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Variational approach to nuclear fluid dynamics
Nuclear Physics A398 (1983) 59-83 @ North-Holland Publishing Company
VARIATIONAL APPROACH TO NUCLEAR FLUID DYNAMICS J. P. DA PROVIDENCIA
and G. HOLZWARTH
Siegen University, Fachbereich 7 - Physik, Adolf-Reichwein-Str., 59 Siegen 21, West Germany
Received 10 August 1982 (Revised 25 October 1982) Abstract: A variational derivation of a fluid-dynamical
formalism for finite Fermi systems is presented which is based on a single determinant as variational function and does not exclude the possibility of transverse flow. Therefore the explicit specification of the time-odd part has to go beyond the local X-approximation, while the time-even part is taken in the generalized scaling form. The necessary boundary conditions are derived from the variation of the lagrangian. The results confirm previous simplified approaches to a remarkable degree for quadrupole modes; for other multipolarities the deviations are much less than might be expected according to a sizeable change in the transverse sound speed.
1. Introduction Recent attempts ’ -5) to formulate a simple, macroscopic description for nuclear collective motion have led to the closed set of fluid-dynamical equations
m&j=$[s], pis+j
= 0.
(2)
Here p(x, t) and j(x, t) are the single-particle density and current density, respectively, m is the bare nucleon mass, E is the total energy expressed as a functional of the scaling field s. In the generalized scaling approach s, the vector field s defines the change in the time-even part b+ of the single-particle density matrix fi through p+
(x,
x’,
t)
=
e”“‘v+v’“‘e~‘““~+v”s”
bo(x, x7,
where s = s(x, t), s’ = s(x), t), and CO is the static ground-state single-particle density matrix. Eq. (1) resembles the Euler equation of hydrodynamics and is a genuinely variational result obtained by varying the quantum-mechanical action 59
60
J. P. da Providencia, C. Holzwarth 1 Variational approach
with respect to the scaling field s. On the other hand, the time-odd part of @is left completely unspecified, except for the auxiliary condition (2f.which relates the tirst moment of fi in p-space, the current j, wit’- ha ? time-derivative of s. Thus (2) is sufficient to guarantee the continuity equate _.I
v* (pd+j)
= 0,
(4)
but by itself is not a necessary condition following from variation of the action. While (4) connects only the longitudinal parts of the current and the scaling field, (2) relates also the transverse components of both fields. Only in the special case where the full single-particle density matrix fi is restricted to the form a =
e’X’“&~
+ e-
WJf,
(5)
i.e. where the time-odd part can be specified by one local function x(x, t), (4) is equivalent to (2) because in this case the flow is irrotational with x as velocity potential, and under this restriction the basic equations (I), (2) are a variational result ‘O). In fact, the first derivation of the “elasticity” pressure tensor for Fermi fluids given by Bertsch ‘) is restricted to this case only. It has, however, been shown ‘) that (2) does hold in a truncation scheme of Landau’s equation for the motion of Fermi fluids if one considers distortions of the local Fermi surface in momentum space only up to multipolarity I,,, = 2. Therefore relation (2) is also implied if one truncates the coupled equations of motion for the moments of the distribution function after the second moment. This method which has been used by Winter 7), Nix and Sierk *), Wong and Azziz 9, to rederive the pressure tensor of nuclear fluid dynamics is, of course, also not a variational method because the truncation is made in equations of motion, i.e. after variation. Inclusion of higher moments in p-space 6, violates relation (2) and leads to considerable changes in the sound speed (e.g. the speed of transverse sound is ci = &I+ for t,,, = 2, but cL = 4~ for i max= 3). It is therefore desirable to develop a fully variational approach to nuclear fluid dynamics which includes also the transverse components of the flow, i.e. goes beyond (5). Thus the purpose _of the present investigation is twofold: Formulate the lagrangian from which equations of motion and boundary conditions may be obtained in a genuinely variational way. Compare the results with the simple set (l), (2) in order to learn about the validity of (2) and see whether the boundary conditions obtained serve to compensate for the changes in the sound speeds. Altogether we hope to obtain in this way a justification of the simplified previous approaches from a substantially deeper point of view and, on the other hand, learn about the limitations of the simple fluid-dynamical set (1) and (2).
61
J. P. da Providencia, G. Holzwarth / Variational approach
2. Lagrangian,
equations of motion and boundary conditions
The following considerations
will be based on the time-dependent I@) = expi (0 + B)(@,),
determinants (6)
where I@,-,)is the HF ground state and Q and p are time-even and time-odd singleparticle operators &= i
Qi = QT,
B=
t
pi=
-B’.
(7)
i=l
i=l
In the generalized scaling approach pi is restricted to the first term in an expansion in powers of momentum fl
Pi = &(Xi, with s as cation of oi = x(x, ansatz for
t)~ji+ji.S(Xi,
t)),
(8)
the variational scaling field. As we have said before, an explicit specifithe variational form of the time-odd part of I@) has to go beyond t) if we want to include also transverse flow. Thus the most simple oi is then
with the variational scalar field x and (symmetrical) tensor field c$,~, where in (9) summation over vector indices ~1,/? is understood and the index h indicates a hermitian form. From comparison with Landau theory6) it should be clear that with (9) we go beyond the I,,, = 2 truncation scheme of the Fermi surface distortion. Apart from total time derivatives the lagrangian L = ih(@lb)-(@H@)
(10)
to second order in Q and P is P) = - f tr([P, $o]d)-(@H@)‘2’,
(11)
where (@Ha) (2) is the second-order change in the total energy. In the classical limit we replace / commutators by Poisson brackets and operators A^ by their Wigner transforms A(x, p) [A, jj] + ih{A, B} = ih(VA . VP-VA.
VB).
(12)
62
To avoid
J. P. da Providencia,
confusion
we denote
G. Holzwarrh
the Wigner
/ Variational approach
transform
of 6 and PO as f and JO, and
for P and Q we have
Qk P) = x(xt t)+h~~p&&> t). The trace in (11) goes over into a phase-space d2’ =
integral
(13b)
JdP so we have for (11)
dP{P,f,}Q-(@H@)‘?
(14)
s The local density
p(x, t) is (up to second
P =
order)
given by
fd3P s
Inserting
=
ss s
=
p()+p’+p’*‘.
fod3p- {fViJd3p+t
({P{P,foll+ {Q~QJbH)d3~ (15)
(13) we obtain
(16)
P(l)= UP,%) PC*’ = )a,(s,a,(Posa))+3a,c~,,(Po(apX Here we have used for the static second
+~P~all~,,)+~a,(~SvPoP~))).
(17)
moment
i.e. fO is taken to be isotropic in momentum space with sharp momentum the first term in (14) reads With this assumption
cut-off at
IpI = pF.
dP{P, fo)Q = s In order to proceed further kinetic part of (@H@)(*) is
(QiTW2’ =
s
d3rPes,(&X ++P;(@,~J,,
the energy change
+@,,)).
(@H@)‘*) has to be specified.
-df~~({P{P'fo}}+{Q{Q,r,,,,.
(19)
The
(20)
63
J. P. da Providencia, G. Holzwarth 1 Variational approach
with m* as an effective mass for the quasiparticles. For the potential part we assume a density-dependent term of the form Caaapa together with a velocitydependent two-body interaction proportional to p. p’/2m*. The second-order term then is
(@V@)‘2)= s +
d3x 1 a,@~p;-‘p’~’ ++a(~ - l)p;- ‘(p’“)‘) cl
sd’x$s
d3pd3p’(f&PI, Q(x,P,)‘& {fok
P'L
Q(xv~'11, (21)
with constant coefficients a, and F,. In accordance with previous notation we introduce the “intrinsic energy functional” E[s] which comprises all s-dependent parts of ( @H@)‘2’: E[s] =
dT 2
+(P{P, fO}} + d3x 1 a&(a d s
- l)p”,- 2(~‘1’)2+ a~“,- ‘&(s,~,&,sa)))
Formally there appears an additional term
on the r.h.s. of (22) which, however, for self-consistent p,, vanishes identically. We shall in the following consider only spherical square densities
PO(x)= P&R -r),
(23)
with the constant p0 satisfying the equilibrium condition for nuclear matter density 6 6p
3 lOm* (
-++&z,p”-’
~
= 0.
(24)
>IPO
Therefore all space integrals are restricted to the domain where r < The expression for the current is obtained from its definition
R.
64
J. P. da Providencia, G. Holzwarth / Variarional approach
Here we encounter surface current
the difficulty
unless
We shall therefore
that with (23) the last term in (25) leads to a pure
we require
impose
(26) as a boundary
condition
4 UP’ Observing (26) in the evaluation of the remaining lagrangian I?’ (14) finally appears as
on the variational
parts
of (20) and
field
(21) the
+Mw+?~ + qJ4,A’+tkv#&d2 + (Q#L,m#qJ,)) (27) Equations of motion and boundary conditions at r = R are now obtained from lagrangian variations of (27) with respect to x, s, and 4,s. Free variation of x inside and at the boundary r = R leads to the continuity equation
and the boundary
condition
[&(Po&+jJl,=, where the expression
= 0,
(29)
(25) for the current
ja =
E9 (8 x+h$(9dbsa+aB4 m a
Pa
1)’
(30)
and the relation 1 -_= m have been used. Free variation
of s inside
$(l+&)
and at the boundary
(31)
r = R leads to the “Euler’‘-type
J. P. da Providencia, G. Holzwarth 1 Variational approach
equation
65
(1) (32)
and the boundary
condition
E&&JLR = 03
(33)
with the vector functional
and the pressure
tensor
The results up to now are identical with corresponding equations in the simplified version of the generalized scaling approach [cf. eqs. (86), (89) in ref. “)I. Relation (2) between S and j, however, is not obtained here. Instead, it is replaced now by a much more complicated equation derived from the c$,~ variation R 6,,LP m
=
w
&,d3x
s0
-
&#@apdb
= 0.
