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The calculation of nuclear rotational states based on the generalized Hartree-Fock approximation
Nuclear Physics A97 (1967) 298--320; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
T H E C A L C U L A T I O N OF N U C L E A R R O T A T I O N A L STATES BASED
ON THE GENERALIZED
HARTREE-FOCK APPROXIMATION
MITSUO SANO t
Western Reserve University, Cleveland, Ohio
and MASAMICHI WAKAI Osaka University, Toyonaka, Osaka
Received 28 November 1966 Abstract: The generalized Hartree-Fock method is applied to a description of rotating nuclei, and
the generalized Hartree-Fock equation is solved with the aid of self-consistent perturbation calculations. Numerical calculations are carried out for the rare-earth nuclei, and the results are compared with experiment. It is shown that good agreement with experiment is obtained.
1. Introduction
The successful method of the theory of superconductivity 1) has been applied to the study of nuclear structure z). It permits a microscopic treatment and has been proved to be an elegant and powerful method in explaining the properties of the ground and low-lying excited states of various nuclei. The moment of inertia which can be defined from the rotational sequences of the low-lying energy levels had been difficult to reproduce for a long time 3). However, it was found that a consideration of the effect of the pairing correlation gives theoretical values of the moment of inertia in satisfactory agreement with experiment 4). The deformed nucleus might have another typical property similar to the superconducting system, because the Coriolis force prevents the formation of coupled pairs. This corresponds to the case of the external magnetic field applied to superconducting electron systems whose pairing correlation is decoupled by the field. It is expected that there is a critical value of the angular momentum, i.e. an upper bound of the angular m o m e n t u m for which the nucleus can be in the superconducting phase. This problem was first pointed out by Mottelson and Valatin 5) and afterwards various numerical calculations were carried out by many authors 6-11). However, none of these authors performed numerical calculations to an extent which is comparable to experimental results. The purposes of the present paper are to provide a description of rotating nuclei by using the procedure of the generalized Hartree-Fock approximation and to perform t Senior Foreign Scientist Fellow of the National Science Foundation. Present address: University of California, Davis, California. 298
NUCLEAR ROTATIONAL STATES
299
detailed numerical calculations of the rotational states. The method of the description is almost the same as previously described 7), but it has been revised so that the selfconsistent calculation is possible within the generalized Hartree-Fock approximation. In sect. 2, the generalized Hartree-Fock equation is derived, and it is solved in sect. 3 with the aid of self-consistent perturbation calculations. The results of the numerical calculation are given in sect. 4. 2. Generalized Hartree-Foek equation in rotating system We take a spherically symmetric shell-model Hamiltonian with chemical potential 2 =
l/~p~ e~ cp ca c~.
(1)
~B76
H e r e , . , fl, etc. represent a complete set of the shell-model states, c+ and c~ are the creation and annihilation operators of a nucleon in the s t a t e , and e~ the energy of a single particle state ~. The potential V is assumed to be spherically symmetric with the matrix elements obeying the antisymmetry condition V~p,a = - Ve=,a = - V,pay = V~=p.
(2)
In investigating the rotational motion in deformed nuclei, it is necessary first to find the nuclear equilibrium shape in the ground state. This problem has been investigated by Baranger et al. 12). They have rewritten the generalized Hartree-Fock equation into a form especially suited for the nuclear system and applied it to the description of the ground state. We shall extend their method to a system rotating with an angular velocity co and obtain a description of intrinsic particles coupled to a rotating field. We first assume that the nuclear equilibrium shape in the ground state is axially symmetric. This assumption is consistent with the results obtained by Baranger et al. and also with experimental evidence. Then the rotational spectrum of the ground band is that of a rotation about the x-axis chosen perpendicular to the symmetry axis z of the nucleus. In the rotating frame of reference, we search for the lowest state of the system with a fixed average value of the angular momentum about the x-axis. In this case we must add to eq. (1) a term i-i
= -coax
= - c o Z < lJxlP>c +ce •
=#
(3)
in order to obtain the lowest state by taking into account the pairing correlation for the auxiliary Hamiltonian Jt ~ = H + H~,,
(4)
we perform the generalized Bogolyubov transformation 13). The generalized Bogolyubov transformation gives a complete set of fermion operators called quasi-particle operators, which are the most general linear combinations of c + and c operators a~ = Z ( A , c ,'
+ +B~,c~,). '
(5)
300
M. S A N O A N D M. W A K A I
The requirement that the transformation (5) is canonical entails the orthogonality relations ~ ( A ~i*A ,j+ B , B ~ ( A =i B ~j + B , Ai ~ )j = 0 , i* , )j = 6,j, E
i i* i (A:A ei* +B:Be) = 6:e ,
Z (A~B , ei. + B=,. Ae) , = O.
i
i
(6)
The idea is to make the new vacuum state Oo(~O) similar to an independent-particle state, in the sense that it is defined by a~(o)Oo(m) = 0.
(7)
We can obtain the generalized Hartree-Fock equations from the requirement that the expectation value of the auxiliary Hamiltonian ~ in the new vacuum state Oo(O9) must have a minimum value. Then A and B satisfy the following equations: i
7 i
*
*
i
*
i
i
i ) B = - Z (F,~ - ooJ,~)B~- E~ A~a A e = Ei B~
-
(8a)
The complex conjugate of these equations becomes
(~ _OB':+
E (F,~-oJ,~)B~" + E A,eAe,. = - E i B ~,., o . i* -- Z i* -- (~,-- +OA: * ooJ~)A, , i, - E ,=A~e*~'eni* = - E, A~. (F~,-e
(8b)
The quantities F and A are defined by A=e = -Ae= = 2 ~ V=p~#c~,
(9a)
76
F~, = F~,* = 4 ~ V~,e~pp6, eo
(9b)
where K~ = -~c~, =
(9c)
i
Pfla = Po5 = (Oolcp-%]Oo> = 2
'*BpB6" '
(9d)
i
We see that F ~ is the potential arising from the density p of the particles and is the familiar Hartree-Fock self-consistent potential. On the other hand, /c~n is the wave function of the Cooper bound pair state and gives rise to the pairing potential A~fl. The expectation value of the Hamiltonian dgCis given by
=
E
Z
E
Z
a
a7
~fl
~'
(lo)
The chemical potential 2 is determined by fixing the average number of particles JV" = ( E c+%> = E P,,"
(11)
N U C L E A R ROTATIONAL STATES
301
The angular velocity co must be determined from the following self-consistency condition: x / I ( I + l ) = (dx> = E J, aP=a - a)J(a)).
(12)
This equation gives a definition of the moment of inertia J ( a ) ) . Using eqs. (8) and (10), we obtain the energy of the system with a fixed angular momentum I W(I) = ( J f +LA/'+coJx) = U(I)+
I(I+1) 2J(I) '
(13)
where
U(I) = ½ 2 (e~ + 2-Ei)B~B~. i i*
(14)
i~t
The energy U(I) is the intrinsic energy of the nuclear system with the angular momentum I. The ground state energy W(0) is obtained by putting I = 0 into eq. (13). Thus the excitation energy corresponding to the angular momentum I is given by
~(I) = W(I)- W(O) = U(I)- U(O) + 1(1+ 1) 2J(I)
(15)
For deformed nuclei, the average single-particle field calculated from the generalized Hartree-Fock equations must be already non-spherical. The last term on the righthand side in eq. (15) has the form which gives the usual rotational energy of a deformed nucleus, while the other terms give the variation of the intrinsic energy due to the rotational motion. It is to be noted that the moment of inertia is also a function of the angular momentum.
3. Perturbation approach 3.1. SELF-CONSISTENT PERTURBATION OF THE GENERALIZED HARTREE-FOCK EQUATION The generalized Hartree-Fock equations must be solved self-consistently by taking into account the conditions (11) and (12). However, it is not so easy to carry out such calculations. In order to investigate the effects of the Coriolis force on the rotational spectrum one may solve eq. (8) by using a perturbation calculation together with some other simplifying assumptions. We thus first assume that the rotation is slow so that the Coriolis force can be treated as a perturbation. In addition, we assume that the pairing potential (9a) is a constant and further that each of the two states degenerate in energy is the time reverse of the other. This is the same simple situation as in the BCS theory a). The pairing potential can now be written as
A~,p = 6~,?A,
(16)
302
M. SANO AND M. WAKAI
with A = - ½ G ~ s,K,, _.,.
(17)
7
Here fi means the state corresponding to the time reversal of state 3 and s~ the phase factor related to the time reversal of the state 7- The state - 7 is defined as having the same quantum numbers as 7, except for the magnetic quantum number which has the opposite sign. With the further substitution,
C'~ = s~B~_~,
(18)
the generalized Hartree-Fock equation (8a) is rewritten as i i i i X)Ao + E Foc,Ay + ACo~-co ~ J~,A~, = E, 7
-
"
i
y
i
7 *
i
i
i
F~.~C~,+ AAo~-co ~ J<, C~
],
= e,
i
C,
(19)
y
The Hartree-Fock potential F is not invariant under the time reversal. In the perturbation calculation, the density matrix p can be expressed in terms of powers of co. Then it is easily able to divide the potential F into two parts, i.e. an invariant part and the other part with opposite sign under the time reversal
F = F(°)+F ',
T F T -1 = F ( ° ) - F '.
(20)
The angular m o m e n t u m operators are odd under the time reversal. Therefore F (°) and F ' are the potentials arising from the densities which have even and odd powers of co, respectively. The form of eq. (19) suggests the introduction of the matrix
M = Mo+M' = \
A,
-(e-2)-F
¢°) + \
0,
F -coJx
"
Eq. (19) is the eigenvalue equation for the matrix M. Here, we define ~ as an eigenvector of M with a positive eigenvalue E, and write it as
C' "
(22)
The procedure of the calculation is the following: We first assume that the values of F (°) and A in M (°) are known for a given co and treat M ' as a small perturbation. We thus re-determine F (°) and A from the self-consistency requirement. For sufficiently small o~, the use of a perturbation calculation may be justified. Near the critical rotation at which the energy gap parameter A vanishes, however, the rotational energy related to marx is not small in comparison with the pairing energy. The use of the perturbation calculation may still be justified even for larger values of the rotational frequency co, provided coarx is smaller than the quasi-particle energies given by eq. (19).
303
NUCLEAR ROTATIONAL STATES
The unperturbed equations can be written M o O i+ = E i + ~ ) i+,
(23a)
mo ~i-
(23b)
= El_ dl)i-,
where E~+ = E, and E~- = -E~. The function ~+ is an eigenvector of Mo with positive eigenvalue E,, and ~bi- an eigenvector of Mo with negative eigenvalue - - E i. It is possible to write them as follows: •
iA(O,,
d?'+ =
\c(O)i],
=
~
A(O)i. ] .
