- Email: [email protected]
Generalized cranking model for collective nuclear motion
Nuclear 0
Physics
North-Holland
A426 (1984) 353-378 Publishing
Company
GENERALIZED CRANKING MODEL FOR COLLECTIVE NUCLEAR MOTION J. KUNZ Theoretical
and J. R. NIX
Division, Los .4lamos National Laboratory, Received
3 January
Los .4lamos, New Mexico 87545, USA 1984
The Inglis cranking model is generalized to take into account effects of any velocity dependence present in the single-particle potential and the reaction of the pairing field to the collective motion. The generalized model is applied to translations, rotations and some special types of vibrations. Some of our results and our numerical calculations are obtained with a harmonic-oscillator single-particle potential. Unlike the inertia calculated with the Inglis cranking model, the inertia calculated with the generalized cranking model is independent of the effective mass and approaches the irrotational value in the limit of large pairing.
Abstract:
1. Introduction
The cranking model has been widely used to calculate the inertia corresponding to collective nuclear motion After the introduction of pairing correlations into the Inglis cranking model ‘) by Belyaev ‘), calculations of the inertia in a modified harmonic-oscillator potential with pairing were performed by Nilsson and Prior “) for rotations and by Bes4) for y-vibrations. The pairing correlations reduced the calculated rotational moment of inertia below the rigid-body value and increased the calculated vibrational inertia above the irrotational value, so that relatively good agreement between theoretical and experimental results was obtained. Later, interest in P-vibrational inertias was spurred in connection with fission deformations. Calculations in a modified harmonic-oscillator potential with pairing yielded inertias several times the irrotational values “) and led to reasonable agreement with experimental spontaneous-fission lifetimes 6). However, some systematic discrepancies between theoretical inertias and experimental results were observed. The Nilsson and Prior results “) for the rotational moment of inertia were too low by about 15 to 20 %. Randrup et al. ‘) needed an overall renormalization factor of 0.8 for the calculated vibrational inertias to reproduce experimental spontaneous-fission lifetimes. Also, it was early recognized that the Inglis cranking model had some intrinsic deficiencies. Prange *) pointed out that the rotational moment of inertia calculated with the Inglis model goes to zero in the limit of large pairing, whereas it should approach the finite irrotational-flow value. Similarly, when pairing correlations are 353
354
J. Km:,
J. R. Ni.r :’ Generalized
cranking model
included, the Inglis pushing model gives for uniform translations below the correct mass of the nucleus ‘) and that also vanishes pairing. In
addition,
problems
arose
with
the
Inglis
cranking
an inertia that lies in the limit of large model
for
velocity-
dependent potentials. Moments of inertia calculated with the Nilsson modified harmonic-oscillator potential did not approach the corresponding rigid-body moments of inertia for high temperatures, but were instead too large “). As alternatives to the Inglis cranking model, there have been other treatments that do not lead to the above unacceptable results. Migdal lo) developed a formalism based upon Green’s functions to calculate rotational moments of inertia. He properly took into account the velocity dependence of a particular potential by introducing the effective mass into the cranking term. Furthermore, he showed that the collective rotation introduces an additional term in the pairing potential that keeps the moment of inertia finite in the limit of large pairing. Belyaev 11) obtained analogous results in his self-consistent-held formalism. The deficiencies of the Inglis cranking model were further pointed out by Hartree-Fock equations took care Thouless and Valatin “), w h ose self-consistent of both problems. Furthermore, their formalism demonstrated the approximation that is necessary to obtain the Inglis formula. Starting from the stationary equation in the rotating frame
[X - oYx, where &’ is the generalized hamiltonian showed that the Inglis formula is obtained
a] = 0, and ~8 is the generalized density, they when the lowest-order correction Xl to
the hamiltonian is neglected. The X’, term, however, takes care of the effective mass and the additional pairing term due to the collective rotation. For quadrup-ole vibrations the omission of the %‘, term in the Inglis cranking formula was pointed out by Baranger and Kumar 13). In this paper we give a time-independent derivation of the generalized cranking formula, starting from the stationary equation of Thouless and Valatin 12) and keeping the X1 term. In sect. 2 we derive a general expression for the inertia containing two additional terms to account for the effective mass and the reaction of the pairing field to the collective motion. We apply these results in sect. 3 to translations, rotations and vibrations. where for vibrations we discuss only specific cases for which the collective momentum operator has a certain form. The two submatrices of &,, namely h, to account for the effective mass and A, to account for the reaction of the pairing field to the collective motion, are discussed in appendices A and C, respectively. Appendix B demonstrates that the inertia is independent of the effective mass because of the hi term, and appendix D contains a semiclassical calculation of the inertia which shows that in the limit of large pairing the inertia approaches the irrotational value.
355
J. Kunz, J. R. Nix 1 Generalized cranking model
2. Time-independent
formalism
The usual perturbative cranking model that was first introduced by Inglis ‘) does not properly take into account the effects of any velocity dependence present in the single-particle potential and completely neglects the reaction of the pairing field to the collective motion. This leads to two unacceptable results: (i) the calculated inertia depends strongly on the effective mass M*, and (ii) it vanishes in the limit of strong pairing. In our generalization of the cranking model we take care of both these deficiencies. We derive the equations of the generalized cranking model by following the time-independent self-consistent Hartree-Fock formalism of Thouless and Valatin I’). Since we perform a perturbative treatment, we take into account only terms linear in the collective motion. The self-consistent Hartree-Fock potential is approximated by a suitable single-particle potential. In our derivation of the additional terms arising from the reaction of the mean held to the collective motion we argue along the lines of Migdal “1 who discussed rotations, and Bohr and Mottelson 14). 2.1. COLLECTIVE
INERTIA
We consider collective nuclear motion that can be treated equation in the moving frame of the form 12) [Z-XI,
by a stationary
a] = 0,
(2.1)
where the square brackets denote the commutator, X is the generalized mean field, l7 is the generalized collective momentum and 8 is the generalized density. They are defined by
where h represents the single-particle spectrum, A is the pairing potential, p is the density matrix and K is the abnormal density matrix. As discussed in more detail in appendix A, the lagrange multiplier i gives the magnitude of the velocity field u1 that describes the transformation from the laboratory frame to the moving frame. Consequently, the collective momentum operator assumes the form 1 n = Jjj (u,.p+p.a,) for the collective motions considered here. To obtain the perturbative cranking model equations we substitute % = X0,+.%,,
.@ = 5?o+w,
(2.2)
356
J. Kunz, J. R. Nix i Generalized cranking model
into eq. (2.1) and retain terms that are linear in Xi and 9,. In contrast, the X, term is neglected in the Inglis cranking model. Our linearization leads to two coupled
matrix
equations
for p, and
namely
K,,
h,p,-p,h,+d,~r-~,d,*+h,p,-p,h,+d,K,*-K,d:
=
J~v,-P,~),
= j.(7CK0+ K,,n*). We next rewrite these equations in terms of matrix 1~) and 1~) of the unperturbed hamiltonian satisfying [Jr,,
(2.3)
(2.4) elements
between
eigenstates
.%,I = 0.
Eq. (2.4) is then solved to obtain an expression for the matrix elements of the abnormal density K~, with use made of the proper relations of matrix elements with their complex conjugate and hermitian conjugate 15). Upon inserting this expression into eq. (2.3) we obtain as our final expression for the normal density matrix
/Jlpv
=
(ix,,.
--.
-h,,)[W,+4(
E2Kr-K,)-_(hr+h,)(p~-pV)]
~---
_Ei-.
v
Ir
&-,[(A
+-.-LL’_L_F----.
+A .k’ +P,II
l)+($,+h,)(K,+K,)]
(2.5)
b
In this and subsequent equations, for notational simplicity we omit on h, A, p and the subscript 0 indicating unperturbed quantities. As usual, E, denotes the quasiparticle energy
K
E, = (h;+A;)‘. Our expression (2.5) for pi contains two additional terms proportional to h, and A, that are not present in the usual cranking result. Finally, we obtain from eq. (2.5) the inertia I corresponding to the collective motion considered by taking the trace of the density matrix with the collective momentum operator ‘*), which leads to
I = &Tr(B?in)
The two additional
terms proportional
= fTr(p,n).
to h, and A, that appear
(2.6)
in eq. (2.6) for the
J. Kunz, J. R. Nix / Generalized cranking model
357
inertia I remove its dependence on the effective mass and keep it finite for large pairing gaps. By setting h, and A, equal to zero, i.e. by equating the effective mass to the nucleon mass and neglecting the reaction of the pairing field to the collective motion, we can easily verify that the resultant expression for the inertia yields the usual cranking expression 6, for a rotating nucleus at finite temperature in the presence of pairing. The determination of the two additional terms is our next concern. 2.2. DETERMINATION
OF h, AND d I
The occurrence of additional terms to account for an effective mass and the reaction of the pairing field to the collective motion has been discussed by Migdal lo) for the case of rotations. In addition, the occurrence of an h, term for non-local potentials has been pointed out by Bohr and Mottelson i4) for several types of collective motion. Here we generalize their arguments to collective motion that can be described by the stationary equation (2.1) with a collective momentum operator of the form (2.2). The single-particle potential, which approximates the self-consistent HartreeFock potential arising from two-body interactions, should not depend on the absolute momenta of the nucleons but on their momenta relative to those of the other nucleons. Therefore, in all momentum-dependent potential terms we make the replacement i”*r4) p -+ p-mu,.
(2.7)
To obtain the h, term we expand the potential in a Taylor series to lowest order in v1 and symmetrize in order to take into account the possible non-commutativity of u, -A and ----- V,V_ This leads to
hl = 4m[vA-V,W,
p)+V,V(r,
p). VJ
In the applications to be made in sect. 3 we specialize to a momentumdependent potential that is quadratic in the momentum. In that case expression (2.8) for h, reduces to h lpv
=
(1-mlm*P~,,,
(2.9)
where the effective mass m* is constant. In general, m* is a position-dependent and momentum-dependent tensor, which is the case for the Niisson modified harmonic-oscillator potential 16). An alternative derivation of eq. (2.9) for a special case is given in appendix A. Finally, we remark that a formally analogous result to eq. (2.9) has been obtained in recent adiabatic time-dependent HartreeFock calculations i7).
J. Kunz, J. R. Nix 1 Generalized
358
cranking model
We now turn to the reaction of the pairing field to the collective motion. In order to evaluate A, we must invoke another equation. For this purpose Migdal lo) suggested the use of either the gap equation or the continuity equation. He employed the latter for the case of rotations. The continuity equation requires that the divergence of the current j,,,(r) in the moving frame vanish, i.e. V. jm(r) = 0, because the density in the moving frame undergoing collective motion is given by
(2.10)
is stationary.
