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On the width of the giant dipole resonance in deformed nuclei
Nuclear Physics A531 (1991) 27-38 North-Holland
N THE WIDT
F
E
A
I NUCLEI
RESONANCE I
B. BUSH and Y. ALHASSID' Centerfor Theoretical Physics, Sloane Physics Laboratory and A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, CT 06511, USA Received 6 August 1990 Abstract: Applying surface dissipation models to the Goldhaber-Teller model, we calculate the dependence of the giant dipole resonance (GDR) width on the nuclear quadrupole deformation. When expressed in units of the spherical width, this width reduces to a purely geometrical elliptic integral. It is shown to be very well approximated by the empirical power law with an exponent of 1.6. This approach utilizes no free parameters and reproduces the experimentally observed width dependence for GDR's built on the ground state of heavy nuclei. The formula derived here plays an important role in a recently developed macroscopic approach to the GDR in hot rotating nuclei.
1. Introduction Recently there has been an extensive experimental study of the properties ofgiant dipole resonances (GDR) built on highly excited nuclei (hot nuclei) produced in heavy-ion reactions 1-3 ). The coupling of the dipole degrees of freedom to the quadrupole deformation degrees of freedom allows us to use the GDR as a probe of nuclear shapes at finite temperature . For the ground state GDR, it is well known that for a deformed nucleus the resonance splits into several components and the frequency of each component is inversely proportional to the length of the semi-axis along which the vibration occurs') . In refs. 6'7) we have developed a macroscopic approach to the GDR in hot rotating nuclei in which the above observation at T=O still holds at finite T. The major new ingredient is the addition of thermal fluctuations at finite temperature which causes broadening of the GDR. However, to reproduce the experimentally observed spectral shapes of the resonance (including at T = 0) it is still necessary to assume that for a given deformation, each component is not sharp but has a certain intrinsic width which depends on the deformation . In our theory 7) we have used the phenomenological expression s) for the width of the jth component T = To(E j .il
E0)s,
(1 .1)
where E,, and ï'. are the energy and width of a_ resonance built on a snherical nucleus. S is a certain exponent and E, is the energy of the vibration along the jth ' A.P. Sloane Fellow. 0375-9474/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
28
B. Bush, Y Alhassid / GDR width
semi-axis which is assumed to be inversely proportional to the length Rj of that semi-axis : Eo exp [--15/4irB
cos (y-~irj)] . 3
(1 .2)
In eq. (1 .2) Ro and E0 are the radius and GDR energy, respectively, of a spherical nucleus. We have also used the Hill-Wheeler parameter P, y of ellipsoidal shapes. he dependence of the width r on deformation P, y then follows from the combination of eqs. (1 .1) and (1 .2) . fills (1 .2) has theoretical justifications, such as hydrodynamical models "), relation (1 .1) have been obtained phenomenologically by studying the ground state G width systematics in heavy nuclei ") . The purpose of this paper is to derive (1 .1) for a nucleus at finite temperature and at a given deformation, using a simple theoretical model. Our derivation is based on the wall formula [see eq. (2.1) below] which was derived in the one-body dissipation theory 9") . This theory is based on the assumption that the nucleon mean free path is long compared to the size of the nucleus and the nucleon is therefore more likely to collide with the boundary of the nucleus than with another nucleon . The model was derived both classically ") and quantum mechanically'`') . It also follows from the Fermi gas solution of the oltzmann transport equation') or as a limit of RPA damping '4). We find that the width rj, when expressed in units of To, can be recast in a purely geometrical expression. This expression is then found to be very well approximated by the power law (1 .1) with an exponent 3 = 1 .6. e original one-body dissipation wall formula, which is characterized by k, = I (in eq (2.1) below) is known to overestimate ") the observed spherical width F0 of the resonance . This is mostly due to several simplifications and idealizations inherent to the model: the model does not take into account the reduction in the nucleon velocity at the surface and the fact that the collisions are not completely random and may retain some memory 15-17) . However, the geometrical structure of this formula is valid under the more general assumption that dissipation is local and restricted to the vicinity of the surface . By expanding 's) the rate of loss of collective energy to leading order in the surface diffuseness (divided by the nuclear radius) one obtains a dissipation formula whose functional form is similar to the wall formula but with a modified coefficient k, [see eq. (2.1) below] . As discussed in refs. 15,16) , a one-body dissipation mechanism with more realistic features can strongly reduce the value of k . An RPA calculation 14,19) indicates that this reduction can lead to a dissipation which is as low as 10% of the original wall formula value. It was further suggested 15,17) that the contribution from two-body collisions will have the same geometrical structure as the wall formula as long as these collisions are confined to the vicinity of the surface where they are expected to be most iii ilac if-acliof di'~. "C aMCM-nUdC0n CURMUS are to the Pauh principle, they become more probable at the surface ") . Also the free nucleon-nucleon cross section is larger near the surface where the kinetic energy
B. Bush, Y. Alhassid / GDR width
29
is smaller. Since our derivation of the power law (1 .1) depends only on the geometrical structure of the wall formula, it remains valid in the more realistic approaches to dissipation described above. It is important to note that the power law (1.1) as a phenomenological law has been verified only for prolate nuclei') (for which ground state GDR's were measured). Since in our applications at finite temperature 7), triaxial shapes play a crucial role, the derivation of (1 .1) for a general triaxial shape as presented in this paper is of great importance. . Analysis of ground-state data gives 8,20) S --1 .9-2.2 in heavy nuclei (A :::- 100). However, this is based on the classical splitting (1.2) and ignores quantummechanical effects . This makes the exponent seem higher. Using quantum models to describe the quadrupole collective degrees of freedom and their coupling `c the dipole, we find additional splitting of the GDR 7). If we then broaden each of these components by the rule (1.1) we find 7) that S -1.6. The value for S that we find from the wall formula is â =1 .6, in excellent agreement with the above phenomenological value. This provides an additional support for using the value of S =1 .6 in the theory 6,') of the GDR built on a hot rotating nucleus rather than using 20) values of S == 1 .9-2.2. 2. Dipole width in surface dissipation models The purpose of this section is to use the geometrical properties which are common to surface dissipation models (described in sect. 1), in order to derive a geometrical expression for the GDR width in a deformed nucleus (in units of the spherical width T0) . 2.1 . SURFACE DISSIPATION
The "wall" formula, derived in the one-body dissipation model 9-"), provides an expression for the classical dissipation function
I J-2É=pv -1 k,
dXy2 .
