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Nuclear Physics A472 (1987) 215-236
Nosh-Holland, Amsterdam
ISOSPIN AND MACROSCOPIC MODELS FOR THE EXCITATION OF GIANT RESONANCES AND OTHER COLLECTIVE STATES*
G.R. SATCHLER
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6373, USA
Received 25 March 1987
Abstract: Recently there has been some confusion over the excitation of “isovector” giant resonances
by isoscalar probes. We present a unified theoretical description of macroscopic models that may
be used to calculate the excitation of giant resonances by inelastic scattering so as to clarify this
and related questions. Particular attention is given to the relative phase of the Coulombic and
hadronic contributions. After describing models for the transition densities that are based upon
certain sum rules, a folding model and a deformed optical potential model for the transition
potentials are discussed. Applications are made to some examples of inelastic cY-particle scattering
and the two models are compared. Folding yields cross sections for the breathing-mode monopole
resonance that are smaller than those from the potential model by almost a factor of two, so that
the measured values appear to require more than 100% depletion of the sum rule. It is suggested
that this might be due to neglect of density dependence in the effective interaction.
1. Introduction
Recently there has been interest ‘-‘) in the excitation of isovector giant resonances,
especialIy the giant dipole, by isoscalar probes, such as the inelastic scattering of
deuterons, alpha particles, and heavier ions with N = Z. These particles are charged
and hence can produce isovector states by Coulomb excitation 3-5), but in addition
production can occur via the isoscalar hadronic interaction if the neutron and proton
transition densities have different shapes 6*7). Particular attention has been paid to
the isovector giant dipole resonance (GDR) because this occurs at an excitation
energy close to that of the isoscalar giant monopole resonance (GMR) and there
was concern that its production could obscure the interpretation of the GMR
excitation.
Here we give a unified theory of the macroscopic models used to describe such
excitations, with especial attention to their use in calculations, because there appears
to be some confusion in the literature about the correct way to do this. For example,
a recent paper’) asserted that cross sections for exciting the GDR by a-particle
scattering could be comparable to those for the GMR. We shall show below that
this is incorrect; see also ref. 9). Further, it was suggested that interference between
hadronic and Coulomb excitation amplitudes could produce GDR angular distribu-
tions similar to those for the GMR. This is also incorrect and results, in part, from
* Research sponsored by the Division of Nuclear Physics, US Department of Energy under contract
DE-ACO5-g40~1400 with Martin Marietta Energy Systems, Inc.
0375-9474/87/~03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
216 G.R. Satchier / Isospin and macroscopic models
use of the wrong relative sign for these two amplitudes. Other authors 435) have also
promulagated this error in phase.
Although the discussion here specifically addresses giant resonances, and macro-
scopic models for their transition densities, it should also illuminate the isoscalar
production of other states in which the neutron and proton excitations are out-of-
phase and hence called “isovector”.
First we discuss the transition densities, and their isospin character, associated
with various excitations of the target nucleus. The corresponding transition potential
experienced by some probe is then the probe-target nucleon interaction potential
folded into the transition density.
2. Transition densities for giant resonances
We discuss here macrosopic models for collective oscillations that do not involve
the nucleon spins. Although especially appropriate for a giant resonance whose
excitation (by definition) exhausts most of the limit imposed by an energy-weighted
sum rule (EWSR) 1o-12), these models are commonly used for any strong, collective
excitation.
2.1. ISOSPIN STRUCTURE AND SUM RULES
The theoretical framework has been described in chapter 14 of a recent book I’);
see also refs. lz313). Macroscopic nuclear oscillations are classified as primarily
“isoscalar” (7 = 0) or primarily “isovector” (7 = 1) according to whether the neutron
and proton density distributions oscillate in- or out-of-phase with each other,
respectively. Probes can also be classified as isoscalar (t =0) or isovector (t = 1).
The target neutrons and protons respond equally to a t = 0 probe, such as the
hadronic field presented by a scattered a-particle. On the other hand, a t = 1 probe
produces equal but opposite responses from neutrons and protons. The Coulomb
field due to a charged particle affects only the target protons; it consists of equal
(but opposite) amounts of t = 0 and t = 1. Clearly, it is the response to a probe with
t that determines the degree to which the excitation is truly isoscalar (t = 0) or
isovector (t = 1). Of course, we expect the strongest response with t = 7. However,
we may have nonvanishing t # TJ response if there are differences between the
neutron and proton transition densities, such as may occur in nuclei with N # 2.
In particular, a so-called “isovector” (7) = 1) state may be excited by an isoscalar
(f = 0) field. (It might be less confusing if alternative names for the states could be
found, such as in-phase (IP) for r] = 0 and out-of-phase (OP) for T= 1.)