The current
of a nucleus
j(r) = ReTr [pi_&)], where;(r) is the matrix of the current operator. On the other hand, the divergence of the current operator can be expressed in terms of the commutator of the spatial density operator B(r) and the hamiltonian H through the operator continuity equation
V.;(r)
Therefore,
with linear
= - k [a(r), H].
terms retained,
condition
(2.10) can be rewritten
as
(2.11) ~~~~~~~~~~~~ol~+~~~~oC~~~~~~,]~-~~{po[p~~~,~~]} = 0. Upon
inserting rr
eq. (2.5) for p, into eq. (2.11), we obtain
A
p-v -
L~~~+~vbw
= c (II,,,-3x,,) I1.Y
+p,-
I)+
ih,+h,jiK,+K,jj(hp-hy)
T-E;
[@dQ(
KIr-K,)+(A,-A,)(pll-pv)](A,-A,) Et-E:
~
P,,(r).
p,,(r)
(2.12)
Since eq. (2.12) for A, depends on position r, it represents an infinite set of equations that must be satisfied in order to obtain the matrix elements Al,_,.
3. Appikaiioas
In this section we demonstrate the importance of taking into account the effective mass and the reaction of the pairing field to the collective motion. In order to be able to calculate the effects of the h, and A, terms analytically, we consider
J. Kunz, J. R. Nix i Generalized
359
cranking model
the simple hamiltonian H = p2/2m*+
V(r),
(3.1)
with constant effective mass m*. Furthermore, in most cases we specialize hamiltonian in eq. (3.1) still further to the harmonic-oscillator potential V(r) = ~m[wf(x2+y2)+cofz2]. For nonzero
temperatures
the normal
density
pV = +[I -(II,/&) and the abnormal
density
the
(3.2)
is given by
tanh (E,/2T)]
is given by K, = i( A,/E,) tanh (E,/2T).
In the limit of zero generalized cranking interest here is the pairing gap A for all to the much simpler
temperature, which we consider in some of our studies, the equations simplify considerably. Furthermore, since our main strong pairing behaviour of the inertia, we take a constant states. Then, eq. (2.6) for the inertia reduces in the limit T + 0 expression lo)
(E,E l=$~_-_Y_-
- h,h, - A2)ln,,12 - Af~j;~ti,, 2E,E,(E,
V.L where we use the notation
+ EL)
(3.3)
’
lo) A, = if(r)lm/m*,
(3.4)
[IC,H].
(3.5)
i = ;
Also, eq. (2.12) for A, [or equivalently simplifies to
for f(r)
in the new notation
of eq. (3.4)]
(3.6)
Upon
multiplying
eq. (3.6) by j(r)
_A/‘,
and integrating
c
%-b
v,,,
2EvE,SE,+ E,)
=
c
we obtain
the relation
(h-hJ21_L,12
v.,, 4&5,,(~,+
E,) ’
(3.7)
360
J. Kunz, J. R. Nix
! Generalized
cranking
model
which is then substituted into the second part of eq. (3.3). This yields as our final expression for the inertia
ZE,E,(E,+
E,)
(3.8)
----’
This equation is employed in our numerical calculations of the inertia, which are performed for a 240Pu nucleus in a harmonic-o~illator potential. In our analytic calculation of the inertia we follow the semiclassical formalism used by Migdal lo) for rotational motion. The key point in this derivation is the introduction of the function
9Sx) =
sinh-‘x x(1 +x2)1 ’
x = (~~-~~)~2~.
As discussed in refs. ‘Ovi4), this exploits the sharp maximum of the denominator E,E,(E,+E,). With this approximation, eq. (3.8) for the inertia leads to
with (3.10) Also, in this approximation eq. (3.6) for the function f(r), pairing field to the motion, becomes
i.e. the reaction of the
We now apply the formalism that we have developed to translational, rotational and vibrational motion. In the last case we investigate the breathing mode and ellipsoidal /3- and y-vibrations. For each kind of collective motion we first consider the effect of the h, term only in the absence of pairing, where we can easily demonstrate that the inertia is independent of m*. Then the effect of A, is investigated in the limit of zero temperature, where we show both in a semiclassical approximation lo) and also numerically that the inertia remains finite for strong pairing fields. In fact, with increasing pairing it either approaches or remains at the irrotational value.
J. Kunz, J. R. Nix
3.1. TRANSLATIONAL
/
Generalized
cranking
361
model
MOTION
For uniform translational motion along collective momentum operator is given by
one axis, for example
1 lr = - VA.p, %
v1 = l:eZ.
the z-axis,
the
(3.12)
The velocity vi. in eq. (3.12) is both irrotational and divergence-free. For translations the inertia of the moving nucleus must rigorously be the total mass M. Since vL is irrotational we can apply the result of appendix B.l and obtain immediately I = c m(v)ef(v)p, Y
= mA = M.
(3.13)
Thus, because the h, term is included, eq. (3.13) gives correctly the mass of the nucleus for the translational inertia. We now turn to the determination of A, in the pushing model. In previous treatments 9), with pairing included the Inghs pushing model has given a mass considerably smaller than the correct value mA. We now demonstrate that with the A, term included the correct value for the mass is recovered. Since the velocity V~ given by eq. (3.12) is irrotational, the function S(r) is proportional to the velocity potential according to appendix C.l. Thus, for the reaction of the pairing field we obtain
A, az, or, in terms of spherical
Then,
the semiclassical
harmonics,
approximation
derived
in appendix
D yields for the inertia
I = I, = mA, which is the correct total mass of the nucleus. We show in fig. la our numerical result for the translational inertia of ‘*‘Pu as a function of the pairing gap A, calculated for a harmonic-oscillator potential with 15 oscillator shells, which corresponds to a maximum principal oscillator quantum number of N = 14. As indicated by the solid line, with increasing A the inertia calculated with our expression (3.8) remains close to its value at A = 0. On the other hand, as indicated by the long-dashed line, the inertia calculated in the Inglis
J. Kunz, J. R. Nix 1 Generalized cranking model
362
“OPU Harmonic oscillator ______-________________-__ \ ~~~~i,;-
0
8
2
10
Pairilg Gap A kleV1 Fig. la. Dependence of the translational inertia upon the pairing gap A for 24”Pu in a harmonicoscillator potential. The solid curve gives the present result calculated in the generalized cranking model with 15 oscillator shells and the long-dashed curve gives the corresponding result calculated in the Inglis, cranking model. The short-dashed curve gives the irrotational result, which exactly equals the nuclear mass.
-3 $j 3oo _
Convergeng with basis size
-
lrrotational -____-____-____-______
0
2
, Number of shells
10
12
Fig. lb. Convergence of the translational inertia with the basis size. The three solid curves give the results calculated in the generalized cranking model with 9, 13 and 17 oscillator shells, respectively, whereas the short-dashed curve gives the exact mass.
-_:_:-_ ‘pL_ _I_..:_&:,, cranking modei approaches zero in the iimit of iarge panmg. Illt; UCVUiLI”II between the result calculated in the generalized cranking model and the exact value of the mass at large pairing gaps arises from a slow convergence in the numerical calculations. The three solid curves in fig. lb show the inertia calculated with 9, 13, and 17 oscillator shells, which correspond to maximum principal oscillator
J. Kunr, J. R. Nix i Generalized
cranking model
363
quantum numbers of N = 8, 12 and 16, respectively. These results illustrate that although the inertia increases towards the proper mass with increasing number of shells taken into account in the calculation, the convergence is indeed slow. 3.2. ROTATIONAL
MOTION
We choose a deformed nucleus whose symmetry axis is the z-axis and which is rotating around the x-axis. The velocity field u1 is then u1 = wxr=w(O,z, it is divergence-free simply
but rotational.
-y);
The corresponding
Since the curl of L’~does not vanish the evaluation -A-^ ~-urup~a~su ~,.-~I:~,.~~ rL,. IIIUIC; LIIS\LIiii
the
c’;iSiZ Of iii~i~iiCJiXl
collective momentum
is
of the inertia is somewhat
VdCiCiij;
DcLdusc iii,. D--“*.^-
l‘lC rL-
G,xaLL --~-^4
form of the potential enters the calculation, we restrict ourselves here to the harmonic-oscillator potential. We first calculate the momentum of inertia without pairing, using an expression derived in appendix B.2, and show that it is independent of the effective mass M*. As seen from eq. (BS), the moment of inertia contains two terms, the first one of which is the rigid-body moment of inertia
Irig =
3
s
p(r)(w x r)’ d3r.
The second term vanishes at zero temperature and equilibrium deformation, but for non-equilibrium deformations it can be either positive or negative, reflecting shell and surface effects. At large temperatures, when the shell effects have vanished, this term yields at the equilibrium deformation a 4”/, reduction of the rigid-body moment of inertia “*19) . The independence of the inertia on the effective mass tensor for the &&son modified harmonic-oscillator potential is shown in refs. 16*19). We now turn to the effect of the A, term. As shown first by Migdal lo) for rotations in the harmonic-oscillator potential, A, a yz, or again in terms of spherical harmonics,
A, a (Y2,+ Y2-l).
364
J. Kunr, J. R. Nix / Generalized cranking model
240Pu Harmonic oscillator
0
0
2
8
10
Pairilg Gap A k?Vl Fig. 2. Dependence of the rotational moment of inertia upon the pairing gap A for 240Pu in a harmonic-oscillator potential at the equilibrium deformation E = 0.318. The solid curve gives the present result calculated in the generalized cranking model with 15 oscillator shells, the long-dashed curve gives the corresponding result calculated in the Inglis cranking model and the short-dashed curve gives the irrotational result.
We rederive this result in appendix C.2. Furthermore, Bohr and Mottelson 14) discussed the occurrence of an additional term to the pairing field with symmetry Y,, in the intrinsic frame for a nuclear shape with symmetry Y,,. Since the above result for A, is proportional to the velocity potential for irrotational flow, in the limit of strong pairing the moment of inertia approaches the classical irrotational result. We demonstrate this in the semiclassical approximation in appendix D. Our numerical results are shown for 240Pu in a harmonic-oscillator potential at its equilibrium deformation in fig. 2. The shortdashed iine gives the ciassicai irrotationai resuit, which increases siightiy with increasing pairing gap because higher-energy states with larger spatial extent are increasingly occupied. As shown by the long-dashed line, the moment of inertia calculated in the Inglis cranking model vanishes in the limit of large pairing. On the other hand, as shown by the solid line, the moment of inertia calculated in the generalized cranking model approaches the irrotational value in the limit of large pairing. For nonequilibrium deformations, shell effects vanish rapidly and the moment of inertia approaches its rigid-body value for very small pairing gaps ; in the limit of large pairing the moment of inertia again aproaches the irrotational value corresponding to the particular deformation considered.
3.3. VIBRATIONAL
MOTION
An adequate treatment of vibrational motion must be time-dependent. A rigorous time-dependent derivation of the generalized cranking formulas will be
J. Km:.
given in ref. *O). There
J. R. Nix i Generalized
the starting
point
where the lowest-order time derivative obtained from the stationary equation
cranking
nwalel
365
is the time-dependent
of the generalized
[MO, a,]
equation
density
‘*)
matrix
$.
is
= 0.