(24)
The solution of eq. (23), as has been shown by Baranger a2), is
A~O)i = W~u~,
C(=°)~
=
W ail ) i ,
(25)
with
ui = v/½(1 + e~/Ei),
v, = x/½(1 - ei/E~),
Ei = ~/e2 + A 2,
(26) (27)
where e i and W~ are the solutions of the eigenvalue equation =
r(°)W i
(28)
This equation involves only the Hartree-Fock potential F (°) and determines the energies e~ of odd-particle states in deformed nuclei. The states i defined by (28) are essentially the Nilsson states in a deformed well 14). We use a Latin subscript i to represent all quantum numbers of the Nilsson state. The Nilsson state ri> is now expanded in terms of state ]~> in the spherical well as li> = Z W~t~>.
(29)
at
The unitary transformation matrix elements W~ are just the eigenvectors of eq. (28) and satisfy the following orthogonality relations:
Y wjw£= 6u, Z
i
i*
w; w~
(30)
= 6~ .
i
The explicit definition of the time reversal is ~ Tli> = I'f) = 2 wii~> = E W~l~> = E W:~i* 1~>,
at
(31)
at
where the time-reflection operator T is a product of a unitary operator and a complex conjugate operator. From eq. (31) W~ satisfies the relation
W/* =
i --i W~ = s i s a W _ a .
304
M. S A N O A N D M. W A K A I
The phase factors s~ and s i satisfy, by definition, the relations 2
1,
s~s_~
- 1,
sis_ i = - 1 .
s~ = 1,
We can now transform the matrix elements of various operators in the spherical shell-model representation to those in the Nilsson representation by using eq. (29). The matrix element of the angular momentum operator is given in the Nilsson representation as
Wd j~a W~ =. Employing the phase convention implied by eq. (31), we readily obtain the relations
= * = - <]l J~]~>.
(32)
We must further mention the fact that the matrix elements ofj~ are real in the representation employed here. In the Nilsson representation, the eigenvector ~ can be expanded as follows:
~i = ~i + ~ (q~S[M'lq~i> q~i+ ~ <~ k[M'[q~j>(q~j[M'[~i> t~k Ej (E,- Ej)(z,- ½ ~ <4~JlM'I~>2 ~ ' + . . . . •
(33)
(El--El) 2
where the sum must run over both the positive and negative energy states given by eq. (23). The matrix elements in eq. (33) are calculated by using eqs. (24) and (25) and become as follows:
<~kilM,l~k> =
- ~oJ~(- v, u~ + u, vk),
- o J,k(u,u +
E~ E~ E~ Ei
> O, > O, < O,
E k > O,
< O,
E k < O,
E k < O,
(34)
E k > O,
where
Jx = Jx-- --1F' .
(35)
Using eq. (34), the terms of eq. (33) that are linear and quadratic in co are easily calculated and are given as follows: first-order terms
-o Jj,
+ e,,,,+ ,;,,) wL
(36a)
- osj,
+E,v,- jv,)wL
(36b)
C(1)~ = ~ ~ - ~ •
E,
-
Ei
NUCLEARROTATIONALSTATES
305
second-order terms
092Jkj Jji A<:2" = ~,Zk(e~ - e~)(F4 - E~,) ["' {e~' + E, + + el ~k + a ~ + ~ ~j}
+ vi{ZE i A + Aek-- Aei}] Wff
2
2e, u, Y' . (e~-E~)
C~2>i
E T-k (E 2 -E,)(E:2
ul {(E,+ E y - 4 E I E
a • ½ (1
~
W~, eiE~
(36c)
/J
2 -E,.~2 + ui{2Ei A - Aek + Ae/}] W)
+
v ~'24& {va(~,-+) t i
__l ~j
0")2j2" Ui {(Ei~-Ej) - 4 E i E , ' ½ (1
. (~}-e~)
~
AE+eleYl} Wi. e,~
(36d)
/
3.2. THE EFFECT OF ROTATION ON PAIR CORRELATION We shall take up to the second-order contribution of e>Jx into account in calculating the gap equation (17). In this equation, terms of first-order in ~o], do not appear because they are odd under the time reversal. As the result of the perturbation calculation, eq. (17) becomes 4
1 (1_o92
(37)
where
Xij - (El q-E j)2 1
EiE i / + Ei+E i
EZEy
Eq. (37) gives an idea of how A depends on ~o. Since the factor in bracket in the right hand side of eq. (37) decreases with increased co, it is clear that El and consequently A has to be decreased in order to keep the value of the right-hand side a constant. Therefore, it is expected that there is a critical value of the angular velocity, coc, above which the nucleus ceases to be superconducting. The critical .angular velocity ~% can be determined from the solution of the eigenvalue problem of the equation obtained from eq. (37) by dropping A. Within the same approximation, the average number of the nucleons given by eq. (11) becomes ,/U = ,Ar(°) + j V "(2) =
• ~, + ~ '
..~
2
l,J
o
(e,+E/
(1
E, Ej /
iJ
~2 (<+E~) ~ , E j ,
.(38)
306
M. SANO A N D M. W A K A I
I f we put A --- 0 in eq. (38), the second term in the right-hand side vanishes and we obtain formulae familiar in the usual Nilsson model without the pairing correlation. 3.3. MOMENT OF INERTIA We can determine from eq. (12) the value of the angular velocity o9 corresponding to a fixed angular m o m e n t u m I and also can obtain the formula for the moment of inertia from that. Taking into account up to the first order in o9 expansion of the density p, we obtain j
= _1 ~ j ~ p ~ ,
=
-
"" E i+Ej
EiEj /
,
~o < 09c .
(39)
If we neglect the term F ' in the matrix element Jjl, this is the expression first obtained by Belyaev 2), which gives the generalized cranking model formula in which the energy denominators contain the quasi-particle energies. However, in our case the moment of inertia is a function of the angular velocity, i.e. the angular m o m e n t u m because of the effect of rotation on the energy gap A. The reduction of the energy gap A results in an increase in the moment of inertia. The expression (39) reduces for A = 0 to the cranking model formula 3)
j = ~ J.u,fJi, ij
~i --
(e, > eF > e~),
09 > ¢o¢,
(40)
8j
where eF means the Fermi energy. This also gives the expression for the moment of inertia associated with the rotation of a rigid body is). The numerical value of the moment of inertia for A = 0 is a few times larger than the value for o) = 0. Therefore, for a sufficiently large value of the angular m o m e n t u m the rotational spectrum deviates greatly from the I(I+ 1) spectrum which is the case with a constant value of the moment of inertia. 3.4. CHANGES OF INTRINSIC ENERGY DUE TO THE ROTATION AND EXCITATION ENERGY The intrinsic energy given by eq. (14) is a function of the angular m o m e n t u m and, as the result of the perturbation calculation, becomes
U(I) = ½ E (e~,+J.-EilB~,B~, ' '* = U(°~(I)+ u(E~(I),
(41)
where
Uw)(I) = ½ Z (~. + 2 - F..'~tq~°)'Rw3i" _ ~ o ~(o)+½ P r(°).(°)-A2/G,
(42a)
NUCLEAR ROTATIONAL STATES
U(2)(I) =
8
L(Ei+Ej)E~E~
~
307
(EI+Ej)E?E~
EiE~. / J
(42b)
Eq. (42a) is o b t a i n e d by using eqs. (23) a n d (17). The s e c o n d - o r d e r terms o f the intrinsic energy given by (42b) b e c o m e zero a b o v e the critical a n g u l a r m o m e n t u m Ic, because o f the energy g a p p a r a m e t e r A = 0. W e n o w discuss some characteristic features o f the change o f the H a r t r e e - F o c k energy due to the Coriolis force with the aid o f some simplifying assumptions. W e
>.i I,~ Z I,d
,~c •
will
J' °I"C '
a - - 'l
!
I
Z
(2_
,
........
~s !
E--
I
,, .........
X i,i
"~i,,
-----'-~.I= 0 SUPER
,
I
~rgA2(o)
NORMAL
Fig. 1. A schematic illustration of the rotational level scheme. The complete spectrum consists of two parts. One is the levels in the superconducting phase shown in the left half of the figure, the other is the levels in the normal phase shown in the right half. The levels depicted by broken lines, in fact, are absent in the presence of the pairing interaction. The system is not found in these states, rather it is found in the superconducting state with energy corresponding to the same angular momentum. The dotted line connects two levels, normal and superconducting, whose energies correspond to the same angular momentum.
assume t h a t (i) the p a i r i n g interaction is a c o n s t a n t G within an energy interval a r o u n d the F e r m i surface a n d equals zero otherwise, (ii) the sum over the Nilsson state i can be a p p r o x i m a t e d b y integral with the single-particle level density set equal to a constant g, (iii) changes o f the H a r t r e e - F o c k self-consistent p o t e n t i a l F with respect to the change o f ~o is small a n d can be neglected. E v e n u n d e r these r a t h e r crude a s s u m p tions a significant d e p a r t u r e f r o m the p r e d i c t i o n s o f the shell m o d e l w i t h o u t the p a i r ing i n t e r a c t i o n is expected a n d q u a l i t a t i v e features o f the p a i r i n g system can b e e x a m ined.
308
M. SANO AND M. WAKAI
Above the critical angular momentum Ic, the change of the intrinsic energy becomes
U(I)-U(O) = U(°)(I) - U(O) ~- ½9A2(0),
I > Ic.
(43)
The energy gap A(0) is that in the ground state. The change of the Hartree-Fock energy given by eq. (43), which is a constant above the transition energy 8c corresponding to the critical angular momentum, is equal to the lowering of the ground state energy due to the pairing interaction. Therefore the excitation energy above the critical angular momentum is given by
o~(I) = U(°)(I)- U(O)+ I(I+ 1) _~ ½gA2(O) + I(I+ 1) 2Jrlg
2'frig
I > Io. '
(44)
=
Below the transition energy the nucleus is in the superconducting phase and the excitation energy is a complicated function of the angular momentum
~(1) = U ( I ) - U(O) + I(I + 1) 2J(I) '
I < Ic.
