 0.0023 0.011 0.576 1.282 0.0024 1.647 7.7 x lo-
T -
16
T -
20
22.5
T-
26
27.5
T -
*1 1I 1 0.130 1.377 0.010 0.008 0.643 1.172 0.167 1.662 0.036
0.406 1.372 0.033 4 x lo-’ 0.481 0.978 0.313 1.623 0.062
0.619 1.366 0.044 0.009 0.447 0.680 0.339 1.671 0.033
0.821 1.366 0.039 0.021 0.435 0.448 0.311 1.703 0.011
0.955 1.378 0.016 0.018 0.484 0.210 0.271 1.726 0.0023
0 0 0.596 0 0.273 1.731 0
i
From the results of ref. 1) and the data in table 3 it can be concluded that the reduced electric quadrupole transition probabilities between various rotational states of even nuclei can be divided into three types: a) Transitions for which the reduced probabilities are of the order of
RELATIVE
TRANSITION
63
PROBABILITIES
unity. Of this type are cascade transitions between levels of the ground rotational band and cascade transitions between “anomalous” rotational levels; examples are the transitions 3 + 22; 42 + 3; 42 + 22. b) Transitions between levels of the ground rotational band and anomalous rotational levels of different spin. Examples are the transitions 3 + 21; 41 + 22, 42 --f 2X; 61 --z 42. The reduced probabilities for such transitions are equal to zero for y = 0 or 30” and are very small for other values of y. c) Transitions between levels of identical spin. Examples are 22 --+ 21, 42 --f 41. The reduced probabilities for such transitions are equal to zero for y = Oand th en significantly increase with increase of y, attaining a maximum (of the order of unity) for y = 30’. The 3 + 41 transition also belongs to this type of transition. 4. Quantum Number K and K Forbiddenness The wave functions (2.3) for rotational states of even nuclei are represented by linear combinations t of the functions (2.4) corresponding to states with a definite value of the total nuclear angular momentum (quantum number K) projection on axis 3 in a coordinate system fixed to the nucleus. The magnitude of the coefficients determining the contribution of various terms to such a linear combination appreciably depends on the parameter y which characterizes the deviation of the shape of the nucleus from axial symmetry. It can be seen from table 2 and ref. 1) that for y < 15 the wave functions of the nuclear rotational states can be approximated by expressions which contain only one value of K. Thus, the wave functions of spin 2 states can be replaced by the approximate functions ypl = 120) = y& = 122) =
5 4 g--g %I~
() (
&
(4.1)
i(qL+x,-P). 1
The wave functions of spin 4 states can be approximated
by the expressions
etc. t Only the function for a rotational state with a spin of 3 corresponds to a definite value of K = 2 (see ref. I)).
64
A.
S. DAVYDOV
AND
V. S. ROSTOVSKY
In those cases (y < 16”) when the rotational motion of the nucleus can be approximately described by wave functions of the (4.1) (4.2) type the rotational state of an even nucleus can be characterized by two quantum numbers, J and K. In this approximation values K = 0 correspond to levels of the “ground rotational band”. Anomalous rotational levels can then be divided into a set of levels with K = 2, 4, 6, . . . The non-vanishing reduced probabilities (in our e2Q,/l&z units} for electric quadrupole transitions between states described by the approximate functions IJK) have the form b,(EZ; JK + J’K)
= 5 cos”y(2JOK/J’K)“,
b,(E2; JK + J’K+2)
= $(l+s,,)
sin2y(2J2K/ J’K+2)2,
b,(E2; JK + J’K-2)
= ~(l+&,)
sin2r(2J-2KJ
J’K-2)2.