(2 .1)
Here 9 is the local normal component of the nuclear surface velocity relative to the nuclear gas inside it, dX is a surface element and the integration extends over the whole nuclear boundary, v is the average nucleon velocity (in the rest frame of the gas) and p is the particle mass density of the Fermi gas of nucleons . The constant ks is unity for the idealized wall formula 9,10) . As we have remarked in sect. 1, eq. (2 .1), with a general coefficient k, whose value is usuaiiy less than unity, is valid for a more general class of surface dissipation models. This is seen by expanding the rate of change of the collective energy to leading order in the surface diffuseness, under the assumption that dissipation take
B. Bush, Y Alhassid / GDR width
30
place in a thin region surrounding the nuclear surface `5). Such an expansion follows closely that of ref. "), which is valid irrespective of the precise microscopic mechanism . In ref.") it was suggested that such models can even include a contribution from two-body collisions near the surface. Our results below will be independent of k, since the dependence on ks will be absorbed in F0 . 'o apply the wall formula (2.1) to the GDR, we describe the latter by the oldhaber-Teller model 1121) in which the neutron cloud oscillates with respect to the proton cloud. ne nucleons are enclosed in a deformed stationary surface. The shape of the surface follows the shape of the nuclear mean-field and is similar for the protons and neutrons . Denoting by Rp and Rn the center of mass of the protons and neutrons, respectively, we have from eq. (2.1), when applied to tie two gases:
s ppej,
)`'+2-k,
d-E (
1
,_,p,,0,,
I
dX
fi)2
.
(2 .2)
he relative velocity of the boundary with respect to the proton and neutron gases are given by -Ap and -A, respectively . In (2.2) h is a unit vector normal to the surface and the subscripts p, n refer to protons and neutrons, respectively . enoting the relative displacement by and assuming the center of mass of the whole nucleus is stationary we find
(23) so that 9 = -1k s "02A
ZN
A
No' + ZIU" A 2 A
-1 dX
(4~ . fi) 2 .
(2A)
ere ~ is a unit vector along the displacement R and po is the spherical mass density R-1 p01 A = m/47r 3 0 with m the nucleon mass. The collective kinetic energy in the Goldhaber-Teller model is given by jt 2 +!Nmjt2n= '(NZIA) Mk2 . T= jZ 2 M p 2
(15)
Choosing R along the jth semi-axis of a deformed nucleus, the equation of motion for Ri is 22) d ( aT
dt aÉj
+
aV a.~ ~ -r . aRj t9R)
(2.6)
where V is the potential energy for the GDR vibration. From eqs. (2.4) and (2.5) we have ZN A
j + mk, ZN(Nl5p+Zbn)Éj+ aV =0 . A A aR i
(23)
B. Bush, Y. Alhassid / GDR width
31
Assuming a quadratic potential we recover from eq. (2.7) the equation of a damped harmonic oscillator with damping width I'; l'; = Fo( I n; d-T ) / 3ITR 2 ,
(2.8)
ZN Z-l Op + N-' O. A R0 .
(2.9)
where T0 = ks
Since, for a spherical surface f n; d2 = 37rRô, 0 is the spherical width . In eq. (2.8) n; is the component of the unit vector normal to the surface along the jth semi-axis. Eq . (2 .8) reduces the dependence of the width T,- on deformation to the evaluation of the purely geometric quantity f n; dl 2.2. SURFACE INTEGRAL FOR AN ELLIPSOID
We now turn to the evaluation of the surface integral in eq. (2.8) for an ellipsoid of semi-axes (RI , R2 , R3 ) and for convenience we choose j = z We define the dimensionless quantity I(R, , R2,
R3) -,
2 y~l/~-1
dl nz/3~R0 ,
(2.10
3 where the integration is over the surface of an ellipsoid with semi-axes R,, R2 , and where R0 is the radius of a spherical nucleus which has the same volume as the deformed one, i.e. R, R2R3
=Rô .
(2.11)
From its definition (2.10), I obeys the following scaling property I(AR, , AR2 , AR 3) = I (R, , R2 , R3 ) .
(2.12)
Choosing A = Ro l , we find that I in eq. (2.10) is given by I(a, b, c) = 3
41r X 22 2 +y2 l 6'+z 2 /c2 _1
where abc =1 ; a, b, c are given by the Hill-Wheeler parameters
R;/ R0
(2.l3)
(j =1, 2, 3), respectively, and in terms of ß
cos (y-3iT)] ,
5/4ir ß
cos (y+3~)] ,
a =exp [,15-5/4 7r b =exp [
dl nz ,
c = exp [ 5/47rß cos y] .
(2 .14)
Using the divergence theorem, we can convert (2.13) to a volume integral I=
3 (abc) _2 / 3 4 ~r
dV (z-
)(z -
n) .
(2.15)
3?
B. Bush, Y. Alhassid / GDR width A c)2 e unit vector normal to the ellipsoid (x/a )2 + (y/b)2 + (Z/ = l, is given by = s/ s where s = (x/a 2, yl b2, Z/ c2) . Thus I is given by 4 2 +Y 2/ 4 / (abc) dx dy d.,. 41rc` (x /a +y /b +Z /c )
f
(2.16)
x2/a`'+y2Jb2+Z`'/c`'-1 .