The transition density of the target nucleus that is involved in the excitation of
one 2”-pole phonon of type n, with projection pn, by a probe with isospin character
t = 0 or 1 is given by the nuclear matrix element
@g(r) =(lm; T&&(r)lO)= (2t+1)-“2g:‘(r)Y;‘(8, #)” (2.1)
G.R. Sarchler / Isospin and macroscopic models 217
of the isoscalar/isovector density operator
(2.2)
where the sums run over the target nucleons. The radial part may then be written
as a sum of neutron and proton parts
g?‘(r) = g?(r)+ (-)““gXr) 2 (2.3)
where, by convention, g” and gp have the same sign. If the neutron and proton
contributions were identical, g” = gp, the 7 = 0 oscillation would be exactly isoscalar
and the 7 = 1 one would be pure isovector, and there would be no response unless
t = 7).
The EWSR that is most often used as a measure of the strength of observed “giant
resonances” is that obtained from the action on the target ground state of the
isoscalar and isovector multipole operators r’Y;“( 13, 4) and rr’Y;l( 6, 4), respec-
tively, where r = + 1 for neutrons, -1 for protons. (The radial dependence is the
same as that for electric radiation in the long wavelength limit, but in general differs
from that for the fields due to other probes.) These two operators give equal and
model-independent sum rule limits ‘om12) if the nuclear forces are independent of
velocity, except for the dipole case. If I = 1, the isoscalar sum vanishes because the
t = 0 operator simply produces a translation of the center of mass.
2.2. GIANT RESONANCES WITH 132
If one of these EWSR limits for 12 2 is exhausted by a transition to a single state
of excitation energy E, in a spherical nucleus, it can be shown lo) that the correspond-
ing transition density has the Tassie form
gj( r) = -a$’ dp,/dr, (2.4)
where p,(r) is the ground-state density distribution for nucleons of type i = n or p.
We see immediately that the response for t f v vanishes if p,(r) = p,(r), as is
approximately the case in nuclei with N = 2. Further, if they have the same shape,
pn/pp = N/Z, then the t # 77 response is (N - Z)/A times that with t = r) for a given
amplitude (Y,.
The parameter (Y, is determined from the sum rule limit to be l”,ll)
(~:=2d(h~/m)(A(r~‘-~)E,)-~, (2Sa)
where m is the nucleon mass and the radial average (ry) is taken over the ground
state density distribution p = pn + pp. This LY, has dimensions of (length)2-‘. Some-
times 11) it is written as
(Y, = /3,R2-‘, (2.6)
218 G.R. Satciifer ,I Isospin and macroscopic models
where PI is dimensionless, by introducing an arbitrary reference length R. If R is
chosen as the nuclear radius, R = c, then this pi is the analog of the deformation
parameter introduced by Bohr and Mottelson ‘*) (BM) when deforming the nuclear
surface r = c. The latter model is frequently used for low-lying collective states and
a single phonon excitation yields 14)
g;(r) = & dpilde, (2.7a)
= -& dpJdr (2.7b)
if pi = pi(r - c); also St = (&c) is the deformation length. Thus it differs from the
Tassie form by omission of the factor of (r/c)‘-‘. The sum rule limit (2.5a) now
becomes ‘J ‘)
s: = (P,c)’ = 1(21-k 1)2 2?rfi2 (r2i-Z) -- (1+2)* AmE, (I’-‘)”
(2Sb)
although we must recognize that it is somewhat inconsistent to assume that all of
this strength is exhausted by one state and yet use the BM transition density (2.7)
rather than the Tassie one (2.4).
If the ground-state density is approximated by a uniform distribution with a
radius R, the two expressions (2Sa) and (2.5b) both reduce to
(2Sc)
This provides the simplest estimate for the sum rule limit on the deformation length
6[. In principle, the equivalent uniform radius R depends upon 1, but perhaps it is
sufficient to use the value obtained from the mean square radius, (r*) = gR*.
2.3. GIANT MONOPOLE OSCILLATIONS
Frequently, the operators that are used r’,‘*) to generate the corresponding EWSR
for isoscalar and isovector monopole excitations (t = 0) are r2 and &, respectively.
If the excitation of a single state exhausts one of these EWSR, the corresponding
transition densities have the form **)
g:(r) = --aJ3pi(r) 3- r dpildrl, (2.8)
with i = n or p. These represent breathing modes ‘“*13). The dimensionless amplitude
is given by
a;=2~(h~/m)(A(r~)E,)-‘. (2.9)
[Beware: eq. (2.9) is normalized to correspond to eq. (2.1), where E= (47r)-“‘.