The rules given at the end of the preceding section reduce in the approximation under consideration to the condition of validity of the equality AK=0
(44
for the most probable transitions. Those transitions which occur when conditions (4.3) are not satisfied are usually called K-forbidden transitions. Since quantum number K is only approximate, rule (3.3) can be important only for nuclei with y < 15’. If y > MO, the exact functions (2.3) should be employed. The results obtained in section 3 permit one to estimate the error which arises when the approximate functions )JK) are substituted for the exact functions (1.3). It should be mentioned that in analyzing the data on the relative transition intensities a number of authors 3--6)were compelled to assign a quantum number K = 2 to some of the levels possessing spins 3 and 2. Such excited states were ascribed by the authors to the so-called y-vibrations, the frequency of the latter being chosen arbitrarily. However, if one views such states as rotational states of a non-axial nucleus a simple explanation of the spacing of the levels and of the relative transition probabilities between them can be proposed. In this case, only a single parameter y which is uniquely defined by the ratio of the energies of two spin 2 levels is involved in the theory (see the following section). 5. Comparison
with Experiment
In order to facilitate comparison of the results with the experimental data, plots of the dependence of parameter y on the ratios of the rotational level energies for various spins are shown in fig. 1. The parameter y can be uniquely determined from the ratio of the energies of two spin 2 levels with the aid of the formula
RELATIVE
E,(2) p= El (2)
TRANSITION
PROBABILITIES
l+dl--_%sinB3y. l-dl-i
sin*3y
66
w
After determining in this way the value of y, one can use fig. 1 to find the position of the rotational levels of even nuclei. It can be seen from fig. 1 that as a result of deviation of the shape of the nucleus from axial symmetry the interval rule 1 : 3.3 : 7 : 12 which is observed in the rotational band of axially symmetric nuclei is violated in the “ground rotational band”. Thus, for example, for y = 30” the levels of the “ground rotational band” should satisfy the interval rule 1 : 2.67 : 6 : 8. In fig. 1 the circles denote the experimental values of the ratio of the excitation energies of Oslso, OS”~~,Dy18*, UBa, Puss8 7), Um “), 0~186,0~18814) and Erl66, Er168l4) nuclei to the energies of the first excited level of these nuclei. The figure shows that the theory yields spin sequences and energy ratios which are in good agreement with the experimental data. The slight difference between the experimental points and theoretical values can be removed by introducing a correction factor
which accounts for the relation between rotation and internal excitation of the nucleus. The magnitude of this correotion factor may serve as a criterion of validity of the adiabatic approximation. It should be noted that deviations of the experimental energy ratios from the interval rule 1 : 3.3 : 7 : 12 was wholly ascribed to violation of adiabaticity when the rotational spectra were analyzed from the viewpoint of axial symmetry of the nucleus. As a consequence the significance of the correction term (6.2) was overestimated. As a matter of fact deviation of axial nuclei from the interval rule is largely due to violation of axial symmetry of the nucleus. In this respect the experimental data on the 2+, 4+, 6+ and 8+ levels of the Osfe* nucleus are especially interesting. The positions of these levels have been determined by ScharffGoldhaber, Alburger, Harbottle and McKeown ‘) and Aten, de Feyfer, Sterk and Wapstra 10) who studied cascade y transitions in the decay of the 10 min 0~190isomer with a spin lo-. The aforementioned authors 9110)note that the experimental spin sequence satisfactorily agrees with the spin sequence of the levels in the HP** nucleus which possesses a pronounced rotational spectrum. However, the observed energy ratios were found to be appreciably different from the theoretical values for axially symmetric nuclei. It is also noted in the papers cited above that even the introduction of the correction (6.2) does not petit one to attain the experimental value 1 : 2.93 : 6.62 : 8.93. Apart from a change in the interval rule for the levels of the ground rotational band, violation of axial symmetry of the nucleus leads to the appear-
A.
06
S. DAVYDOV
AND
V.
S. ROSTOVSKY
ante of new rotational (“anomalous”) levels. Such are, for example, the second levels with spins of 2 and 4 in the Oslo0 nucleus. These levels are not noticeable in decay of the 10 minute OsIQOisomer but exhibit themselves in K-capture decay of Ir lQo. The peculiarities of the spectrum of the OPO nucleus are related to the large value of y = 21.4’ which corresponds to the experimental energy ratio &(2)/J&(2) = 3.15. With the value y = 21.4 the reduced probabilities and relative electric quadrupole transition probabilities can be calculated for the Oslo0 nucleus. The values are given in table 4; TABLE
4
ReIative transition probabilities between various levels in the Osl*O nucleus
E2 transitions