Since the usual cranking formula obtained in time-dependent perturbation theory *‘) is formally identical to that obtained in a stationary approach, one expects the generalized cranking formula to also hold for the time-dependent case. In fact, as shown in ref. *O), eq. (3.6) for A, is obtained in the time-dependent formalism. However. since there is no stationary frame, the time-dependent continuity equation must be invoked instead of eq. (2.10). Thus, condition (2.11) is replaced by Trip,Cp(*),hojj+Tr(po[p(r),h,jj
= ihTr[P,p(r)j,
which leads to eq. (3.6). In addition, expression (3.8) for the inertia is obtained in the time-dependent approach. However, the full time-dependent expression contains an additional pairingvibration coupling term to the inertia 2.4), which does not arise in the time-independent treatment presented here. For strong pairing this pairing-vibration coupling term vanishes along with the usual cranking term. Since our main concern here is to demonstrate that the inertia remains finite and in fact approaches the irrotational value for strong pairing, we neglect the pairing3 L _..._..__ .l_. r-. _-_l!-r!_ vibration couphng term* It shouid be kept in -I_ mmo, nowever. rna~ ior reansric pairing strength its contribution to the inertia is quite discussed in detail in ref. *‘). For vibrations the collective momentum operator
signihcant.
causes a change in the collective coordinate a of the nucleus. consider those cases that satisfy the equality
nlv) = ; (01. p+ where
the single-particle
wave
function
p’
DJV),
Iv) is an eigenstate
This
will be
In this subsection
we
(3.14)
of the unperturbed
J. Kunz. J. R. Nix / Generalized cranking mu&l
366
hamiltonian H. If pairing correlations are present, 71also acts on the occupation numbers U, and V,, which leads to the pairing-vibration coupling term not taken into account here. We investigate the breathing mode, @-vibrations and y-vibrations, speciabzing to the harmonic-oscillator potential for /?- and y-vibrations. First, we determine for each of the three cases the velocity u1 as defined by eq. (3.14). For the breathing mode we obtain 22) _
ik
flv) = ;g(r.p+p. r)(v),
(3.15)
where the dimensionless parameter b describes the scaling of the coordinates due to the expansion or contraction. The velocity 22)
is irrotatianal but not divergence-free. For p vibrations in the harmonic-oscillator ~rameter F;that defines the frequen~es by
potential,
we use the collective
w, = og = W&)( 1 + SE), 0,
=
o,(e)( L -_4&).
This choice leads to 22*23) ih z Iv> = c,VQ 1p(v), where c, is the deformatjon~e~ndent
and Q is the quad~pole
constant
operator Q = 22’ -x2 -y2.
Thus, the velocity
is both irrotational
and divergence-free.
(3.16)
J. Kunz, J. R. Nix / Generalized
For y-vibrations is related
in the harmonic-oscillator
to the frequencies
by means
367
cranking model
potential,
the collective
parameter
7
of 24)
0, = WOexp [ - (5/4n)*jcos
(y -&)I,
oy = WOexp [ - (5/4K)*/?COs (y - $r)], 0: = ti, exp [ - (5/4~)+pcos The volume-conservation
y].
condition
defines the relation between the deformation-dependent constant frequency GO, whereas the two deformation related to each other to first order by
frequency parameters
We and the G and /? are
E = 3(5/47t)ffi. Then
we obtain
the equality ih
$ Iv) =
VQ - plv),
(3.17)
with Q = -$(5/4n)+/Qsiny(2zZ-x2
-~2)+J3~~~~(~~2-x2)].
Thus, the velocity
is again
both irrotational
and divergence-free.
Since for all three cases considered the velocities vL are irrotational, according to eq. (B.4) of appendix B.l the corresponding inertias are independent of m* and in the absence of pairing are equal to the irrotational values. To determine the reaction of the pairing field d, to the collective motion for the three cases considered, we apply eq. (C.2) of appendix C.l. This leads to breathing
mode
p-vibrations & a [sin y Y,, - (s)+ cos y( Y,, -k Y2_ ,)I,
y-vibrations.
According to appendix D, in the semiclassical approximation the inertias are equal to the corresponding irrotational values for all values of the pairing strength.
J. Kunz, J. R. Nix / Generalized cranking model
368
=Of+J
% I 300 2 S ‘.
Harmoticosciliator
-
5 200 =.,
0
Resent
‘1
E
P 1+ z loo9 a
\ ‘1
:
Irrot~~o_ry!_____z ____-----
____-..------
‘\
‘1c\
. . __-“F
fi
---_ /
I
I
/
2
0
,
8
Paif
Gap
~ 10
A !iieV,
Fig. 3. Dependence of the br~thing inertia upon the pairing gap d for 2rloPu in a ha~onic~~~lator potential at the equilibrium deformation E = 0.318. The three curves are analogous to those in fig. 2.
Finally, we show in figs. 3-5 our numerical results for 240Pu in a harmonicoscillator potentiaf with 15 osciflator shells. These results for the breathing, fivibratian~l and ~-vibmtiona~ inertias are similar to those shown in fig. la for the translational inertia. For B- and g-vibrations we obtain the classical irrotational values shown in figs. 4 and 5 by making a transformation from collective coordinates that define the deformation of the matter distribution to the coordinates E and y that define the deformation of the single-particle potential 20*25). In all cases, the i~otationa1 inertias shown by the short-dashed lines increase slightly with increasing pairing gap because higher-energy states I
40
s _ 3 2 30 S .2
1
/
1
,
/
2mPu
akin
/
oscillator IrtotaConal ________------- __--i
___--------
2 20 -_ i5 -- .\ ‘l S ” ._ z & 10 5 -_ 6 & 0’ 0
I
Present ‘r
’ 2
‘N
(
. .
’
-.
--__?b (
’
----_ ’
’ 8
10
PairiigGap A kleV1 Fig. 4. Dependence of the ~-vib~tiona1 inertia upon the pairing gap d for 240pU in a harmonicoscillator potential at the equilibrium deformation E = 0.318. The three curves are analogous to those in fig. 2.
369
J. Kunz, J. R. Nix / Generalized cranking model 2.0
/
‘;
f 3 3S .P
1.5
-
1.0
-
?z S ‘% & 0.5
-
2
’
I
1
I
240Pu Harmonicoscillator lrrotational _____------.
I\
-
.\
._
_-““”
6 0.0,
0
_____---=
Present
‘. ‘.
? *
_____------
--\_ C
’ 2
’
’
I
’ 8
10
PairiigGap A !MeVl Fig.
5. Dependence
oscillator
potential
of the y-vibrational at the equilibrium
inertia
deformation
upon
the pairing
gap A for z40Pu in a harmonic-
E = 0.318. The three curves are analogous to those in fig. 2.
with larger spatial extent are increasingly occupied. As shown by the longdashed lines, the inertias calculated in the Inglis cranking model approach zero in the limit of large pairing. On the other hand, as shown by the solid lines, the inertias calculated in the generalized cranking model remain close to the irrotational values. The deviations at large pairing gaps arise from a slow convergence in the numerical calculations.
4. conclusions In this paper we have generalized the Inglis cranking model for collective nuclear motion by properly taking into account the velocity dependence of single-particle potentials through an effective mass in the cranking term and the reaction of the pairing field to the collective motion. Both effects manifest themselves in the X1 term in a self-consistent treatment 12), but are neglected in the Inglis cranking model. As has been shown previously 16), whereas the rotational moment of inertia calculated with the Nilsson modified harmonic-oscillator potential is larger than the rigid-body value when the effective-mass term h, is omitted, inclusion of this term reduces the calculated moment of inertia to approximately the rigid-body value. Inclusion of the effective mass term for P-vibrational inertias may account for the renormalization factor of 0.8 that was originally needed to reproduce experimental spontaneous-fission lifetimes ‘). The reaction of the pairing gap to the collective motion A, has been shown to be essential to obtain a finite inertia in the limit of large pairing. In fact, for large
370
J. Kunz, J. R. Nix / Generalized cranking model
pairing the inertia approaches its irrotational value. This result has been shown both analytically in a semiclassical approximatioh and numerically, where remaining deviations were attributed to a slow convergence. Taking the d, contribution into account for rotational moments of inertia in a moditied harmonic-oscillator potential would increase the calculated values, which were originally 3, smaller than the experimental values by about 15 to 20”/. However, this increase would be partly cancelled by effects arising from the effective mass. For vibrations at realistic pairing strength, the A, contribution would be small compared to the large pairing-vibration coupling term. We applied our generalized cranking model to translations, rotations and some special types of vibrations. In these applications we have restricted ourselves to cases where the collective momentum operator assumes a certain form consistent with our time-independent derivation. The application to the time-dependent vibrational motion where no stationary frame exists needs special consideration. The general time-dependent derivation and its application to /I-vibrations will be given in a forthcoming paper ‘O). Finally, we remark that our results for the effective-mass term are analogous to those obtained in adiabatic timedependent Hartree-Fock calculations 17). It would be interesting to compare our A, terms, where the reaction of the gap to the collective motion is proportional to the spherical harmonics describing the collective motion considered, to corresponding adiabatic time-dependent HartreeFock-Bogoliubov results.
Appendix ALTERNATIVE
DERIVATION
OF h, FOR SPECIAL
A CASE
As discussed in sect. 2, when the single-particle potential is velocity-dependent our expression for the collective inertia contains an additional term proportional to h, to account for this. We determine h, by generalizing the derivation of ref. 16) for rotations to collective motion that can be described by the collective momentum operator 71 defined by eq. (2.2). The velocity oL appearing in this equation characterizes the transformation from the laboratory frame to the moving frame. As an alternative to our general derivation in sect. 2, we present here a very transparent derivation of the h, term for the classical example in which the singleparticle potential contains a velocity-dependent term au* in addition to a spatial term V(r). Then the lagrangian for a nucleus at rest is L = +md
- V(r)-au'.
For a nucleus performing collective motion the lagrangian must be modified because the potential inside the nucleus should be independent of the motion of the
J. Kunz, J. R. Nix 1 Generalized cranking model
371
nucleus as a whole, i.e. the potential should be galilean-invariant. Consequently, the potential should not depend on the absolute velocity of the nucleons, but on the velocity relative to the other nucleons. The lagrangian for a nucleus in motion is then L = $?ll? - V(r) - cd,
where u’ = u-u01 is the velocity in the moving frame. To lowest order in the velocity u1 the lagrangian in the moving frame becomes L = $rl*d2 +mu’ . uj,- V(r),
where m* = m-2a is the effective mass. In terms of the canonical momentum p’ = m*u’+mu, the hamiltonian
in the moving frame is H = p‘ /2m + V(r) - (m/m*)u, . p’,
(A.1)
where we again retain only terms linear in u> Since the last term in eq. (A.l) is the sum of the usual cranking term - ul. p’ and the additional term h,, it follows that h,-v,*p
= -(m/m*)u,.p,
64.2)
where we now omit the prime on p. From eq. (A.2) we obtain h, = (1 -m/m*)u,.p.
(A.3)
In the quanta1 case we must take into account the possiblity that p and u1 may not commute, i.e. that V *u2. may be nonzero. In this case we must make the replacement u,.p
-+$u,.p+p.u,).
Then expression (A.3) for h, agrees with the general result (2.9) obtained in sect. 2.