(45)
The transition from the super to normal phase is illustrated in fig. 1. The left half of this figure represents the levels of the rotating nuclear system in the superconducting phase, while the remaining part represents those expected for the normal phase. The broken lines join the levels with the energies corresponding to the same angular momentum. The difference in the energy of the corresponding levels decreases with the increase of the angular momentum as the correlated nucleon pairs are broken by the Coriolis interaction, or in other words, as the number of elementary excitations increases. The difference vanishes at the critical angular momentum where the energy gap also vanishes, and above this critical angular momentum the normal phase is reached. 4. Results of the numerical calculations
4.1. METHOD OF CALCULATION We provided a description of rotating nuclei by using the procedure of the generalized Hartree-Fock approximation and obtained the generalized Hartree-Fock equation (8). It appears to be difficult to solve eq. (8) under the requirement of selfconsistency. Then we carried out the calculations to solve the generalized HartreeFock equation (8) by using the procedure of the self-consistent perturbation method. The self-consistent perturbation method mentioned in the sect. 3 differs in several respects from that of the usual perturbation calculation. In the former case, the calculation should be self-consistent within the generalized Hartree-Fock approximation. Then the changes of the Hartree-Fock potential F and the pairing potential A are taken into account within the procedures of calculation self-consistently. The formalism can be used for any nucleon-nucleon potential. A critical question is the choice
N U C L E A R R O T A T I O N A L STATES
309
of the nucleon-nucleon potential. We have already assumed the pairing potential as a constant. This gives the same simplification as in the BCS theory. The pairing-plusquadrupole model has been proposed by the Copenhagen school for the purpose of producing pairing effects and quadrupole deformations. This model has fairly well succeeded in explaining experiments at low energy, in spite of the fact that these forces do not look like the real nucleon-nucleon interaction. The strength of the pairing interaction on the basis of the odd-even mass difference has been analysed systematically, and the moment of inertia and the gyromagnetic factor have been calculated by Nilsson and Prior and by Griffin and Rich 4). The nuclear equilibrium deformations based on the pairing-plus-quadrupole model also have been investigated in detail by B6s and Szymanski 16) and by Baranger and Kumar 12). These authors have not only succeeded in explaining qualitative features of the ground and low-lying states in the deformed region, but they have carried out rather successfully a number of quantitative tests. We adopt the pairing-plus-quadrupole forces. Then eq. (28) which determines the single-particle state becomes
ei6ij = e ~ ° ) - 2 6 i j - Z Z q~/" QM.
(46)
M
The first part dji°) is spherically symmetric and is expressed in the Nilsson model by (0) _
ji
1
2M
(pa)ji+½M(.oo(rZ)ji_hO)ot¢[2(l
" S)_l~lZ]ji '
(47)
where M is the nucleon mass, o) o the harmonic oscillator frequency and ~c and # the constants which are chosen in order to have the correct level sequence in spherical nuclei. The last term in eq. (46) denotes the matrix elements of the deformed part by a product of the single-particle and the total nuclear quadrupole moments, and X is the coupling constant of the quadrupole force. The single-particle and the total nuclear quadrupole moments are given by
q~ = x/Js6-n(jlr 2 Y2MIi),
(48)
QM = ~/-16n Z
(49)
po
We now proceed as follows: In the numerical calculation, we consider only the axial symmetric fl-deformation of the nucleus and will not consider any effect of the 7-deformation. Therefore, only a M = 0 component is taken into account in eq. (46). The coupling constant Z must be determined so as to give the mass quadrupole moment or the nuclear equilibrium deformation in the ground state. In the ground state the quadrupole moment becomes
Qo = x/-167z Ev] • i
(50)
310
M. SANOAND M. WAKAI
Therefore, Nilsson's d e f o r m a t i o n p a r a m e t e r fi is related to the q u a d r u p o l e matrix element Qo a n d the coupling c o n s t a n t X by the e q u a t i o n ZOo = -~fi Mo92 .
(51)
The experimentally k n o w n d e f o r m a t i o n parameters 3 are listed in table 1. Eq. (46) should be diagonalized exactly by including the matrix elements between states b e l o n g i n g to different shells. I n this diagonalization, however, matrix elements between single-particle states of different principal q u a n t u m n u m b e r N are p u t equal to TABLE 1 Deformation parameters Nucleus xs~Sm 15~Sm a54Gd 156Gd aSSGd X~°Gd lS°Dy X6ZDy l~4Dy X64Er 166Er lnSEr XT°Er
Nucleus 0.27 0.32 0.26 0.30 0.33 0.33 0.28 0.30 0.31 0.29 0.30 0.30 0.29
17°Yb 172Yb l¢4Yb 17eYb lV6Hf XTSHf lS°Hf as°W ls2W 184W ls6W ls4Os
0.29 0.29 0.29 0.28 0.27 0.25 0.24 0.23 0.23 0.21 0.20 0.20
TABLE 2 Parameters defining the single-particle level spectrum employed in the calculations
proton
neutron
Shells
K
#
N= 3 4 5
0.05 0.05 0.05
0.45 0.55 0.55
N= 4 5 6
0.05 0.05 0.05
0.45 0.45 0.45
Energy shifts (hto0)
h¥: --0.075 others: +0.10
i¥: unchanged others: +0.15
zero. This is the a p p r o x i m a t i o n used by Nilsson 14), who then chose the three parameters entering i n this m o d e l o n the following basis. The first, hogo = 41A - * MeV, is the frequency associated with the h a r m o n i c oscillator potential. This corresponds to choosing the nuclear radius R = 1.2 A ¢ fm. The second parameter ~ measures the
NUCLEAR ROTATIONAL STATES
311
strength of the spin-orbit potential. It is chosen to be 0.050 in order to reproduce the level sequence in spherical nuclei. # depends on the value of the ! 2 term; Mottelson and Nilsson 17) have taken # = 0.45 and have been able to account fairly well for the properties of odd-neutron levels in deformed nuclei with 150 < A < 190 and A < 220. For the odd-proton levels some changes are necessary due to the Coulomb interaction. The parameters employed in the present paper are listed in table 2. With these parameters, there is an excellent correspondence between the empirical levels and those predicted within one nuclear shell. However, Nilsson's energy levels corresponding to neighbouring shells have to be somewhat shifted in order to explain the odd-particle spectra. A few ad hoc changes have been made in the level scheme when we solve eq. (46). These are indicated in column five in table 2. In the calculation, we include all states of the N = 3, 4, 5 shells for protons (46 levels) and all states of the N = 4, 5, 6 shells for neutrons (64 levels). The strength of the pairing interaction must be chosen so as to reproduce data of the odd-even mass differences. We tried to calculate the energy gap parameter A with the values Gp = 24A -1 MeV, G, = 18A -1 MeV, 17A -1 MeV and 16A -1 MeV. The perturbation term F ' was neglected in the numerical calculations. 4.2. RESULTS OF THE NUMERICAL CALCULATIONS In the calculation, we chose the quadrupole interaction strength • so as to give the equilibrium deformation ~ in the ground state listed in table 1. It is then clear that we are able to reproduce experiment of the intrinsic quadrupole moments. Figs. 2 and 3 show comparison of the values of Ap and An corresponding to constant values Gp = 24A - t MeV and G n = 18A -1 MeV with the empirical values. The experimental values are those obtained from the even-odd mass difference by Nilsson and Prior 4). It is found that the values Gp = 24A -1 MeV and Gn = 18A -1 MeV reproduce rather well the empirical trends and give the best fit. At both ends of the rare-earth region, the experimental values are not quite well reproduced except for the energy gaps of neutrons at the beginning of the region. In order to get better agreements with the experiments, it may be needed to re-arrange the energy shifts of single-particle levels listed in table 2, but we have not yet tried it. Other values for the pairing interaction strength, Gn = 17A- 1 MeV and 1 6 / i - ~ MeV, were found to result in an average decrease of An of the order to magnitude of 10 to 20 ~ . The energy gap parameter A decreases with increasing angular m o m e n t u m ! and becomes zero above the critical angular m o m e n t u m lc. The reduction of the energy gap results in an increase in the moment of inertia. Especially the moment of inertia increases rapidly as the energy gap approaches zero, and it reaches a value of the moment of inertia of the rigid rotation at the critial angular momentum. In the rareearth region, the energy gap parameter Ap for protons is a little larger than that for neutrons. Therefore, the reduction of the energy gap parameter for neutrons is faster than that for protons. There are two critical values of the angular m o m e n t u m corresponding to the neutron and proton energy gaps, and even after the neutron state
312
M. SANO AND M. WAKAI
2.0 exp ....
•
>
1.0
th
%
--
<3 " ~ " Os
(.9 ~"
0.5--
uJ
0
I 150
I
I
I
160
I70
180
ATOMIC MASS
NUMBER
I
190
A
Fig. 2. The p r o t o n energy gap parameter Zip plotted versus atomic mass n u m b e r A. The experimental points are f r o m Nilsson and Prior 4). The dashed lines represent the theoretical values for the pairing interaction strength Gp = 24A -1 MeV. 20 exp
....
th
1.0
,8 Yb
W
Hf
o5
0
I 150
I 160
I 170 ATOMIC
MASS
NUMBER
! [80
i 190
A
Fig. 3. T h e n e u t r o n g a p p a r a m e t e r Z I n p l o t t e d v e r s u s a t o m i c m a s s n u m b e r A . The experimental points are taken f r o m Nilsson and Prior 4). The dashed lines represent the theoretical values for the pairing interaction strength Gn = 18A -1 MeV.
313
NUCLEAR ROTATIONAL STATES
PROTON ....
60-
NEUTRON
\
40-o
~
m
.J
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~
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MASS
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Fig. 4. The critical angular m o m e n t u m plotted versus atomic mass n u m b e r A. The full lines represent the critical angular m o m e n t a for the p r o t o n system and the dashed lines those for the neutron system. The parameters e m p l o y e d in the calculation are the values o f Gp = 24A-1 MeV and Gn = 18A-1 MeV.
z
< ~:
,;',
5
•
. ~
\ '
0
I 150
/
,..,,, '
..'', /
,, I
\
;'t
,, ,,
"
"', "v""
I 160
I 170 ATOMIC
.,
t
MASS
NUMBER
"'-- ....
/
.';o~
I IBO A
Fig. 5. The phase transition energy plotted versus atomic mass n u m b e r A. The full lines represent the phase transition energies for the p r o t o n system and the dashed lines those for the neutron system. The parameters e m p l o y e d in the calculation are the values o f G p = 24A -a MeV and Gn = 18A -I MeV.