Changing the coordinate (x, y, Z) to (f, °q, C) where
e = (x1a )2;
we find
I = 3 (a -c)-2/3 b 47r
ac
+(a one
f
+ ~ +zC ,
e, n4
b) .
s shown in appendu I-
= (y/ )2;
-0
de d
C = (Z/ C
c)2
,
(2 .l7)
(el a2+ r1/b2+C/e2)3/2 (2.18)
, the triple integral in (2.18) can be reduced to a single
3 a2b` r' 2 (abc)2/3 o dt
I
t . b) t)+(a<-' (a2 + t)3(b2 + t)(c`'+
(2.19)
is easily checked that for a sphere a = b = c we obtain in (2.19) I =1. I is thus a purely geometrical factor that describes the width h for a deformed nucleus in units of the spherical width I'® . e integral in (2.19) is an elliptic integral. To transform it to a canonical form which we use the substitution t = a`'tg 2 gives =/2 3 a 2b2 sin2 Jcos jp I=(2 .20) (b2+a2 tg2 )1/2(C2+a2 tg2 ) , /2 +(aHb) . 2 (abc)2/3 J® Finally, substituting (1- a`/ C2)112 sin = sin 0, we obtain t
ac 1/2 a c b2 - a2 I = ( a c)`l3 (c`'-a2)3/2 D arccos +(a<->b), (~c , b -c2 -a2 where is given by 23) sin2 do . D(O, k) = foo (1- k2 sin2 ofi l'/2
(2 .21)
(2.22)
is a linear combination of the elliptic integrals of the first and second kind, E(, k) and E(0, k) (2 .23) (0, k) = CE(o, k) - E(0, k)]/k 2 . In eq. (2.22), we have assumed a , b c. Otherwise, should be replaced by corresponding analytic continuations.
ß. Bush, Y. Alhassid / GDR width y=_1301
y=-,,00
10
33
Y= - 140°
--150
rY
r,
ro
\
r°
\
.\
v
r,
o
Y
(Power Low) 01
-0 4 -0 .2
0 02
04
-0 .4 -0.2
0
0.2
0.4
13
-0.4 -02
0
02
04
-04 -02
0
02
04
Fig. 1. Tr /I~ (solid lint'), r, ./FO (long-dashed line) and 1 _/ro (short-dashed line) versus deformation /3 for several values of y (-120°, -130°, -140°, and -150°). Top panel : from the one-body dissipation width formula (2.24) and (2.19). Bottom panel: from the power law (l.l) and (1.2) using S =1.6.
From (2.8) and (2 .10) we have TZ
=
l'o I(a,
b, c),
(2.24a)
where I is given by (2.19) and a, b, c are functions of ß, y as in (2.14). T,r and r,, are obtained by a suitable permutation of a, b, c TX =ToI(b, c, a),
(2.24b)
Ty = ToI(c, a, b) .
(2.24c)
Eqs. (2.24) determine the exact dependence of I' on deformation in the one-body dissipation model. This dependence is shown in fig. 1 (top panel) where the widths T;/To (i = x, y, z) are shown as a function of the deformation ß, for several values of y (-120°, -130°, -140° and -150°). In sect. 3, we shall show that the power law (1 .1) is in good agreement with (2.24) for realistic values of ß. The dependence of T;/To on deformation which follows from the power law [using eq. (1.2)] is shown in the bottom panel of fig. 1 using 8 =1 .6. 3.
over Law
In this section we show that the power law (1 .1) is a good approximation to the width equations derived in sect. 2 .
B. Bush, Y. Alhassid / GDR width
34 3.1 . EXPONENT S
e shall first show that eq. (1 .1) which can be rewritten as (3 r;=ro( o/ ;)' ,
.1)
is a good approximation to (2.24) for small enough deformations ß, and derive the value of S. In subsect. 3.2 we shall verify how well eq. (3.1) describes eq. (2.27) globally for a realistic range of deformations . e take, without loss of generality, j = d an assume ß to be small enough so that the deviation from a sphere is small, i.e. b--ca=v,
c=1+u,
(3 .2)
were u, v 1 . Since abc =1, u and v determine the most general small ellipsoidal deviation from a sphere (including triaxial) . (3.1) reads, to first order in u, v,
Thus, to establish eq. (3 .1) we have to show that to first order in u and v, I in eq. (2.19) depends only on u and find the value of S. e expansion of (2.19) to first order in a'-, b', c' gives t 1,2 2 2 c`'-5 a +b 21 = C2 C -3 dt(,+t)7/2 2 4 ), fo where we have used I(1, 1, 1) =1. sing (3 .2), we find to first order in u and v I==
1-SU .
(3 .5)
is is just the required equation (3.3) with S=s=1 .6 .
(3.6)
us, we have established the power law (1 .1) for small deformations with an exponent of S =1 .6. 3.2 . NUMERICAL VERIFICATION OF THE POWER LAW
n subsect. 3.1 we showed that (1 .1) holds in the one-body dissipation model with S = 1 .6 for small deformation ß and any y. It turns out that (1 .1) is actually a good approximation over the whole range of realistic deformations . To show that, we have calculated I in (2.19) numerically for -0.4, ß , 0.4 and several y's. A least square fit of (1 .1) to the calculated I give exponents in the range S - 1.6006-1 .6044 with correlation coefficients better than 0.99 (see table 1 for details).
B. Bush, Y. Athassid / GDR width
35
TABLE 1
The slope, intercept and correlation ofa linear least squares fit to in (ri ll',) (calculated from eqs . (2.24) and (2.14)) versus In (E;/ Eo) (calculated from (1 .2)) for i = x, y, z For each value of y (-120°, -130°, -140° and -150°) the linear fit was made for ß in the range -0.4-_ (3 _ 0.4 Component
Slope (S)
Intercept
Correlation
y = -120°
x y z
1.6037 1.6007 1.6037
-0.0054 -0.0082 -0.0054
0.9993 0.9996 0.9993
y = -130"
x y z
1.6041 1.6008 1.6031
-0.0049 -0.0080 -0.0060
0.9988 0.9996 0.9995
y = -140°
x y z
1.6044 1.6012 1.6025
-0.0046 -0.0077 -0.0066
0.9960 0.9996 0.9996
y = -150°
x y z
1.6018 1.6018
-0.0072 -0.0072
0.9996 0.9996
1hig. 2_ Bottom : T;ll'o calculated from eq. (2.24) versus E;1 E, calculated from eq. (1 .2) on a log-log plot for i = x, y, z in the range -0.6 -_,O -_ 0.6 and for several values of y (-120°, -130°, -140° and -150°). We used solid, long-dashed and short-dashed lines for i = x, y and z, respectively. If the power law (1.1) holds, then In ( ;/lo) = 8 In (El E(,), i.e. l',/FO versus E;/EO is a straight line with slope 5 and zero intercept. Such lines with S = 1 .6 are shown by the dotted lines. Top : ß versus E,/ E (i = x, y, z) from eq. (1 .2) .