372
J. Kunz, J. R. Nix ! Generalized
cranking
model
Appendix B NUCLEAR
INERTIA
WITHOUT
PAIRING
To calculate the collective nuclear inertia I in the absence of the pairing interaction, we use the commutator of an operator C with the hamiltonian N that is defined by R = [C, H-J
(B-1)
This permits us to close the cranking sum and thereby obtain the inertia 1. We determine the operator C in eq. (B.1) for a hamiltonian that is of the form l-l = p2/2m*+
V(r).
In the absence of pairing the expression (2.5) for the normal density reduces to
Apart from the factor m/m*,the inertia then assumes the familiar form
03.2)
B.l. IRROTATIONAL
VELOCITY
FIELD vi
When u1 is irrotational, we can satisfy eq. (B.l) with an operator C that depends only on position. In fact, for this case, C is proportional to the velocity potential. Eq. (B.l) leads to
from which we make the identification vI = (ilil/m*)VC. We next use the comp~t~ess inertia I in eq. (B.2) as
EB.3)
of the set of wave functions Iv) to rewrite the
1 = C (mlm*Kvl[C, n]Iv)~~. Y Evaluation of the commutator
then yields for the inertia the irrotational
value (B.4)
J. Kunz, J. R. Nix / Generalized
313
cranking model
whichis independent of the effective mass m*, as expected. Had we not included the h, term, the inertia would have differed from the irrotational value in eq. (B.4) by the factor m*Jm. B.2. ROTATIONAL
VELOCITY
FIELD
D;
For a rotational velocity field u1 the operator C depends on momentum in addition to position, which leads to a dependence on the special choice of the potential k’(r). Here we consider only a rotating nucleus, for which uj. =
0
x
We then specialize to a harmonic-oscillator H = p2/2m* With this hamiltonian given by 26) Cc-i
the operator
of
r. potential, with
+~m[c$(x’
+y2)+otz2].
C satisfying the commutation
relation (B.l) is
4 [bm*(wf+dyz+ &P,P,].
After closing the cranking sum, we obtain for the moment of inertia I =
Cp,. Y
m(vl(y2+z2)lv)+
& z
(vl(~~52-dz2)Iv) . x
1
(B.5)
The first term is the rigid-body moment of inertia, whereas the second term contains the shell effects. The second term vanishes in the ground state of a nucleus at its equilibrium deformation, where the condition
is satisfied ‘s). Appendix C CALCULATION
OF f(r)
We determine the reaction of the pairing field d, to the collective motion by considering the function f(r) that is related to A, through eq. (3.4). The function f(r) is determined from eq. (3.6), which was obtained from the continuity equation and must be satisfied for every position r.
374 C.1. IRROTATIONAL
J. Kunz, J. R. Nix / Generalized cranking model VELOCITY
For an irrotational
FIELD cI
velocity field u1 the solution for f’(r) is of the form f’(r) =
-irC,
Cl)
where x is a constant and the operator C is defined by eq. (B.l). To prove this, we express ri in eq. (3.5) with the help of eq. (B.l) and insert the ansatz (C.l), which leads to
Obviously, eq. (3.6) is satisfied if u = -24. Thus, f(r) for irrotational
velocity u1 is given by f(r)
= i2AC(r),
(C.2)
where C(r) is proportional to the velocity potential given by eq. (B.3). Using solution (C.2) for f(r) in the commutator expression (B.l), we obtain the relation 2A217rpvIz= ;lf,,12(h, - II,)*.
Thus, expression (3.8) for the inertia reduces to
(C.3)
C.2. ROTATIONAL
VELOCITY
FIELD I~
As discussed in appendix B.2, for a rotational velocity field u1 the operator C depends also on momentum, so that ansatz (C.l) cannot be made. In the following, we again specialize to rotations for a harmonic-oscillator potential. In the limit of large pairing strength one intuitively expects the inertia to approach the irrotational result. This suggests an ansatz for f(r) of the type f‘(r) = -icfC(r),
where c is the velocity potential for irrotational
(C.4) motion of a rotating nucleus 14).
J. Kunz, J. R. Nix / Generalized
cranking model
315
This relationship is $
- ihAvc = -AyV(yz), _
111*
with ^r’= (0: - o;)/(wf + a;). Note that c does not satisfy eq. (B.l). On the other hand, for the harmonicoscillator potential one has the relation lo) m*(flo,’- o$)yz.
7i =
Thus, the ansatz j(r) = /Vlic
(C.5)
is equivalent to eq. (C.4), yielding the relation a =
p12(o; + wf).
We now insert ansatz (C.5) into eq. (3.6), multiply by A(r) and integrate, which leads to
033) There are two kinds of non-vanishing corresponding energy differences are
Ik-$1
=
matrix
elements Jlp, for which the
h(o,-w,)
= ho_,
AN=0
h(w,+w,)
= ho,,
AN = 2,
(C.7)
where N denotes the principal oscillator quantum number. Upon introducing the two constants
co= c
Ifv,l’ E,E,h%+ E,,)’
d1;‘!iO
12
=
lL,,l’ c v.,, E,E,(E, + E,) AN=2
and inserting them into eq. (C.6), we obtain
U-4)
376
J. Kunz. 1. R. Nix i Generalized cranking model
Thus, our ansatz (C.5) for f(r) satisfies eq. (3.6), with p given by eq. (C.8). Ey. (3.8) for the inertia then becomes
Appendix D SE~rCLASS~CAL EVALUATION OF THE INERTIA In order to evaluate the inertia given by eq. (3.9), we must first determine I, from eq. (3.10). For this purpose we use semiclassical methods, in analogy with Migdal’s calculation for rotations lo). To lowest order in h, expression (3.10) for I0 is transformed by use of standard methods L5,27) into
d3rd3pA?S(e-e,).
P.1)
In this appendix we denote by ti the collective momentum defined by eq. (2.2) to distinguish it from the transcendental number conventionally denoted by rt. Also, we use the definition h, = e,-e,, where e, is the single-particle energy and eF is the Fermi energy. By performing the integration over dQ, in eq. (D.l) we obtain
10 = $$A--
s
d3rdp3p4u:(r)6(e-e,).
Since energy and mamentum are related by e = p2/2m* + V(r),
we evaluate this integral by substituting de =
(plm*)dp,
which leads to 1, =;
sd3rA
Since in the ~micta~ica1 approximation
3nvl3
Gtd.
(D-2)
the Fermi momentum pF and density p(r)
J. Kunr, J. R. Nix i Generalized
are related
377
cranking model
by “) PF
eq. (D.2) can be rewritten
=
h[3n2p(r)]f,
3
s
as
10 =
d3rp(r)u:(r).
(D.3)
Note that expression (D.3) is independent of the effective mass irrotational velocity field uj,(r) the inertia I,, is simply the classical inertia and for rotations it is simply the classical rigid-body moment With the help of eq. (D.3) for I, we now calculate the total inertia (3.9). For an irrotational velocity field DA(r) our result for f(r) appendix C.l. is
m*. For an irrotational of inertia. given by eq. obtained in
17Tvfl12 = 11;.,12((h,.-h,)/2A)2. Insertion of this relation into eq. (3.9) causes the two terms in the square bracket the inertia remains at the to cancel, with the result that I = I,. Therefore, irrotational value also when pairing correlations are present. Finally, we calculate the moment of inertia for nucleons with pairing in a harmonic-oscillator potential. For this purpose we use our result from appendix C.2 for j’(r), where expression (C.7) for /? is approximated by lo)
with
The quantities ok are defined by eq. (C.7). Following Migdal’s arguments further, we find for the inertia
I=
1
0
&9-4+9+~1 UJf+COz,
+
the expression
@_+g+)2w2_o: @_oZ +g+o:)(oz_
+fB’,, 1 ’
lo)
(D.4)
which we investigate for the two limiting cases of small and large pairing strength. For small A we find that g + 0 and therefore I + I,; i.e. the inertia approaches the rigid-body moment of inertia when the pairing vanishes. For large A we find that g + 1, which causes the first two terms in the square bracket of eq. (D.4) to cancel,
378
J. Kunz, J. R. Nix / Generalized cranking model
leaving us with 1 + Upon inserting eq. (C.7) for o,
1 _hM O(W2 +o:y*
into this relation, we obtain
1 -+ I,
wf-sf 2 w;+o:, .
( 1
Thus, in the limit of large pairing the inertia approaches the classical irrotational result 14).
1) D. R. Inglis, Phys. Rev. 96 (1954) 10.59; 97 (1955) 701 2) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No I I (1959) 3) S. G. Nilsson and 0. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. I6 (1961) 4) D. R. Bits, Mat. Fys. Medd. Dan. Vid. Selsk. 33, No. 2 (1961) 5) A. Sobiczewski, 2. Szymatiski, S. Wycech, S. G. Nilsson, J. R. Nix, C. F. Tsang, C. Gustafson, P.
M&her and B. Nilsson, Nucl. Phys. A131 (1969) 67 6) M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky and C. Y. Wong, Rev. Mod.
Phys. 44 (1972) 320 7) J. Randrup, S. E. Larsson, P. Miiller, S. G. Nilsson, K. Pomorski and A. Sobiczewski, Phys. Rev. 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
18) 19) 20) 21) 22) 23) 24) 25) 26) 27)
Cl3 (1976) 229 R. E. Prange, Nucl. Phys. 22 (1961) 283; 28 (1961) 369, 376 D. J. Rowe, Nuclear collective motion (Methuen, London, 1970) A. B. Migdal, Nuci. Phys. 13 (1959) 655 S. T. Belyaev, Nucl. Phys. 64 (1965) 17; Collective excitations in nuclei (Gordon and Breach, New York, 1968) D. J. Thouless and J. G. Valatin, Nucl. Phys. 31 (1962) 21 I M. Baranger and K. Kumar, Nucl. Phys. A122 (1968) 241 A. Bohr and B. R. Mottelson, Nuclear structure, vol. 2 (~n~min, Reading, Mass., 1975) P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) J. Kunz, Nucl. Phys. A363 (1981) 1 M. J. Giannoni, F. Moreau, P. Quentin, D. Vautherin, M. Veneroni and D. M. Brink, Phys. Lett. 65B ( 1976) 305 ; D. M, Brink and M. DiToro, Nucl. Phys, A372 (1981) I51 M. Durand, J. Kunz and P. Schuck, On the semiclassical description of rotating nuclei, Nucl. Phys., to be published J. Kunz and U. MO& Nucl. Phys. A406 (1983) 269 J. Kunz and J. R. Nix, Calculation of the nuclear inertia in a generalized cranking model, Nucl. Phys., to be publish~ L. Wilets, Theories of nuclear fission (Clarendon, Oxford, 1964) K. K. Kan, Ph.D. thesis, University of Maryland (1975) A. Schuh, J. Kunz and U. Mosel, Flow patterns for collective quadrupole vibrations in heavy nuclei, Nucl. Phys. A412 (1984) 34 E. R. Marshalek and J. 0. Rasmussen, Nucl. Phys. 43 (1953) 438 P. MBller and J. R. Nix, Nucl. Phys. A296 (1978) 289 M. Radomski, Phys. Rev. Cl4 (1976) 1704 A. Bohr and IS. R. Mottelson, Nuclear structure, vol. 1 (Benjamin, Reading, Mass., 1969)
Physics
North-Holland
A426 (1984) 353-378 Publishing
Company
GENERALIZED CRANKING MODEL FOR COLLECTIVE NUCLEAR MOTION J. KUNZ Theoretical
and J. R. NIX
Division, Los .4lamos National Laboratory, Received
3 January
Los .4lamos, New Mexico 87545, USA 1984
The Inglis cranking model is generalized to take into account effects of any velocity dependence present in the single-particle potential and the reaction of the pairing field to the collective motion. The generalized model is applied to translations, rotations and some special types of vibrations. Some of our results and our numerical calculations are obtained with a harmonic-oscillator single-particle potential. Unlike the inertia calculated with the Inglis cranking model, the inertia calculated with the generalized cranking model is independent of the effective mass and approaches the irrotational value in the limit of large pairing.