314
M. SANO AND M. WAKAI
b e c o m e s n o r m a l t h e p r o t o n is a b l e to r e m a i n in t h e s u p e r c o n d u c t i n g state f o r a while. T h e results a r e s h o w n in fig. 4. T h e differences b e t w e e n t h e p r o t o n a n d n e u t r o n g a p p a r a m e t e r s a r e n o t so l a r g e a n d a r e at m o s t a f e w h u n d r e d keV. I n spite o f t h e s e s m a l l values, t h e differences o f t h e critical a n g u l a r m o m e n t u m c o r r e s p o n d i n g to p r o t o n s a n d
Dy160 x
exp
I0
I I
I I
8 I I I i (I-t,l) . ~ / 2]'fix / ! I
w on.klJ
/
Z 0
21(I)
/
I I /
t~J
/
I I /
I
int
,,xf ,.,,.X~ x 0 ~
I0
20 ANGULAR MOMENTUM I
I
50
Fig. 6. The excitation energy for 16°Dy plotted as a function of the angular momentum. This is the results of the calculation with the values of Gp = 24A -x MeV and Gn = 17A -1 MeV, and the critical angular momentum is I e = 20 for neutrons and 1e = 34 for protons; e l n t represents the changes of the Hartree-Fock energy (41) due to the excitation of intrinsic particles. The dashed lines represent the rotational energy with a fixed moment of inertia J~lx, for which is determined so as to give the position of the first excited state. The crosses denote the observed energy levels. n e u t r o n s b e c o m e l a r g e as a result o f t h e l a r g e v a l u e s o f t h e r i g i d m o m e n t o f i n e r t i a f o r b o t h p r o t o n s a n d n e u t r o n s . T h e t r a n s i t i o n e n e r g y c o r r e s p o n d i n g to t h e critical ang u l a r m o m e n t u m is s h o w n in fig. 5.
NUCLEAR ROTATIONAL STATES
315
Above the transition energy the nucleus is in the shell-model phase, for which the pairing correlation is absent. Below the transition energy the nucleus is in the superconducting phase, and the nuclear properties are very different from those of the normal state. The moment of inertia is an especially typical one and is very different from the rigid-body value expected for the shell model. The rigid-body value of the moment of inertia is a few times larger than the value determined so as to give the position of the first excited state. Then the rotational energy of the deformed nucleus is a few times larger than that of the rigid body in the low excited energy region. The moment of inertia gradually approaches the rigid-body value as the energy gap parameter A decreases with increasing angular m o m e n t u m I and reaches to its value at the critical angular m o m e n t u m I c. Therefore, the rotational energy part becomes that of the rigid body above the critical angular m o m e n t u m and largely deviates from that obtained with a fixed moment of inertia, which is determined from the first excited level. The rotational energy for 16°Dy is given in fig. 6 as a function of the angular momentum. This is the result of the calculation with the values Gp = 24A- 1 MeV and Gn = 17A -1 MeV. The critical angular m o m e n t u m is Ic = 20 for neutrons and I¢ = 34 for protons. Here @int gives the change of the Hartree-Fock energy (41) due to the excitation of intrinsic particles. This value increases with the angular m o m e n t u m as the correlated nucleon pairs are broken by the Coriolis force and reaches a value corresponding to the ground state correlation energy due to the pairing interaction at the critical angular momentum. The excitation energy corresponding to the angular m o m e n t u m I is given by eq. (15). This value fairly deviates still from the rotational energy with a fixed moment of inertia o¢'fix, which is determined so as to give the position of the first excited state. The excitation energy levels are shown in figs. 7 and 8 and are compared with the experiments. The energy spectra shown in the left-hand side of the figure for each nucleus are the experimental energy levels as), and A and B on the right-hand side show the theoretical results. We took the pairing interaction strengths as Gp = 24A- ~ MeV and Gn = 18A- 1 MeV in case A, and also as Gp = 24A- ~ MeV and G, = 17AMeV in case B. The results of our calculation explain the general features of experimental levels fairly well, but there still remain several defects. In general, the theoretical energy levels are a little higher than the observed levels. There are two reasons which give rise to such disagreement with the experiments. One is that the moment of inertia is a little small to be fitted to the first excited level. Even if the displacement from the position of the first excited state is small, it gives rise to large deviations from the expected positions of high angular m o m e n t u m states. To calculate the expected position of the high angular m o m e n t u m state, the moment of inertia must be determined so as to give the position of the first excited state. The results of the calculation in case B agree with the experiments much better than those in case A. However, the energy gap parameter values in case B are about 10 ~ smaller than the experiments. Then the latter case is more desirable than the former case to determine the pairing interaction strength so as to reproduce the empirical values of the gap parameter. The
316
M. SANO AND M. WAKAI i 3
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Fig. 7. R o t a t i o n a l e n e r g y s p e c t r a in M e V . T h e e n e r g y s p e c t r a s h o w n in t h e l e f t - h a n d side o f t h e f i g u r e for e a c h n u c l e u s are t h e e x p e r i m e n t a l e n e r g y levels, a n d A a n d B i n t h e r i g h t - h a n d s i d e s h o w t h e t h e o r e t i c a l results. T h e p a i r i n g i n t e r a c t i o n s t r e n g t h s are Gp = 24A -1 M e V a n d Gn ~ 18A -1 M e V i n case A , a n d Gp = 24A -1 M e V a n d Gn = 17A 1 M e V in case B. I
3-m
i I
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I
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W 1 8 2 0sl84
NUCLEAR ROTATIONAL STATES
317
moments of inertia given by case A lead to values about 10-20 ~ smaller than the empirical values which are determined from the positions of the first excited states. In the first 2 + state, the reduction of the energy gap parameter due to the Coriolis force is negligibly small, and also the excitation energy of the intrinsic particles is negligible. These mean that the calculated levels lie about 10-20 ~ higher than the data for the first excited states. Fairly good agreement with the experiments might be obtained in calculations in which the higher-order correction terms of the moment of inertia are taken into account. One other discrepancy with experiments is seen in the higher excited states when good agreement with the first excited state is assured, namely that the predicted levels are a little higher than the observed ones except for the W and Os nuclei. This tendency is especially remarkable in the first half of the rare-earth region. This might imply that the reduction of the energy gap is too small as to obtain good agreement with the experiments and that it is necessary to take into account effects of the collective vibrational motion on the rotational state. The nuclear equilibrium shape is determined according to the balance of the pairing and quadrupole forces. The pairing force has the effect of keeping the nuclear shape spherically, and on the other hand the quadrupole force causes the quadrupole deformation of the nucleus. Now, the reduction of the energy gap parameter due to the Coriolis force breaks the balance of these forces, and therefore the equilibrium deformation which gives the minimum energy changes as a function of the angular momentum. For instance, the value of the equilibrium deformation parameter 6 for 152Gd gradually increases as a function of the angular momentum, and its increasing ratio is 3.1 ~ at I = 10 and 7.4 ~ at I = 20. However, the equilibrium deformation does not increase gradually always. A typical example is shown in fig. 9 for is°Hr. The total energy corresponding to each angular m o m e n t u m was plotted as a function of 6. In this case, the Coulomb interaction between protons E¢ has been taken into account by Ec=3
-
4
+
1 2 )2 -
2
Z2e 2 +
_ _
(52)
R
The curves of the energy surface have much less curvature than that without the correction of the Coulomb energy. Therefore, it is easier to shift the minimum point of the energy surface. It might play a much more important role in our calculations. The results shown in fig. 9, calculated by using the parameters of case A, were plotted for each second the level belonging to the ground state rotational band. The values of the critical angular momenta for 18°Hf are comparatively small, I~ = 12 for neutrons and Io = 24 for protons. The shape of the energy surface changes gradually as the angular m o m e n t u m increases. The minimum point is at 6 = 0.24 for the ground state, but the energy surface corresponding to I = 16 has two minimum points at 6 = 0.23 and 0.32. The absolute values of the energy at these minimum points are almost the same. The minima are very shallow and may be washed out by the fluctuations. How-
318
M. SANO AND M. WAKAI
ever, they are stable minimum points within the generalized Hartree-Fock approximation. Above the / = 18, the energy of the minimum point corresponding to the large deformation becomes lower than that corresponding to the small deformation. Particularly, the energy surface around the minimum point corresponding to the small deformation becomes almost flat, and the solutions of the generalized Hartree-Fock
Hf 180 20
5 _~ ,o lw
0
I
I
I
0.20
030
0.40
DEFORMATION PARAMETER 8
Fig. 9. The energy surfaces plotted as a function of the deformation parameter 6. The results are plotted for each second level belonging to the ground state rotational band.
equation at this point may be unstable. As the state at this point becomes unstable, the nucleus will prefer to take the state corresponding to the minimum point at the large deformation. If the change from the small deformation to the large one suddenly takes place at the angular momentum ! = 20, the energy of the state with I = 20 becomes almost equal to that of the state with I = 18. The information on the instability of states cannot be obtained from the generalized Hartree-Fock equation,
NUCLEAR
ROTATIONAL
319
STATES
because it is derived from the requirement that the expectation value of the Hamiltonian ~ is stationary with respect to arbitrary first-order variations of p and x. In order to verify whether the solutions of the generalized Hartree-Fock equation are stable or not, we must examine the sign of the second variation of the expectation value of the Hamiltonian 7).
Hf 180
/ ,
!
t
//
~E
/
/
I
z(z+,l/
/
/
/
/
,.'
/
/
~6
x4
x
x/x 0 -'-"'x/
I I0
[ 20 ANGULAR MOMENTUM I
I 30
Fig. 10. The rotational spectrum for lS°Hfplotted as a function of the angular momentum.
The strength of the quadrupole force which gives the nuclear deformation 6 = 0.24 for 18°Hf is Z = 0.03 MeV. This value causes the rotational state of the ground band to be unstable at I = 18. This means that a large change of the nuclear deformation takes place suddenly at the angular m o m e n t u m I = 18. The change of the rotational spectrum is shown in fig. 10. The dashed line represents the rotational energies with a
320
M . SANO A N D M. W A K A I
fixed moment of inertia Jflx which is determined so as to give the position of the first excited state. The dot-and-dash line shows the rotational levels in which the antipairing effect due to the Coriolis force is taken into a6count, but the nuclear deformation is fixed at 6 = 0.24. The full line gives the rotational levels in which both the anti-pairing effect and the change o f the nuclear deformation are taken into account, and these states are all stable. We can see the discontinuitous change of the rotational level at I = 18, in which the intrinsic particle state becomes unstable. One of us (M.S.) would like to thank Professor L. S. Kisslinger for the hospitality accorded him during his stay at Western Reserve University and also to thank the National Science Foundation for award of the fellowship.