B. Bush, Y. AJhassid / GDR width
36
o show the quality of agreement between the ore-body dissipation width formula and the power law over a larger range of deformations (r.8i = 0.6), we plotted in ;/E0 Both scales are logarithmic, E ;/FO for eqs. (2 .24) versus . fig. 2 (bottom panel) I' so that an exact power law should give a straight line with slope 8 and zero intercept. (See dotted lines in fig. 2 for 8 =1 .6.) Notice that when -0.6, ß = 0.6, the corre;/E, [eq. (1 .2)] can be read from the top panel of fig. 2. We sponding ranges of E see that for large deformations there are deviations from a power law and that the latter always overestimate the geometrical surface dissipation widths. 3.3. WIDTH T.
e G lZ width Eo for a spherical nucleus is given, in the surface dissipation models by Z-' Ep `9- N- ' On
(3.7)
o
For a cold Fermi gas of nucleons v = ~ vF, where (2EF/
VF
is the Fermi velocity
(VF =
)''').
For the nuch~us Er where Z = 68 and N = 98, we find E ( P ) = 29.2 MeV, SF) _ 37.2 eV and taking o = rO A' /' with ro =1.2 fm we find To (T = 0) = 5.88 MeV for = (i.e. in the original one-body dissipation model). A phenomenological fit to the ground-state R cross section in "Er gives) To = 3 .64 MeV, so that we overestimate the observed width by about 2 eV, which is expected in the one-body dissipation theory "). However, as mentioned in sect. 1 a more realistic modeling of surface dissipation will result in a value for ks which is less than unity' s ) . Here a value of k,, = 0.62 will reproduce the experimental width exactly. .
o c usions
e have derived, using surface dissipation models, a purely geometrical expression for the GDR width (at a fixed deformation) in units of the spherical width. It was shown to follow closely a power law F.i = FO(E/EO) s with an exponent of 8 =1 .6. This result is proven for general triaxial shapes. The e-:ponent agrees well with values found phenomenologically for heavy nuclei (A :,:- 100) through a fit to ground-state GDR cross sections. The above power law with 8 =1 .6 plays a major role in a macroscopic theory of the GDR at finite temperature that we have recently developed. This work was supported in part by the US Department of Energy, Contract No. -AC02-76ER03074.
B. Bush, Y. Athassid / GDR width
37
Appendix A We show here how to reduce the triple integral (2.18) to the single one (2.19). The integral (2.18) is of the general form dx;
llixP' -'
Using the integral representation of the T function, we rewrite (A.1) as ao t r-1 S Y; s 1 x a0
i
_
0
ao
ao
ao
t
r-1
dt O 1-Y- x; e-y- qi`i` Il xiPB - ' dxi , (A.2) T(r) ~ where O is the step function; O (y) = 0 for y < 0 and ®(y) =1 for y > 0. Representing O(y) in terms of its Laplace transform 1 / s we have es l_Ex; :°° 1 (A.3) i)= 2 ds ( ) O(1-y_x s ,ri
10
I0,>
Exchanging the order of integration in (A.3), we can now do the x; integrals =
o
dt
ds es tr-1 T(Pi) . -i o 27ri s T(r) i (s+q;t)p
(A.4)
Substituting t -3- st in (A.4) and rotating the integration axis (s is purely imaginary) we obtain °~ ds Sr_1-EPids ~° dt t r - ' T(Pi) (A.5) . T(r) ; (1+gi t) Pi _ . 2ei o Using t°`-1 as the inverse Laplace transform of s`T(a), we can do the s integration to obtain j !li T(Pi) tr-1 . dt P. T(r)T(1+~ pi -r) o lli (1+qit) The integral in (2.18) is a special case of (A.1) where P1=2, q,=1la 2 ,
P2
= P3 == 2
de d q dS
e.1n4--o
= 2ira bc 3
0
q3
q2=1/b2,
We then find e+71+e--1
r=2,
2
( e / r%y ) '/2
(ela +rl/b
dt (a2+
=1ic 2 .