Abstract:
1. Introduction
The cranking model has been widely used to calculate the inertia corresponding to collective nuclear motion After the introduction of pairing correlations into the Inglis cranking model ‘) by Belyaev ‘), calculations of the inertia in a modified harmonic-oscillator potential with pairing were performed by Nilsson and Prior “) for rotations and by Bes4) for y-vibrations. The pairing correlations reduced the calculated rotational moment of inertia below the rigid-body value and increased the calculated vibrational inertia above the irrotational value, so that relatively good agreement between theoretical and experimental results was obtained. Later, interest in P-vibrational inertias was spurred in connection with fission deformations. Calculations in a modified harmonic-oscillator potential with pairing yielded inertias several times the irrotational values “) and led to reasonable agreement with experimental spontaneous-fission lifetimes 6). However, some systematic discrepancies between theoretical inertias and experimental results were observed. The Nilsson and Prior results “) for the rotational moment of inertia were too low by about 15 to 20 %. Randrup et al. ‘) needed an overall renormalization factor of 0.8 for the calculated vibrational inertias to reproduce experimental spontaneous-fission lifetimes. Also, it was early recognized that the Inglis cranking model had some intrinsic deficiencies. Prange *) pointed out that the rotational moment of inertia calculated with the Inglis model goes to zero in the limit of large pairing, whereas it should approach the finite irrotational-flow value. Similarly, when pairing correlations are 353
354
J. Km:,
J. R. Ni.r :’ Generalized
cranking model
included, the Inglis pushing model gives for uniform translations below the correct mass of the nucleus ‘) and that also vanishes pairing. In
addition,
problems
arose
with
the
Inglis
cranking
an inertia that lies in the limit of large model
for
velocity-
dependent potentials. Moments of inertia calculated with the Nilsson modified harmonic-oscillator potential did not approach the corresponding rigid-body moments of inertia for high temperatures, but were instead too large “). As alternatives to the Inglis cranking model, there have been other treatments that do not lead to the above unacceptable results. Migdal lo) developed a formalism based upon Green’s functions to calculate rotational moments of inertia. He properly took into account the velocity dependence of a particular potential by introducing the effective mass into the cranking term. Furthermore, he showed that the collective rotation introduces an additional term in the pairing potential that keeps the moment of inertia finite in the limit of large pairing. Belyaev 11) obtained analogous results in his self-consistent-held formalism. The deficiencies of the Inglis cranking model were further pointed out by Hartree-Fock equations took care Thouless and Valatin “), w h ose self-consistent of both problems. Furthermore, their formalism demonstrated the approximation that is necessary to obtain the Inglis formula. Starting from the stationary equation in the rotating frame
[X - oYx, where &’ is the generalized hamiltonian showed that the Inglis formula is obtained
a] = 0, and ~8 is the generalized density, they when the lowest-order correction Xl to
the hamiltonian is neglected. The X’, term, however, takes care of the effective mass and the additional pairing term due to the collective rotation. For quadrup-ole vibrations the omission of the %‘, term in the Inglis cranking formula was pointed out by Baranger and Kumar 13). In this paper we give a time-independent derivation of the generalized cranking formula, starting from the stationary equation of Thouless and Valatin 12) and keeping the X1 term. In sect. 2 we derive a general expression for the inertia containing two additional terms to account for the effective mass and the reaction of the pairing field to the collective motion. We apply these results in sect. 3 to translations, rotations and vibrations. where for vibrations we discuss only specific cases for which the collective momentum operator has a certain form. The two submatrices of &,, namely h, to account for the effective mass and A, to account for the reaction of the pairing field to the collective motion, are discussed in appendices A and C, respectively. Appendix B demonstrates that the inertia is independent of the effective mass because of the hi term, and appendix D contains a semiclassical calculation of the inertia which shows that in the limit of large pairing the inertia approaches the irrotational value.
355
J. Kunz, J. R. Nix 1 Generalized cranking model
2. Time-independent
formalism
The usual perturbative cranking model that was first introduced by Inglis ‘) does not properly take into account the effects of any velocity dependence present in the single-particle potential and completely neglects the reaction of the pairing field to the collective motion. This leads to two unacceptable results: (i) the calculated inertia depends strongly on the effective mass M*, and (ii) it vanishes in the limit of strong pairing. In our generalization of the cranking model we take care of both these deficiencies. We derive the equations of the generalized cranking model by following the time-independent self-consistent Hartree-Fock formalism of Thouless and Valatin I’). Since we perform a perturbative treatment, we take into account only terms linear in the collective motion. The self-consistent Hartree-Fock potential is approximated by a suitable single-particle potential. In our derivation of the additional terms arising from the reaction of the mean held to the collective motion we argue along the lines of Migdal “1 who discussed rotations, and Bohr and Mottelson 14). 2.1. COLLECTIVE
INERTIA
We consider collective nuclear motion that can be treated equation in the moving frame of the form 12) [Z-XI,
by a stationary
a] = 0,
(2.1)
where the square brackets denote the commutator, X is the generalized mean field, l7 is the generalized collective momentum and 8 is the generalized density. They are defined by
where h represents the single-particle spectrum, A is the pairing potential, p is the density matrix and K is the abnormal density matrix. As discussed in more detail in appendix A, the lagrange multiplier i gives the magnitude of the velocity field u1 that describes the transformation from the laboratory frame to the moving frame. Consequently, the collective momentum operator assumes the form 1 n = Jjj (u,.p+p.a,) for the collective motions considered here. To obtain the perturbative cranking model equations we substitute % = X0,+.%,,
.@ = 5?o+w,
(2.2)
356
J. Kunz, J. R. Nix i Generalized cranking model
into eq. (2.1) and retain terms that are linear in Xi and 9,. In contrast, the X, term is neglected in the Inglis cranking model. Our linearization leads to two coupled
matrix
equations
for p, and
namely
K,,
h,p,-p,h,+d,~r-~,d,*+h,p,-p,h,+d,K,*-K,d:
=
J~v,-P,~),
= j.(7CK0+ K,,n*). We next rewrite these equations in terms of matrix 1~) and 1~) of the unperturbed hamiltonian satisfying [Jr,,
(2.3)
(2.4) elements
between
eigenstates
.%,I = 0.
Eq. (2.4) is then solved to obtain an expression for the matrix elements of the abnormal density K~, with use made of the proper relations of matrix elements with their complex conjugate and hermitian conjugate 15). Upon inserting this expression into eq. (2.3) we obtain as our final expression for the normal density matrix
/Jlpv
=
(ix,,.
--.
-h,,)[W,+4(
E2Kr-K,)-_(hr+h,)(p~-pV)]
~---
_Ei-.
v
Ir
&-,[(A
+-.-LL’_L_F----.
+A .k’ +P,II
l)+($,+h,)(K,+K,)]
(2.5)
b
In this and subsequent equations, for notational simplicity we omit on h, A, p and the subscript 0 indicating unperturbed quantities. As usual, E, denotes the quasiparticle energy
K
E, = (h;+A;)‘. Our expression (2.5) for pi contains two additional terms proportional to h, and A, that are not present in the usual cranking result. Finally, we obtain from eq. (2.5) the inertia I corresponding to the collective motion considered by taking the trace of the density matrix with the collective momentum operator ‘*), which leads to
I = &Tr(B?in)
The two additional
terms proportional
= fTr(p,n).
to h, and A, that appear
(2.6)
in eq. (2.6) for the
J. Kunz, J. R. Nix / Generalized cranking model
357
inertia I remove its dependence on the effective mass and keep it finite for large pairing gaps. By setting h, and A, equal to zero, i.e. by equating the effective mass to the nucleon mass and neglecting the reaction of the pairing field to the collective motion, we can easily verify that the resultant expression for the inertia yields the usual cranking expression 6, for a rotating nucleus at finite temperature in the presence of pairing. The determination of the two additional terms is our next concern. 2.2. DETERMINATION
OF h, AND d I
The occurrence of additional terms to account for an effective mass and the reaction of the pairing field to the collective motion has been discussed by Migdal lo) for the case of rotations. In addition, the occurrence of an h, term for non-local potentials has been pointed out by Bohr and Mottelson i4) for several types of collective motion. Here we generalize their arguments to collective motion that can be described by the stationary equation (2.1) with a collective momentum operator of the form (2.2). The single-particle potential, which approximates the self-consistent HartreeFock potential arising from two-body interactions, should not depend on the absolute momenta of the nucleons but on their momenta relative to those of the other nucleons. Therefore, in all momentum-dependent potential terms we make the replacement i”*r4) p -+ p-mu,.
(2.7)
To obtain the h, term we expand the potential in a Taylor series to lowest order in v1 and symmetrize in order to take into account the possible non-commutativity of u, -A and ----- V,V_ This leads to
hl = 4m[vA-V,W,
p)+V,V(r,
p). VJ
In the applications to be made in sect. 3 we specialize to a momentumdependent potential that is quadratic in the momentum. In that case expression (2.8) for h, reduces to h lpv
=
(1-mlm*P~,,,
(2.9)
where the effective mass m* is constant. In general, m* is a position-dependent and momentum-dependent tensor, which is the case for the Niisson modified harmonic-oscillator potential 16). An alternative derivation of eq. (2.9) for a special case is given in appendix A. Finally, we remark that a formally analogous result to eq. (2.9) has been obtained in recent adiabatic time-dependent HartreeFock calculations i7).
J. Kunz, J. R. Nix 1 Generalized
358
cranking model
We now turn to the reaction of the pairing field to the collective motion. In order to evaluate A, we must invoke another equation. For this purpose Migdal lo) suggested the use of either the gap equation or the continuity equation. He employed the latter for the case of rotations. The continuity equation requires that the divergence of the current j,,,(r) in the moving frame vanish, i.e. V. jm(r) = 0, because the density in the moving frame undergoing collective motion is given by
(2.10)
is stationary.