(36)
Y! surface
The variation leading
in the interior
region
is unconstrained
therefore
Ems must
vanish,
to
WO$
+j&+
J&PO&
+.i,)
+
(37) The variation on the surface is restricted by (26). Let us therefore I = R in a form which automatically satisfies (26),
and then
allow
for arbitrary
variations
write
c$,@at
c!I$.~ at r = R in (36). This leads to the
66
boundary
J. P. da Providencia, G. Holzwarrh 1 Variational approach
condition &
must vanish
that the tensor 1 1 1 = rDla - - XaX,r,B - -r2 x Px YI- w + - x,xpx,x,l-,, rz r4
at the surface
[r,,]*= R
=
0.
(40)
In the system of basis vectors x, e”‘, eC2), with e(l) and eC2) orthogonal tensor raa has only transverse components because according to (39)
to x, the
Therefore the tensorial boundary condition (40) obtained from the restricted riation of #,@ at r = R leads only to two equations to be satisfied at r = R [e~)r,Be~J]r =
R
=
0
(4 j = 1, 21,
va-
(41)
or, explicitly, [e$)(x,d,~,D + x,(d,+,, + a,~,,))ebj)-~xs(3ay~a~+
~~~~~~~~~~~ e(j)] = 0.
(42)
r=R
This completes the set of field equations (28), (32), (37) and boundary conditions (26), (29), (33) and (42) as derived from the lagrangian (27). It is interesting to note that the function x has been completely absorbed by the current and no longer appears explicitly. Much more interesting is to notice that scaling field s and tensor field $,s may be readily eliminated from the equations of motion so that we can immediately find the general form of the current. We shall see, however, that the same does not hold in the boundary conditions. Let us first contract (37) with ~3, and then take the curl of the resulting vector. This leads to d(p,(V Together with (28) this relation and we obtain wave equations current
x S)+(l
+$m/m*)(V
= ctd(V.j),
(V x jj = c-fd(V x j), sound
= 0.
(43)
is sufficient to eliminate the s-field from (32), (34) for the longitudinal and transverse parts of the
(V.j)
with the longitudinal
xj))
(44a) (44b)
velocity
Wa)
J. P. da Providencia, G. Holzwarth / Variational approach
67
and the transverse velocity cz T
l p:
5 mm* (
l+Q.
>
Wb)
Evidently (45a) agrees with the simple generalized scaling result while the transverse sound speed (45b) differs from the scaling result by the term 8m/7m* in the brackets [cf. eqs. (74) and (77) in ref. “)I. In fact, (45b) coincides with the transverse sound velocity obtained in the I 2 3 truncation scheme of Landau’s equation [cf. eq. (54) in ref. “)I. Comparison with eq. (63b) of ref. 6, shows that with inclusion of 4 aB in (9) we have allowed for 1 = 3 distortions of the local Fermi sphere. This extension remains, however, ineffective for the longitudinal components as long as the third-rank tensor q,sv in eq. (63a) of ref. 6, is excluded: because eq. (63a) of ref.(j) with ‘pap, 5 0 implies the relation voO= ~~~$3 which in connection with the continuity equation eq. (34a) of ref. 6, cuts off eq. (34~) of ref. 6, as indicated for 1 5 2. Therefore the present scheme represents a genuine extension only for the transverse components. For these the sizable modification is already evident from comparing (43) with (2); it should be noted, however, that (43) relates the transverse parts of S and j only up to terms which satisfy the Laplace equation and it is just this freedom which is fixed by the additional boundary conditions, In the boundary conditions it is not possible to eliminate the fields s and $,s. Thus for a complete solution we have to explicitly construct s and 4ola. This is in marked contrast to the simple version of nuclear fluid dynamics where in (32) j is simply replaced by -p,S according to (2) so that the boundary condition (33) is sufficient to determine the eigenmodes completely. General non-singular solutions of (44) are
V .j = C alJi,(kr)X’,,, lm
with spherical Bessel functions jl, spherical harmonics &, and &,,, and constants a, I(m = 0) we obtain for the electric (E) modes
b, c, d. Separating different parities for a given multipolarity
V xi a j,(k,r)Y,,O, i.e. j, =
Wl(kr) + ylrf)Y,,+ P(Vxjl(kTr)Y,d,
and for the purely transverse magnetic (M) modes V-j = 0,
Vxj = c~l+l(kTr)k;l+,o+d~l-,(kTr)k;l-,o~
(46)
68
J. P. da Providencia, G. Holzwarth / Variational approach
i.e. j,
=
jl(kTr)Y,lo.
(47)
Obviously, in the finite system the E-current may contain an additional r’ term in its longitudinal part, while for the M-modes such an extension is not possible. In the following two sections we shall discuss the two types of modes separately.
3. Magnetic modes For transverse
with traceless
modes
(47) the expression
4,s. Therefore
we obtain
(30) for the current
from (37) (by contracting
reduces
to
with a,)
(49)
while the continuity
equation
(28) requires v.s=o.
This is sufficient r’k;,,
to determine
term in addition
(50)
the form of the scaling
to the part which is proportional
iop,s =
field which may contain to the current
((1 +~~)jl(k,r)+cr’)~lo.
(We have assumed a harmonic time dependence condition (29) is trivially fulfilled. The boundary the constant c = c(k,R):
an
(47)
(51)
e -ior.) With (51) the boundary condition (33) relates k,R with
or (52)
J. P. da Providencia, G. Holzwarth 1 Variational approach
Evidently,
for 1 = 1 k, is determined
by the boundary
69
condition (53)
in accordance
with the earlier
result
[eq. (92) in ref. ‘)I. This must be so because
(53) is a consequence of angular momentum Thus for 1 = 1 the current is fixed, independent
conservation as discussed in ref. “). of the value of the constant c in the
scaling field. In the following we shall exclude the case 1 = 1. In order to use the boundary condition (42) we have to construct +,s from (37) which for transverse modes reads
the tensor
field
or with (51)
(54)
The general symmetric, traceless solution may be written in the form parity ( - )I+ ’
of (54) for multipolarity
I(m = 0) and
with ql(r)
= a,j,(k,r)+a,r’+a,r’+‘,
(p2(r) = ad+.
The four constants condition
a, are determined
by (54), (48) with
(47) and
the boundary
(26) as
5m 1 a, = -21 PoPFk,’
a
2-
--
2(21+ 3)a, + (4 + I)a, = 0,
where we have used (52) to obtain
a4=
7m* c(k,R) --p, POP: 1+2
this form of a2. The boundary
(56) condition
(42) for
J. P. da Providencia,
70
G. Holzwarth
/ Variational
approach
transverse modes simplifies to
Let us take the two vectors e”’ in the form &)
=
q(i) x
x,
i = 1, 2,
(58)
with fixed vectors 9”‘. Then (59) acts only on angular variables. Therefore e(“. V(X+#J~~)contains the same radial function as x+$,~ which, however, must vanish at r = R because of (26). Thus (57) reduces to e$)((d,
-
2)~#~,,)f#, = 03
or with (55) [(e@). V)(rd,-
(i -j)],,,
3)cp,(r)(k;,,),)e~'+
= 0.
Again the operator e”). V does not affect the radial dependence, therefore we finally obtain q,(R)
= 0
to fix the transverse wavenumber k,. With the use of (56) and (52) this is easily brought into the form (with x E kTR)
A+ l(xYx
1 21+1 (1 -x8X)jl(Xr = -%(Z+2)(21+3)(45+7F1)’ Before we relation like modes: The on s, so that
proceed to a discussion of (61) it is very instructive to notice that a (61) is indeed necessary to ensure the orthogonality of the resulting proof relies on the functional (34) being a hermitian operator acting for two solutions of (32) (distinguished by indices 1 and 2) we have -io,
s2. s
or
(61).
jld3x = L s2. Y[s,]d3x ms
= - io,
. s
s,‘12d3X, (32
s1
.j2d3x,
s
(62)
71
J. P. da Providencia, G. Holzwarth / Variational approach
In order to have orthogonality
s
s, . jpd3x
a 6,,
if
eJ: z wt,
(63)
we must require that the integrals in (62) be symmetric in the indices 1 and 2:
(64) For magnetic modes we insert current and scaling field (51) into (64) and obtain
s R
c2
s R
2dr = ci
j,(kp)r)r’+
0
j,(k$%)r’+ ‘dr,
0
or, with ci = c(kyR) from (52), we conclude that the ratio
s R
j,(kr)r’+2dr/c(kR)
0
must be independent of k. This, however, is just the 1.h.s. of (61) if we remember the property of Bessel functions
s R
j,(kr)r’+‘dr = R’+3 &.A+ ,(kR).
