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Dynamic inertia tensor for a hot rotating nucleus
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A573 (1994) 47-60
Dynamic inertia tensor for a hot rotating nucleus Frank A. Dodaro, Alan L. Goodman Physics Department, Tulane University, New Orleans, Louisiana 70118, USA
(Received 9 August 1993;revised 6 December 1993)
Abstract A general formula is derived for the dynamic inertia tensor Y(‘) of a hot rotating nucleus. This formula includes both mean field and pairing effects within the cranked finite-temperature Hartree-Fock-Bogoliubov formalism. The intrinsic shape can have an arbitrary orientation relative to the rotation axis, and the angular momentum can be large. Special case limits include finite-temperature generalizations of the independent-particle model and pairing moments of inertia. The dynamic inertia tensor is interpreted as the static susceptibility of a many-body system to a uniform external field.
1. Introduction
The dynamic inertia tensor 9(*) gives the nuclear response to a rotational perturbation and continues to be a valuable probe in nuclear structure physics. Theoretic studies of 9(*) typically involve some form of the Inglis cranking model [1,2] where a rotating nuclear potential approximates the effect of rotation on the microscopic structure. Recently [31, a formula for 9 (*) has been derived which is valid at arbitrary magnitude and orientation of the rotational frequency vector w. This formula permits microscopic study of nuclear inertial properties within the independentparticle model, but it does not include pair correlations or finite-temperature. However, at nominal spin and normal deformation, nuclear pairing effects become important. The interplay between single-particle, pairing and rotational degrees of freedom may play an important role in the inertial properties of identical bands observed at normal deformation [4]. In order to study such systems, pairing effects should be included in the dynamic inertia tensor formula, as well as arbitrary magnitude and direction of W. 0375-9474/94/$07.00 0 1994 SSDZ 0375-9474(93)E0688-5
Elsevier Science B.V. All rights reserved
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F.A. Dodaro, A. L. Goodman /Nuclear
Physics A573 (1994) 47-60
Another area of interest has been the study of hot rotating nuclei. Alhassid et al. [S-8] have used the macroscopic Landau model to study shape transitions in hot rotating nuclei and properties of the giant dipole resonance (GDR) y-ray decay. They have also derived a finite-temperature generalization of the Inglis cranking moment of inertia formula (without pairing) that is valid for o = 0 [6,9]. Goodman [lo] has studied statistical orientation fluctuations of the nuclear shape relative to the rotation axis with the microscopic i13,2 model. It is desirable to have a dynamic inertia tensor formula that includes both thermal and pairing effects at finite-rotation (w # 0). Such a formula would provide a microscopic justification and/or test of the Landau model inertial parameters over the entire temperature and rotational frequency range. The 3 (2) formula would also give a better understanding of the relative influence of microscopic degrees of freedom (single-particle, pairing, rotational, thermal) on nuclear inertial properties. In sect. 2, we derive a general expression for the dynamic inertia tensor of a hot rotating nucleus. This expression is evaluated in the finite-temperature mean field appro~mation in sect. 3, which extends the 4 (2) formula of ref. [3f to include thermal and pairing effects. The various limiting forms of Y(*) are presented in sect. 4, along with a numerical example for the finite-temperature pairing limit.
2, Cranked hot nuclei We describe a hot rotating nucleus using quantum statistical mechanics with the three-dimensional cranking model. The rotational frequency vector w can have any orientation relative to the intrinsic shape. Since our result will be evaluated in the mean-field approximation (sect. 31, we use the grand canonical ensemble in the rotating frame. The equilibrium state minimizes the grand potential in the rotating frame [ll] ~~(w,T,~)=E-TS-~N--U~I,
(2.1)
where E =
(ri,>,
S = (-
kL&,
(2.2) (2.3)
N=
(2.4)
Ii = (4).
(2.5)
Expectation (6)
values are given by = Tr(dd),
P-6)
F.A. Dodaro, A.L. Goodman /Nuclear
Physics A573 (1994) 47-60
49
where the density operator 8 is 6 = Z-l e-Pl+, Z = Tr( e-@)
(2.7) 9
(2.8)
fjL&-pl\j-~.‘j,
(2.9)
p = l/kT.
(2.10)
Here 8, is the many-body hamiltonian of the non-rotating nucleus. The grand canonical traces are taken over eigenstates of the constrained hamiltonian fi’. The grand partition function (2.8) is Z(w, T, p) = ze-PEhcW+), (I
(2.11)
where E&(w, II) are the eigenvalues of fi’. The nuclear spin I is treated semiclassically as the expectation of the spin operator i The nuclear rotation axis and frequency are defined by o, where the magnitude of o is adjusted to conserve the average spin &=J(J+
I).
(2.12)
i=l
Now, the dynamic inertia tensor YC2’ is defined by [lo]
(2.13) or (2.14) where (2.14) follows from (2.5) and (2.6). The statistical thermodynamic tion of the Hellmann-Feynman theorem [12] gives
generaliza-
(2.15) It follows from (2.13) that
@(to,
T, p) = -
(2.16)
We now derive an expression for 9 (2) by substituting from (2.7)-(2.10) into (2.14). Derivatives of exponential operators are computed using a theorem from Wilcox [ 131 1 da
-&(*)
,-(l-+-b_, ah
7
(2.17)
50
F.A. Dodaro, A. L. Goodman /Nuclear
Physics A573 (1994) 47-60
where the operator d(h) depends continuously algebra we obtain the final result
=
‘du
on the parameter
e-U’hJ)(w,T,pj,
A. After some
(2.18)
where (2. 8) involves a change of integration variable and (2.19)
Ax=.&(.().
Alhassid et al. [6] has obtained a similar result for the w = 0 limit. Eq. (2.18) gives an expression for the dynamic inertia tensor at finite-temperature and finite-rotation. Some general properties of Yc2’ are: (1) Yc2) is real and symmetric. This property follows from (2.16) and the assumed analyticity of the grand potential. (2) 9c2) is closely connected to the spin correlation function between 4 and 4. For the special case where 4 commutes with the constrained hamiltonian, then [& fir]= [A& I?‘] = 0 and (2.18) becomes Q’(w,
T, p) = (AxAJ;)(ku = PC AJ:A.&,,,p,,
(2.20)
=P((A4)2)cw.r,~,.