The current
of a nucleus
j(r) = ReTr [pi_&)], where;(r) is the matrix of the current operator. On the other hand, the divergence of the current operator can be expressed in terms of the commutator of the spatial density operator B(r) and the hamiltonian H through the operator continuity equation
V.;(r)
Therefore,
with linear
= - k [a(r), H].
terms retained,
condition
(2.10) can be rewritten
as
(2.11) ~~~~~~~~~~~~ol~+~~~~oC~~~~~~,]~-~~{po[p~~~,~~]} = 0. Upon
inserting rr
eq. (2.5) for p, into eq. (2.11), we obtain
A
p-v -
L~~~+~vbw
= c (II,,,-3x,,) I1.Y
+p,-
I)+
ih,+h,jiK,+K,jj(hp-hy)
T-E;
[@dQ(
KIr-K,)+(A,-A,)(pll-pv)](A,-A,) Et-E:
~
P,,(r).
p,,(r)
(2.12)
Since eq. (2.12) for A, depends on position r, it represents an infinite set of equations that must be satisfied in order to obtain the matrix elements Al,_,.
3. Appikaiioas
In this section we demonstrate the importance of taking into account the effective mass and the reaction of the pairing field to the collective motion. In order to be able to calculate the effects of the h, and A, terms analytically, we consider
J. Kunz, J. R. Nix i Generalized
359
cranking model
the simple hamiltonian H = p2/2m*+
V(r),
(3.1)
with constant effective mass m*. Furthermore, in most cases we specialize hamiltonian in eq. (3.1) still further to the harmonic-oscillator potential V(r) = ~m[wf(x2+y2)+cofz2]. For nonzero
temperatures
the normal
density
pV = +[I -(II,/&) and the abnormal
density
the
(3.2)
is given by
tanh (E,/2T)]
is given by K, = i( A,/E,) tanh (E,/2T).
In the limit of zero generalized cranking interest here is the pairing gap A for all to the much simpler
temperature, which we consider in some of our studies, the equations simplify considerably. Furthermore, since our main strong pairing behaviour of the inertia, we take a constant states. Then, eq. (2.6) for the inertia reduces in the limit T + 0 expression lo)
(E,E l=$~_-_Y_-
- h,h, - A2)ln,,12 - Af~j;~ti,, 2E,E,(E,
V.L where we use the notation
+ EL)
(3.3)
’
lo) A, = if(r)lm/m*,
(3.4)
[IC,H].
(3.5)
i = ;
Also, eq. (2.12) for A, [or equivalently simplifies to
for f(r)
in the new notation
of eq. (3.4)]
(3.6)
Upon
multiplying
eq. (3.6) by j(r)
_A/‘,
and integrating
c
%-b
v,,,
2EvE,SE,+ E,)
=
c
we obtain
the relation
(h-hJ21_L,12
v.,, 4&5,,(~,+
E,) ’
(3.7)
360
J. Kunz, J. R. Nix
! Generalized
cranking
model
which is then substituted into the second part of eq. (3.3). This yields as our final expression for the inertia
ZE,E,(E,+
E,)
(3.8)
----’
This equation is employed in our numerical calculations of the inertia, which are performed for a 240Pu nucleus in a harmonic-o~illator potential. In our analytic calculation of the inertia we follow the semiclassical formalism used by Migdal lo) for rotational motion. The key point in this derivation is the introduction of the function
9Sx) =
sinh-‘x x(1 +x2)1 ’
x = (~~-~~)~2~.
As discussed in refs. ‘Ovi4), this exploits the sharp maximum of the denominator E,E,(E,+E,). With this approximation, eq. (3.8) for the inertia leads to
with (3.10) Also, in this approximation eq. (3.6) for the function f(r), pairing field to the motion, becomes
i.e. the reaction of the
We now apply the formalism that we have developed to translational, rotational and vibrational motion. In the last case we investigate the breathing mode and ellipsoidal /3- and y-vibrations. For each kind of collective motion we first consider the effect of the h, term only in the absence of pairing, where we can easily demonstrate that the inertia is independent of m*. Then the effect of A, is investigated in the limit of zero temperature, where we show both in a semiclassical approximation lo) and also numerically that the inertia remains finite for strong pairing fields. In fact, with increasing pairing it either approaches or remains at the irrotational value.
J. Kunz, J. R. Nix
3.1. TRANSLATIONAL
/
Generalized
cranking
361
model
MOTION
For uniform translational motion along collective momentum operator is given by
one axis, for example
1 lr = - VA.p, %
v1 = l:eZ.
the z-axis,
the
(3.12)
The velocity vi. in eq. (3.12) is both irrotational and divergence-free. For translations the inertia of the moving nucleus must rigorously be the total mass M. Since vL is irrotational we can apply the result of appendix B.l and obtain immediately I = c m(v)ef(v)p, Y
= mA = M.
(3.13)
Thus, because the h, term is included, eq. (3.13) gives correctly the mass of the nucleus for the translational inertia. We now turn to the determination of A, in the pushing model. In previous treatments 9), with pairing included the Inghs pushing model has given a mass considerably smaller than the correct value mA. We now demonstrate that with the A, term included the correct value for the mass is recovered. Since the velocity V~ given by eq. (3.12) is irrotational, the function S(r) is proportional to the velocity potential according to appendix C.l. Thus, for the reaction of the pairing field we obtain
A, az, or, in terms of spherical
Then,
the semiclassical
harmonics,
approximation
derived
in appendix
D yields for the inertia
I = I, = mA, which is the correct total mass of the nucleus. We show in fig. la our numerical result for the translational inertia of ‘*‘Pu as a function of the pairing gap A, calculated for a harmonic-oscillator potential with 15 oscillator shells, which corresponds to a maximum principal oscillator quantum number of N = 14. As indicated by the solid line, with increasing A the inertia calculated with our expression (3.8) remains close to its value at A = 0. On the other hand, as indicated by the long-dashed line, the inertia calculated in the Inglis
J. Kunz, J. R. Nix 1 Generalized cranking model
362
“OPU Harmonic oscillator ______-________________-__ \ ~~~~i,;-
0
8
2
10
Pairilg Gap A kleV1 Fig. la. Dependence of the translational inertia upon the pairing gap A for 24”Pu in a harmonicoscillator potential. The solid curve gives the present result calculated in the generalized cranking model with 15 oscillator shells and the long-dashed curve gives the corresponding result calculated in the Inglis, cranking model. The short-dashed curve gives the irrotational result, which exactly equals the nuclear mass.
-3 $j 3oo _
Convergeng with basis size
-
lrrotational -____-____-____-______
0
2
, Number of shells
10
12
Fig. lb. Convergence of the translational inertia with the basis size. The three solid curves give the results calculated in the generalized cranking model with 9, 13 and 17 oscillator shells, respectively, whereas the short-dashed curve gives the exact mass.
-_:_:-_ ‘pL_ _I_..:_&:,, cranking modei approaches zero in the iimit of iarge panmg. Illt; UCVUiLI”II between the result calculated in the generalized cranking model and the exact value of the mass at large pairing gaps arises from a slow convergence in the numerical calculations. The three solid curves in fig. lb show the inertia calculated with 9, 13, and 17 oscillator shells, which correspond to maximum principal oscillator
J. Kunr, J. R. Nix i Generalized
cranking model
363
quantum numbers of N = 8, 12 and 16, respectively. These results illustrate that although the inertia increases towards the proper mass with increasing number of shells taken into account in the calculation, the convergence is indeed slow. 3.2. ROTATIONAL
MOTION
We choose a deformed nucleus whose symmetry axis is the z-axis and which is rotating around the x-axis. The velocity field u1 is then u1 = wxr=w(O,z, it is divergence-free simply
but rotational.
-y);
The corresponding
Since the curl of L’~does not vanish the evaluation -A-^ ~-urup~a~su ~,.-~I:~,.~~ rL,. IIIUIC; LIIS\LIiii
the
c’;iSiZ Of iii~i~iiCJiXl
collective momentum
is
of the inertia is somewhat
VdCiCiij;
DcLdusc iii,. D--“*.^-
l‘lC rL-
G,xaLL --~-^4
form of the potential enters the calculation, we restrict ourselves here to the harmonic-oscillator potential. We first calculate the momentum of inertia without pairing, using an expression derived in appendix B.2, and show that it is independent of the effective mass M*. As seen from eq. (BS), the moment of inertia contains two terms, the first one of which is the rigid-body moment of inertia
Irig =
3
s
p(r)(w x r)’ d3r.
The second term vanishes at zero temperature and equilibrium deformation, but for non-equilibrium deformations it can be either positive or negative, reflecting shell and surface effects. At large temperatures, when the shell effects have vanished, this term yields at the equilibrium deformation a 4”/, reduction of the rigid-body moment of inertia “*19) . The independence of the inertia on the effective mass tensor for the &&son modified harmonic-oscillator potential is shown in refs. 16*19). We now turn to the effect of the A, term. As shown first by Migdal lo) for rotations in the harmonic-oscillator potential, A, a yz, or again in terms of spherical harmonics,
A, a (Y2,+ Y2-l).
364
J. Kunr, J. R. Nix / Generalized cranking model
240Pu Harmonic oscillator
0
0
2
8
10
Pairilg Gap A k?Vl Fig. 2. Dependence of the rotational moment of inertia upon the pairing gap A for 240Pu in a harmonic-oscillator potential at the equilibrium deformation E = 0.318. The solid curve gives the present result calculated in the generalized cranking model with 15 oscillator shells, the long-dashed curve gives the corresponding result calculated in the Inglis cranking model and the short-dashed curve gives the irrotational result.
We rederive this result in appendix C.2. Furthermore, Bohr and Mottelson 14) discussed the occurrence of an additional term to the pairing field with symmetry Y,, in the intrinsic frame for a nuclear shape with symmetry Y,,. Since the above result for A, is proportional to the velocity potential for irrotational flow, in the limit of strong pairing the moment of inertia approaches the classical irrotational result. We demonstrate this in the semiclassical approximation in appendix D. Our numerical results are shown for 240Pu in a harmonic-oscillator potential at its equilibrium deformation in fig. 2. The shortdashed iine gives the ciassicai irrotationai resuit, which increases siightiy with increasing pairing gap because higher-energy states with larger spatial extent are increasingly occupied. As shown by the long-dashed line, the moment of inertia calculated in the Inglis cranking model vanishes in the limit of large pairing. On the other hand, as shown by the solid line, the moment of inertia calculated in the generalized cranking model approaches the irrotational value in the limit of large pairing. For nonequilibrium deformations, shell effects vanish rapidly and the moment of inertia approaches its rigid-body value for very small pairing gaps ; in the limit of large pairing the moment of inertia again aproaches the irrotational value corresponding to the particular deformation considered.
3.3. VIBRATIONAL
MOTION
An adequate treatment of vibrational motion must be time-dependent. A rigorous time-dependent derivation of the generalized cranking formulas will be
J. Km:.
given in ref. *O). There
J. R. Nix i Generalized
the starting
point
where the lowest-order time derivative obtained from the stationary equation
cranking
nwalel
365
is the time-dependent
of the generalized
[MO, a,]
equation
density
‘*)
matrix
$.
is
= 0.