21+ 41 + 22 + 22-+ 22 4 3 + 3+ 3 --f 61 -+ 42 + 42 + 42 + 42+ 42 +
0 21 21 0 41 21 22 41 41 22 41 21 3 61
Transition energy &eV) 186 360 400 586 40 604 204 244 600 640
580 940 336 80
Reduced probabilities in ePQ,a/16n units 0.934 1.33 0.467 0.066 0.071 0.18 1.60 0.516 1.6 0.461 0.319 0.007 0.813 0.06
Relative probability
1 39.3 21.5 21.9 3.5 x 10-b 69 2.9 2.14 251 100 100 24.6 16.5 9 x LO-’
transitions referring to a theoretical level of spin 3 (which so far has not been observed experimentally) are also included. Using OP” as an example we have demonstrated that the theory permits one to calculate the relative electric quadrupole transition probabilities between all rotational states. The values for the relative transition intensities thus obtained yield a qualitative explanation of the observed decay schemes of the excited states of the Osreo nucleus. For example, decay of the isomer state (spin 10-J of the OP” nucleus occurs only as a result of a series of cascade transitions via rotational levels of the ground rotational band without noticeable excitation of anomalous rotational levels. The 4+ and 2+ anomalous rotational levels are excited in K capture of the Irlao nucleus. From table 4 it follows that by emitting a E2 y quantum a nucleus can go over from an excited state corresponding to the 4+ anomalous rotational level to either a rotational state of spin 4 in the ground rotational band with subsequent cascade emission of two y quanta or (with about the same probability) to a state 2+ corresponding to the anomalous rotational band. From an anoma-
I&LATIVE
~mNsmoR
67
PRCIEUAYUL~T~E~
lous spin 2 level the transition can occur either to a state of spin 2 of the ground rotational band or (with about equal probability) directly to the ground state. According to the theory the OslSOnucleus should possess an excited state (about 790 keV) with a spin of 3. However, as can be seen from the table, excitation of this rotational state is not probable inasmuch as transition to other rotational states are much more probable from the (spin 4 and 6) levels located above it. This may explain why this rotational state hitherto has not yet been detected. A comparison between the theoretical values of the transition probabilities and experimental values has been presented in ref. 1). Since this paper was published, new experimental data on the ratio of the cascade (22 + 21) and direct (22 + 0) reduced probabilities have appeared. The experimental data of McGowan 11)and Nathan 4) and the theoretical values obtained on basis of the formula b(E2; 22 --f 21) b(E2; 22 + 0)
=
20 sin2 3y
7{9-8 sin*3y---3-2
sins3yld9-8
sin*3y}
are given in table 5. Ratio of reduced quadrupole
Nuclei
transition probabilities for various nuclei
E,(2) E,(2)
Y0
16.4 8.11 8.02 4.08 2.98 2.37 2.23 6.0
10.3 14.0 14.1 19.3 21.4 26.4 20.3 16.2
b(E2; 22 + 21)
b(E2; 22 -+ 21)
b(E2; 22 + 0)
b(E2; 21 3 0)
them
*
1.7 2.0 3.0 4.5 7.1 21 40 2.7
exp 1.8 1.94 2.30 2.7 9.7 9.1 30 2.38
them
exp
0.062 -
0.014 -
0.18 0.32 0.60 0.90 1.29 0.16
0.12 0.19 0.62 0.68 2.8 0.12
The parameter y was computed from the ratio (6.1). The experimental data lr) and theoretical values of b(E2; 22 -+ 21)
20 sins3y
b(E2; 21+ 0) = 7{[9--8 sin*3y+ 13-2 sins(3y)]2/9-8
sin*3y}
for the ratio of the probabilities of transitions (22 -+ 21) and (21+ 0) are given in the same table. It can be concluded from tabIe 6 that the theory yields a unique relation between the indicated reduced transition probability ratios and ratio of the energies of both spin 2 levels.
68
A,
S.
DAVYDOV
AND
V.
53.
ROSTOVSKY
Nathan and Waggoner “) have presented experimental values for the ratio of the intensities of the E2 transitions 3 + 41 and 3 + 21 in the Smls2 nucleus which imply that the experimental ratio of the reduced probabilities b(E2; 3 + 41) = 1.88.
i b(E2; 3 + 21) Iexp
From the ratio E2(2)/E,(2) = 8.9 for the Smfb2 nucleus it follows that y = 13.5’ and the theory thus yields b(E2; 3 + 41) b(E2; 3 +
21) theor
= 1.37,
which is in good agreement with the experimental ratio. References 1) A. S. Davydov and G. F. Filippov JETP 35 (1968) 440; (lQ58) 703; Nuclear Physics 8 (1968) 237 2) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1962); A. Bohr and B. Mottelson, Mat, Fys. Meld. Dan. Vid. Selsk. 27, No. 16 (1953) 3) 0. Nathan and M. Waggoner, Nuclear Physics 2 (1966/67) 548 4) 0. Nathan, Nuclear Physics 4 (1957) 126 5) J. Juliano and F. Stephens, Phys. Rev. 108 (1958) 341 6) G. D. Hickman and M. Wiedenbeck, Phys. Rev. 111 (1958) 539 7) B. S. DBelepov and L. K. Peker, Decay Schemes of Radioactive Nuclei, Acad. Sci. USSR (1968) 8) D. Strominger, J, M. Hollander and G. T. Seaborg, Revs. Mod. Phys. 30 (1968) 686 9) G. Scharff-Goldhaber, D. Alburger, G. Harbottle and M. McKeown, Bull. Amer. Phys. Sot. 2 (1957) 26; preprint (1958) 10) A. Aten, G. de Feyfer, M. Sterk and A. Wapstra, Physica 21 (1965) 740, 990 11) F. McGowan, Report at International Conference on Nuclear Physics, Paris (July, 1968); F. McGowan and Stelson, Bull. Am. Phys. Sot. 3 (1968) 228 12) R. M. Diamond and J. M. Hollander, Nuclear Physics 8 (1958) 143 13) A. S. Davydov and W. S. Rostovsky, JETP 36 (1959) no. 6 14) K. P. Jacob, J. Mihelich, B. Harmatz and T, Handley, Bull. Am. Phys. Sot. 3 (1968) 558