(2.21)
and
J$“(w,
T, P)
In sect. 3, the mean-field approximation is used. We will then consider the special case of cranking a nucleus about a symmetry axis. So from (2.21) we see that statistical fluctuations provide a mechanism for a finite-temperature nucleus to respond to rotation about a symmetry axis. Such a response is generally prohibited at zero temperature since statistical fluctuations in a conserved quantum number will vanish exponentially as T + 0 (assuming the ground state is non-degenerate), so that (2.21) vanishes at zero temperature. This is consistent with the result for a zero temperature nucleu;.skanked about a symmetry axis [3]. 0. This property follows from (3) In the high temperature limit, Y(‘) (2.18) and is consistent with the fact that arbitrarily large temperatures destroy spin correlations. (4) The expression in (2.18) has the form of a susceptibility tensor [13]. This follows from the form of the constrained hamiltonian fi’ and allows us to interpret Yc2’ as the isothermal static susceptibility of the nuclear spin Z to a uniform rotational field o. Consider the general case of a system with constrained hamiltonian fiLfi,+i-A.&
(2.22)
F.A. Dodaro, A.L. Goodman /Nuclear
where fi, constrains expectation isothermal tensor by
Physics A573 (1994) 47-60
51
is the intrinsic many-body hamiltonian, the chemical potential /_L the expectation of N, and the field components hi constrain the of 6. If we assume that di are single-body operators, then the system static susceptibility tensor x is formally related to the dynamic inertia
(2.23) Therefore, the results presented in sect. 3 may be used to calculate the mean field susceptibility for many-fermion systems that couple linearly to an external field. 3. Mean-field approximation The expression for Y(*) given in (2.18) is rather formal, so we need to make additional approximations in order to obtain a more useful result. The cranked finite-temperature Hartree-Fock-Bogoliubov (FTHFBC) theory [ll] is capable of describing single-particle, pairing, thermal and rotational aspects of nuclear structure. Since this is the most general of the finite-temperature mean-field theories, we will evaluate (2.18) within the FTHFBC approximation. The FTHFBC theory approximates a system of interacting particles {&i> in thermal equilibrium by a statistical ensemble of non-interacting quasiparticles {a:]. The quasiparticle operators are related to the particle operators by a general Bogoliubov transformation
(3-l) The transformation matrices are found by solving the nonlinear FTHFBC equations. These equations are derived and discussed in ref. [ll]. A complete solution is specified by the matrices U, I/ and the set of quasiparticle energies {Ei]. These energies are associated with the creation operators {#} and represent elementary excitations of the system. At temperature T, the excitations occur with probabilities {fi} given by 1 fi = epEi + 1
.
(3.2)
The solution to the FTHFBC equations minimizes the grand potential in the rotating frame (2.1), where the chemical potential /.L and rotational frequency w are adjusted to conserve average particle number and spin, using equations (2.4) and (2.12). Expectation values are computed as in (2.6)-(2.101, except that the grand canonical traces are evaluated in the Fock space of quasiparticles. For example, 2 = Tr( e-O*) = Fl({ni) %
le-pri’l{ni}),
(3.3)
F.A. Dodaro, A.L. Goodman /Nuclear
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Physics A573 (1994) 47-60
= 0 or 1 denotes a complete set of quasiparticle where (ni} = (n(;‘), n$‘, ...Irz$) occupation numbers, and the trace in (3.3) is performed without restriction on the total number of quasiparticles. In the HFB approximation, the constrained hamiltonian is diagonal in the quasiparticle space (3.4) where E, is the quasiparticle vacuum energy. We use (3.4) to facilitate the evaluation of (2.18) by writing the spin operators in terms of the quasiparticle operators. From (3.1) we get (3.5)
4= (.T,)O+C(J)” mm 1
a^+ a^ m
m
m +
c [(J,)z,a^t,a^, + (JJf,$it,@,+ h.c.] ,
(3.6)
(m,n) where the sum in (3.5) is unrestricted and the sum on (m, n) in (3.6) is over ordered pairs (m < n). The expansion coefficients are given by (Ji)’ = Tr(JiVtV)(normal (.&)i, = (u*.@-
trace),
(3.7)
V*J;Q)&
(3.8)
(.Ji)Zn = (u*J,V+ - V*Ji*U+)&
(3.9)
where the tilde signifies transpose. Now we rewrite the dynamic inertia tensor from (2.18) as -a;?)(~,
T, p) =p /Dldrr (e@‘x
ePUa~<.> - (J)(h)).
(3.10)
i
We use (3.4) and (3.6) to write the integrand in (3.10) as (e 0p*& e-@AT)
= (Ji)“(J7.) + (Jj)o~(Ji),Sf,,(&a^,> m
(3.11)
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Physics A573 (1994) 47-60
53
where (3.11) contains the only nonvanishing terms in the FI’HFB approximation. The expectation values in (3.11) are evaluated using (3.4) and Wick’s theorem. As an example,
(egP$ht,kn e-~P~&~~,) = = eM4,,
-En)
e@(Em-En)(
dt,b,&f&,)
IfAl -fn)~mr~ns +frf?AwAr~*
(3.12)
The dynamic inertia tensor in the FTHFBC approximation is computed evaluating (3.11), substituting back into (3.10), and performing the integration (T. After some algebra we obtain
by on
>:!2)-rrHFBC(W, T, P) 11
=P
C
(JiIEn(Jj)~,.fm(l
-fm>
m
+
C
( [(Ji)~n(Jj)‘,‘~ +C.C.]PAi +
[(Ji)zn(Jj)2
+c*c*]p~~)~
(md
(3.13) with diagonal elements given by S!2)-rrHFBC(~, II
T, p) =pcI(J,);,1*f,(l
-f,)
m +2
(3.14) {l~J,)~~l*~~~+I~~i~~~l*~~O,)~
C o?l,n)
where the sum (m, n) is over ordered pairs, the spin coefficients (3.8)-(3.9), and the thermal weighting coefficients are given by pll =
mn
p
= mn
fin -fn
(3.15)
E,,-E,’
1-fm -fn E,+E,
are given in
’
(3.16)
Eq. (3.13) gives an expression for the dynamic inertia tensor at the point (0, T, p) of the intensive variable phase space in terms of the solution to the FTHFBC equations at that point. The dependence of .Y(*) upon the intensive variables is therefore implicit in (3.13). It is easily shown from (3.81, (3.9), and (3.13) that Yc2) is real and symmetric. The first sum in (3.13) contributes only at finite-temperature and is associated with statistical fluctuations of the spin operators. The second sum involves first-order scattering processes permitted by the cranking term --w *i These processes are the scattering of a single quasiparticle between states (P”) and the creation/annihilation of two-quasiparticle states (P*‘). In the high temperature limit, the quasiparticle occupation probabilities fi -+ i, sothe second sum in (3.13) vanishes. The first sum tends to zero with inverse temperature.
F.A. Dodaro, A.L. Goodman /Nuclear
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Physics A573 (1994) 47-60
If we crank a nucleus about a symmetry axis (say, the i axis), then the operator 4 is associated with a conserved symmetry and it can be shown [141 that the second sum in (3.13) vanishes term-by-term. This leaves statistical fluctuations in 4 as the only contribution to Y(*), which is consistent with (2.21). Finally we note that for systems having a vanishing spin expectation at w = 0, T, IL) =%9”‘(0,
y’*)(w,
(3.17)
T, p),
where Y(‘) is the kinematic inertia tensor [lo]. So if we evaluate (3.13) for the FTHFB solution to a non-rotating nucleus, we get the finite-temperature kinematic inertia tensor at w = 0. In cases where this tensor is diagonal, the diagonal elements are the nuclear principal moments of inertia for slow rotation.