+~/c2)
(A.7)
31 2
t ll2 / t) 3 2
(b 2 + t) 112 (C
2
+ t)
112
"
(
A.8 )
38
B. Bush, Y Alhassid / GDR width eferenus
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 2 ,1 1, 22) 23)
J& GaardhA A.M. Bruce and B. Herskind, Nucl. Phys. A482 (1988) 121c P. Thirolf, D. Habs, D. Schwalm, R.D. Fisher and V. Metag, Nucl. Phys. A 2 (1988) 93c K.A. Snover, Nucl. Phys. A482 (1988) 13c A. Bohr and B.M. Mottelson, Nuclear structure, vol . 11 (Benjamin, Reading, 1975) ch. 6 J.M. Eisenberg and W. Greiner, Nuclear theory, vol. I (North-Holland, Amsterdam, 1975) chs. 10 and 11 Y. Alhassid, B. Bush and S. Levit, Phys. Rev. Lea . 61 (1988) 1926 Y. Alhassid and B. Bush, Nucl. Phys. A509 (1990) 461 M. Danos and W. Greiner, Phys. Lett. 8 (1964) 113 ; M.G. Huber, M. Danos, H.J. Weber and W. Greiner, Phys. Rev. 155 (1967) 1073 J. Blockt, Y. Boneh, J.R. Nix, J. Rundrup, M. Robel, A.J. Sierk and W.J. Swiatecki, Ann of Phys. 113 (1978) 330 D.H.E. Gross, Nucl. Phys. A240 (1975) 472 W.D. Myers, W.J. Swiatecki, T. Kotlama, L.J. El-Jaick and E.R. Hilf, Phys. Rev. C15 (1977) 2032 A.J. Sierk, S.E. Koonin and J.R. Nix, Phys. Rev . C17 (1978) 646 R.W. Hass, J. of Phys. G5 (1979) LIOI C. Yannouleas, Nucl. Phys. A439 (1985) 336 J.R. Nix and A.J. Sierk, Proc. Int. School Seminar on heavy ion physics, Dubna 1986, Joint Institute for Nuclear Research Report JINR-D7-87-68, p.453 (1987) ; J.R. Nix, Nuel. Phys. A502 (1989) 609c G.D. Adeev, I.I. Gonchar, V.V. Pashkevich, N.I . Pischasov and 0.1. Serdyuk, Sov. J. Part. Nucl. 19 (1988) 529 J.R. Nix, private communication (1990) W. Swiatecki, Prog. Part. Nucl. Phys. 4 (1980) 383 J.J. Griffin and M. Dworzecka, Nucl. Phys. A455 (1986) 61 M. Gallardo, F.J. Luis and R.A. Broglia, Phys. Lett. B141 (1987) 222 M. Goldhaber and E. Teller, Phys . Rev. C74 (1948) 1046 H. Goldstein, Classical mechanics (Addison-Wesley, New York, 1981) J. Mathews and R.L. Walker, Mathematical methods of physics (Benjamin, Reading, MA, 1970)
N THE WIDT
F
E
A
I NUCLEI
RESONANCE I
B. BUSH and Y. ALHASSID' Centerfor Theoretical Physics, Sloane Physics Laboratory and A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, CT 06511, USA Received 6 August 1990 Abstract: Applying surface dissipation models to the Goldhaber-Teller model, we calculate the dependence of the giant dipole resonance (GDR) width on the nuclear quadrupole deformation. When expressed in units of the spherical width, this width reduces to a purely geometrical elliptic integral. It is shown to be very well approximated by the empirical power law with an exponent of 1.6. This approach utilizes no free parameters and reproduces the experimentally observed width dependence for GDR's built on the ground state of heavy nuclei. The formula derived here plays an important role in a recently developed macroscopic approach to the GDR in hot rotating nuclei.
1. Introduction Recently there has been an extensive experimental study of the properties ofgiant dipole resonances (GDR) built on highly excited nuclei (hot nuclei) produced in heavy-ion reactions 1-3 ). The coupling of the dipole degrees of freedom to the quadrupole deformation degrees of freedom allows us to use the GDR as a probe of nuclear shapes at finite temperature . For the ground state GDR, it is well known that for a deformed nucleus the resonance splits into several components and the frequency of each component is inversely proportional to the length of the semi-axis along which the vibration occurs') . In refs. 6'7) we have developed a macroscopic approach to the GDR in hot rotating nuclei in which the above observation at T=O still holds at finite T. The major new ingredient is the addition of thermal fluctuations at finite temperature which causes broadening of the GDR. However, to reproduce the experimentally observed spectral shapes of the resonance (including at T = 0) it is still necessary to assume that for a given deformation, each component is not sharp but has a certain intrinsic width which depends on the deformation . In our theory 7) we have used the phenomenological expression s) for the width of the jth component T = To(E j .il
E0)s,
(1 .1)
where E,, and ï'. are the energy and width of a_ resonance built on a snherical nucleus. S is a certain exponent and E, is the energy of the vibration along the jth ' A.P. Sloane Fellow. 0375-9474/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
28
B. Bush, Y Alhassid / GDR width
semi-axis which is assumed to be inversely proportional to the length Rj of that semi-axis : Eo exp [--15/4irB
cos (y-~irj)] . 3
(1 .2)
In eq. (1 .2) Ro and E0 are the radius and GDR energy, respectively, of a spherical nucleus. We have also used the Hill-Wheeler parameter P, y of ellipsoidal shapes. he dependence of the width r on deformation P, y then follows from the combination of eqs. (1 .1) and (1 .2) . fills (1 .2) has theoretical justifications, such as hydrodynamical models "), relation (1 .1) have been obtained phenomenologically by studying the ground state G width systematics in heavy nuclei ") . The purpose of this paper is to derive (1 .1) for a nucleus at finite temperature and at a given deformation, using a simple theoretical model. Our derivation is based on the wall formula [see eq. (2.1) below] which was derived in the one-body dissipation theory 9") . This theory is based on the assumption that the nucleon mean free path is long compared to the size of the nucleus and the nucleon is therefore more likely to collide with the boundary of the nucleus than with another nucleon . The model was derived both classically ") and quantum mechanically'`') . It also follows from the Fermi gas solution of the oltzmann transport equation') or as a limit of RPA damping '4). We find that the width rj, when expressed in units of To, can be recast in a purely geometrical expression. This expression is then found to be very well approximated by the power law (1 .1) with an exponent 3 = 1 .6. e original one-body dissipation wall formula, which is characterized by k, = I (in eq (2.1) below) is known to overestimate ") the observed spherical width F0 of the resonance . This is mostly due to several simplifications and idealizations inherent to the model: the model does not take into account the reduction in the nucleon velocity at the surface and the fact that the collisions are not completely random and may retain some memory 15-17) . However, the geometrical structure of this formula is valid under the more general assumption that dissipation is local and restricted to the vicinity of the surface . By expanding 's) the rate of loss of collective energy to leading order in the surface diffuseness (divided by the nuclear radius) one obtains a dissipation formula whose functional form is similar to the wall formula but with a modified coefficient k, [see eq. (2.1) below] . As discussed in refs. 15,16) , a one-body dissipation mechanism with more realistic features can strongly reduce the value of k . An RPA calculation 14,19) indicates that this reduction can lead to a dissipation which is as low as 10% of the original wall formula value. It was further suggested 15,17) that the contribution from two-body collisions will have the same geometrical structure as the wall formula as long as these collisions are confined to the vicinity of the surface where they are expected to be most iii ilac if-acliof di'~. "C aMCM-nUdC0n CURMUS are to the Pauh principle, they become more probable at the surface ") . Also the free nucleon-nucleon cross section is larger near the surface where the kinetic energy
B. Bush, Y. Alhassid / GDR width
29
is smaller. Since our derivation of the power law (1 .1) depends only on the geometrical structure of the wall formula, it remains valid in the more realistic approaches to dissipation described above. It is important to note that the power law (1.1) as a phenomenological law has been verified only for prolate nuclei') (for which ground state GDR's were measured). Since in our applications at finite temperature 7), triaxial shapes play a crucial role, the derivation of (1 .1) for a general triaxial shape as presented in this paper is of great importance. . Analysis of ground-state data gives 8,20) S --1 .9-2.2 in heavy nuclei (A :::- 100). However, this is based on the classical splitting (1.2) and ignores quantummechanical effects . This makes the exponent seem higher. Using quantum models to describe the quadrupole collective degrees of freedom and their coupling `c the dipole, we find additional splitting of the GDR 7). If we then broaden each of these components by the rule (1.1) we find 7) that S -1.6. The value for S that we find from the wall formula is â =1 .6, in excellent agreement with the above phenomenological value. This provides an additional support for using the value of S =1 .6 in the theory 6,') of the GDR built on a hot rotating nucleus rather than using 20) values of S == 1 .9-2.2. 2. Dipole width in surface dissipation models The purpose of this section is to use the geometrical properties which are common to surface dissipation models (described in sect. 1), in order to derive a geometrical expression for the GDR width in a deformed nucleus (in units of the spherical width T0) . 2.1 . SURFACE DISSIPATION
The "wall" formula, derived in the one-body dissipation model 9-"), provides an expression for the classical dissipation function
I J-2É=pv -1 k,
dXy2 .