Since the usual cranking formula obtained in time-dependent perturbation theory *‘) is formally identical to that obtained in a stationary approach, one expects the generalized cranking formula to also hold for the time-dependent case. In fact, as shown in ref. *O), eq. (3.6) for A, is obtained in the time-dependent formalism. However. since there is no stationary frame, the time-dependent continuity equation must be invoked instead of eq. (2.10). Thus, condition (2.11) is replaced by Trip,Cp(*),hojj+Tr(po[p(r),h,jj
= ihTr[P,p(r)j,
which leads to eq. (3.6). In addition, expression (3.8) for the inertia is obtained in the time-dependent approach. However, the full time-dependent expression contains an additional pairingvibration coupling term to the inertia 2.4), which does not arise in the time-independent treatment presented here. For strong pairing this pairing-vibration coupling term vanishes along with the usual cranking term. Since our main concern here is to demonstrate that the inertia remains finite and in fact approaches the irrotational value for strong pairing, we neglect the pairing3 L _..._..__ .l_. r-. _-_l!-r!_ vibration couphng term* It shouid be kept in -I_ mmo, nowever. rna~ ior reansric pairing strength its contribution to the inertia is quite discussed in detail in ref. *‘). For vibrations the collective momentum operator
signihcant.
causes a change in the collective coordinate a of the nucleus. consider those cases that satisfy the equality
nlv) = ; (01. p+ where
the single-particle
wave
function
p’
DJV),
Iv) is an eigenstate
This
will be
In this subsection
we
(3.14)
of the unperturbed
J. Kunz. J. R. Nix / Generalized cranking mu&l
366
hamiltonian H. If pairing correlations are present, 71also acts on the occupation numbers U, and V,, which leads to the pairing-vibration coupling term not taken into account here. We investigate the breathing mode, @-vibrations and y-vibrations, speciabzing to the harmonic-oscillator potential for /?- and y-vibrations. First, we determine for each of the three cases the velocity u1 as defined by eq. (3.14). For the breathing mode we obtain 22) _
ik
flv) = ;g(r.p+p. r)(v),
(3.15)
where the dimensionless parameter b describes the scaling of the coordinates due to the expansion or contraction. The velocity 22)
is irrotatianal but not divergence-free. For p vibrations in the harmonic-oscillator ~rameter F;that defines the frequen~es by
potential,
we use the collective
w, = og = W&)( 1 + SE), 0,
=
o,(e)( L -_4&).
This choice leads to 22*23) ih z Iv> = c,VQ 1p(v), where c, is the deformatjon~e~ndent
and Q is the quad~pole
constant
operator Q = 22’ -x2 -y2.
Thus, the velocity
is both irrotational
and divergence-free.
(3.16)
J. Kunz, J. R. Nix / Generalized
For y-vibrations is related
in the harmonic-oscillator
to the frequencies
by means
367
cranking model
potential,
the collective
parameter
7
of 24)
0, = WOexp [ - (5/4n)*jcos
(y -&)I,
oy = WOexp [ - (5/4K)*/?COs (y - $r)], 0: = ti, exp [ - (5/4~)+pcos The volume-conservation
y].
condition
defines the relation between the deformation-dependent constant frequency GO, whereas the two deformation related to each other to first order by
frequency parameters
We and the G and /? are
E = 3(5/47t)ffi. Then
we obtain
the equality ih
$ Iv) =
VQ - plv),
(3.17)
with Q = -$(5/4n)+/Qsiny(2zZ-x2
-~2)+J3~~~~(~~2-x2)].
Thus, the velocity
is again
both irrotational
and divergence-free.
Since for all three cases considered the velocities vL are irrotational, according to eq. (B.4) of appendix B.l the corresponding inertias are independent of m* and in the absence of pairing are equal to the irrotational values. To determine the reaction of the pairing field d, to the collective motion for the three cases considered, we apply eq. (C.2) of appendix C.l. This leads to breathing
mode
p-vibrations & a [sin y Y,, - (s)+ cos y( Y,, -k Y2_ ,)I,
y-vibrations.
According to appendix D, in the semiclassical approximation the inertias are equal to the corresponding irrotational values for all values of the pairing strength.
J. Kunz, J. R. Nix / Generalized cranking model
368
=Of+J
% I 300 2 S ‘.
Harmoticosciliator
-
5 200 =.,
0
Resent
‘1
E
P 1+ z loo9 a
\ ‘1
:
Irrot~~o_ry!_____z ____-----
____-..------
‘\
‘1c\
. . __-“F
fi
---_ /
I
I
/
2
0
,
8
Paif
Gap
~ 10
A !iieV,
Fig. 3. Dependence of the br~thing inertia upon the pairing gap d for 2rloPu in a ha~onic~~~lator potential at the equilibrium deformation E = 0.318. The three curves are analogous to those in fig. 2.
Finally, we show in figs. 3-5 our numerical results for 240Pu in a harmonicoscillator potentiaf with 15 osciflator shells. These results for the breathing, fivibratian~l and ~-vibmtiona~ inertias are similar to those shown in fig. la for the translational inertia. For B- and g-vibrations we obtain the classical irrotational values shown in figs. 4 and 5 by making a transformation from collective coordinates that define the deformation of the matter distribution to the coordinates E and y that define the deformation of the single-particle potential 20*25). In all cases, the i~otationa1 inertias shown by the short-dashed lines increase slightly with increasing pairing gap because higher-energy states I
40
s _ 3 2 30 S .2
1
/
1
,
/
2mPu
akin
/
oscillator IrtotaConal ________------- __--i
___--------
2 20 -_ i5 -- .\ ‘l S ” ._ z & 10 5 -_ 6 & 0’ 0
I
Present ‘r
’ 2
‘N
(
. .
’
-.
--__?b (
’
----_ ’
’ 8
10
PairiigGap A kleV1 Fig. 4. Dependence of the ~-vib~tiona1 inertia upon the pairing gap d for 240pU in a harmonicoscillator potential at the equilibrium deformation E = 0.318. The three curves are analogous to those in fig. 2.
369
J. Kunz, J. R. Nix / Generalized cranking model 2.0
/
‘;
f 3 3S .P
1.5
-
1.0
-
?z S ‘% & 0.5
-
2
’
I
1
I
240Pu Harmonicoscillator lrrotational _____------.
I\
-
.\
._
_-““”
6 0.0,
0
_____---=
Present
‘. ‘.
? *
_____------
--\_ C
’ 2
’
’
I
’ 8
10
PairiigGap A !MeVl Fig.
5. Dependence
oscillator
potential
of the y-vibrational at the equilibrium
inertia
deformation
upon
the pairing
gap A for z40Pu in a harmonic-
E = 0.318. The three curves are analogous to those in fig. 2.
with larger spatial extent are increasingly occupied. As shown by the longdashed lines, the inertias calculated in the Inglis cranking model approach zero in the limit of large pairing. On the other hand, as shown by the solid lines, the inertias calculated in the generalized cranking model remain close to the irrotational values. The deviations at large pairing gaps arise from a slow convergence in the numerical calculations.
4. conclusions In this paper we have generalized the Inglis cranking model for collective nuclear motion by properly taking into account the velocity dependence of single-particle potentials through an effective mass in the cranking term and the reaction of the pairing field to the collective motion. Both effects manifest themselves in the X1 term in a self-consistent treatment 12), but are neglected in the Inglis cranking model. As has been shown previously 16), whereas the rotational moment of inertia calculated with the Nilsson modified harmonic-oscillator potential is larger than the rigid-body value when the effective-mass term h, is omitted, inclusion of this term reduces the calculated moment of inertia to approximately the rigid-body value. Inclusion of the effective mass term for P-vibrational inertias may account for the renormalization factor of 0.8 that was originally needed to reproduce experimental spontaneous-fission lifetimes ‘). The reaction of the pairing gap to the collective motion A, has been shown to be essential to obtain a finite inertia in the limit of large pairing. In fact, for large
370
J. Kunz, J. R. Nix / Generalized cranking model
pairing the inertia approaches its irrotational value. This result has been shown both analytically in a semiclassical approximatioh and numerically, where remaining deviations were attributed to a slow convergence. Taking the d, contribution into account for rotational moments of inertia in a moditied harmonic-oscillator potential would increase the calculated values, which were originally 3, smaller than the experimental values by about 15 to 20”/. However, this increase would be partly cancelled by effects arising from the effective mass. For vibrations at realistic pairing strength, the A, contribution would be small compared to the large pairing-vibration coupling term. We applied our generalized cranking model to translations, rotations and some special types of vibrations. In these applications we have restricted ourselves to cases where the collective momentum operator assumes a certain form consistent with our time-independent derivation. The application to the time-dependent vibrational motion where no stationary frame exists needs special consideration. The general time-dependent derivation and its application to /I-vibrations will be given in a forthcoming paper ‘O). Finally, we remark that our results for the effective-mass term are analogous to those obtained in adiabatic timedependent Hartree-Fock calculations 17). It would be interesting to compare our A, terms, where the reaction of the gap to the collective motion is proportional to the spherical harmonics describing the collective motion considered, to corresponding adiabatic time-dependent HartreeFock-Bogoliubov results.
Appendix ALTERNATIVE
DERIVATION
OF h, FOR SPECIAL
A CASE
As discussed in sect. 2, when the single-particle potential is velocity-dependent our expression for the collective inertia contains an additional term proportional to h, to account for this. We determine h, by generalizing the derivation of ref. 16) for rotations to collective motion that can be described by the collective momentum operator 71 defined by eq. (2.2). The velocity oL appearing in this equation characterizes the transformation from the laboratory frame to the moving frame. As an alternative to our general derivation in sect. 2, we present here a very transparent derivation of the h, term for the classical example in which the singleparticle potential contains a velocity-dependent term au* in addition to a spatial term V(r). Then the lagrangian for a nucleus at rest is L = +md
- V(r)-au'.
For a nucleus performing collective motion the lagrangian must be modified because the potential inside the nucleus should be independent of the motion of the
J. Kunz, J. R. Nix 1 Generalized cranking model
371
nucleus as a whole, i.e. the potential should be galilean-invariant. Consequently, the potential should not depend on the absolute velocity of the nucleons, but on the velocity relative to the other nucleons. The lagrangian for a nucleus in motion is then L = $?ll? - V(r) - cd,
where u’ = u-u01 is the velocity in the moving frame. To lowest order in the velocity u1 the lagrangian in the moving frame becomes L = $rl*d2 +mu’ . uj,- V(r),
where m* = m-2a is the effective mass. In terms of the canonical momentum p’ = m*u’+mu, the hamiltonian
in the moving frame is H = p‘ /2m + V(r) - (m/m*)u, . p’,
(A.1)
where we again retain only terms linear in u> Since the last term in eq. (A.l) is the sum of the usual cranking term - ul. p’ and the additional term h,, it follows that h,-v,*p
= -(m/m*)u,.p,
64.2)
where we now omit the prime on p. From eq. (A.2) we obtain h, = (1 -m/m*)u,.p.
(A.3)
In the quanta1 case we must take into account the possiblity that p and u1 may not commute, i.e. that V *u2. may be nonzero. In this case we must make the replacement u,.p
-+$u,.p+p.u,).
Then expression (A.3) for h, agrees with the general result (2.9) obtained in sect. 2.