4. Limiting
forms
We now examine some limiting forms for Y(*) corresponding to special case limits of FTHFBC theory. At zero temperature, FTHFBC theory reduces to cranked HFB (HFBC) theory. In this case the occupation probabilities fi -+ 0 and (3.13) becomes ~WHFy.#j,
p) =
IJ
(JJ;JJ,)~; + C.C.
c
E,+E,
b?l,n)
i
(4.1)
’
Eq. (4.1) gives the dynamic inertia tensor for a rotating nucleus at zero temperature. In this limit the quasiparticle scattering processes are restricted to the creation of two-quasiparticle states from the vacuum. Now we consider the independent-particle model (IPM) limit of .Y(*). In this case we have
q(w) = i
e,?< co)
Ei(O) > P
ei(w)
ci(W)
(4.2)
and Ei(m) = I Ei(m) -CL I>
(4.3)
where e!(w) are creation operators for the single-particle orbitals 1i(w)) in the rotating frame. The associated orbital energies are ci(O)* Eq. (4.2) defines the quasiparticle transformation matrices U, V in (3.1). Substituting from (4.2) and (4.3) into (3.13) gives &+IPM+, IJ
T, p) = c (m(w)
I& 1n(o))(n(o)
1J3.I m(o)>
mn
frill-fn
X i
1
%I(~)-%2(o) ’
(4.4)
F-4. Dodaro, A.L. Goodman /Nuclear
X i
Physics A573 (1994) 47-60
55
frill-fn %(W)- %I(@) ’
I
(4.5)
where the sum in (4.4) is unrestricted, the second sum in (4.5) is over ordered pairs, and the occupation probabilities are given by the Fermi-Dirac function fiC
1 eP(&J-pL, + 1 .
(4.6)
Eq. (4.5) gives the dynamic inertia tensor for a system of independent nucleons at finite-temperature and finite-rotation. If we take the zero temperature limit of (4.9, we obtain Goodman’s formula 131for the dynamic inertia tensor at finite-rotation &?-iPM( w, T = 0) [I =
r
(m(o) c ‘mtW> E,(W)>E&J)>Ci(W) -Ei(W)
I( I i(w)> + C.C.
I ,
(4.7)
where EJW) is the Fermi energy. Alternatively, we can arrive at (4.7) by taking the IPM limit of (4.1). It has been demonstrated that at zero temperature, the IPM dynamic inertia tensor can exhibit a resonance when two of the single-particle energies approach each other at the Fermi surface [lOI. This occurs because of a vanishing denominator in Eq. (4.7). Such a resonance disappears at moderate temperatures, even though the level crossing still exists in the single-particle spectrum. The reason for this is that the thermal weighting factor in (4.5) has a finite limit at a degeneracy
frill-.fn + Pfm(l -fm>. i En-enI 1En‘Em
(4.8)
Soat finite-temperature, statistical fluctuations provide a mechanism for “thermal washing” of a resonance and will generally smooth out the rotational dependence of Yc2). If we take the w = 0 limit of (4.5), we set Ii(o)) + Ii> Ei(W) + Ei,
(4.9)
where I i> and li are the single-particle orbitals and energies of the non-rotating system. The form of (4.5) remains unchanged and we get the finite-temperature kinematic inertia tensor at w = 0, with diagonal elements giving Alhassid’s finitetemperature generalization [6] of the Inglis cranking moment of inertia [1,2]. Now we examine the pairing limit of (3.13). We take w = 0 so that we can consider a pair-correlated system with time-reversal symmetry. Let the constrained
56
FA. Dodaro, A.L. Goodman /Nuclear
hamiltonian
Physics A573 (1994) 47-60
at w = 0 be of the pairing type
I$-/&=
c (EY- /.L)(@, V>O -
+ &?,)
c G,,,,@,t&,?,, VV’>O
(4.10)
where the particle model space is spanned by pairs of nucleons in time-reversed orbits {Q,,
= (&
eb),,,,
I;>=flv>,
E,=E,,
(4.11) (4.12)
and E, are the single-nucleon energies. This case corresponds to the finite-temperature BCS (FTBCS) limit [ll] of FTHFB theory. The BCS quasiparticle transformation is a^t=u et-u Y YY
c_ ““7
(4.13)
fite-t + .JJ c V= u “V ““.
(4.14)
The pair occupation parameters u,, U, satisfy of + u,” = 1, where 0,’ is the probability that the paired state VP is occupied. Eqs. (4.13) and (4.14) define the quasiparticle transformation matrices U, V in (3.1). Substituting from (4.13) and (4.14) into (3.13) gives the finite-temperature dynamic inertia tensor for a pair-correlated system at w = 0 ~j2)-ra’*(T,/J)=
c (~VIJ;IY~~~Y~IJ;IV)+(VI~I~~~~ij~I~l~~+C.C. ) vv’>O
x { ( U,Uvr+ U,U”r)2P~;~ + (u,u,, - u,u,!)2Pvy),
(4.15)
with diagonal elements @)-rair(T,/L)=2
c (I(ulJ:Iv’~l~+I(vIJ:I~~)l~) vu’>0
x { (U”U,! + u,u,t)2P,1,+ (u,u,, - u,u,.)2P”y},
(4.16)
where P’l and Pzo are given in (3.15) and (3.16). Eq. (4.15) gives the inertial properties of a finite-temperature pair-correlated system in the slow rotation limit. The coherence factors (U,ZQ + u,u,,) and (u,u,, - u,u,,) are quantum amplitudes for the single-quasiparticle scattering and two-quasiparticle creation/annihilation processes, respectively. As is the case with BCS theory [151, paired states that are nearest to the Fermi surface give the largest contribution to the sum in (4.15). For the P1’ term, states VV’ that are on the same side of the Fermi surface comprise the largest contribution. For the P2’ term, states VV’that are on opposite sides of the Fermi surface comprise the largest contribution.