(2 .1)
Here 9 is the local normal component of the nuclear surface velocity relative to the nuclear gas inside it, dX is a surface element and the integration extends over the whole nuclear boundary, v is the average nucleon velocity (in the rest frame of the gas) and p is the particle mass density of the Fermi gas of nucleons . The constant ks is unity for the idealized wall formula 9,10) . As we have remarked in sect. 1, eq. (2 .1), with a general coefficient k, whose value is usuaiiy less than unity, is valid for a more general class of surface dissipation models. This is seen by expanding the rate of change of the collective energy to leading order in the surface diffuseness, under the assumption that dissipation take
B. Bush, Y Alhassid / GDR width
30
place in a thin region surrounding the nuclear surface `5). Such an expansion follows closely that of ref. "), which is valid irrespective of the precise microscopic mechanism . In ref.") it was suggested that such models can even include a contribution from two-body collisions near the surface. Our results below will be independent of k, since the dependence on ks will be absorbed in F0 . 'o apply the wall formula (2.1) to the GDR, we describe the latter by the oldhaber-Teller model 1121) in which the neutron cloud oscillates with respect to the proton cloud. ne nucleons are enclosed in a deformed stationary surface. The shape of the surface follows the shape of the nuclear mean-field and is similar for the protons and neutrons . Denoting by Rp and Rn the center of mass of the protons and neutrons, respectively, we have from eq. (2.1), when applied to tie two gases:
s ppej,
)`'+2-k,
d-E (
1
,_,p,,0,,
I
dX
fi)2
.
(2 .2)
he relative velocity of the boundary with respect to the proton and neutron gases are given by -Ap and -A, respectively . In (2.2) h is a unit vector normal to the surface and the subscripts p, n refer to protons and neutrons, respectively . enoting the relative displacement by and assuming the center of mass of the whole nucleus is stationary we find
(23) so that 9 = -1k s "02A
ZN
A
No' + ZIU" A 2 A
-1 dX
(4~ . fi) 2 .
(2A)
ere ~ is a unit vector along the displacement R and po is the spherical mass density R-1 p01 A = m/47r 3 0 with m the nucleon mass. The collective kinetic energy in the Goldhaber-Teller model is given by jt 2 +!Nmjt2n= '(NZIA) Mk2 . T= jZ 2 M p 2
(15)
Choosing R along the jth semi-axis of a deformed nucleus, the equation of motion for Ri is 22) d ( aT
dt aÉj
+
aV a.~ ~ -r . aRj t9R)
(2.6)
where V is the potential energy for the GDR vibration. From eqs. (2.4) and (2.5) we have ZN A
j + mk, ZN(Nl5p+Zbn)Éj+ aV =0 . A A aR i
(23)
B. Bush, Y. Alhassid / GDR width
31
Assuming a quadratic potential we recover from eq. (2.7) the equation of a damped harmonic oscillator with damping width I'; l'; = Fo( I n; d-T ) / 3ITR 2 ,
(2.8)
ZN Z-l Op + N-' O. A R0 .
(2.9)
where T0 = ks
Since, for a spherical surface f n; d2 = 37rRô, 0 is the spherical width . In eq. (2.8) n; is the component of the unit vector normal to the surface along the jth semi-axis. Eq . (2 .8) reduces the dependence of the width T,- on deformation to the evaluation of the purely geometric quantity f n; dl 2.2. SURFACE INTEGRAL FOR AN ELLIPSOID
We now turn to the evaluation of the surface integral in eq. (2.8) for an ellipsoid of semi-axes (RI , R2 , R3 ) and for convenience we choose j = z We define the dimensionless quantity I(R, , R2,
R3) -,
2 y~l/~-1
dl nz/3~R0 ,
(2.10
3 where the integration is over the surface of an ellipsoid with semi-axes R,, R2 , and where R0 is the radius of a spherical nucleus which has the same volume as the deformed one, i.e. R, R2R3
=Rô .
(2.11)
From its definition (2.10), I obeys the following scaling property I(AR, , AR2 , AR 3) = I (R, , R2 , R3 ) .
(2.12)
Choosing A = Ro l , we find that I in eq. (2.10) is given by I(a, b, c) = 3
41r X 22 2 +y2 l 6'+z 2 /c2 _1
where abc =1 ; a, b, c are given by the Hill-Wheeler parameters
R;/ R0
(2.l3)
(j =1, 2, 3), respectively, and in terms of ß
cos (y-3iT)] ,
5/4ir ß
cos (y+3~)] ,
a =exp [,15-5/4 7r b =exp [
dl nz ,
c = exp [ 5/47rß cos y] .