372
J. Kunz, J. R. Nix ! Generalized
cranking
model
Appendix B NUCLEAR
INERTIA
WITHOUT
PAIRING
To calculate the collective nuclear inertia I in the absence of the pairing interaction, we use the commutator of an operator C with the hamiltonian N that is defined by R = [C, H-J
(B-1)
This permits us to close the cranking sum and thereby obtain the inertia 1. We determine the operator C in eq. (B.1) for a hamiltonian that is of the form l-l = p2/2m*+
V(r).
In the absence of pairing the expression (2.5) for the normal density reduces to
Apart from the factor m/m*,the inertia then assumes the familiar form
03.2)
B.l. IRROTATIONAL
VELOCITY
FIELD vi
When u1 is irrotational, we can satisfy eq. (B.l) with an operator C that depends only on position. In fact, for this case, C is proportional to the velocity potential. Eq. (B.l) leads to
from which we make the identification vI = (ilil/m*)VC. We next use the comp~t~ess inertia I in eq. (B.2) as
EB.3)
of the set of wave functions Iv) to rewrite the
1 = C (mlm*Kvl[C, n]Iv)~~. Y Evaluation of the commutator
then yields for the inertia the irrotational
value (B.4)
J. Kunz, J. R. Nix / Generalized
313
cranking model
whichis independent of the effective mass m*, as expected. Had we not included the h, term, the inertia would have differed from the irrotational value in eq. (B.4) by the factor m*Jm. B.2. ROTATIONAL
VELOCITY
FIELD
D;
For a rotational velocity field u1 the operator C depends on momentum in addition to position, which leads to a dependence on the special choice of the potential k’(r). Here we consider only a rotating nucleus, for which uj. =
0
x
We then specialize to a harmonic-oscillator H = p2/2m* With this hamiltonian given by 26) Cc-i
the operator
of
r. potential, with
+~m[c$(x’
+y2)+otz2].
C satisfying the commutation
relation (B.l) is
4 [bm*(wf+dyz+ &P,P,].
After closing the cranking sum, we obtain for the moment of inertia I =
Cp,. Y
m(vl(y2+z2)lv)+
& z
(vl(~~52-dz2)Iv) . x
1
(B.5)
The first term is the rigid-body moment of inertia, whereas the second term contains the shell effects. The second term vanishes in the ground state of a nucleus at its equilibrium deformation, where the condition
is satisfied ‘s). Appendix C CALCULATION
OF f(r)
We determine the reaction of the pairing field d, to the collective motion by considering the function f(r) that is related to A, through eq. (3.4). The function f(r) is determined from eq. (3.6), which was obtained from the continuity equation and must be satisfied for every position r.
374 C.1. IRROTATIONAL
J. Kunz, J. R. Nix / Generalized cranking model VELOCITY
For an irrotational
FIELD cI
velocity field u1 the solution for f’(r) is of the form f’(r) =
-irC,
Cl)
where x is a constant and the operator C is defined by eq. (B.l). To prove this, we express ri in eq. (3.5) with the help of eq. (B.l) and insert the ansatz (C.l), which leads to
Obviously, eq. (3.6) is satisfied if u = -24. Thus, f(r) for irrotational
velocity u1 is given by f(r)
= i2AC(r),
(C.2)
where C(r) is proportional to the velocity potential given by eq. (B.3). Using solution (C.2) for f(r) in the commutator expression (B.l), we obtain the relation 2A217rpvIz= ;lf,,12(h, - II,)*.
Thus, expression (3.8) for the inertia reduces to
(C.3)
C.2. ROTATIONAL
VELOCITY
FIELD I~
As discussed in appendix B.2, for a rotational velocity field u1 the operator C depends also on momentum, so that ansatz (C.l) cannot be made. In the following, we again specialize to rotations for a harmonic-oscillator potential. In the limit of large pairing strength one intuitively expects the inertia to approach the irrotational result. This suggests an ansatz for f(r) of the type f‘(r) = -icfC(r),
where c is the velocity potential for irrotational
(C.4) motion of a rotating nucleus 14).
J. Kunz, J. R. Nix / Generalized
cranking model
315
This relationship is $
- ihAvc = -AyV(yz), _
111*
with ^r’= (0: - o;)/(wf + a;). Note that c does not satisfy eq. (B.l). On the other hand, for the harmonicoscillator potential one has the relation lo) m*(flo,’- o$)yz.
7i =
Thus, the ansatz j(r) = /Vlic
(C.5)
is equivalent to eq. (C.4), yielding the relation a =
p12(o; + wf).
We now insert ansatz (C.5) into eq. (3.6), multiply by A(r) and integrate, which leads to
033) There are two kinds of non-vanishing corresponding energy differences are
Ik-$1
=
matrix
elements Jlp, for which the
h(o,-w,)
= ho_,
AN=0
h(w,+w,)
= ho,,
AN = 2,
(C.7)
where N denotes the principal oscillator quantum number. Upon introducing the two constants
co= c
Ifv,l’ E,E,h%+ E,,)’
d1;‘!iO
12
=
lL,,l’ c v.,, E,E,(E, + E,) AN=2
and inserting them into eq. (C.6), we obtain
U-4)
376
J. Kunz. 1. R. Nix i Generalized cranking model
Thus, our ansatz (C.5) for f(r) satisfies eq. (3.6), with p given by eq. (C.8). Ey. (3.8) for the inertia then becomes
Appendix D SE~rCLASS~CAL EVALUATION OF THE INERTIA In order to evaluate the inertia given by eq. (3.9), we must first determine I, from eq. (3.10). For this purpose we use semiclassical methods, in analogy with Migdal’s calculation for rotations lo). To lowest order in h, expression (3.10) for I0 is transformed by use of standard methods L5,27) into
d3rd3pA?S(e-e,).
P.1)
In this appendix we denote by ti the collective momentum defined by eq. (2.2) to distinguish it from the transcendental number conventionally denoted by rt. Also, we use the definition h, = e,-e,, where e, is the single-particle energy and eF is the Fermi energy. By performing the integration over dQ, in eq. (D.l) we obtain
10 = $$A--
s
d3rdp3p4u:(r)6(e-e,).
Since energy and mamentum are related by e = p2/2m* + V(r),
we evaluate this integral by substituting de =
(plm*)dp,
which leads to 1, =;
sd3rA
Since in the ~micta~ica1 approximation
3nvl3
Gtd.
(D-2)
the Fermi momentum pF and density p(r)
J. Kunr, J. R. Nix i Generalized
are related
377
cranking model
by “) PF
eq. (D.2) can be rewritten
=
h[3n2p(r)]f,
3
s
as
10 =
d3rp(r)u:(r).
(D.3)
Note that expression (D.3) is independent of the effective mass irrotational velocity field uj,(r) the inertia I,, is simply the classical inertia and for rotations it is simply the classical rigid-body moment With the help of eq. (D.3) for I, we now calculate the total inertia (3.9). For an irrotational velocity field DA(r) our result for f(r) appendix C.l. is
m*. For an irrotational of inertia. given by eq. obtained in
17Tvfl12 = 11;.,12((h,.-h,)/2A)2. Insertion of this relation into eq. (3.9) causes the two terms in the square bracket the inertia remains at the to cancel, with the result that I = I,. Therefore, irrotational value also when pairing correlations are present. Finally, we calculate the moment of inertia for nucleons with pairing in a harmonic-oscillator potential. For this purpose we use our result from appendix C.2 for j’(r), where expression (C.7) for /? is approximated by lo)
with
The quantities ok are defined by eq. (C.7). Following Migdal’s arguments further, we find for the inertia
I=
1
0
&9-4+9+~1 UJf+COz,
+
the expression
@_+g+)2w2_o: @_oZ +g+o:)(oz_
+fB’,, 1 ’
lo)
(D.4)
which we investigate for the two limiting cases of small and large pairing strength. For small A we find that g + 0 and therefore I + I,; i.e. the inertia approaches the rigid-body moment of inertia when the pairing vanishes. For large A we find that g + 1, which causes the first two terms in the square bracket of eq. (D.4) to cancel,
378
J. Kunz, J. R. Nix / Generalized cranking model
leaving us with 1 + Upon inserting eq. (C.7) for o,
1 _hM O(W2 +o:y*
into this relation, we obtain
1 -+ I,
wf-sf 2 w;+o:, .
( 1
Thus, in the limit of large pairing the inertia approaches the classical irrotational result 14).
1) D. R. Inglis, Phys. Rev. 96 (1954) 10.59; 97 (1955) 701 2) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No I I (1959) 3) S. G. Nilsson and 0. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. I6 (1961) 4) D. R. Bits, Mat. Fys. Medd. Dan. Vid. Selsk. 33, No. 2 (1961) 5) A. Sobiczewski, 2. Szymatiski, S. Wycech, S. G. Nilsson, J. R. Nix, C. F. Tsang, C. Gustafson, P.
M&her and B. Nilsson, Nucl. Phys. A131 (1969) 67 6) M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky and C. Y. Wong, Rev. Mod.
Phys. 44 (1972) 320 7) J. Randrup, S. E. Larsson, P. Miiller, S. G. Nilsson, K. Pomorski and A. Sobiczewski, Phys. Rev. 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
18) 19) 20) 21) 22) 23) 24) 25) 26) 27)
Cl3 (1976) 229 R. E. Prange, Nucl. Phys. 22 (1961) 283; 28 (1961) 369, 376 D. J. Rowe, Nuclear collective motion (Methuen, London, 1970) A. B. Migdal, Nuci. Phys. 13 (1959) 655 S. T. Belyaev, Nucl. Phys. 64 (1965) 17; Collective excitations in nuclei (Gordon and Breach, New York, 1968) D. J. Thouless and J. G. Valatin, Nucl. Phys. 31 (1962) 21 I M. Baranger and K. Kumar, Nucl. Phys. A122 (1968) 241 A. Bohr and B. R. Mottelson, Nuclear structure, vol. 2 (~n~min, Reading, Mass., 1975) P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) J. Kunz, Nucl. Phys. A363 (1981) 1 M. J. Giannoni, F. Moreau, P. Quentin, D. Vautherin, M. Veneroni and D. M. Brink, Phys. Lett. 65B ( 1976) 305 ; D. M, Brink and M. DiToro, Nucl. Phys, A372 (1981) I51 M. Durand, J. Kunz and P. Schuck, On the semiclassical description of rotating nuclei, Nucl. Phys., to be published J. Kunz and U. MO& Nucl. Phys. A406 (1983) 269 J. Kunz and J. R. Nix, Calculation of the nuclear inertia in a generalized cranking model, Nucl. Phys., to be publish~ L. Wilets, Theories of nuclear fission (Clarendon, Oxford, 1964) K. K. Kan, Ph.D. thesis, University of Maryland (1975) A. Schuh, J. Kunz and U. Mosel, Flow patterns for collective quadrupole vibrations in heavy nuclei, Nucl. Phys. A412 (1984) 34 E. R. Marshalek and J. 0. Rasmussen, Nucl. Phys. 43 (1953) 438 P. MBller and J. R. Nix, Nucl. Phys. A296 (1978) 289 M. Radomski, Phys. Rev. Cl4 (1976) 1704 A. Bohr and IS. R. Mottelson, Nuclear structure, vol. 1 (Benjamin, Reading, Mass., 1969)