F.A. Dodaro, A.L. Goodman /Nuclear
Physics AS73 (1994) 47-60
57
If we take the zero-temperature limit of (4.16), then fi + 0 and we get the pairing moment of inertia formula [16,171 @)-Pair(T=0)=2
c (I(ul~/v’)I*+ vu’>0
l(~l~lF’)l*)
- v,u,,)* x(U.VdE,,+E,t *
(4.17)
Alternatively, we can recover the pairing moment of inertia formula by taking the BCS limit of (4.1). In order to better understand the interplay between single-particle, pairing and thermal effects in (4.15), we consider a simple numerical example. We take the case of a constant pairing force in a single j-shell [18] Al=
c
(Em - /_L)@?,,, - G
i?l20
c
@&&,,,
,
(4.18)
mm’>0
where m is the projection of spin on the z-axis. The single-nucleon energies E, are taken as the eigenvalues of an axially symmetric potential with quadrupole deformation 3m2 -j( E,
=
K
i
j + 1)
j(i+l)
I
(4.19)
’
where K is an energy unit proportional to the deformation (K - 2 MeV for normal deformation). Since the constrained hamiltonian (4.18) has complex conjugation symmetry and axial symmetry about the z-axis, we can show that Yc2) is diagonal with &*) =9(*) =9, and ,a;!‘) = 9,, . The moments q,, 9, are the principal momen;; of iilrtia for slow rotation parallel, perpendicular to the symmetry axis. The FTBCS solution to (4.18) is obtained by solving the gap equation l=;Gc
tanh( $E,) E m m>O
.
The quasiparticle energies E, and pair occupation parameters
(4.20) u,, v, are given
by
- p)* + A'] J%= [(%I
“*,
(4.21) (4.22)
v;=1-u;.
(4.23)
The pair gap A is a measure of pairing correlations present in the system. Above a critical temperature T,, Eq. (4.20) ceases to have a solution and the pair gap vanishes. At this temperature, the system undergoes a phase transition from a paired state to a normal state. The chemical potential P is adjusted to achieve the
58
F.A. Dodaro, A.L. Goodman /Nuclear
Physics A573 (1994) 47-60
j=11/2
kT/G Fig, 1. The pair gap A as a function of temperature for fiied values of c = K/G.
desired expectation value for the particle number. We consider a half-filled shell with j = y. Such a configuration roughly corresponds to proton filling in the lh 11,2 shell. We take the pairing force constant G as the energy scale and introduce the parameter u = K/G as the relative strength of the deformation potential. Fig. 1 gives the pair gap A as a function of temperature for fixed values of u. For u = 0, the deformation potential vanishes and all of the single-particle energies 14.19) are degenerate (E, = ,U = 0). This configuration corresponds to a half-filled shell of degenerate pairs, which is the most favorable for pairing correlations. As a result, the g = 0 pair gap curve envelops all curves with nonzero c in Fig. 1. Pairing correlations can be reduced (or destroyed) by either raising the temperature or increasing CT.For a given LT,raising the temperature adds uncorrelated energy to the nucleons and inhibits the ability of pairs to interact. For a given temperature, increasing (T achieves the same effect by increasing the energy separation between pairs. In Fig. 2 the z-component of Eq. (4.16) has been evaluated to obtain -PI, the moment of inertia for slow rotation about an axis perpendicuIar to the symmet~ axis. Curves of Yl are plotted as a function of temperature for fixed values of u. The curves show an increase in YL with temperature for T < Tc and a discontinuity in the first derivative at T,. This behavior can be understood by considering the thermal dependence of the pair gap. As T increases towards Tc, a decreasing pair gap reduces the quasiparticle energies E, which occur in the energy denominators of P1’ and PzO in (4.16). This accounts for the sharp rise in Y, near T,. For T > T,, one can show by an argument similar to Belyaev [17] that the pairing inertia tensor (4.15) reduces to the w = 0 limit of the independent-particle model inertia tensor (4.4). So for T > T,, 9, becomes the moment of inertia for a system of independent nucleons with a doubly degenerate spectrum. We expect
FA. Dodaro, A.L. Goodman /Nuclear
Physics A573 (1994) 47-60
59
kT/G Fig. 2. The moment of inertia 9,
as a function of temperature for fixed values of D = K/G.
to be smooth in this region and at high temperature (kT x+ G) to tend towards zero with inverse temperature. The u = 0 curve provides an upper bound on 9, in this region. In Fig. 3 the zz-component of Eq. (4.16) has been evaluated to obtain q,, the moment of inertia for slow rotation about the symmetry axis. Curves of 3, are plotted as a function of temperature for fixed values of u. Since fZ is associated
9,
25
20
j=11/2 N=6
5
1.0
2.1D
kT/G
Fig. 3. The moment of inertia 3, as a function of temperature for fiied values of c = K/G.
FA. Dodaro, A.L. Goodman /Nuclear
60
Physics A573 (1994) 47-60
with a conserved symmetry, we have from (2.21) that 3, is equal to the squared fluctuation of .I, multiplied by p. Hence 3, is nonzero only at finite-temperature and all of the curves in Fig. 3 pass through the origin. We also note that for CT= 0, the hamiltonian in (4.18) is spherically symmetric so that 3, =Y, and the c = 0 curves in Figs. 2 and 3 are identical. For (+> 0, the curves of 3, are similar in structure to those of 4, , although the values of 4, are somewhat smaller. Both moments of inertia show a decreased variation with temperature as u is increased, which is consistent with the ability of deformation to inhibit pairing effects.
5. Conclusions
We have generalized the formula for the dynamic inertia tensor of a nucleus to include thermal and pairing effects. This generalized formula for high spins and arbitrary orientations of the rotational frequency dynamic inertia tensor in the cranking approximation has been interpreted isothermal static susceptibility of the nuclear spin to a uniform rotational
rotating is valid w. The as the field.
This work was supported by the Louisiana State Board of Regents LEQSF Grant no. GF-10 and the National Science Foundation.
under
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18]
D.R. In&, Phys. Rev. 96 (1954) 1059 D.R. Inglis, Phys. Rev. 103 (1956) 1786 A.L. Goodman, Phys. Rev. C45 (1992) 1649 J. Zhang and L.L. Riedinger, Phys. Rev. Lett. 69 (1992) 3448 Y. Alhassid, S. Levit and J. Zingman, Phys. Rev. Lett. 57 (1986) 539 Y. Alhassid, J. Zingman and S. Levit, Nucl. Phys. A469 (1987) 205 Y. Alhassid and B. Bush, Phys. Rev. Lett. 65 (1990) 2527 Y. Alhassid and B. Bush, Nucl. Phys. A531 (1991) 39 Y. Alhassid and B. Bush, Nucl. Phys. A549 (1992) 12 A.L. Goodman, Nucl. Phys. A542 (1992) 237 A.L. Goodman, Nucl. Phys. A352 (1981) 30 S. Golden, Chem. Phys. Lett. 31 (1975) 195 R.M. Wilcox, J. Math. Phys. 8 (1967) 962 J.L. Egido, Phys. Rev. Lett. 61 (1988) 767 J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175 S.G. Nilsson and 0. Prior, Mat. Fys. Medd. Dan Vid. Selsk. 32, no. 16 (1961) ST. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, no. 11 (1959) A.L. Goodman, Phys. Rev. C29 (1984) 1887
Nuclear Physics A573 (1994) 47-60
Dynamic inertia tensor for a hot rotating nucleus Frank A. Dodaro, Alan L. Goodman Physics Department, Tulane University, New Orleans, Louisiana 70118, USA
(Received 9 August 1993;revised 6 December 1993)
Abstract A general formula is derived for the dynamic inertia tensor Y(‘) of a hot rotating nucleus. This formula includes both mean field and pairing effects within the cranked finite-temperature Hartree-Fock-Bogoliubov formalism. The intrinsic shape can have an arbitrary orientation relative to the rotation axis, and the angular momentum can be large. Special case limits include finite-temperature generalizations of the independent-particle model and pairing moments of inertia. The dynamic inertia tensor is interpreted as the static susceptibility of a many-body system to a uniform external field.