0
Thus we may be sure that the boundary condition (61) guarantees orthogonality (63) for modes belonging to different eigenvalues. For the most interesting magnetic mode, the 1 = 2 twist mode, the 1.h.s. of (61) is plotted in fig. 1. The poles, indicated by vertical lines, evidently coincide with the x-values given by the boundary condition in the simplified scaling model (1 - xa,)jr(x) = 0
(65)
[cf. eq. (92) in ref. “)I. For I = 2 and F, = 0 the r.h.s. of (61) is equal to -$$, indicated by the horizontal line in fig. 1. Obviously the solutions of (61) coincide very closely with the solutions of (65) except for the lowest one. Therefore the frequencies o = c,k, for the higher solutions fully reflect the increase of the transverse sound speed (45b), while the wavenumbers are essentially unchanged. The lowest solution, however, which is the physically important one, because it carries almost all the M2 strength, is strongly shifted to smaller values of k,, namely (for F, = 0) to k,R = 1.712
(66)
J. P. da Providencia, G. Holzwarth / Variational approach
X=kTR
Fig. I. This figure illustrates equation (61) for I = 2. The plotted curve is the left-hand side of (61) (j3(x)/x)/(I -x?Jj2(x). Vertical lines denote the zeros of the denominator. The full horizontal line at -0.2679 gives the value of the right-hand side for I = 2 and F, = 0. The lowest intersection occurs at ,Y = 1.712 while the lowest pole lies at x = 2.501.
as compared to the lowest solution of the scaling model condition (65) which lies at (k,R),, = 2.501. Multiplying these wavenumbers by the corresponding sound velocities
we obtain
(for F, = 0)
k, i&o&
1.7124? = 2301,,4
1002 ’
’
We therefore arrive at the remarkable result that the frequency of the collective magnetic mode remains practically unchanged when one goes from the simple generalized scaling model to this variational formulation including the tensor field + or, in other words, when one includes also octupole deformation of the Fermi 4 surface. In fact, the correspondence goes still deeper: We have plotted in fig. 2 the resulting radial dependence for the transverse current and the scaling field iop,s from (51). Obviously the term $m/m* in front of the Bessel function is essentially cancelled by the cr’ term so as to approximately restore the relation (2) zop,s
XJ.
(67)
It is clear that such a cancellation can only take place in a finite system, and only for the lowest mode. For higher k, values the nodes of the Bessel function lie inside the nucleus and the r’ term obviously cannot cancel a numerical factor in
J. P. da Providencia, G. Holzwarth / Variational approach
4
2
I” 2-
Fig. 2. The radial
functions
hw,=8,l77MeV
6
73
t R
fm
A=206
of the vector fields j and - p,S according 2- state (arbitrary units).
to UQ. (51) for the first excited
front of the Bessel function. This confirms our earlier suggestion 6, that it may be the finiteness of the system which renders the higher distortions of the Fermi surface unimportant when we consider truncation schemes of the Landau-Vlasov equation. 4. Electric modes The general form of the scaling field s for electric modes is determined by (28) and (43). With (46) we obtain for j and s j =
Vtil(k,r)+ylr’)Y,o+BV xjl(kTr)k;l~v +qm/m*)V x (~j[i,(k,r)+6,rf’2)~lo.
iop,s = V(j,(k,r)+6,r’)I;,+(l Longitudinal
(68)
and transverse wavenumbers are related through c,k,
= c,k,
= CO.
(69)
Thus in (68) there are 4 unknown constants yr, /?, 6,, 6, and one wavenumber, k, say, to be determined. The “Euler” equation (32), (34) which is 02j = (&
-&)V(V.
iop,s)-
&
d(i0p,s)
(70)
14
J. P. da Providencia, G. Holzwarth / Variational approach
links the r’ term in j with the r’+’ term in s: k+r,J The boundary
condition
(29) connects
(yi -S,)$R’-im((1 The boundary vector
condition
= - iS,m2(21+
3).
(71)
all four constants:
+$r/m*)62R’+2 for the pressure
+~m/m*fij,(k,R))
= 0.
(33) leads to two equations
because
(72) in the
XJ,, = (P+(r)Y,I+,,+P-(r)I;I-l,)p both radial
functions
P*(r)
must vanish
P+(R) = 0,
at r = R, P_(R)
= 0.
(7% b)
As in the case of the magnetic modes the four equations (71) (72) (73a, b) are sufficient to determine the constants yi, /I, 6,, 6, as functions of k,R, the wavenumber itself, however, remains undetermined. We then have to construct 4,s from (37) and impose the remaining boundary conditions (26) and (42). Inspection shows that (37) remains unchanged under a transformation
where f is an arbitrary function. The same holds for the boundary However, (26) is not invariant under (74) but goes over into
~~A&R Because f is arbitrary
this means
condition
= &lfWLR.
(42).
(75)
that the only useful part of (26) is (76)
with e”’ from (58) while the component
of x,4,@ parallel
We shall therefore does not impose any restriction on c#J,@. then need not take care of the freedom (74) any further.
to x,
replace
(26) by (76) and
J. P. da Providencia, G. Holzwarth 1 Variational approach
A special solution of (37) for j and s from (68) is +$J = a,F,+a,F
a, #+ 2
F=
-3Y1 -~I)-------
2(21+ 3)
&I
+
(77)
We therefore write
where $hog)is the general solution of the homogeneous equation
+(a,a,gy + asay4$ + dg? -+(a,a,@
+ 6,,a,(+a,4gj
+ a,rb:o,')) = 0.
(79)
The most general form of 4:;) is
4:oB) = (x,xsw(~) + 6,8~(~) + (Xmas+x,a,pw + a,a,zw, where, however, X remains undetermined and may be omitted (cf. (74)). A severe limitation on the actual form of W and Y is imposed on @) by the connection of 4 aS with the current j through (30): With the abbreviation
(30) requires with (46) and (77)
(83) From (80) we have for u u = r(4+ra,)w+(V(3+ra,)+xd)Y+VdZ, and therefore (83) requires W(x) =
w,r’Y,,,
Y(x) =
(yo++y2++2)Ylo,
.I. P. da Providencia, G. Holzwarth / Variational approach
76
with constants w,,, y, connected to ijz through (/+4)w,+2(21+3)y,
= ,s ”
(85)
while y, remains undetermined. Inserting (80) with (84) into (79) restricts Z to the form Z(x)
with z0 undetermined
=
(z,~‘+z,v’+2+Z4r~+4f~0,
(86)
and 3 independent equations relating W, Y and 2: (22 + 31)w, + 16(21+ 3)y, = 0, -w,+(21
+l)yz + 18(2/+5)z,
(19+l)y,+9(21+3)z,
= 0,
= 0.
(87)
So far we have obtained 4 equations (85, 87) for the altogether 6 constants wO,y,,, p2, zO, z2, z4 characterizing the homogeneous solution (80). Finally $a@= ~~~)+~~~~ must be subject to three boundary conditions, namely (76}, and (42) (for i = j, and i # j). These three relations fix the remaining (twofold) freedom in (p$ and termine wavenumber Their form a lengthy will be here their does allow a transparent as the case But in present the conditions (73), and ensure orthogonality of belongto eigenvalues which however, most checked cally the discussed the section.
5. Results Numerical results presented in this section are obtained with a potential energy functional of the form (88)
C aoPd = azp2 + a2 +*P’ +*, d with a z- -
and F, = 0.
-2 x 3075.8 MeV . fm3,
a2+* =&x20216.4
MeV+fm3+i
J. P. da Providencia, G. Holzwarth / Variational approach
Magnetic modes. These are, of course, not affected by the choice F, dependence is easily obtained from (61).
71
(88) while the
(i) I” = I+ : As we have remarked already it is a consequence of (52) that the wavenumbers k, for the unnatural parity dipole modes in the present variational formalism coincide with the corresponding wavenumbers in the simple formulation (1) (2) of nuclear fluid dynamics (NFD). Therefore the change in the transverse sound speed in the two different approaches as given by (45b) in comparison with eq. (77) in ref. 4), namely (for m = m*)
is fully reflected
in the frequencies
of the l+ modes
while the radial form of the current is unchanged. As was discussed previously in ref. 5), the boundary condition (53) leads to vanishing Ml strength for these convectional magnetic dipole modes. (ii) I” = 2- : We have seen in sect. 3 that the lowest M2 state which carries 98 % of the total M2 strength reproduces with remarkable accuracy the NFD result. At the same time it is clear from fig. 1 that for all the higher excited 2- modes the situation resembles the l+ case: The intersection of the horizontal line in fig. 1 with the function (jl+ ,/x)/(1 - xa,)j, occurs very close to the poles which describe the NFD solutions. From (52) then follows that c(k,R) is almost zero, which means that according to (51) current and scaling field differ by a factor of 9. Similarly, the frequencies of the higher modes differ from the corresponding NFD values by a factor of fi. Again, however, these higher modes carry very little M2 strength (cf. table 1). (iii) I” = 3+ : In the magnetic octupole mode we find even for the lowest excited state a significant difference between scaling and current field (see fig. 3), i.e. the factor y in front of the Bessel function in iop,s is only partly compensated by the cr3 term in (51). We find a factor of about 1.65 between iop,s and j, which is reflected also in the frequency of the lower 3+ state (which carries 95 yO of the total M3 strength) 03+
=
Jm
(03+)NFD.