References 1) J. Bardeen, L. N. Cooper and J. R. Schieffer, Phys. Rev. 108 (1957) 1175; N. N. Bogolyubov, Nuovo Cim. 7 (1958) 794; J. G. Valatin, Nuovo Cim. 7 (1958) 843 2) A. Bohr, B. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936; S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 11 (1959); L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 9 (1960); Revs. Mod. Phys. 35 (1963) 853 3) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953); D. R. Inglis, Phys. Rev. 96 (1954) 1059, 97 (1955) 701 4) S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 16 (1961); J. J. Griffin and M. Rich, Phys. Rev. 118 (1960) 850 5) B. R. Mottelson and J. G. Valatin, Phys. Rev. Lett. 5 (1960) 511; J. G. Valatin, Lectures in theoretical physics, 1V (Wiley, New York, 1962) 6) K. Y. Chan and J. G. Valatin, Nuclear Physics 82 (1966) 222; K. Y. Chan, Nuclear Physics 85 (1966) 261 7) M. Sano and M. Wakai, Nuclear Physics 67 (1965) 481 8) T. Kammuri, Progr. Theor. Phys. 33 (1965) 456 9) E. R. Marshalek, Phys. Rev. 139 (1965) B770 10) T. Udagawa and R. K. Sheline, Phys. Rev. 147 (1966) 671 11) Yu. T. Grin and A. I. Larkin, Soy. J. Nucl. Phys. 2 (1966) 27 12) M. Baranger, Phys. Rev. 122 (1961) 992; M. Baranger and K. Kumar, Nuclear Physics 62 (1965) 113 13) N. N. Bogolyubov, Usp. Fiz. Nauk 67 (1959) 549 14) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) 15) B. R. Mottelson, Cours de l'ecole d'dt6 de physique th6orique des houches, 1958 (Dunod, Paris, 1959) 16) D. R. B6s and Z. Szymanski, Nuclear Physics 28 (1961) 42 17) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1, No. 8 (1959) 18) H. Morinaga and P. C. Gugelot, Nuclear Physics 46 (1963) 210; H. Morinaga, Nuclear Physics 75 (1966) 385; K. Kotajima and D. Vinciguerna, Phys. Lett. 8 (1964) 68; F. S. Stephens, N. Lark and R. M. Diamond, Phys. Rev. Lett. 12 (1964) 225
T H E C A L C U L A T I O N OF N U C L E A R R O T A T I O N A L STATES BASED
ON THE GENERALIZED
HARTREE-FOCK APPROXIMATION
MITSUO SANO t
Western Reserve University, Cleveland, Ohio
and MASAMICHI WAKAI Osaka University, Toyonaka, Osaka
Received 28 November 1966 Abstract: The generalized Hartree-Fock method is applied to a description of rotating nuclei, and
the generalized Hartree-Fock equation is solved with the aid of self-consistent perturbation calculations. Numerical calculations are carried out for the rare-earth nuclei, and the results are compared with experiment. It is shown that good agreement with experiment is obtained.
1. Introduction
The successful method of the theory of superconductivity 1) has been applied to the study of nuclear structure z). It permits a microscopic treatment and has been proved to be an elegant and powerful method in explaining the properties of the ground and low-lying excited states of various nuclei. The moment of inertia which can be defined from the rotational sequences of the low-lying energy levels had been difficult to reproduce for a long time 3). However, it was found that a consideration of the effect of the pairing correlation gives theoretical values of the moment of inertia in satisfactory agreement with experiment 4). The deformed nucleus might have another typical property similar to the superconducting system, because the Coriolis force prevents the formation of coupled pairs. This corresponds to the case of the external magnetic field applied to superconducting electron systems whose pairing correlation is decoupled by the field. It is expected that there is a critical value of the angular momentum, i.e. an upper bound of the angular m o m e n t u m for which the nucleus can be in the superconducting phase. This problem was first pointed out by Mottelson and Valatin 5) and afterwards various numerical calculations were carried out by many authors 6-11). However, none of these authors performed numerical calculations to an extent which is comparable to experimental results. The purposes of the present paper are to provide a description of rotating nuclei by using the procedure of the generalized Hartree-Fock approximation and to perform t Senior Foreign Scientist Fellow of the National Science Foundation. Present address: University of California, Davis, California. 298
NUCLEAR ROTATIONAL STATES
299
detailed numerical calculations of the rotational states. The method of the description is almost the same as previously described 7), but it has been revised so that the selfconsistent calculation is possible within the generalized Hartree-Fock approximation. In sect. 2, the generalized Hartree-Fock equation is derived, and it is solved in sect. 3 with the aid of self-consistent perturbation calculations. The results of the numerical calculation are given in sect. 4. 2. Generalized Hartree-Foek equation in rotating system We take a spherically symmetric shell-model Hamiltonian with chemical potential 2 =
l/~p~ e~ cp ca c~.
(1)
~B76
H e r e , . , fl, etc. represent a complete set of the shell-model states, c+ and c~ are the creation and annihilation operators of a nucleon in the s t a t e , and e~ the energy of a single particle state ~. The potential V is assumed to be spherically symmetric with the matrix elements obeying the antisymmetry condition V~p,a = - Ve=,a = - V,pay = V~=p.
(2)
In investigating the rotational motion in deformed nuclei, it is necessary first to find the nuclear equilibrium shape in the ground state. This problem has been investigated by Baranger et al. 12). They have rewritten the generalized Hartree-Fock equation into a form especially suited for the nuclear system and applied it to the description of the ground state. We shall extend their method to a system rotating with an angular velocity co and obtain a description of intrinsic particles coupled to a rotating field. We first assume that the nuclear equilibrium shape in the ground state is axially symmetric. This assumption is consistent with the results obtained by Baranger et al. and also with experimental evidence. Then the rotational spectrum of the ground band is that of a rotation about the x-axis chosen perpendicular to the symmetry axis z of the nucleus. In the rotating frame of reference, we search for the lowest state of the system with a fixed average value of the angular momentum about the x-axis. In this case we must add to eq. (1) a term i-i
= -coax
= - c o Z < lJxlP>c +ce •
=#
(3)
in order to obtain the lowest state by taking into account the pairing correlation for the auxiliary Hamiltonian Jt ~ = H + H~,,
(4)
we perform the generalized Bogolyubov transformation 13). The generalized Bogolyubov transformation gives a complete set of fermion operators called quasi-particle operators, which are the most general linear combinations of c + and c operators a~ = Z ( A , c ,'
+ +B~,c~,). '
(5)
300
M. S A N O A N D M. W A K A I
The requirement that the transformation (5) is canonical entails the orthogonality relations ~ ( A ~i*A ,j+ B , B ~ ( A =i B ~j + B , Ai ~ )j = 0 , i* , )j = 6,j, E
i i* i (A:A ei* +B:Be) = 6:e ,
Z (A~B , ei. + B=,. Ae) , = O.
i
i
(6)
The idea is to make the new vacuum state Oo(~O) similar to an independent-particle state, in the sense that it is defined by a~(o)Oo(m) = 0.
(7)
We can obtain the generalized Hartree-Fock equations from the requirement that the expectation value of the auxiliary Hamiltonian ~ in the new vacuum state Oo(O9) must have a minimum value. Then A and B satisfy the following equations: i
7 i
*
*
i
*
i
i
i ) B = - Z (F,~ - ooJ,~)B~- E~ A~a A e = Ei B~
-
(8a)
The complex conjugate of these equations becomes
(~ _OB':+
E (F,~-oJ,~)B~" + E A,eAe,. = - E i B ~,., o . i* -- Z i* -- (~,-- +OA: * ooJ~)A, , i, - E ,=A~e*~'eni* = - E, A~. (F~,-e
(8b)
The quantities F and A are defined by A=e = -Ae= = 2 ~ V=p~#c~,
(9a)
76
F~, = F~,* = 4 ~ V~,e~pp6, eo
(9b)
where K~ = -~c~, =
(9c)
i
Pfla = Po5 = (Oolcp-%]Oo> = 2
'*BpB6" '
(9d)
i
We see that F ~ is the potential arising from the density p of the particles and is the familiar Hartree-Fock self-consistent potential. On the other hand, /c~n is the wave function of the Cooper bound pair state and gives rise to the pairing potential A~fl. The expectation value of the Hamiltonian dgCis given by
=
E
Z
E
Z
a
a7
~fl
~'
(lo)
The chemical potential 2 is determined by fixing the average number of particles JV" = ( E c+%> = E P,,"
(11)
N U C L E A R ROTATIONAL STATES
301
The angular velocity co must be determined from the following self-consistency condition: x / I ( I + l ) = (dx> = E J, aP=a - a)J(a)).
(12)
This equation gives a definition of the moment of inertia J ( a ) ) . Using eqs. (8) and (10), we obtain the energy of the system with a fixed angular momentum I W(I) = ( J f +LA/'+coJx) = U(I)+
I(I+1) 2J(I) '
(13)
where
U(I) = ½ 2 (e~ + 2-Ei)B~B~. i i*
(14)
i~t
The energy U(I) is the intrinsic energy of the nuclear system with the angular momentum I. The ground state energy W(0) is obtained by putting I = 0 into eq. (13). Thus the excitation energy corresponding to the angular momentum I is given by
~(I) = W(I)- W(O) = U(I)- U(O) + 1(1+ 1) 2J(I)
(15)
For deformed nuclei, the average single-particle field calculated from the generalized Hartree-Fock equations must be already non-spherical. The last term on the righthand side in eq. (15) has the form which gives the usual rotational energy of a deformed nucleus, while the other terms give the variation of the intrinsic energy due to the rotational motion. It is to be noted that the moment of inertia is also a function of the angular momentum.
3. Perturbation approach 3.1. SELF-CONSISTENT PERTURBATION OF THE GENERALIZED HARTREE-FOCK EQUATION The generalized Hartree-Fock equations must be solved self-consistently by taking into account the conditions (11) and (12). However, it is not so easy to carry out such calculations. In order to investigate the effects of the Coriolis force on the rotational spectrum one may solve eq. (8) by using a perturbation calculation together with some other simplifying assumptions. We thus first assume that the rotation is slow so that the Coriolis force can be treated as a perturbation. In addition, we assume that the pairing potential (9a) is a constant and further that each of the two states degenerate in energy is the time reverse of the other. This is the same simple situation as in the BCS theory a). The pairing potential can now be written as
A~,p = 6~,?A,
(16)
302
M. SANO AND M. WAKAI
with A = - ½ G ~ s,K,, _.,.
(17)
7
Here fi means the state corresponding to the time reversal of state 3 and s~ the phase factor related to the time reversal of the state 7- The state - 7 is defined as having the same quantum numbers as 7, except for the magnetic quantum number which has the opposite sign. With the further substitution,
C'~ = s~B~_~,
(18)
the generalized Hartree-Fock equation (8a) is rewritten as i i i i X)Ao + E Foc,Ay + ACo~-co ~ J~,A~, = E, 7
-
"
i
y
i
7 *
i
i
i
F~.~C~,+ AAo~-co ~ J<, C~
],
= e,
i
C,
(19)
y
The Hartree-Fock potential F is not invariant under the time reversal. In the perturbation calculation, the density matrix p can be expressed in terms of powers of co. Then it is easily able to divide the potential F into two parts, i.e. an invariant part and the other part with opposite sign under the time reversal
F = F(°)+F ',
T F T -1 = F ( ° ) - F '.