1. Introduction
The dynamic inertia tensor 9(*) gives the nuclear response to a rotational perturbation and continues to be a valuable probe in nuclear structure physics. Theoretic studies of 9(*) typically involve some form of the Inglis cranking model [1,2] where a rotating nuclear potential approximates the effect of rotation on the microscopic structure. Recently [31, a formula for 9 (*) has been derived which is valid at arbitrary magnitude and orientation of the rotational frequency vector w. This formula permits microscopic study of nuclear inertial properties within the independentparticle model, but it does not include pair correlations or finite-temperature. However, at nominal spin and normal deformation, nuclear pairing effects become important. The interplay between single-particle, pairing and rotational degrees of freedom may play an important role in the inertial properties of identical bands observed at normal deformation [4]. In order to study such systems, pairing effects should be included in the dynamic inertia tensor formula, as well as arbitrary magnitude and direction of W. 0375-9474/94/$07.00 0 1994 SSDZ 0375-9474(93)E0688-5
Elsevier Science B.V. All rights reserved
48
F.A. Dodaro, A. L. Goodman /Nuclear
Physics A573 (1994) 47-60
Another area of interest has been the study of hot rotating nuclei. Alhassid et al. [S-8] have used the macroscopic Landau model to study shape transitions in hot rotating nuclei and properties of the giant dipole resonance (GDR) y-ray decay. They have also derived a finite-temperature generalization of the Inglis cranking moment of inertia formula (without pairing) that is valid for o = 0 [6,9]. Goodman [lo] has studied statistical orientation fluctuations of the nuclear shape relative to the rotation axis with the microscopic i13,2 model. It is desirable to have a dynamic inertia tensor formula that includes both thermal and pairing effects at finite-rotation (w # 0). Such a formula would provide a microscopic justification and/or test of the Landau model inertial parameters over the entire temperature and rotational frequency range. The 3 (2) formula would also give a better understanding of the relative influence of microscopic degrees of freedom (single-particle, pairing, rotational, thermal) on nuclear inertial properties. In sect. 2, we derive a general expression for the dynamic inertia tensor of a hot rotating nucleus. This expression is evaluated in the finite-temperature mean field appro~mation in sect. 3, which extends the 4 (2) formula of ref. [3f to include thermal and pairing effects. The various limiting forms of Y(*) are presented in sect. 4, along with a numerical example for the finite-temperature pairing limit.
2, Cranked hot nuclei We describe a hot rotating nucleus using quantum statistical mechanics with the three-dimensional cranking model. The rotational frequency vector w can have any orientation relative to the intrinsic shape. Since our result will be evaluated in the mean-field approximation (sect. 31, we use the grand canonical ensemble in the rotating frame. The equilibrium state minimizes the grand potential in the rotating frame [ll] ~~(w,T,~)=E-TS-~N--U~I,
(2.1)
where E =
(ri,>,
S = (-
kL&,
(2.2) (2.3)
N=
(2.4)
Ii = (4).
(2.5)
Expectation (6)
values are given by = Tr(dd),
P-6)
F.A. Dodaro, A.L. Goodman /Nuclear
Physics A573 (1994) 47-60
49
where the density operator 8 is 6 = Z-l e-Pl+, Z = Tr( e-@)
(2.7) 9
(2.8)
fjL&-pl\j-~.‘j,
(2.9)
p = l/kT.
(2.10)
Here 8, is the many-body hamiltonian of the non-rotating nucleus. The grand canonical traces are taken over eigenstates of the constrained hamiltonian fi’. The grand partition function (2.8) is Z(w, T, p) = ze-PEhcW+), (I
(2.11)
where E&(w, II) are the eigenvalues of fi’. The nuclear spin I is treated semiclassically as the expectation of the spin operator i The nuclear rotation axis and frequency are defined by o, where the magnitude of o is adjusted to conserve the average spin &=J(J+
I).
(2.12)
i=l
Now, the dynamic inertia tensor YC2’ is defined by [lo]
(2.13) or (2.14) where (2.14) follows from (2.5) and (2.6). The statistical thermodynamic tion of the Hellmann-Feynman theorem [12] gives
generaliza-
(2.15) It follows from (2.13) that
@(to,
T, p) = -
(2.16)
We now derive an expression for 9 (2) by substituting from (2.7)-(2.10) into (2.14). Derivatives of exponential operators are computed using a theorem from Wilcox [ 131 1 da
-&(*)
,-(l-+-b_, ah
7
(2.17)
50
F.A. Dodaro, A. L. Goodman /Nuclear
Physics A573 (1994) 47-60
where the operator d(h) depends continuously algebra we obtain the final result
=
‘du
on the parameter
e-U’hJ)(w,T,pj,
A. After some
(2.18)
where (2. 8) involves a change of integration variable and (2.19)
Ax=.&(.().
Alhassid et al. [6] has obtained a similar result for the w = 0 limit. Eq. (2.18) gives an expression for the dynamic inertia tensor at finite-temperature and finite-rotation. Some general properties of Yc2’ are: (1) Yc2) is real and symmetric. This property follows from (2.16) and the assumed analyticity of the grand potential. (2) 9c2) is closely connected to the spin correlation function between 4 and 4. For the special case where 4 commutes with the constrained hamiltonian, then [& fir]= [A& I?‘] = 0 and (2.18) becomes Q’(w,
T, p) = (AxAJ;)(ku = PC AJ:A.&,,,p,,
(2.20)
=P((A4)2)cw.r,~,.