All higher excited 3+ modes follow the same pattern as shown by the l+ and higher excited 2- states. Electric modes: Although the evaluation of the natural parity solution is slightly more involved than for magnetic modes the emerging pattern resembles the one we found above. We present results in terms of the unambiguous radial functions S+(I)
78
J. P. da Providencia, G. Holzwarth 1 Variational approach TABLE I Results
r:
ho
21 2;
8.16 23.28
3: 3:
12.61 27.56
1; 1; 1; 14
for the B(EI) and B(MI) values
B(I”) 3175 64 361 13
RI”)
ho 8.18 34.03
3800 38
x lo3 x IO3
16.16 40.26
285 8
x IO3 x IO3
0 10.98 21.46 25.02
0 9.2 x IO-’ 45.0x lo-3 15.3x 1o-3
0 11.75 25.64 35.85
2: 2: 2:
3.60 15.62 26.76
1690 114 0.1
8.60 15.82 29.59
3; 3;
12.74 20.78
70.2 x lo3 16.4 x IO3
13.13 22.89
76.9 x lo3 10.2 x lo3
4: 4:
16.28 26.02
3.1 x IO6 I .2 x IO6
17.26 29.33
3.7 x IO6 0.67 x lo6
0 46.5 x 1O-3 23.1 x 1O-3 1.6 x lO-3 1691 114 1.5
For the states listed in the first column energies (in MeV) and B(EI) or B(Mn values (in e2. fm*’ or & fm”-*) are given. The second and third columns show the results obtained from nuclear fluid dynamics as given by eqs. (I) and (2) the fourth and fifth columns show results of the present variational calculation. For the isoscalar dipole we replaced the excitation operator r’y1, by j,(qr) q,,,, with qR = 4.49.
I”= 3+ Fig. 3. The radial
hw,=16,l6MeV
functions
of j and
-p,,J
A=206 according
to eq. (51) for the first excited
3+ state.
J. P. da Providencia, G. Holzwarth / Variaiional approach
79
and j, (Y) defined by
Ax) = j+(r)%+ 1o W+j-(r)Y,I-lOW
Plb)
(The longitudinal and transverse radial functions of (68) are not without ambiguity because of the identity Vr’Y,, = i,/m V x r’Y,,,.) (i) 1” = 0’ : Monopole modes are purely longitudinal. In this case the NFD equations (l), (2) are a genuinely variational result, therefore the present approach coincides with NFD. (ii) I” = I-, 2+, 3-, 4’: The 1owest l- mode is the uniform translation which occurs at o = 0 in both formulations. In figs. 4a-c we present scaling and velocity fields for the lowest excited states with I” = 2+, 3-, 4+, which may be classified as surface modes because they show very little compression in the nuclear interior. It is most remarkable to which extent the NFD relation (2) is recovered for the quadrupole mode. In other words, for I = 2 the system does not take advantage of the additional freedom which is available in the present variational approach. Correspondingly, the lowest I = 2 frequency coincides with an accuracy of the order 10e3 with the NFD result. For I = 3 and 1 = 4 we do not find such perfect agreement. Although scaling and velocity field still show a rather similar pattern they no longer coincide. The resulting frequencies lie about 10 % above the NFD
Fig. 4a
J. P. da Providencia, G. Holzwarrh 1 Variational approach
80
/--,
/
,,-‘X\
‘\’ \
/ /I ;i’ 1’ // /
-&Be/
j-
w
6
l%3-
hq13,l
MeV
R
fm
A=208
Fig. 4b
/
/’
/’ // /
/’
--%A \\
\
/
‘6
\
W
‘R fm
“+.
1=4+
‘Y.\~ Tiw,=17,3MeV
A= 208
-P.f+ ‘t,_ \j+
‘\\ ‘\ Fig. 4c Fig. 4. The radial functions ofj and -p,,S according to eqs. (91) for the first excited 2+(a), 3-(b), states. The full lines show the density change in the nuclear interior given by - p,V 5.
4+(c)
J. P. da Providencia, G. Holzwarth / Variational approach
81
Fig. 5. Same as fig. 4 for the second excited 2+ state.
values. The same holds also for higher excited 1 = 2, 3, 4 states. As an example we show in fig. 5 the second excited (compressive) 2’ state, which obviously is still reasonably approximated by the NFD relation (2) while the frequency is reproduced within 1.3 %. Rather strong changes are observed for the higher excited (compressive) dipole states. Table 1 shows that NFD produces a low-lying rather weak state, with two strong states at 21.5 MeV and 25.0 MeV while in the present approach the low state is missing and the strength is contained in two modes lying below the NFD states. This again points to the fact that results of macroscopic approaches should be taken with great caution if they show strong fragmentation of the total strength. In such cases one should perhaps consider only energy-weighted moments as eventually
reliable
quantities.
6. Conclusion In the present article we have given a variational derivation of a fluid-dynamical formalism for finite Fermi systems which is based on a single determinant as a variational function and does not exclude the possibility of transverse flow from the outset. In order to obtain a genuinely variational approach it has been nec-
82
J. P. da Providencia,
essary to give an explicit density
matrix
lized scaling
G. Holzwarth
specification
of the time-odd
while for the time-even form used previously.
/ Variational
approach
part fi_ of the single-particle
part we have adhered
To allow for transverse
to the simple generaflow then requires
the
inclusion of a tensor field 4,s(x, t) as simplest possibility of going beyond the assumption of local x(x, t). This is in contrast to the previous simplified formulation of nuclear fluid dynamics in which the time-odd part p_ of the singleparticle density matrix was not specified at all except for the condition (2) which relates the first moment of p_ (namely the current) to the time-derivative of the scaling field s. Although at present we do not have a variational derivation of (2) it seemed quite natural to require that the time derivative of the displacement field s should be the velocity. The present variational approach, however, leads to a relation (43) between S and j which formally differs strongly from (2). But the numerical analysis has shown that for the lowest excited strongly collective electric and magnetic quadrupole states the simple relation (2) is numerically reestablished to a remarkable degree. Therefore previous conclusions about the occurence of transverse flow are supported by the present variational approach. For higher excited states and higher angular momentum (2) is no more valid but we still observe a definite tendency to reestablish relation (2). It is not clear at present whether deviations from (2) in this variational formulation might not have their origin in the very limited generalized scaling form of the time-even part of the density matrix. It might well be that extension of (8) to include the next order term 4,srp,psp, might reestablish (2) also for the octupole vibration. Such an extension seems desirable also from another point of view: Fluid dynamics as based on generalized scaling (3) seems unable to reproduce lowlying quadrupole and octupole states which we would expect in a microscopic theory as due to AN = 0 (for 2+) and AN = 1 (for 33) excitations (N: principal quantum number). It has been shown long ago by Stringari 2, in a harmonic oscillator model that with the above extension one can in fact obtain these lowlying states. This might be useful not so much for the description of low-lying states (which generally depend sensitively on the precise shell structure of the individual nucleus considered) but instead to eliminate admixtures of low-lying states from the giant states. Such admixtures might be due to the restricted form of P (8) in the generalized scaling approach. But within the limitations of generalized scaling (3) we may conclude that the variational approach supports the essential results as they were obtained by simply equating the time derivative of the scaling field with the velocity. Specifically we recover the occurence of transverse flow in both magnetic and electric modes. Most remarkably the explicit inclusion of octupole distortions in the local Fermi surface does not alter the results for a finite nucleus in a very significant way although the transverse sound speed is changed by a factor of 77. A similar compensation of changes in wavenumber and sound velocity has been discussed in ref. 11) for the comparison between first and zero sound in the case of the breathing mode.
J. P. da Providencia, G. Holzwarth / Variational approach
83
The authors would like to express their gratitude to Professor J. da Providencia for his interest in this work and many helpful discussions.
References 1) G. F. Bertsch, Ann. of Phys. 86 (1974) 138; Nucl. Phys. A249 (1975) 253 2) S. Stringari, Nucl. Phys. A279 (1977) 454; A325 (1979) 199 3) H. Sagawa and G. Holzwarth, Prog. Theor. Phys. 59 (1978) 1213; G. Holzwarth and G. Eckart, Z. Phys. A284 (1978) 291 4) G. Eckart, G. Holzwarth and J. P. da Providencia, Nucl. Phys. A364 (1981) 1 5) G. Holzwarth and G. Eckart, Nucl. Phys. A325 (1979) 1 6) T. Yukawa and G. Holzwarth, Nucl. Phys. A364 (1981) 29 7) J. Winter, LMU Mtinchen preprint 1980, submitted to Ann. of Phys.; and Proc workshop on semiclassical methods, Grenoble 1981 (ILL report Grenoble 1981, no. 15) 8) J. R. Nix and A. J. Sierk, Phys. Rev. C21 (1980) 396 9) C. Y. Wong and N. Azziz, Phys. Rev. C24 (1981) 2290 10) F. E. Serr, G. F. Bertsch and J. Borysowicz, Phys. Lett. 92B (1980) 241 11) B. K. Jennings and A. D. Jackson, Phys. Reports 66 (1980) 141
VARIATIONAL APPROACH TO NUCLEAR FLUID DYNAMICS J. P. DA PROVIDENCIA
and G. HOLZWARTH
Siegen University, Fachbereich 7 - Physik, Adolf-Reichwein-Str., 59 Siegen 21, West Germany
Received 10 August 1982 (Revised 25 October 1982) Abstract: A variational derivation of a fluid-dynamical
formalism for finite Fermi systems is presented which is based on a single determinant as variational function and does not exclude the possibility of transverse flow. Therefore the explicit specification of the time-odd part has to go beyond the local X-approximation, while the time-even part is taken in the generalized scaling form. The necessary boundary conditions are derived from the variation of the lagrangian. The results confirm previous simplified approaches to a remarkable degree for quadrupole modes; for other multipolarities the deviations are much less than might be expected according to a sizeable change in the transverse sound speed.