(20)
The angular m o m e n t u m operators are odd under the time reversal. Therefore F (°) and F ' are the potentials arising from the densities which have even and odd powers of co, respectively. The form of eq. (19) suggests the introduction of the matrix
M = Mo+M' = \
A,
-(e-2)-F
¢°) + \
0,
F -coJx
"
Eq. (19) is the eigenvalue equation for the matrix M. Here, we define ~ as an eigenvector of M with a positive eigenvalue E, and write it as
C' "
(22)
The procedure of the calculation is the following: We first assume that the values of F (°) and A in M (°) are known for a given co and treat M ' as a small perturbation. We thus re-determine F (°) and A from the self-consistency requirement. For sufficiently small o~, the use of a perturbation calculation may be justified. Near the critical rotation at which the energy gap parameter A vanishes, however, the rotational energy related to marx is not small in comparison with the pairing energy. The use of the perturbation calculation may still be justified even for larger values of the rotational frequency co, provided coarx is smaller than the quasi-particle energies given by eq. (19).
303
NUCLEAR ROTATIONAL STATES
The unperturbed equations can be written M o O i+ = E i + ~ ) i+,
(23a)
mo ~i-
(23b)
= El_ dl)i-,
where E~+ = E, and E~- = -E~. The function ~+ is an eigenvector of Mo with positive eigenvalue E,, and ~bi- an eigenvector of Mo with negative eigenvalue - - E i. It is possible to write them as follows: •
iA(O,,
d?'+ =
\c(O)i],
=
~
A(O)i. ] .
(24)
The solution of eq. (23), as has been shown by Baranger a2), is
A~O)i = W~u~,
C(=°)~
=
W ail ) i ,
(25)
with
ui = v/½(1 + e~/Ei),
v, = x/½(1 - ei/E~),
Ei = ~/e2 + A 2,
(26) (27)
where e i and W~ are the solutions of the eigenvalue equation =
r(°)W i
(28)
This equation involves only the Hartree-Fock potential F (°) and determines the energies e~ of odd-particle states in deformed nuclei. The states i defined by (28) are essentially the Nilsson states in a deformed well 14). We use a Latin subscript i to represent all quantum numbers of the Nilsson state. The Nilsson state ri> is now expanded in terms of state ]~> in the spherical well as li> = Z W~t~>.
(29)
at
The unitary transformation matrix elements W~ are just the eigenvectors of eq. (28) and satisfy the following orthogonality relations:
Y wjw£= 6u, Z
i
i*
w; w~
(30)
= 6~ .
i
The explicit definition of the time reversal is ~ Tli> = I'f) = 2 wii~> = E W~l~> = E W:~i* 1~>,
at
(31)
at
where the time-reflection operator T is a product of a unitary operator and a complex conjugate operator. From eq. (31) W~ satisfies the relation
W/* =
i --i W~ = s i s a W _ a .
304
M. S A N O A N D M. W A K A I
The phase factors s~ and s i satisfy, by definition, the relations 2
1,
s~s_~
- 1,
sis_ i = - 1 .
s~ = 1,
We can now transform the matrix elements of various operators in the spherical shell-model representation to those in the Nilsson representation by using eq. (29). The matrix element of the angular momentum operator is given in the Nilsson representation as
Wd j~a W~ =
(32)
We must further mention the fact that the matrix elements ofj~ are real in the representation employed here. In the Nilsson representation, the eigenvector ~ can be expanded as follows:
~i = ~i + ~ (q~S[M'lq~i> q~i+ ~ <~ k[M'[q~j>(q~j[M'[~i> t~k Ej (E,- Ej)(z,- ½ ~ <4~JlM'I~>2 ~ ' + . . . . •
(33)
(El--El) 2
where the sum must run over both the positive and negative energy states given by eq. (23). The matrix elements in eq. (33) are calculated by using eqs. (24) and (25) and become as follows:
<~kilM,l~k> =
- ~oJ~(- v, u~ + u, vk),
- o J,k(u,u +
E~ E~ E~ Ei
> O, > O, < O,
E k > O,
< O,
E k < O,
E k < O,
(34)
E k > O,
where
Jx = Jx-- --1F' .
(35)
Using eq. (34), the terms of eq. (33) that are linear and quadratic in co are easily calculated and are given as follows: first-order terms
-o Jj,
+ e,,,,+ ,;,,) wL
(36a)
- osj,
+E,v,- jv,)wL
(36b)
C(1)~ = ~ ~ - ~ •
E,
-
Ei
NUCLEARROTATIONALSTATES
305
second-order terms
092Jkj Jji A<:2" = ~,Zk(e~ - e~)(F4 - E~,) ["' {e~' + E, + + el ~k + a ~ + ~ ~j}
+ vi{ZE i A + Aek-- Aei}] Wff
2
2e, u, Y' . (e~-E~)
C~2>i
E T-k (E 2 -E,)(E:2
ul {(E,+ E y - 4 E I E
a • ½ (1
~
W~, eiE~
(36c)
/J
2 -E,.~2 + ui{2Ei A - Aek + Ae/}] W)
+
v ~'24& {va(~,-+) t i
__l ~j
0")2j2" Ui {(Ei~-Ej) - 4 E i E , ' ½ (1
. (~}-e~)
~
AE+eleYl} Wi. e,~
(36d)
/
3.2. THE EFFECT OF ROTATION ON PAIR CORRELATION We shall take up to the second-order contribution of e>Jx into account in calculating the gap equation (17). In this equation, terms of first-order in ~o], do not appear because they are odd under the time reversal. As the result of the perturbation calculation, eq. (17) becomes 4
1 (1_o92
(37)
where
Xij - (El q-E j)2 1
EiE i / + Ei+E i
EZEy
Eq. (37) gives an idea of how A depends on ~o. Since the factor in bracket in the right hand side of eq. (37) decreases with increased co, it is clear that El and consequently A has to be decreased in order to keep the value of the right-hand side a constant. Therefore, it is expected that there is a critical value of the angular velocity, coc, above which the nucleus ceases to be superconducting. The critical .angular velocity ~% can be determined from the solution of the eigenvalue problem of the equation obtained from eq. (37) by dropping A. Within the same approximation, the average number of the nucleons given by eq. (11) becomes ,/U = ,Ar(°) + j V "(2) =
• ~, + ~ '
..~
2
l,J
o
(e,+E/
(1
E, Ej /
iJ
~2 (<+E~) ~ , E j ,
.(38)
306
M. SANO A N D M. W A K A I
I f we put A --- 0 in eq. (38), the second term in the right-hand side vanishes and we obtain formulae familiar in the usual Nilsson model without the pairing correlation. 3.3. MOMENT OF INERTIA We can determine from eq. (12) the value of the angular velocity o9 corresponding to a fixed angular m o m e n t u m I and also can obtain the formula for the moment of inertia from that. Taking into account up to the first order in o9 expansion of the density p, we obtain j
= _1 ~ j ~ p ~ ,
=
-
"" E i+Ej
EiEj /
,
~o < 09c .
(39)
If we neglect the term F ' in the matrix element Jjl, this is the expression first obtained by Belyaev 2), which gives the generalized cranking model formula in which the energy denominators contain the quasi-particle energies. However, in our case the moment of inertia is a function of the angular velocity, i.e. the angular m o m e n t u m because of the effect of rotation on the energy gap A. The reduction of the energy gap A results in an increase in the moment of inertia. The expression (39) reduces for A = 0 to the cranking model formula 3)
j = ~ J.u,fJi, ij
~i --
(e, > eF > e~),
09 > ¢o¢,
(40)
8j
where eF means the Fermi energy. This also gives the expression for the moment of inertia associated with the rotation of a rigid body is). The numerical value of the moment of inertia for A = 0 is a few times larger than the value for o) = 0. Therefore, for a sufficiently large value of the angular m o m e n t u m the rotational spectrum deviates greatly from the I(I+ 1) spectrum which is the case with a constant value of the moment of inertia. 3.4. CHANGES OF INTRINSIC ENERGY DUE TO THE ROTATION AND EXCITATION ENERGY The intrinsic energy given by eq. (14) is a function of the angular m o m e n t u m and, as the result of the perturbation calculation, becomes
U(I) = ½ E (e~,+J.-EilB~,B~, ' '* = U(°~(I)+ u(E~(I),
(41)
where
Uw)(I) = ½ Z (~. + 2 - F..'~tq~°)'Rw3i" _ ~ o ~(o)+½ P r(°).(°)-A2/G,
(42a)
NUCLEAR ROTATIONAL STATES
U(2)(I) =
8
L(Ei+Ej)E~E~
~
307
(EI+Ej)E?E~
EiE~. / J
(42b)
Eq. (42a) is o b t a i n e d by using eqs. (23) a n d (17). The s e c o n d - o r d e r terms o f the intrinsic energy given by (42b) b e c o m e zero a b o v e the critical a n g u l a r m o m e n t u m Ic, because o f the energy g a p p a r a m e t e r A = 0. W e n o w discuss some characteristic features o f the change o f the H a r t r e e - F o c k energy due to the Coriolis force with the aid o f some simplifying assumptions. W e
>.i I,~ Z I,d
,~c •
will
J' °I"C '
a - - 'l
!
I
Z
(2_
,
........
~s !
E--
I
,, .........
X i,i
"~i,,
-----'-~.I= 0 SUPER
,
I
~rgA2(o)
NORMAL
Fig. 1. A schematic illustration of the rotational level scheme. The complete spectrum consists of two parts. One is the levels in the superconducting phase shown in the left half of the figure, the other is the levels in the normal phase shown in the right half. The levels depicted by broken lines, in fact, are absent in the presence of the pairing interaction. The system is not found in these states, rather it is found in the superconducting state with energy corresponding to the same angular momentum. The dotted line connects two levels, normal and superconducting, whose energies correspond to the same angular momentum.
assume t h a t (i) the p a i r i n g interaction is a c o n s t a n t G within an energy interval a r o u n d the F e r m i surface a n d equals zero otherwise, (ii) the sum over the Nilsson state i can be a p p r o x i m a t e d b y integral with the single-particle level density set equal to a constant g, (iii) changes o f the H a r t r e e - F o c k self-consistent p o t e n t i a l F with respect to the change o f ~o is small a n d can be neglected. E v e n u n d e r these r a t h e r crude a s s u m p tions a significant d e p a r t u r e f r o m the p r e d i c t i o n s o f the shell m o d e l w i t h o u t the p a i r ing i n t e r a c t i o n is expected a n d q u a l i t a t i v e features o f the p a i r i n g system can b e e x a m ined.