(2.21)
and
J$“(w,
T, P)
In sect. 3, the mean-field approximation is used. We will then consider the special case of cranking a nucleus about a symmetry axis. So from (2.21) we see that statistical fluctuations provide a mechanism for a finite-temperature nucleus to respond to rotation about a symmetry axis. Such a response is generally prohibited at zero temperature since statistical fluctuations in a conserved quantum number will vanish exponentially as T + 0 (assuming the ground state is non-degenerate), so that (2.21) vanishes at zero temperature. This is consistent with the result for a zero temperature nucleu;.skanked about a symmetry axis [3]. 0. This property follows from (3) In the high temperature limit, Y(‘) (2.18) and is consistent with the fact that arbitrarily large temperatures destroy spin correlations. (4) The expression in (2.18) has the form of a susceptibility tensor [13]. This follows from the form of the constrained hamiltonian fi’ and allows us to interpret Yc2’ as the isothermal static susceptibility of the nuclear spin Z to a uniform rotational field o. Consider the general case of a system with constrained hamiltonian fiLfi,+i-A.&
(2.22)
F.A. Dodaro, A.L. Goodman /Nuclear
where fi, constrains expectation isothermal tensor by
Physics A573 (1994) 47-60
51
is the intrinsic many-body hamiltonian, the chemical potential /_L the expectation of N, and the field components hi constrain the of 6. If we assume that di are single-body operators, then the system static susceptibility tensor x is formally related to the dynamic inertia
(2.23) Therefore, the results presented in sect. 3 may be used to calculate the mean field susceptibility for many-fermion systems that couple linearly to an external field. 3. Mean-field approximation The expression for Y(*) given in (2.18) is rather formal, so we need to make additional approximations in order to obtain a more useful result. The cranked finite-temperature Hartree-Fock-Bogoliubov (FTHFBC) theory [ll] is capable of describing single-particle, pairing, thermal and rotational aspects of nuclear structure. Since this is the most general of the finite-temperature mean-field theories, we will evaluate (2.18) within the FTHFBC approximation. The FTHFBC theory approximates a system of interacting particles {&i> in thermal equilibrium by a statistical ensemble of non-interacting quasiparticles {a:]. The quasiparticle operators are related to the particle operators by a general Bogoliubov transformation
(3-l) The transformation matrices are found by solving the nonlinear FTHFBC equations. These equations are derived and discussed in ref. [ll]. A complete solution is specified by the matrices U, I/ and the set of quasiparticle energies {Ei]. These energies are associated with the creation operators {#} and represent elementary excitations of the system. At temperature T, the excitations occur with probabilities {fi} given by 1 fi = epEi + 1
.
(3.2)
The solution to the FTHFBC equations minimizes the grand potential in the rotating frame (2.1), where the chemical potential /.L and rotational frequency w are adjusted to conserve average particle number and spin, using equations (2.4) and (2.12). Expectation values are computed as in (2.6)-(2.101, except that the grand canonical traces are evaluated in the Fock space of quasiparticles. For example, 2 = Tr( e-O*) = Fl({ni) %
le-pri’l{ni}),
(3.3)
F.A. Dodaro, A.L. Goodman /Nuclear
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Physics A573 (1994) 47-60
= 0 or 1 denotes a complete set of quasiparticle where (ni} = (n(;‘), n$‘, ...Irz$) occupation numbers, and the trace in (3.3) is performed without restriction on the total number of quasiparticles. In the HFB approximation, the constrained hamiltonian is diagonal in the quasiparticle space (3.4) where E, is the quasiparticle vacuum energy. We use (3.4) to facilitate the evaluation of (2.18) by writing the spin operators in terms of the quasiparticle operators. From (3.1) we get (3.5)
4= (.T,)O+C(J)” mm 1
a^+ a^ m
m
m +
c [(J,)z,a^t,a^, + (JJf,$it,@,+ h.c.] ,
(3.6)
(m,n) where the sum in (3.5) is unrestricted and the sum on (m, n) in (3.6) is over ordered pairs (m < n). The expansion coefficients are given by (Ji)’ = Tr(JiVtV)(normal (.&)i, = (u*.@-
trace),
(3.7)
V*J;Q)&
(3.8)
(.Ji)Zn = (u*J,V+ - V*Ji*U+)&
(3.9)
where the tilde signifies transpose. Now we rewrite the dynamic inertia tensor from (2.18) as -a;?)(~,
T, p) =p /Dldrr (e@‘x
ePUa~<.> - (J)(h)).
(3.10)
i
We use (3.4) and (3.6) to write the integrand in (3.10) as (e 0p*& e-@AT)
= (Ji)“(J7.) + (Jj)o~(Ji),Sf,,(&a^,> m
(3.11)
F.A. Dodaro, A.L. Goodman /Nuclear
Physics A573 (1994) 47-60
53
where (3.11) contains the only nonvanishing terms in the FI’HFB approximation. The expectation values in (3.11) are evaluated using (3.4) and Wick’s theorem. As an example,
(egP$ht,kn e-~P~&~~,) = = eM4,,
-En)
e@(Em-En)(
dt,b,&f&,)
IfAl -fn)~mr~ns +frf?AwAr~*
(3.12)
The dynamic inertia tensor in the FTHFBC approximation is computed evaluating (3.11), substituting back into (3.10), and performing the integration (T. After some algebra we obtain
by on
>:!2)-rrHFBC(W, T, P) 11
=P
C
(JiIEn(Jj)~,.fm(l
-fm>
m
+
C
( [(Ji)~n(Jj)‘,‘~ +C.C.]PAi +
[(Ji)zn(Jj)2
+c*c*]p~~)~
(md
(3.13) with diagonal elements given by S!2)-rrHFBC(~, II
T, p) =pcI(J,);,1*f,(l
-f,)
m +2
(3.14) {l~J,)~~l*~~~+I~~i~~~l*~~O,)~
C o?l,n)
where the sum (m, n) is over ordered pairs, the spin coefficients (3.8)-(3.9), and the thermal weighting coefficients are given by pll =
mn
p
= mn
fin -fn
(3.15)
E,,-E,’
1-fm -fn E,+E,
are given in
’
(3.16)
Eq. (3.13) gives an expression for the dynamic inertia tensor at the point (0, T, p) of the intensive variable phase space in terms of the solution to the FTHFBC equations at that point. The dependence of .Y(*) upon the intensive variables is therefore implicit in (3.13). It is easily shown from (3.81, (3.9), and (3.13) that Yc2) is real and symmetric. The first sum in (3.13) contributes only at finite-temperature and is associated with statistical fluctuations of the spin operators. The second sum involves first-order scattering processes permitted by the cranking term --w *i These processes are the scattering of a single quasiparticle between states (P”) and the creation/annihilation of two-quasiparticle states (P*‘). In the high temperature limit, the quasiparticle occupation probabilities fi -+ i, sothe second sum in (3.13) vanishes. The first sum tends to zero with inverse temperature.