1. Introduction Recent attempts ’ -5) to formulate a simple, macroscopic description for nuclear collective motion have led to the closed set of fluid-dynamical equations
m&j=$[s], pis+j
= 0.
(2)
Here p(x, t) and j(x, t) are the single-particle density and current density, respectively, m is the bare nucleon mass, E is the total energy expressed as a functional of the scaling field s. In the generalized scaling approach s, the vector field s defines the change in the time-even part b+ of the single-particle density matrix fi through p+
(x,
x’,
t)
=
e”“‘v+v’“‘e~‘““~+v”s”
bo(x, x7,
where s = s(x, t), s’ = s(x), t), and CO is the static ground-state single-particle density matrix. Eq. (1) resembles the Euler equation of hydrodynamics and is a genuinely variational result obtained by varying the quantum-mechanical action 59
60
J. P. da Providencia, C. Holzwarth 1 Variational approach
with respect to the scaling field s. On the other hand, the time-odd part of @is left completely unspecified, except for the auxiliary condition (2f.which relates the tirst moment of fi in p-space, the current j, wit’- ha ? time-derivative of s. Thus (2) is sufficient to guarantee the continuity equate _.I
v* (pd+j)
= 0,
(4)
but by itself is not a necessary condition following from variation of the action. While (4) connects only the longitudinal parts of the current and the scaling field, (2) relates also the transverse components of both fields. Only in the special case where the full single-particle density matrix fi is restricted to the form a =
e’X’“&~
+ e-
WJf,
(5)
i.e. where the time-odd part can be specified by one local function x(x, t), (4) is equivalent to (2) because in this case the flow is irrotational with x as velocity potential, and under this restriction the basic equations (I), (2) are a variational result ‘O). In fact, the first derivation of the “elasticity” pressure tensor for Fermi fluids given by Bertsch ‘) is restricted to this case only. It has, however, been shown ‘) that (2) does hold in a truncation scheme of Landau’s equation for the motion of Fermi fluids if one considers distortions of the local Fermi surface in momentum space only up to multipolarity I,,, = 2. Therefore relation (2) is also implied if one truncates the coupled equations of motion for the moments of the distribution function after the second moment. This method which has been used by Winter 7), Nix and Sierk *), Wong and Azziz 9, to rederive the pressure tensor of nuclear fluid dynamics is, of course, also not a variational method because the truncation is made in equations of motion, i.e. after variation. Inclusion of higher moments in p-space 6, violates relation (2) and leads to considerable changes in the sound speed (e.g. the speed of transverse sound is ci = &I+ for t,,, = 2, but cL = 4~ for i max= 3). It is therefore desirable to develop a fully variational approach to nuclear fluid dynamics which includes also the transverse components of the flow, i.e. goes beyond (5). Thus the purpose _of the present investigation is twofold: Formulate the lagrangian from which equations of motion and boundary conditions may be obtained in a genuinely variational way. Compare the results with the simple set (l), (2) in order to learn about the validity of (2) and see whether the boundary conditions obtained serve to compensate for the changes in the sound speeds. Altogether we hope to obtain in this way a justification of the simplified previous approaches from a substantially deeper point of view and, on the other hand, learn about the limitations of the simple fluid-dynamical set (1) and (2).
61
J. P. da Providencia, G. Holzwarth / Variational approach
2. Lagrangian,
equations of motion and boundary conditions
The following considerations
will be based on the time-dependent I@) = expi (0 + B)(@,),
determinants (6)
where I@,-,)is the HF ground state and Q and p are time-even and time-odd singleparticle operators &= i
Qi = QT,
B=
t
pi=
-B’.
(7)
i=l
i=l
In the generalized scaling approach pi is restricted to the first term in an expansion in powers of momentum fl
Pi = &(Xi, with s as cation of oi = x(x, ansatz for
t)~ji+ji.S(Xi,
t)),
(8)
the variational scaling field. As we have said before, an explicit specifithe variational form of the time-odd part of I@) has to go beyond t) if we want to include also transverse flow. Thus the most simple oi is then
with the variational scalar field x and (symmetrical) tensor field c$,~, where in (9) summation over vector indices ~1,/? is understood and the index h indicates a hermitian form. From comparison with Landau theory6) it should be clear that with (9) we go beyond the I,,, = 2 truncation scheme of the Fermi surface distortion. Apart from total time derivatives the lagrangian L = ih(@lb)-(@H@)
(10)
to second order in Q and P is P) = - f tr([P, $o]d)-(@H@)‘2’,
(11)
where (@Ha) (2) is the second-order change in the total energy. In the classical limit we replace / commutators by Poisson brackets and operators A^ by their Wigner transforms A(x, p) [A, jj] + ih{A, B} = ih(VA . VP-VA.
VB).
(12)
62
To avoid
J. P. da Providencia,
confusion
we denote
G. Holzwarrh
the Wigner
/ Variational approach
transform
of 6 and PO as f and JO, and
for P and Q we have
Qk P) = x(xt t)+h~~p&&> t). The trace in (11) goes over into a phase-space d2’ =
integral
(13b)
JdP so we have for (11)
dP{P,f,}Q-(@H@)‘?
(14)
s The local density
p(x, t) is (up to second
P =
order)
given by
fd3P s
Inserting
=
ss s
=
p()+p’+p’*‘.
fod3p- {fViJd3p+t
({P{P,foll+ {Q~QJbH)d3~ (15)
(13) we obtain
(16)
P(l)= UP,%) PC*’ = )a,(s,a,(Posa))+3a,c~,,(Po(apX Here we have used for the static second
+~P~all~,,)+~a,(~SvPoP~))).
(17)
moment
i.e. fO is taken to be isotropic in momentum space with sharp momentum the first term in (14) reads With this assumption
cut-off at
IpI = pF.
dP{P, fo)Q = s In order to proceed further kinetic part of (@H@)(*) is
(QiTW2’ =
s
d3rPes,(&X ++P;(@,~J,,
the energy change
+@,,)).
(@H@)‘*) has to be specified.
-df~~({P{P'fo}}+{Q{Q,r,,,,.
(19)
The
(20)
63
J. P. da Providencia, G. Holzwarth 1 Variational approach
with m* as an effective mass for the quasiparticles. For the potential part we assume a density-dependent term of the form Caaapa together with a velocitydependent two-body interaction proportional to p. p’/2m*. The second-order term then is
(@V@)‘2)= s +
d3x 1 a,@~p;-‘p’~’ ++a(~ - l)p;- ‘(p’“)‘) cl
sd’x$s
d3pd3p’(f&PI, Q(x,P,)‘& {fok
P'L
Q(xv~'11, (21)
with constant coefficients a, and F,. In accordance with previous notation we introduce the “intrinsic energy functional” E[s] which comprises all s-dependent parts of ( @H@)‘2’: E[s] =
dT 2
+(P{P, fO}} + d3x 1 a&(a d s
- l)p”,- 2(~‘1’)2+ a~“,- ‘&(s,~,&,sa)))
Formally there appears an additional term
on the r.h.s. of (22) which, however, for self-consistent p,, vanishes identically. We shall in the following consider only spherical square densities
PO(x)= P&R -r),
(23)
with the constant p0 satisfying the equilibrium condition for nuclear matter density 6 6p
3 lOm* (
-++&z,p”-’
~
= 0.
(24)
>IPO
Therefore all space integrals are restricted to the domain where r < The expression for the current is obtained from its definition
R.
64
J. P. da Providencia, G. Holzwarth / Variarional approach
Here we encounter surface current
the difficulty
unless
We shall therefore
that with (23) the last term in (25) leads to a pure
we require
impose
(26) as a boundary
condition
4 UP’ Observing (26) in the evaluation of the remaining lagrangian I?’ (14) finally appears as
on the variational
parts
of (20) and
field
(21) the
+Mw+?~ + qJ4,A’+tkv#&d2 + (Q#L,m#qJ,)) (27) Equations of motion and boundary conditions at r = R are now obtained from lagrangian variations of (27) with respect to x, s, and 4,s. Free variation of x inside and at the boundary r = R leads to the continuity equation
and the boundary
condition
[&(Po&+jJl,=, where the expression
= 0,
(29)
(25) for the current
ja =
E9 (8 x+h$(9dbsa+aB4 m a
Pa
1)’
(30)
and the relation 1 -_= m have been used. Free variation
of s inside
$(l+&)
and at the boundary
(31)
r = R leads to the “Euler’‘-type
J. P. da Providencia, G. Holzwarth 1 Variational approach
equation
65
(1) (32)
and the boundary
condition
E&&JLR = 03
(33)
with the vector functional
and the pressure
tensor
The results up to now are identical with corresponding equations in the simplified version of the generalized scaling approach [cf. eqs. (86), (89) in ref. “)I. Relation (2) between S and j, however, is not obtained here. Instead, it is replaced now by a much more complicated equation derived from the c$,~ variation R 6,,LP m
=
w
&,d3x
s0
-
&#@apdb
= 0.