(2 .14)
Using the divergence theorem, we can convert (2.13) to a volume integral I=
3 (abc) _2 / 3 4 ~r
dV (z-
)(z -
n) .
(2.15)
3?
B. Bush, Y. Alhassid / GDR width A c)2 e unit vector normal to the ellipsoid (x/a )2 + (y/b)2 + (Z/ = l, is given by = s/ s where s = (x/a 2, yl b2, Z/ c2) . Thus I is given by 4 2 +Y 2/ 4 / (abc) dx dy d.,. 41rc` (x /a +y /b +Z /c )
f
(2.16)
x2/a`'+y2Jb2+Z`'/c`'-1 .
Changing the coordinate (x, y, Z) to (f, °q, C) where
e = (x1a )2;
we find
I = 3 (a -c)-2/3 b 47r
ac
+(a one
f
+ ~ +zC ,
e, n4
b) .
s shown in appendu I-
= (y/ )2;
-0
de d
C = (Z/ C
c)2
,
(2 .l7)
(el a2+ r1/b2+C/e2)3/2 (2.18)
, the triple integral in (2.18) can be reduced to a single
3 a2b` r' 2 (abc)2/3 o dt
I
t . b) t)+(a<-' (a2 + t)3(b2 + t)(c`'+
(2.19)
is easily checked that for a sphere a = b = c we obtain in (2.19) I =1. I is thus a purely geometrical factor that describes the width h for a deformed nucleus in units of the spherical width I'® . e integral in (2.19) is an elliptic integral. To transform it to a canonical form which we use the substitution t = a`'tg 2 gives =/2 3 a 2b2 sin2 Jcos jp I=(2 .20) (b2+a2 tg2 )1/2(C2+a2 tg2 ) , /2 +(aHb) . 2 (abc)2/3 J® Finally, substituting (1- a`/ C2)112 sin = sin 0, we obtain t
ac 1/2 a c b2 - a2 I = ( a c)`l3 (c`'-a2)3/2 D arccos +(a<->b), (~c , b -c2 -a2 where is given by 23) sin2 do . D(O, k) = foo (1- k2 sin2 ofi l'/2
(2 .21)
(2.22)
is a linear combination of the elliptic integrals of the first and second kind, E(, k) and E(0, k) (2 .23) (0, k) = CE(o, k) - E(0, k)]/k 2 . In eq. (2.22), we have assumed a , b c. Otherwise, should be replaced by corresponding analytic continuations.
ß. Bush, Y. Alhassid / GDR width y=_1301
y=-,,00
10
33
Y= - 140°
--150
rY
r,
ro
\
r°
\
.\
v
r,
o
Y
(Power Low) 01
-0 4 -0 .2
0 02
04
-0 .4 -0.2
0
0.2
0.4
13
-0.4 -02
0
02
04
-04 -02
0
02
04
Fig. 1. Tr /I~ (solid lint'), r, ./FO (long-dashed line) and 1 _/ro (short-dashed line) versus deformation /3 for several values of y (-120°, -130°, -140°, and -150°). Top panel : from the one-body dissipation width formula (2.24) and (2.19). Bottom panel: from the power law (l.l) and (1.2) using S =1.6.
From (2.8) and (2 .10) we have TZ
=
l'o I(a,
b, c),
(2.24a)
where I is given by (2.19) and a, b, c are functions of ß, y as in (2.14). T,r and r,, are obtained by a suitable permutation of a, b, c TX =ToI(b, c, a),
(2.24b)
Ty = ToI(c, a, b) .
(2.24c)
Eqs. (2.24) determine the exact dependence of I' on deformation in the one-body dissipation model. This dependence is shown in fig. 1 (top panel) where the widths T;/To (i = x, y, z) are shown as a function of the deformation ß, for several values of y (-120°, -130°, -140° and -150°). In sect. 3, we shall show that the power law (1 .1) is in good agreement with (2.24) for realistic values of ß. The dependence of T;/To on deformation which follows from the power law [using eq. (1.2)] is shown in the bottom panel of fig. 1 using 8 =1 .6. 3.
over Law
In this section we show that the power law (1 .1) is a good approximation to the width equations derived in sect. 2 .
B. Bush, Y. Alhassid / GDR width
34 3.1 . EXPONENT S
e shall first show that eq. (1 .1) which can be rewritten as (3 r;=ro( o/ ;)' ,
.1)
is a good approximation to (2.24) for small enough deformations ß, and derive the value of S. In subsect. 3.2 we shall verify how well eq. (3.1) describes eq. (2.27) globally for a realistic range of deformations . e take, without loss of generality, j = d an assume ß to be small enough so that the deviation from a sphere is small, i.e. b--ca=v,
c=1+u,
(3 .2)
were u, v 1 . Since abc =1, u and v determine the most general small ellipsoidal deviation from a sphere (including triaxial) . (3.1) reads, to first order in u, v,
Thus, to establish eq. (3 .1) we have to show that to first order in u and v, I in eq. (2.19) depends only on u and find the value of S. e expansion of (2.19) to first order in a'-, b', c' gives t 1,2 2 2 c`'-5 a +b 21 = C2 C -3 dt(,+t)7/2 2 4 ), fo where we have used I(1, 1, 1) =1. sing (3 .2), we find to first order in u and v I==
1-SU .
(3 .5)
is is just the required equation (3.3) with S=s=1 .6 .
(3.6)
us, we have established the power law (1 .1) for small deformations with an exponent of S =1 .6. 3.2 . NUMERICAL VERIFICATION OF THE POWER LAW
n subsect. 3.1 we showed that (1 .1) holds in the one-body dissipation model with S = 1 .6 for small deformation ß and any y. It turns out that (1 .1) is actually a good approximation over the whole range of realistic deformations . To show that, we have calculated I in (2.19) numerically for -0.4, ß , 0.4 and several y's. A least square fit of (1 .1) to the calculated I give exponents in the range S - 1.6006-1 .6044 with correlation coefficients better than 0.99 (see table 1 for details).