F.A. Dodaro, A.L. Goodman /Nuclear
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Physics A573 (1994) 47-60
If we crank a nucleus about a symmetry axis (say, the i axis), then the operator 4 is associated with a conserved symmetry and it can be shown [141 that the second sum in (3.13) vanishes term-by-term. This leaves statistical fluctuations in 4 as the only contribution to Y(*), which is consistent with (2.21). Finally we note that for systems having a vanishing spin expectation at w = 0, T, IL) =%9”‘(0,
y’*)(w,
(3.17)
T, p),
where Y(‘) is the kinematic inertia tensor [lo]. So if we evaluate (3.13) for the FTHFB solution to a non-rotating nucleus, we get the finite-temperature kinematic inertia tensor at w = 0. In cases where this tensor is diagonal, the diagonal elements are the nuclear principal moments of inertia for slow rotation.
4. Limiting
forms
We now examine some limiting forms for Y(*) corresponding to special case limits of FTHFBC theory. At zero temperature, FTHFBC theory reduces to cranked HFB (HFBC) theory. In this case the occupation probabilities fi -+ 0 and (3.13) becomes ~WHFy.#j,
p) =
IJ
(JJ;JJ,)~; + C.C.
c
E,+E,
b?l,n)
i
(4.1)
’
Eq. (4.1) gives the dynamic inertia tensor for a rotating nucleus at zero temperature. In this limit the quasiparticle scattering processes are restricted to the creation of two-quasiparticle states from the vacuum. Now we consider the independent-particle model (IPM) limit of .Y(*). In this case we have
q(w) = i
e,?< co)
Ei(O) > P
ei(w)
ci(W)
(4.2)
and Ei(m) = I Ei(m) -CL I>
(4.3)
where e!(w) are creation operators for the single-particle orbitals 1i(w)) in the rotating frame. The associated orbital energies are ci(O)* Eq. (4.2) defines the quasiparticle transformation matrices U, V in (3.1). Substituting from (4.2) and (4.3) into (3.13) gives &+IPM+, IJ
T, p) = c (m(w)
I& 1n(o))(n(o)
1J3.I m(o)>
mn
frill-fn
X i
1
%I(~)-%2(o) ’
(4.4)
F-4. Dodaro, A.L. Goodman /Nuclear
X i
Physics A573 (1994) 47-60
55
frill-fn %(W)- %I(@) ’
I
(4.5)
where the sum in (4.4) is unrestricted, the second sum in (4.5) is over ordered pairs, and the occupation probabilities are given by the Fermi-Dirac function fiC
1 eP(&J-pL, + 1 .
(4.6)
Eq. (4.5) gives the dynamic inertia tensor for a system of independent nucleons at finite-temperature and finite-rotation. If we take the zero temperature limit of (4.9, we obtain Goodman’s formula 131for the dynamic inertia tensor at finite-rotation &?-iPM( w, T = 0) [I =
r
I( I i(w)> + C.C.
I ,
(4.7)
where EJW) is the Fermi energy. Alternatively, we can arrive at (4.7) by taking the IPM limit of (4.1). It has been demonstrated that at zero temperature, the IPM dynamic inertia tensor can exhibit a resonance when two of the single-particle energies approach each other at the Fermi surface [lOI. This occurs because of a vanishing denominator in Eq. (4.7). Such a resonance disappears at moderate temperatures, even though the level crossing still exists in the single-particle spectrum. The reason for this is that the thermal weighting factor in (4.5) has a finite limit at a degeneracy
frill-.fn + Pfm(l -fm>. i En-enI 1En‘Em
(4.8)
Soat finite-temperature, statistical fluctuations provide a mechanism for “thermal washing” of a resonance and will generally smooth out the rotational dependence of Yc2). If we take the w = 0 limit of (4.5), we set Ii(o)) + Ii> Ei(W) + Ei,
(4.9)
where I i> and li are the single-particle orbitals and energies of the non-rotating system. The form of (4.5) remains unchanged and we get the finite-temperature kinematic inertia tensor at w = 0, with diagonal elements giving Alhassid’s finitetemperature generalization [6] of the Inglis cranking moment of inertia [1,2]. Now we examine the pairing limit of (3.13). We take w = 0 so that we can consider a pair-correlated system with time-reversal symmetry. Let the constrained
56
FA. Dodaro, A.L. Goodman /Nuclear
hamiltonian
Physics A573 (1994) 47-60
at w = 0 be of the pairing type
I$-/&=
c (EY- /.L)(@, V>O -
+ &?,)
c G,,,,@,t&,?,, VV’>O
(4.10)
where the particle model space is spanned by pairs of nucleons in time-reversed orbits {Q,,
= (&
eb),,,,
I;>=flv>,
E,=E,,
(4.11) (4.12)
and E, are the single-nucleon energies. This case corresponds to the finite-temperature BCS (FTBCS) limit [ll] of FTHFB theory. The BCS quasiparticle transformation is a^t=u et-u Y YY
c_ ““7
(4.13)
fite-t + .JJ c V= u “V ““.
(4.14)
The pair occupation parameters u,, U, satisfy of + u,” = 1, where 0,’ is the probability that the paired state VP is occupied. Eqs. (4.13) and (4.14) define the quasiparticle transformation matrices U, V in (3.1). Substituting from (4.13) and (4.14) into (3.13) gives the finite-temperature dynamic inertia tensor for a pair-correlated system at w = 0 ~j2)-ra’*(T,/J)=
c (~VIJ;IY~~~Y~IJ;IV)+(VI~I~~~~ij~I~l~~+C.C. ) vv’>O
x { ( U,Uvr+ U,U”r)2P~;~ + (u,u,, - u,u,!)2Pvy),
(4.15)
with diagonal elements @)-rair(T,/L)=2
c (I(ulJ:Iv’~l~+I(vIJ:I~~)l~) vu’>0
x { (U”U,! + u,u,t)2P,1,+ (u,u,, - u,u,.)2P”y},
(4.16)
where P’l and Pzo are given in (3.15) and (3.16). Eq. (4.15) gives the inertial properties of a finite-temperature pair-correlated system in the slow rotation limit. The coherence factors (U,ZQ + u,u,,) and (u,u,, - u,u,,) are quantum amplitudes for the single-quasiparticle scattering and two-quasiparticle creation/annihilation processes, respectively. As is the case with BCS theory [151, paired states that are nearest to the Fermi surface give the largest contribution to the sum in (4.15). For the P1’ term, states VV’ that are on the same side of the Fermi surface comprise the largest contribution. For the P2’ term, states VV’that are on opposite sides of the Fermi surface comprise the largest contribution.