(36)
Y! surface
The variation leading
in the interior
region
is unconstrained
therefore
Ems must
vanish,
to
WO$
+j&+
J&PO&
+.i,)
+
(37) The variation on the surface is restricted by (26). Let us therefore I = R in a form which automatically satisfies (26),
and then
allow
for arbitrary
variations
write
c$,@at
c!I$.~ at r = R in (36). This leads to the
66
boundary
J. P. da Providencia, G. Holzwarrh 1 Variational approach
condition &
must vanish
that the tensor 1 1 1 = rDla - - XaX,r,B - -r2 x Px YI- w + - x,xpx,x,l-,, rz r4
at the surface
[r,,]*= R
=
0.
(40)
In the system of basis vectors x, e”‘, eC2), with e(l) and eC2) orthogonal tensor raa has only transverse components because according to (39)
to x, the
Therefore the tensorial boundary condition (40) obtained from the restricted riation of #,@ at r = R leads only to two equations to be satisfied at r = R [e~)r,Be~J]r =
R
=
0
(4 j = 1, 21,
va-
(41)
or, explicitly, [e$)(x,d,~,D + x,(d,+,, + a,~,,))ebj)-~xs(3ay~a~+
~~~~~~~~~~~ e(j)] = 0.
(42)
r=R
This completes the set of field equations (28), (32), (37) and boundary conditions (26), (29), (33) and (42) as derived from the lagrangian (27). It is interesting to note that the function x has been completely absorbed by the current and no longer appears explicitly. Much more interesting is to notice that scaling field s and tensor field $,s may be readily eliminated from the equations of motion so that we can immediately find the general form of the current. We shall see, however, that the same does not hold in the boundary conditions. Let us first contract (37) with ~3, and then take the curl of the resulting vector. This leads to d(p,(V Together with (28) this relation and we obtain wave equations current
x S)+(l
+$m/m*)(V
= ctd(V.j),
(V x jj = c-fd(V x j), sound
= 0.
(43)
is sufficient to eliminate the s-field from (32), (34) for the longitudinal and transverse parts of the
(V.j)
with the longitudinal
xj))
(44a) (44b)
velocity
Wa)
J. P. da Providencia, G. Holzwarth / Variational approach
67
and the transverse velocity cz T
l p:
5 mm* (
l+Q.
>
Wb)
Evidently (45a) agrees with the simple generalized scaling result while the transverse sound speed (45b) differs from the scaling result by the term 8m/7m* in the brackets [cf. eqs. (74) and (77) in ref. “)I. In fact, (45b) coincides with the transverse sound velocity obtained in the I 2 3 truncation scheme of Landau’s equation [cf. eq. (54) in ref. “)I. Comparison with eq. (63b) of ref. 6, shows that with inclusion of 4 aB in (9) we have allowed for 1 = 3 distortions of the local Fermi sphere. This extension remains, however, ineffective for the longitudinal components as long as the third-rank tensor q,sv in eq. (63a) of ref. 6, is excluded: because eq. (63a) of ref.(j) with ‘pap, 5 0 implies the relation voO= ~~~$3 which in connection with the continuity equation eq. (34a) of ref. 6, cuts off eq. (34~) of ref. 6, as indicated for 1 5 2. Therefore the present scheme represents a genuine extension only for the transverse components. For these the sizable modification is already evident from comparing (43) with (2); it should be noted, however, that (43) relates the transverse parts of S and j only up to terms which satisfy the Laplace equation and it is just this freedom which is fixed by the additional boundary conditions, In the boundary conditions it is not possible to eliminate the fields s and $,s. Thus for a complete solution we have to explicitly construct s and 4ola. This is in marked contrast to the simple version of nuclear fluid dynamics where in (32) j is simply replaced by -p,S according to (2) so that the boundary condition (33) is sufficient to determine the eigenmodes completely. General non-singular solutions of (44) are
V .j = C alJi,(kr)X’,,, lm
with spherical Bessel functions jl, spherical harmonics &, and &,,, and constants a, I(m = 0) we obtain for the electric (E) modes
b, c, d. Separating different parities for a given multipolarity
V xi a j,(k,r)Y,,O, i.e. j, =
Wl(kr) + ylrf)Y,,+ P(Vxjl(kTr)Y,d,
and for the purely transverse magnetic (M) modes V-j = 0,
Vxj = c~l+l(kTr)k;l+,o+d~l-,(kTr)k;l-,o~
(46)
68
J. P. da Providencia, G. Holzwarth / Variational approach
i.e. j,
=
jl(kTr)Y,lo.
(47)
Obviously, in the finite system the E-current may contain an additional r’ term in its longitudinal part, while for the M-modes such an extension is not possible. In the following two sections we shall discuss the two types of modes separately.
3. Magnetic modes For transverse
with traceless
modes
(47) the expression
4,s. Therefore
we obtain
(30) for the current
from (37) (by contracting
reduces
to
with a,)
(49)
while the continuity
equation
(28) requires v.s=o.
This is sufficient r’k;,,
to determine
term in addition
(50)
the form of the scaling
to the part which is proportional
iop,s =
field which may contain to the current
((1 +~~)jl(k,r)+cr’)~lo.
(We have assumed a harmonic time dependence condition (29) is trivially fulfilled. The boundary the constant c = c(k,R):
an
(47)
(51)
e -ior.) With (51) the boundary condition (33) relates k,R with
or (52)
J. P. da Providencia, G. Holzwarth 1 Variational approach
Evidently,
for 1 = 1 k, is determined
by the boundary
69
condition (53)
in accordance
with the earlier
result
[eq. (92) in ref. ‘)I. This must be so because
(53) is a consequence of angular momentum Thus for 1 = 1 the current is fixed, independent
conservation as discussed in ref. “). of the value of the constant c in the
scaling field. In the following we shall exclude the case 1 = 1. In order to use the boundary condition (42) we have to construct +,s from (37) which for transverse modes reads
the tensor
field
or with (51)
(54)
The general symmetric, traceless solution may be written in the form parity ( - )I+ ’
of (54) for multipolarity
I(m = 0) and
with ql(r)
= a,j,(k,r)+a,r’+a,r’+‘,
(p2(r) = ad+.
The four constants condition
a, are determined
by (54), (48) with
(47) and
the boundary
(26) as
5m 1 a, = -21 PoPFk,’
a
2-
--
2(21+ 3)a, + (4 + I)a, = 0,
where we have used (52) to obtain
a4=
7m* c(k,R) --p, POP: 1+2
this form of a2. The boundary
(56) condition
(42) for
J. P. da Providencia,
70
G. Holzwarth
/ Variational
approach
transverse modes simplifies to
Let us take the two vectors e”’ in the form &)
=
q(i) x
x,
i = 1, 2,
(58)
with fixed vectors 9”‘. Then (59) acts only on angular variables. Therefore e(“. V(X+#J~~)contains the same radial function as x+$,~ which, however, must vanish at r = R because of (26). Thus (57) reduces to e$)((d,
-
2)~#~,,)f#, = 03
or with (55) [(e@). V)(rd,-
(i -j)],,,
3)cp,(r)(k;,,),)e~'+
= 0.
Again the operator e”). V does not affect the radial dependence, therefore we finally obtain q,(R)
= 0
to fix the transverse wavenumber k,. With the use of (56) and (52) this is easily brought into the form (with x E kTR)
A+ l(xYx
1 21+1 (1 -x8X)jl(Xr = -%(Z+2)(21+3)(45+7F1)’ Before we relation like modes: The on s, so that
proceed to a discussion of (61) it is very instructive to notice that a (61) is indeed necessary to ensure the orthogonality of the resulting proof relies on the functional (34) being a hermitian operator acting for two solutions of (32) (distinguished by indices 1 and 2) we have -io,
s2. s
or
(61).
jld3x = L s2. Y[s,]d3x ms
= - io,
. s
s,‘12d3X, (32
s1
.j2d3x,
s
(62)
71
J. P. da Providencia, G. Holzwarth / Variational approach
In order to have orthogonality
s
s, . jpd3x
a 6,,
if
eJ: z wt,
(63)
we must require that the integrals in (62) be symmetric in the indices 1 and 2:
(64) For magnetic modes we insert current and scaling field (51) into (64) and obtain
s R
c2
s R
2dr = ci
j,(kp)r)r’+
0
j,(k$%)r’+ ‘dr,
0
or, with ci = c(kyR) from (52), we conclude that the ratio
s R
j,(kr)r’+2dr/c(kR)
0
must be independent of k. This, however, is just the 1.h.s. of (61) if we remember the property of Bessel functions
s R
j,(kr)r’+‘dr = R’+3 &.A+ ,(kR).