B. Bush, Y. Athassid / GDR width
35
TABLE 1
The slope, intercept and correlation ofa linear least squares fit to in (ri ll',) (calculated from eqs . (2.24) and (2.14)) versus In (E;/ Eo) (calculated from (1 .2)) for i = x, y, z For each value of y (-120°, -130°, -140° and -150°) the linear fit was made for ß in the range -0.4-_ (3 _ 0.4 Component
Slope (S)
Intercept
Correlation
y = -120°
x y z
1.6037 1.6007 1.6037
-0.0054 -0.0082 -0.0054
0.9993 0.9996 0.9993
y = -130"
x y z
1.6041 1.6008 1.6031
-0.0049 -0.0080 -0.0060
0.9988 0.9996 0.9995
y = -140°
x y z
1.6044 1.6012 1.6025
-0.0046 -0.0077 -0.0066
0.9960 0.9996 0.9996
y = -150°
x y z
1.6018 1.6018
-0.0072 -0.0072
0.9996 0.9996
1hig. 2_ Bottom : T;ll'o calculated from eq. (2.24) versus E;1 E, calculated from eq. (1 .2) on a log-log plot for i = x, y, z in the range -0.6 -_,O -_ 0.6 and for several values of y (-120°, -130°, -140° and -150°). We used solid, long-dashed and short-dashed lines for i = x, y and z, respectively. If the power law (1.1) holds, then In ( ;/lo) = 8 In (El E(,), i.e. l',/FO versus E;/EO is a straight line with slope 5 and zero intercept. Such lines with S = 1 .6 are shown by the dotted lines. Top : ß versus E,/ E (i = x, y, z) from eq. (1 .2) .
B. Bush, Y. AJhassid / GDR width
36
o show the quality of agreement between the ore-body dissipation width formula and the power law over a larger range of deformations (r.8i = 0.6), we plotted in ;/E0 Both scales are logarithmic, E ;/FO for eqs. (2 .24) versus . fig. 2 (bottom panel) I' so that an exact power law should give a straight line with slope 8 and zero intercept. (See dotted lines in fig. 2 for 8 =1 .6.) Notice that when -0.6, ß = 0.6, the corre;/E, [eq. (1 .2)] can be read from the top panel of fig. 2. We sponding ranges of E see that for large deformations there are deviations from a power law and that the latter always overestimate the geometrical surface dissipation widths. 3.3. WIDTH T.
e G lZ width Eo for a spherical nucleus is given, in the surface dissipation models by Z-' Ep `9- N- ' On
(3.7)
o
For a cold Fermi gas of nucleons v = ~ vF, where (2EF/
VF
is the Fermi velocity
(VF =
)''').
For the nuch~us Er where Z = 68 and N = 98, we find E ( P ) = 29.2 MeV, SF) _ 37.2 eV and taking o = rO A' /' with ro =1.2 fm we find To (T = 0) = 5.88 MeV for = (i.e. in the original one-body dissipation model). A phenomenological fit to the ground-state R cross section in "Er gives) To = 3 .64 MeV, so that we overestimate the observed width by about 2 eV, which is expected in the one-body dissipation theory "). However, as mentioned in sect. 1 a more realistic modeling of surface dissipation will result in a value for ks which is less than unity' s ) . Here a value of k,, = 0.62 will reproduce the experimental width exactly. .
o c usions
e have derived, using surface dissipation models, a purely geometrical expression for the GDR width (at a fixed deformation) in units of the spherical width. It was shown to follow closely a power law F.i = FO(E/EO) s with an exponent of 8 =1 .6. This result is proven for general triaxial shapes. The e-:ponent agrees well with values found phenomenologically for heavy nuclei (A :,:- 100) through a fit to ground-state GDR cross sections. The above power law with 8 =1 .6 plays a major role in a macroscopic theory of the GDR at finite temperature that we have recently developed. This work was supported in part by the US Department of Energy, Contract No. -AC02-76ER03074.
B. Bush, Y. Athassid / GDR width
37
Appendix A We show here how to reduce the triple integral (2.18) to the single one (2.19). The integral (2.18) is of the general form dx;
llixP' -'
Using the integral representation of the T function, we rewrite (A.1) as ao t r-1 S Y; s 1 x a0
i
_
0
ao
ao
ao
t
r-1
dt O 1-Y- x; e-y- qi`i` Il xiPB - ' dxi , (A.2) T(r) ~ where O is the step function; O (y) = 0 for y < 0 and ®(y) =1 for y > 0. Representing O(y) in terms of its Laplace transform 1 / s we have es l_Ex; :°° 1 (A.3) i)= 2 ds ( ) O(1-y_x s ,ri
10
I0,>
Exchanging the order of integration in (A.3), we can now do the x; integrals =
o
dt
ds es tr-1 T(Pi) . -i o 27ri s T(r) i (s+q;t)p
(A.4)
Substituting t -3- st in (A.4) and rotating the integration axis (s is purely imaginary) we obtain °~ ds Sr_1-EPids ~° dt t r - ' T(Pi) (A.5) . T(r) ; (1+gi t) Pi _ . 2ei o Using t°`-1 as the inverse Laplace transform of s`T(a), we can do the s integration to obtain j !li T(Pi) tr-1 . dt P. T(r)T(1+~ pi -r) o lli (1+qit) The integral in (2.18) is a special case of (A.1) where P1=2, q,=1la 2 ,
P2
= P3 == 2
de d q dS
e.1n4--o
= 2ira bc 3
0
q3
q2=1/b2,
We then find e+71+e--1
r=2,
2
( e / r%y ) '/2
(ela +rl/b
dt (a2+
=1ic 2 .
+~/c2)
(A.7)
31 2
t ll2 / t) 3 2
(b 2 + t) 112 (C
2
+ t)
112
"
(
A.8 )
38
B. Bush, Y Alhassid / GDR width eferenus
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 2 ,1 1, 22) 23)
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