308
M. SANO AND M. WAKAI
Above the critical angular momentum Ic, the change of the intrinsic energy becomes
U(I)-U(O) = U(°)(I) - U(O) ~- ½9A2(0),
I > Ic.
(43)
The energy gap A(0) is that in the ground state. The change of the Hartree-Fock energy given by eq. (43), which is a constant above the transition energy 8c corresponding to the critical angular momentum, is equal to the lowering of the ground state energy due to the pairing interaction. Therefore the excitation energy above the critical angular momentum is given by
o~(I) = U(°)(I)- U(O)+ I(I+ 1) _~ ½gA2(O) + I(I+ 1) 2Jrlg
2'frig
I > Io. '
(44)
=
Below the transition energy the nucleus is in the superconducting phase and the excitation energy is a complicated function of the angular momentum
~(1) = U ( I ) - U(O) + I(I + 1) 2J(I) '
I < Ic.
(45)
The transition from the super to normal phase is illustrated in fig. 1. The left half of this figure represents the levels of the rotating nuclear system in the superconducting phase, while the remaining part represents those expected for the normal phase. The broken lines join the levels with the energies corresponding to the same angular momentum. The difference in the energy of the corresponding levels decreases with the increase of the angular momentum as the correlated nucleon pairs are broken by the Coriolis interaction, or in other words, as the number of elementary excitations increases. The difference vanishes at the critical angular momentum where the energy gap also vanishes, and above this critical angular momentum the normal phase is reached. 4. Results of the numerical calculations
4.1. METHOD OF CALCULATION We provided a description of rotating nuclei by using the procedure of the generalized Hartree-Fock approximation and obtained the generalized Hartree-Fock equation (8). It appears to be difficult to solve eq. (8) under the requirement of selfconsistency. Then we carried out the calculations to solve the generalized HartreeFock equation (8) by using the procedure of the self-consistent perturbation method. The self-consistent perturbation method mentioned in the sect. 3 differs in several respects from that of the usual perturbation calculation. In the former case, the calculation should be self-consistent within the generalized Hartree-Fock approximation. Then the changes of the Hartree-Fock potential F and the pairing potential A are taken into account within the procedures of calculation self-consistently. The formalism can be used for any nucleon-nucleon potential. A critical question is the choice
N U C L E A R R O T A T I O N A L STATES
309
of the nucleon-nucleon potential. We have already assumed the pairing potential as a constant. This gives the same simplification as in the BCS theory. The pairing-plusquadrupole model has been proposed by the Copenhagen school for the purpose of producing pairing effects and quadrupole deformations. This model has fairly well succeeded in explaining experiments at low energy, in spite of the fact that these forces do not look like the real nucleon-nucleon interaction. The strength of the pairing interaction on the basis of the odd-even mass difference has been analysed systematically, and the moment of inertia and the gyromagnetic factor have been calculated by Nilsson and Prior and by Griffin and Rich 4). The nuclear equilibrium deformations based on the pairing-plus-quadrupole model also have been investigated in detail by B6s and Szymanski 16) and by Baranger and Kumar 12). These authors have not only succeeded in explaining qualitative features of the ground and low-lying states in the deformed region, but they have carried out rather successfully a number of quantitative tests. We adopt the pairing-plus-quadrupole forces. Then eq. (28) which determines the single-particle state becomes
ei6ij = e ~ ° ) - 2 6 i j - Z Z q~/" QM.
(46)
M
The first part dji°) is spherically symmetric and is expressed in the Nilsson model by (0) _
ji
1
2M
(pa)ji+½M(.oo(rZ)ji_hO)ot¢[2(l
" S)_l~lZ]ji '
(47)
where M is the nucleon mass, o) o the harmonic oscillator frequency and ~c and # the constants which are chosen in order to have the correct level sequence in spherical nuclei. The last term in eq. (46) denotes the matrix elements of the deformed part by a product of the single-particle and the total nuclear quadrupole moments, and X is the coupling constant of the quadrupole force. The single-particle and the total nuclear quadrupole moments are given by
q~ = x/Js6-n(jlr 2 Y2MIi),
(48)
QM = ~/-16n Z
(49)
po
We now proceed as follows: In the numerical calculation, we consider only the axial symmetric fl-deformation of the nucleus and will not consider any effect of the 7-deformation. Therefore, only a M = 0 component is taken into account in eq. (46). The coupling constant Z must be determined so as to give the mass quadrupole moment or the nuclear equilibrium deformation in the ground state. In the ground state the quadrupole moment becomes
Qo = x/-167z E
(50)
310
M. SANOAND M. WAKAI
Therefore, Nilsson's d e f o r m a t i o n p a r a m e t e r fi is related to the q u a d r u p o l e matrix element Qo a n d the coupling c o n s t a n t X by the e q u a t i o n ZOo = -~fi Mo92 .
(51)
The experimentally k n o w n d e f o r m a t i o n parameters 3 are listed in table 1. Eq. (46) should be diagonalized exactly by including the matrix elements between states b e l o n g i n g to different shells. I n this diagonalization, however, matrix elements between single-particle states of different principal q u a n t u m n u m b e r N are p u t equal to TABLE 1 Deformation parameters Nucleus xs~Sm 15~Sm a54Gd 156Gd aSSGd X~°Gd lS°Dy X6ZDy l~4Dy X64Er 166Er lnSEr XT°Er
Nucleus 0.27 0.32 0.26 0.30 0.33 0.33 0.28 0.30 0.31 0.29 0.30 0.30 0.29
17°Yb 172Yb l¢4Yb 17eYb lV6Hf XTSHf lS°Hf as°W ls2W 184W ls6W ls4Os
0.29 0.29 0.29 0.28 0.27 0.25 0.24 0.23 0.23 0.21 0.20 0.20
TABLE 2 Parameters defining the single-particle level spectrum employed in the calculations
proton
neutron
Shells
K
#
N= 3 4 5
0.05 0.05 0.05
0.45 0.55 0.55
N= 4 5 6
0.05 0.05 0.05
0.45 0.45 0.45
Energy shifts (hto0)
h¥: --0.075 others: +0.10
i¥: unchanged others: +0.15
zero. This is the a p p r o x i m a t i o n used by Nilsson 14), who then chose the three parameters entering i n this m o d e l o n the following basis. The first, hogo = 41A - * MeV, is the frequency associated with the h a r m o n i c oscillator potential. This corresponds to choosing the nuclear radius R = 1.2 A ¢ fm. The second parameter ~ measures the
NUCLEAR ROTATIONAL STATES
311
strength of the spin-orbit potential. It is chosen to be 0.050 in order to reproduce the level sequence in spherical nuclei. # depends on the value of the ! 2 term; Mottelson and Nilsson 17) have taken # = 0.45 and have been able to account fairly well for the properties of odd-neutron levels in deformed nuclei with 150 < A < 190 and A < 220. For the odd-proton levels some changes are necessary due to the Coulomb interaction. The parameters employed in the present paper are listed in table 2. With these parameters, there is an excellent correspondence between the empirical levels and those predicted within one nuclear shell. However, Nilsson's energy levels corresponding to neighbouring shells have to be somewhat shifted in order to explain the odd-particle spectra. A few ad hoc changes have been made in the level scheme when we solve eq. (46). These are indicated in column five in table 2. In the calculation, we include all states of the N = 3, 4, 5 shells for protons (46 levels) and all states of the N = 4, 5, 6 shells for neutrons (64 levels). The strength of the pairing interaction must be chosen so as to reproduce data of the odd-even mass differences. We tried to calculate the energy gap parameter A with the values Gp = 24A -1 MeV, G, = 18A -1 MeV, 17A -1 MeV and 16A -1 MeV. The perturbation term F ' was neglected in the numerical calculations. 4.2. RESULTS OF THE NUMERICAL CALCULATIONS In the calculation, we chose the quadrupole interaction strength • so as to give the equilibrium deformation ~ in the ground state listed in table 1. It is then clear that we are able to reproduce experiment of the intrinsic quadrupole moments. Figs. 2 and 3 show comparison of the values of Ap and An corresponding to constant values Gp = 24A - t MeV and G n = 18A -1 MeV with the empirical values. The experimental values are those obtained from the even-odd mass difference by Nilsson and Prior 4). It is found that the values Gp = 24A -1 MeV and Gn = 18A -1 MeV reproduce rather well the empirical trends and give the best fit. At both ends of the rare-earth region, the experimental values are not quite well reproduced except for the energy gaps of neutrons at the beginning of the region. In order to get better agreements with the experiments, it may be needed to re-arrange the energy shifts of single-particle levels listed in table 2, but we have not yet tried it. Other values for the pairing interaction strength, Gn = 17A- 1 MeV and 1 6 / i - ~ MeV, were found to result in an average decrease of An of the order to magnitude of 10 to 20 ~ . The energy gap parameter A decreases with increasing angular m o m e n t u m ! and becomes zero above the critical angular m o m e n t u m lc. The reduction of the energy gap results in an increase in the moment of inertia. Especially the moment of inertia increases rapidly as the energy gap approaches zero, and it reaches a value of the moment of inertia of the rigid rotation at the critial angular momentum. In the rareearth region, the energy gap parameter Ap for protons is a little larger than that for neutrons. Therefore, the reduction of the energy gap parameter for neutrons is faster than that for protons. There are two critical values of the angular m o m e n t u m corresponding to the neutron and proton energy gaps, and even after the neutron state
312
M. SANO AND M. WAKAI
2.0 exp ....
•
>
1.0
th
%
--
<3 " ~ " Os
(.9 ~"
0.5--
uJ
0
I 150
I
I
I
160
I70
180
ATOMIC MASS
NUMBER
I
190
A
Fig. 2. The p r o t o n energy gap parameter Zip plotted versus atomic mass n u m b e r A. The experimental points are f r o m Nilsson and Prior 4). The dashed lines represent the theoretical values for the pairing interaction strength Gp = 24A -1 MeV. 20 exp
....
th
1.0
,8 Yb
W
Hf
o5
0
I 150
I 160
I 170 ATOMIC
MASS
NUMBER
! [80
i 190
A
Fig. 3. T h e n e u t r o n g a p p a r a m e t e r Z I n p l o t t e d v e r s u s a t o m i c m a s s n u m b e r A . The experimental points are taken f r o m Nilsson and Prior 4). The dashed lines represent the theoretical values for the pairing interaction strength Gn = 18A -1 MeV.