0
Thus we may be sure that the boundary condition (61) guarantees orthogonality (63) for modes belonging to different eigenvalues. For the most interesting magnetic mode, the 1 = 2 twist mode, the 1.h.s. of (61) is plotted in fig. 1. The poles, indicated by vertical lines, evidently coincide with the x-values given by the boundary condition in the simplified scaling model (1 - xa,)jr(x) = 0
(65)
[cf. eq. (92) in ref. “)I. For I = 2 and F, = 0 the r.h.s. of (61) is equal to -$$, indicated by the horizontal line in fig. 1. Obviously the solutions of (61) coincide very closely with the solutions of (65) except for the lowest one. Therefore the frequencies o = c,k, for the higher solutions fully reflect the increase of the transverse sound speed (45b), while the wavenumbers are essentially unchanged. The lowest solution, however, which is the physically important one, because it carries almost all the M2 strength, is strongly shifted to smaller values of k,, namely (for F, = 0) to k,R = 1.712
(66)
J. P. da Providencia, G. Holzwarth / Variational approach
X=kTR
Fig. I. This figure illustrates equation (61) for I = 2. The plotted curve is the left-hand side of (61) (j3(x)/x)/(I -x?Jj2(x). Vertical lines denote the zeros of the denominator. The full horizontal line at -0.2679 gives the value of the right-hand side for I = 2 and F, = 0. The lowest intersection occurs at ,Y = 1.712 while the lowest pole lies at x = 2.501.
as compared to the lowest solution of the scaling model condition (65) which lies at (k,R),, = 2.501. Multiplying these wavenumbers by the corresponding sound velocities
we obtain
(for F, = 0)
k, i&o&
1.7124? = 2301,,4
1002 ’
’
We therefore arrive at the remarkable result that the frequency of the collective magnetic mode remains practically unchanged when one goes from the simple generalized scaling model to this variational formulation including the tensor field + or, in other words, when one includes also octupole deformation of the Fermi 4 surface. In fact, the correspondence goes still deeper: We have plotted in fig. 2 the resulting radial dependence for the transverse current and the scaling field iop,s from (51). Obviously the term $m/m* in front of the Bessel function is essentially cancelled by the cr’ term so as to approximately restore the relation (2) zop,s
XJ.
(67)
It is clear that such a cancellation can only take place in a finite system, and only for the lowest mode. For higher k, values the nodes of the Bessel function lie inside the nucleus and the r’ term obviously cannot cancel a numerical factor in
J. P. da Providencia, G. Holzwarth / Variational approach
4
2
I” 2-
Fig. 2. The radial
functions
hw,=8,l77MeV
6
73
t R
fm
A=206
of the vector fields j and - p,S according 2- state (arbitrary units).
to UQ. (51) for the first excited
front of the Bessel function. This confirms our earlier suggestion 6, that it may be the finiteness of the system which renders the higher distortions of the Fermi surface unimportant when we consider truncation schemes of the Landau-Vlasov equation. 4. Electric modes The general form of the scaling field s for electric modes is determined by (28) and (43). With (46) we obtain for j and s j =
Vtil(k,r)+ylr’)Y,o+BV xjl(kTr)k;l~v +qm/m*)V x (~j[i,(k,r)+6,rf’2)~lo.
iop,s = V(j,(k,r)+6,r’)I;,+(l Longitudinal
(68)
and transverse wavenumbers are related through c,k,
= c,k,
= CO.
(69)
Thus in (68) there are 4 unknown constants yr, /?, 6,, 6, and one wavenumber, k, say, to be determined. The “Euler” equation (32), (34) which is 02j = (&
-&)V(V.
iop,s)-
&
d(i0p,s)
(70)
14
J. P. da Providencia, G. Holzwarth / Variational approach
links the r’ term in j with the r’+’ term in s: k+r,J The boundary
condition
(29) connects
(yi -S,)$R’-im((1 The boundary vector
condition
= - iS,m2(21+
3).
(71)
all four constants:
+$r/m*)62R’+2 for the pressure
+~m/m*fij,(k,R))
= 0.
(33) leads to two equations
because
(72) in the
XJ,, = (P+(r)Y,I+,,+P-(r)I;I-l,)p both radial
functions
P*(r)
must vanish
P+(R) = 0,
at r = R, P_(R)
= 0.
(7% b)
As in the case of the magnetic modes the four equations (71) (72) (73a, b) are sufficient to determine the constants yi, /I, 6,, 6, as functions of k,R, the wavenumber itself, however, remains undetermined. We then have to construct 4,s from (37) and impose the remaining boundary conditions (26) and (42). Inspection shows that (37) remains unchanged under a transformation
where f is an arbitrary function. The same holds for the boundary However, (26) is not invariant under (74) but goes over into
~~A&R Because f is arbitrary
this means
condition
= &lfWLR.
(42).
(75)
that the only useful part of (26) is (76)
with e”’ from (58) while the component
of x,4,@ parallel
We shall therefore does not impose any restriction on c#J,@. then need not take care of the freedom (74) any further.
to x,
replace
(26) by (76) and
J. P. da Providencia, G. Holzwarth 1 Variational approach
A special solution of (37) for j and s from (68) is +$J = a,F,+a,F
a, #+ 2
F=
-3Y1 -~I)-------
2(21+ 3)
&I
+
(77)
We therefore write
where $hog)is the general solution of the homogeneous equation
+(a,a,gy + asay4$ + dg? -+(a,a,@
+ 6,,a,(+a,4gj
+ a,rb:o,')) = 0.
(79)
The most general form of 4:;) is
4:oB) = (x,xsw(~) + 6,8~(~) + (Xmas+x,a,pw + a,a,zw, where, however, X remains undetermined and may be omitted (cf. (74)). A severe limitation on the actual form of W and Y is imposed on @) by the connection of 4 aS with the current j through (30): With the abbreviation
(30) requires with (46) and (77)
(83) From (80) we have for u u = r(4+ra,)w+(V(3+ra,)+xd)Y+VdZ, and therefore (83) requires W(x) =
w,r’Y,,,
Y(x) =
(yo++y2++2)Ylo,
.I. P. da Providencia, G. Holzwarth / Variational approach
76
with constants w,,, y, connected to ijz through (/+4)w,+2(21+3)y,
= ,s ”
(85)
while y, remains undetermined. Inserting (80) with (84) into (79) restricts Z to the form Z(x)
with z0 undetermined
=
(z,~‘+z,v’+2+Z4r~+4f~0,
(86)
and 3 independent equations relating W, Y and 2: (22 + 31)w, + 16(21+ 3)y, = 0, -w,+(21
+l)yz + 18(2/+5)z,
(19+l)y,+9(21+3)z,
= 0,
= 0.
(87)
So far we have obtained 4 equations (85, 87) for the altogether 6 constants wO,y,,, p2, zO, z2, z4 characterizing the homogeneous solution (80). Finally $a@= ~~~)+~~~~ must be subject to three boundary conditions, namely (76}, and (42) (for i = j, and i # j). These three relations fix the remaining (twofold) freedom in (p$ and termine wavenumber Their form a lengthy will be here their does allow a transparent as the case But in present the conditions (73), and ensure orthogonality of belongto eigenvalues which however, most checked cally the discussed the section.
5. Results Numerical results presented in this section are obtained with a potential energy functional of the form (88)
C aoPd = azp2 + a2 +*P’ +*, d with a z- -
and F, = 0.
-2 x 3075.8 MeV . fm3,
a2+* =&x20216.4
MeV+fm3+i
J. P. da Providencia, G. Holzwarth / Variational approach
Magnetic modes. These are, of course, not affected by the choice F, dependence is easily obtained from (61).
71
(88) while the
(i) I” = I+ : As we have remarked already it is a consequence of (52) that the wavenumbers k, for the unnatural parity dipole modes in the present variational formalism coincide with the corresponding wavenumbers in the simple formulation (1) (2) of nuclear fluid dynamics (NFD). Therefore the change in the transverse sound speed in the two different approaches as given by (45b) in comparison with eq. (77) in ref. 4), namely (for m = m*)
is fully reflected
in the frequencies
of the l+ modes
while the radial form of the current is unchanged. As was discussed previously in ref. 5), the boundary condition (53) leads to vanishing Ml strength for these convectional magnetic dipole modes. (ii) I” = 2- : We have seen in sect. 3 that the lowest M2 state which carries 98 % of the total M2 strength reproduces with remarkable accuracy the NFD result. At the same time it is clear from fig. 1 that for all the higher excited 2- modes the situation resembles the l+ case: The intersection of the horizontal line in fig. 1 with the function (jl+ ,/x)/(1 - xa,)j, occurs very close to the poles which describe the NFD solutions. From (52) then follows that c(k,R) is almost zero, which means that according to (51) current and scaling field differ by a factor of 9. Similarly, the frequencies of the higher modes differ from the corresponding NFD values by a factor of fi. Again, however, these higher modes carry very little M2 strength (cf. table 1). (iii) I” = 3+ : In the magnetic octupole mode we find even for the lowest excited state a significant difference between scaling and current field (see fig. 3), i.e. the factor y in front of the Bessel function in iop,s is only partly compensated by the cr3 term in (51). We find a factor of about 1.65 between iop,s and j, which is reflected also in the frequency of the lower 3+ state (which carries 95 yO of the total M3 strength) 03+
=
Jm
(03+)NFD.