F.A. Dodaro, A.L. Goodman /Nuclear
Physics AS73 (1994) 47-60
57
If we take the zero-temperature limit of (4.16), then fi + 0 and we get the pairing moment of inertia formula [16,171 @)-Pair(T=0)=2
c (I(ul~/v’)I*+ vu’>0
l(~l~lF’)l*)
- v,u,,)* x(U.VdE,,+E,t *
(4.17)
Alternatively, we can recover the pairing moment of inertia formula by taking the BCS limit of (4.1). In order to better understand the interplay between single-particle, pairing and thermal effects in (4.15), we consider a simple numerical example. We take the case of a constant pairing force in a single j-shell [18] Al=
c
(Em - /_L)@?,,, - G
i?l20
c
@&&,,,
,
(4.18)
mm’>0
where m is the projection of spin on the z-axis. The single-nucleon energies E, are taken as the eigenvalues of an axially symmetric potential with quadrupole deformation 3m2 -j( E,
=
K
i
j + 1)
j(i+l)
I
(4.19)
’
where K is an energy unit proportional to the deformation (K - 2 MeV for normal deformation). Since the constrained hamiltonian (4.18) has complex conjugation symmetry and axial symmetry about the z-axis, we can show that Yc2) is diagonal with &*) =9(*) =9, and ,a;!‘) = 9,, . The moments q,, 9, are the principal momen;; of iilrtia for slow rotation parallel, perpendicular to the symmetry axis. The FTBCS solution to (4.18) is obtained by solving the gap equation l=;Gc
tanh( $E,) E m m>O
.
The quasiparticle energies E, and pair occupation parameters
(4.20) u,, v, are given
by
- p)* + A'] J%= [(%I
“*,
(4.21) (4.22)
v;=1-u;.
(4.23)
The pair gap A is a measure of pairing correlations present in the system. Above a critical temperature T,, Eq. (4.20) ceases to have a solution and the pair gap vanishes. At this temperature, the system undergoes a phase transition from a paired state to a normal state. The chemical potential P is adjusted to achieve the
58
F.A. Dodaro, A.L. Goodman /Nuclear
Physics A573 (1994) 47-60
j=11/2
kT/G Fig, 1. The pair gap A as a function of temperature for fiied values of c = K/G.
desired expectation value for the particle number. We consider a half-filled shell with j = y. Such a configuration roughly corresponds to proton filling in the lh 11,2 shell. We take the pairing force constant G as the energy scale and introduce the parameter u = K/G as the relative strength of the deformation potential. Fig. 1 gives the pair gap A as a function of temperature for fixed values of u. For u = 0, the deformation potential vanishes and all of the single-particle energies 14.19) are degenerate (E, = ,U = 0). This configuration corresponds to a half-filled shell of degenerate pairs, which is the most favorable for pairing correlations. As a result, the g = 0 pair gap curve envelops all curves with nonzero c in Fig. 1. Pairing correlations can be reduced (or destroyed) by either raising the temperature or increasing CT.For a given LT,raising the temperature adds uncorrelated energy to the nucleons and inhibits the ability of pairs to interact. For a given temperature, increasing (T achieves the same effect by increasing the energy separation between pairs. In Fig. 2 the z-component of Eq. (4.16) has been evaluated to obtain -PI, the moment of inertia for slow rotation about an axis perpendicuIar to the symmet~ axis. Curves of Yl are plotted as a function of temperature for fixed values of u. The curves show an increase in YL with temperature for T < Tc and a discontinuity in the first derivative at T,. This behavior can be understood by considering the thermal dependence of the pair gap. As T increases towards Tc, a decreasing pair gap reduces the quasiparticle energies E, which occur in the energy denominators of P1’ and PzO in (4.16). This accounts for the sharp rise in Y, near T,. For T > T,, one can show by an argument similar to Belyaev [17] that the pairing inertia tensor (4.15) reduces to the w = 0 limit of the independent-particle model inertia tensor (4.4). So for T > T,, 9, becomes the moment of inertia for a system of independent nucleons with a doubly degenerate spectrum. We expect
FA. Dodaro, A.L. Goodman /Nuclear
Physics A573 (1994) 47-60
59
kT/G Fig. 2. The moment of inertia 9,
as a function of temperature for fixed values of D = K/G.
to be smooth in this region and at high temperature (kT x+ G) to tend towards zero with inverse temperature. The u = 0 curve provides an upper bound on 9, in this region. In Fig. 3 the zz-component of Eq. (4.16) has been evaluated to obtain q,, the moment of inertia for slow rotation about the symmetry axis. Curves of 3, are plotted as a function of temperature for fixed values of u. Since fZ is associated
9,
25
20
j=11/2 N=6
5
1.0
2.1D
kT/G
Fig. 3. The moment of inertia 3, as a function of temperature for fiied values of c = K/G.
FA. Dodaro, A.L. Goodman /Nuclear
60
Physics A573 (1994) 47-60
with a conserved symmetry, we have from (2.21) that 3, is equal to the squared fluctuation of .I, multiplied by p. Hence 3, is nonzero only at finite-temperature and all of the curves in Fig. 3 pass through the origin. We also note that for CT= 0, the hamiltonian in (4.18) is spherically symmetric so that 3, =Y, and the c = 0 curves in Figs. 2 and 3 are identical. For (+> 0, the curves of 3, are similar in structure to those of 4, , although the values of 4, are somewhat smaller. Both moments of inertia show a decreased variation with temperature as u is increased, which is consistent with the ability of deformation to inhibit pairing effects.
5. Conclusions
We have generalized the formula for the dynamic inertia tensor of a nucleus to include thermal and pairing effects. This generalized formula for high spins and arbitrary orientations of the rotational frequency dynamic inertia tensor in the cranking approximation has been interpreted isothermal static susceptibility of the nuclear spin to a uniform rotational
rotating is valid w. The as the field.
This work was supported by the Louisiana State Board of Regents LEQSF Grant no. GF-10 and the National Science Foundation.
under
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18]
D.R. In&, Phys. Rev. 96 (1954) 1059 D.R. Inglis, Phys. Rev. 103 (1956) 1786 A.L. Goodman, Phys. Rev. C45 (1992) 1649 J. Zhang and L.L. Riedinger, Phys. Rev. Lett. 69 (1992) 3448 Y. Alhassid, S. Levit and J. Zingman, Phys. Rev. Lett. 57 (1986) 539 Y. Alhassid, J. Zingman and S. Levit, Nucl. Phys. A469 (1987) 205 Y. Alhassid and B. Bush, Phys. Rev. Lett. 65 (1990) 2527 Y. Alhassid and B. Bush, Nucl. Phys. A531 (1991) 39 Y. Alhassid and B. Bush, Nucl. Phys. A549 (1992) 12 A.L. Goodman, Nucl. Phys. A542 (1992) 237 A.L. Goodman, Nucl. Phys. A352 (1981) 30 S. Golden, Chem. Phys. Lett. 31 (1975) 195 R.M. Wilcox, J. Math. Phys. 8 (1967) 962 J.L. Egido, Phys. Rev. Lett. 61 (1988) 767 J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175 S.G. Nilsson and 0. Prior, Mat. Fys. Medd. Dan Vid. Selsk. 32, no. 16 (1961) ST. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, no. 11 (1959) A.L. Goodman, Phys. Rev. C29 (1984) 1887