313
NUCLEAR ROTATIONAL STATES
PROTON ....
60-
NEUTRON
\
40-o
~
m
.J
~
Gd
~
"
~
z <[ 20--
t~s#
~%
##
•
•
%
##-.
....
\'.-"
,'i\ - - --#
0
l t5O
I 160
I 170
ATOMIC
MASS
NUMBER
--
~Os
I 180 A
Fig. 4. The critical angular m o m e n t u m plotted versus atomic mass n u m b e r A. The full lines represent the critical angular m o m e n t a for the p r o t o n system and the dashed lines those for the neutron system. The parameters e m p l o y e d in the calculation are the values o f Gp = 24A-1 MeV and Gn = 18A-1 MeV.
z
< ~:
,;',
5
•
. ~
\ '
0
I 150
/
,..,,, '
..'', /
,, I
\
;'t
,, ,,
"
"', "v""
I 160
I 170 ATOMIC
.,
t
MASS
NUMBER
"'-- ....
/
.';o~
I IBO A
Fig. 5. The phase transition energy plotted versus atomic mass n u m b e r A. The full lines represent the phase transition energies for the p r o t o n system and the dashed lines those for the neutron system. The parameters e m p l o y e d in the calculation are the values o f G p = 24A -a MeV and Gn = 18A -I MeV.
314
M. SANO AND M. WAKAI
b e c o m e s n o r m a l t h e p r o t o n is a b l e to r e m a i n in t h e s u p e r c o n d u c t i n g state f o r a while. T h e results a r e s h o w n in fig. 4. T h e differences b e t w e e n t h e p r o t o n a n d n e u t r o n g a p p a r a m e t e r s a r e n o t so l a r g e a n d a r e at m o s t a f e w h u n d r e d keV. I n spite o f t h e s e s m a l l values, t h e differences o f t h e critical a n g u l a r m o m e n t u m c o r r e s p o n d i n g to p r o t o n s a n d
Dy160 x
exp
I0
I I
I I
8 I I I i (I-t,l) . ~ / 2]'fix / ! I
w on.klJ
/
Z 0
21(I)
/
I I /
t~J
/
I I /
I
int
,,xf ,.,,.X~ x 0 ~
I0
20 ANGULAR MOMENTUM I
I
50
Fig. 6. The excitation energy for 16°Dy plotted as a function of the angular momentum. This is the results of the calculation with the values of Gp = 24A -x MeV and Gn = 17A -1 MeV, and the critical angular momentum is I e = 20 for neutrons and 1e = 34 for protons; e l n t represents the changes of the Hartree-Fock energy (41) due to the excitation of intrinsic particles. The dashed lines represent the rotational energy with a fixed moment of inertia J~lx, for which is determined so as to give the position of the first excited state. The crosses denote the observed energy levels. n e u t r o n s b e c o m e l a r g e as a result o f t h e l a r g e v a l u e s o f t h e r i g i d m o m e n t o f i n e r t i a f o r b o t h p r o t o n s a n d n e u t r o n s . T h e t r a n s i t i o n e n e r g y c o r r e s p o n d i n g to t h e critical ang u l a r m o m e n t u m is s h o w n in fig. 5.
NUCLEAR ROTATIONAL STATES
315
Above the transition energy the nucleus is in the shell-model phase, for which the pairing correlation is absent. Below the transition energy the nucleus is in the superconducting phase, and the nuclear properties are very different from those of the normal state. The moment of inertia is an especially typical one and is very different from the rigid-body value expected for the shell model. The rigid-body value of the moment of inertia is a few times larger than the value determined so as to give the position of the first excited state. Then the rotational energy of the deformed nucleus is a few times larger than that of the rigid body in the low excited energy region. The moment of inertia gradually approaches the rigid-body value as the energy gap parameter A decreases with increasing angular m o m e n t u m I and reaches to its value at the critical angular m o m e n t u m I c. Therefore, the rotational energy part becomes that of the rigid body above the critical angular m o m e n t u m and largely deviates from that obtained with a fixed moment of inertia, which is determined from the first excited level. The rotational energy for 16°Dy is given in fig. 6 as a function of the angular momentum. This is the result of the calculation with the values Gp = 24A- 1 MeV and Gn = 17A -1 MeV. The critical angular m o m e n t u m is Ic = 20 for neutrons and I¢ = 34 for protons. Here @int gives the change of the Hartree-Fock energy (41) due to the excitation of intrinsic particles. This value increases with the angular m o m e n t u m as the correlated nucleon pairs are broken by the Coriolis force and reaches a value corresponding to the ground state correlation energy due to the pairing interaction at the critical angular momentum. The excitation energy corresponding to the angular m o m e n t u m I is given by eq. (15). This value fairly deviates still from the rotational energy with a fixed moment of inertia o¢'fix, which is determined so as to give the position of the first excited state. The excitation energy levels are shown in figs. 7 and 8 and are compared with the experiments. The energy spectra shown in the left-hand side of the figure for each nucleus are the experimental energy levels as), and A and B on the right-hand side show the theoretical results. We took the pairing interaction strengths as Gp = 24A- ~ MeV and Gn = 18A- 1 MeV in case A, and also as Gp = 24A- ~ MeV and G, = 17AMeV in case B. The results of our calculation explain the general features of experimental levels fairly well, but there still remain several defects. In general, the theoretical energy levels are a little higher than the observed levels. There are two reasons which give rise to such disagreement with the experiments. One is that the moment of inertia is a little small to be fitted to the first excited level. Even if the displacement from the position of the first excited state is small, it gives rise to large deviations from the expected positions of high angular m o m e n t u m states. To calculate the expected position of the high angular m o m e n t u m state, the moment of inertia must be determined so as to give the position of the first excited state. The results of the calculation in case B agree with the experiments much better than those in case A. However, the energy gap parameter values in case B are about 10 ~ smaller than the experiments. Then the latter case is more desirable than the former case to determine the pairing interaction strength so as to reproduce the empirical values of the gap parameter. The
316
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NUCLEAR ROTATIONAL STATES
317
moments of inertia given by case A lead to values about 10-20 ~ smaller than the empirical values which are determined from the positions of the first excited states. In the first 2 + state, the reduction of the energy gap parameter due to the Coriolis force is negligibly small, and also the excitation energy of the intrinsic particles is negligible. These mean that the calculated levels lie about 10-20 ~ higher than the data for the first excited states. Fairly good agreement with the experiments might be obtained in calculations in which the higher-order correction terms of the moment of inertia are taken into account. One other discrepancy with experiments is seen in the higher excited states when good agreement with the first excited state is assured, namely that the predicted levels are a little higher than the observed ones except for the W and Os nuclei. This tendency is especially remarkable in the first half of the rare-earth region. This might imply that the reduction of the energy gap is too small as to obtain good agreement with the experiments and that it is necessary to take into account effects of the collective vibrational motion on the rotational state. The nuclear equilibrium shape is determined according to the balance of the pairing and quadrupole forces. The pairing force has the effect of keeping the nuclear shape spherically, and on the other hand the quadrupole force causes the quadrupole deformation of the nucleus. Now, the reduction of the energy gap parameter due to the Coriolis force breaks the balance of these forces, and therefore the equilibrium deformation which gives the minimum energy changes as a function of the angular momentum. For instance, the value of the equilibrium deformation parameter 6 for 152Gd gradually increases as a function of the angular momentum, and its increasing ratio is 3.1 ~ at I = 10 and 7.4 ~ at I = 20. However, the equilibrium deformation does not increase gradually always. A typical example is shown in fig. 9 for is°Hr. The total energy corresponding to each angular m o m e n t u m was plotted as a function of 6. In this case, the Coulomb interaction between protons E¢ has been taken into account by Ec=3
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The curves of the energy surface have much less curvature than that without the correction of the Coulomb energy. Therefore, it is easier to shift the minimum point of the energy surface. It might play a much more important role in our calculations. The results shown in fig. 9, calculated by using the parameters of case A, were plotted for each second the level belonging to the ground state rotational band. The values of the critical angular momenta for 18°Hf are comparatively small, I~ = 12 for neutrons and Io = 24 for protons. The shape of the energy surface changes gradually as the angular m o m e n t u m increases. The minimum point is at 6 = 0.24 for the ground state, but the energy surface corresponding to I = 16 has two minimum points at 6 = 0.23 and 0.32. The absolute values of the energy at these minimum points are almost the same. The minima are very shallow and may be washed out by the fluctuations. How-
318
M. SANO AND M. WAKAI
ever, they are stable minimum points within the generalized Hartree-Fock approximation. Above the / = 18, the energy of the minimum point corresponding to the large deformation becomes lower than that corresponding to the small deformation. Particularly, the energy surface around the minimum point corresponding to the small deformation becomes almost flat, and the solutions of the generalized Hartree-Fock
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equation at this point may be unstable. As the state at this point becomes unstable, the nucleus will prefer to take the state corresponding to the minimum point at the large deformation. If the change from the small deformation to the large one suddenly takes place at the angular momentum ! = 20, the energy of the state with I = 20 becomes almost equal to that of the state with I = 18. The information on the instability of states cannot be obtained from the generalized Hartree-Fock equation,
NUCLEAR
ROTATIONAL
319
STATES
because it is derived from the requirement that the expectation value of the Hamiltonian ~ is stationary with respect to arbitrary first-order variations of p and x. In order to verify whether the solutions of the generalized Hartree-Fock equation are stable or not, we must examine the sign of the second variation of the expectation value of the Hamiltonian 7).
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The strength of the quadrupole force which gives the nuclear deformation 6 = 0.24 for 18°Hf is Z = 0.03 MeV. This value causes the rotational state of the ground band to be unstable at I = 18. This means that a large change of the nuclear deformation takes place suddenly at the angular m o m e n t u m I = 18. The change of the rotational spectrum is shown in fig. 10. The dashed line represents the rotational energies with a
320
M . SANO A N D M. W A K A I
fixed moment of inertia Jflx which is determined so as to give the position of the first excited state. The dot-and-dash line shows the rotational levels in which the antipairing effect due to the Coriolis force is taken into a6count, but the nuclear deformation is fixed at 6 = 0.24. The full line gives the rotational levels in which both the anti-pairing effect and the change o f the nuclear deformation are taken into account, and these states are all stable. We can see the discontinuitous change of the rotational level at I = 18, in which the intrinsic particle state becomes unstable. One of us (M.S.) would like to thank Professor L. S. Kisslinger for the hospitality accorded him during his stay at Western Reserve University and also to thank the National Science Foundation for award of the fellowship.
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