All higher excited 3+ modes follow the same pattern as shown by the l+ and higher excited 2- states. Electric modes: Although the evaluation of the natural parity solution is slightly more involved than for magnetic modes the emerging pattern resembles the one we found above. We present results in terms of the unambiguous radial functions S+(I)
78
J. P. da Providencia, G. Holzwarth 1 Variational approach TABLE I Results
r:
ho
21 2;
8.16 23.28
3: 3:
12.61 27.56
1; 1; 1; 14
for the B(EI) and B(MI) values
B(I”) 3175 64 361 13
RI”)
ho 8.18 34.03
3800 38
x lo3 x IO3
16.16 40.26
285 8
x IO3 x IO3
0 10.98 21.46 25.02
0 9.2 x IO-’ 45.0x lo-3 15.3x 1o-3
0 11.75 25.64 35.85
2: 2: 2:
3.60 15.62 26.76
1690 114 0.1
8.60 15.82 29.59
3; 3;
12.74 20.78
70.2 x lo3 16.4 x IO3
13.13 22.89
76.9 x lo3 10.2 x lo3
4: 4:
16.28 26.02
3.1 x IO6 I .2 x IO6
17.26 29.33
3.7 x IO6 0.67 x lo6
0 46.5 x 1O-3 23.1 x 1O-3 1.6 x lO-3 1691 114 1.5
For the states listed in the first column energies (in MeV) and B(EI) or B(Mn values (in e2. fm*’ or & fm”-*) are given. The second and third columns show the results obtained from nuclear fluid dynamics as given by eqs. (I) and (2) the fourth and fifth columns show results of the present variational calculation. For the isoscalar dipole we replaced the excitation operator r’y1, by j,(qr) q,,,, with qR = 4.49.
I”= 3+ Fig. 3. The radial
hw,=16,l6MeV
functions
of j and
-p,,J
A=206 according
to eq. (51) for the first excited
3+ state.
J. P. da Providencia, G. Holzwarth / Variaiional approach
79
and j, (Y) defined by
Ax) = j+(r)%+ 1o W+j-(r)Y,I-lOW
Plb)
(The longitudinal and transverse radial functions of (68) are not without ambiguity because of the identity Vr’Y,, = i,/m V x r’Y,,,.) (i) 1” = 0’ : Monopole modes are purely longitudinal. In this case the NFD equations (l), (2) are a genuinely variational result, therefore the present approach coincides with NFD. (ii) I” = I-, 2+, 3-, 4’: The 1owest l- mode is the uniform translation which occurs at o = 0 in both formulations. In figs. 4a-c we present scaling and velocity fields for the lowest excited states with I” = 2+, 3-, 4+, which may be classified as surface modes because they show very little compression in the nuclear interior. It is most remarkable to which extent the NFD relation (2) is recovered for the quadrupole mode. In other words, for I = 2 the system does not take advantage of the additional freedom which is available in the present variational approach. Correspondingly, the lowest I = 2 frequency coincides with an accuracy of the order 10e3 with the NFD result. For I = 3 and 1 = 4 we do not find such perfect agreement. Although scaling and velocity field still show a rather similar pattern they no longer coincide. The resulting frequencies lie about 10 % above the NFD
Fig. 4a
J. P. da Providencia, G. Holzwarrh 1 Variational approach
80
/--,
/
,,-‘X\
‘\’ \
/ /I ;i’ 1’ // /
-&Be/
j-
w
6
l%3-
hq13,l
MeV
R
fm
A=208
Fig. 4b
/
/’
/’ // /
/’
--%A \\
\
/
‘6
\
W
‘R fm
“+.
1=4+
‘Y.\~ Tiw,=17,3MeV
A= 208
-P.f+ ‘t,_ \j+
‘\\ ‘\ Fig. 4c Fig. 4. The radial functions ofj and -p,,S according to eqs. (91) for the first excited 2+(a), 3-(b), states. The full lines show the density change in the nuclear interior given by - p,V 5.
4+(c)
J. P. da Providencia, G. Holzwarth / Variational approach
81
Fig. 5. Same as fig. 4 for the second excited 2+ state.
values. The same holds also for higher excited 1 = 2, 3, 4 states. As an example we show in fig. 5 the second excited (compressive) 2’ state, which obviously is still reasonably approximated by the NFD relation (2) while the frequency is reproduced within 1.3 %. Rather strong changes are observed for the higher excited (compressive) dipole states. Table 1 shows that NFD produces a low-lying rather weak state, with two strong states at 21.5 MeV and 25.0 MeV while in the present approach the low state is missing and the strength is contained in two modes lying below the NFD states. This again points to the fact that results of macroscopic approaches should be taken with great caution if they show strong fragmentation of the total strength. In such cases one should perhaps consider only energy-weighted moments as eventually
reliable
quantities.
6. Conclusion In the present article we have given a variational derivation of a fluid-dynamical formalism for finite Fermi systems which is based on a single determinant as a variational function and does not exclude the possibility of transverse flow from the outset. In order to obtain a genuinely variational approach it has been nec-
82
J. P. da Providencia,
essary to give an explicit density
matrix
lized scaling
G. Holzwarth
specification
of the time-odd
while for the time-even form used previously.
/ Variational
approach
part fi_ of the single-particle
part we have adhered
To allow for transverse
to the simple generaflow then requires
the
inclusion of a tensor field 4,s(x, t) as simplest possibility of going beyond the assumption of local x(x, t). This is in contrast to the previous simplified formulation of nuclear fluid dynamics in which the time-odd part p_ of the singleparticle density matrix was not specified at all except for the condition (2) which relates the first moment of p_ (namely the current) to the time-derivative of the scaling field s. Although at present we do not have a variational derivation of (2) it seemed quite natural to require that the time derivative of the displacement field s should be the velocity. The present variational approach, however, leads to a relation (43) between S and j which formally differs strongly from (2). But the numerical analysis has shown that for the lowest excited strongly collective electric and magnetic quadrupole states the simple relation (2) is numerically reestablished to a remarkable degree. Therefore previous conclusions about the occurence of transverse flow are supported by the present variational approach. For higher excited states and higher angular momentum (2) is no more valid but we still observe a definite tendency to reestablish relation (2). It is not clear at present whether deviations from (2) in this variational formulation might not have their origin in the very limited generalized scaling form of the time-even part of the density matrix. It might well be that extension of (8) to include the next order term 4,srp,psp, might reestablish (2) also for the octupole vibration. Such an extension seems desirable also from another point of view: Fluid dynamics as based on generalized scaling (3) seems unable to reproduce lowlying quadrupole and octupole states which we would expect in a microscopic theory as due to AN = 0 (for 2+) and AN = 1 (for 33) excitations (N: principal quantum number). It has been shown long ago by Stringari 2, in a harmonic oscillator model that with the above extension one can in fact obtain these lowlying states. This might be useful not so much for the description of low-lying states (which generally depend sensitively on the precise shell structure of the individual nucleus considered) but instead to eliminate admixtures of low-lying states from the giant states. Such admixtures might be due to the restricted form of P (8) in the generalized scaling approach. But within the limitations of generalized scaling (3) we may conclude that the variational approach supports the essential results as they were obtained by simply equating the time derivative of the scaling field with the velocity. Specifically we recover the occurence of transverse flow in both magnetic and electric modes. Most remarkably the explicit inclusion of octupole distortions in the local Fermi surface does not alter the results for a finite nucleus in a very significant way although the transverse sound speed is changed by a factor of 77. A similar compensation of changes in wavenumber and sound velocity has been discussed in ref. 11) for the comparison between first and zero sound in the case of the breathing mode.
J. P. da Providencia, G. Holzwarth / Variational approach
83
The authors would like to express their gratitude to Professor J. da Providencia for his interest in this work and many helpful discussions.
References 1) G. F. Bertsch, Ann. of Phys. 86 (1974) 138; Nucl. Phys. A249 (1975) 253 2) S. Stringari, Nucl. Phys. A279 (1977) 454; A325 (1979) 199 3) H. Sagawa and G. Holzwarth, Prog. Theor. Phys. 59 (1978) 1213; G. Holzwarth and G. Eckart, Z. Phys. A284 (1978) 291 4) G. Eckart, G. Holzwarth and J. P. da Providencia, Nucl. Phys. A364 (1981) 1 5) G. Holzwarth and G. Eckart, Nucl. Phys. A325 (1979) 1 6) T. Yukawa and G. Holzwarth, Nucl. Phys. A364 (1981) 29 7) J. Winter, LMU Mtinchen preprint 1980, submitted to Ann. of Phys.; and Proc workshop on semiclassical methods, Grenoble 1981 (ILL report Grenoble 1981, no. 15) 8) J. R. Nix and A. J. Sierk, Phys. Rev. C21 (1980) 396 9) C. Y. Wong and N. Azziz, Phys. Rev. C24 (1981) 2290 10) F. E. Serr, G. F. Bertsch and J. Borysowicz, Phys. Lett. 92B (1980) 241 11) B. K. Jennings and A. D. Jackson, Phys. Reports 66 (1980) 141