Most DWBA or coupled-channels computer programs follow this convention.] We
G. R Sa~~h~eT / Isospin and macro.~co~ic models 219
note that the form (2.8) conserves the number of neutrons or protons, as any monopole transition density must,
J g;(r)r2dr=0. (2.10)
2.4. GIANT DIPOLE RESONANCES
Dipole oscillations (I = 1) require specia1 consideration because of the need to keep the center of mass (c.m.) unchanged. This imposes a relation between the ampiitudes of the neutron and proton oscillations, such that the isoscalar dipole moment (expectation value of r) will vanish for both r] = 0 and n = 1 types. In terms
of eq. (2.3), this means
J g;(r)r3drf(-)” gp(r)r’dr=O. J (2.11)
This does not necessarily imply that g:‘(r) = 0. Indeed, the isoscalar (1= 0) response of the isovector (q = 1) GDR is a topic of interest here. In addition, there may be other oscillations Is) with associated transition densities that satisfy eq. (2.11) but which have nonvanishing transition amplitudes for probes, such as the hadronic field of a scattered particle, whose radial dependence differs from r.
When the excitation of one n = 1 phonon exhausts the sum for the f = 1 dipole (expectation value of or), the associated transition density is lo) that of the Gold- haber-Teller (CT) model fa special case of the Tassie form, eq. (2.4)],
g;(r) = -*, zzdp, 2Ndp, A dr'
g%r) = --%-y dr 3
with the amplitude (“deformation length”) given by
(2.12)
(2.13)
Integration by parts shows immediately that (2.12) satisfies the c.m. constraint (2.11) if n = 1. We note also that if pn and p,, have the same radial shape, so that
p,(r)lp,(r) = N/Z. then g;(r) = sP( ) r and the isoscalar response g;‘(r) vanishes. One would expect this to be approximately true for a nucleus with N = 2, but different p,, , pp shapes (e.g., different radii) could occur when N > Z and allow the excitation of the isovector GDR by an isoscalar hadronic field ‘). For illustration, assume that the distributions p,(r) depend only upon the differences (r-c,), and take the radii c”,~ = c( 1 f&x), where x = (N - Z)jA. Then, to lowest order,
p,,,(r) -?(I * ax)Plr- c(l *Srx)], (2.14)
220 G. R. Satchler / Isospin and macroscopic models
where P(I) = pn( r) + p,(r). Here e = 1 - y and y = 1 results in pn = p,, at small r,
while y = 0 gives pn/pP = N/Z. With this ansatz and eq. (2.12), the isoscalar transition
density is
g:“(r) = g?(r) -g!(r)
(2.15)
which vanishes for y = 0, or c, = cP. The two terms have opposite signs and the
transition density changes sign near the nuclear surface. At larger radii, the second
derivative dominates for conventional density distributions and results in g:‘(r)
being positive if c,> cP. At small radii it is negative. For example, at large radii the
Fermi distribution becomes
c-r drhvw Q ,
( >
where a is the surface diffuseness parameter. This gives
(2.16)
(2.17)
which is positive if c > 3a.
2.5. ISOSPIN SPLITTING
The “classical” models of oscillations just discussed do not take account of the
quantization of isospin. In an actual quanta1 system with N # 2, or ground state
isospin To = $( N - 2) # 0, an “isovector” (7 = 1) vibration results in states with
isospin T = To and To+ 1. (The state with T = To- 1 is only realized as an analogue
in the nucleus with N’ = N - 1, 2’ = 2 + 1, which can be reached by a charge
exchange reaction.) The isovector sum rule strength is shared between these com-
ponents. In the absence of coupling between the isospin and the spatial motion,
the T = To and To+ 1 states would be degenerate and share the strength in the ratio
To: 1, respectively. However, there are major coupling effects I*). The degeneracy
of the two states is raised, with the splitting proportional to (To+ 1) in lowest order.
The sum rule strength is also redistributed between the two component states. To
lowest order in To, the strengths become proportional to “)
(‘TT,~To1T00)2{~o+~,[T(T+1)-T~(To+1)-2]}2,
1
(To/ To+ l)(mo-2mh2 if T= To
= (l/T,+-l)(m,+2T,m,)* ifT=T,+l, (3.18)
where m, and ml are independent of To. The values of ml/m0 for the GDR are
discussed by BM [see also ref. “)I; since m,/m,< 0, the T = To excitation is
enhanced, and the To+ 1 one is inhibited, by the coupling.
G.R. Satchler / Isospin and macroscopic models 221
3. Transition potentials
The transition potentials UyA( r) experienced by a scattered particle when it excites
a 2’-pole oscillation of the q-type in the target nucleus can be written in the same
form (2.1) as the transition density,
Uz(r) = (21+ I)-“‘G:‘(r) Yy”(e, 4)*. (3.1)
We discuss here two models, frequently used, for constructing the multipole form
factors G?‘(r). One is the folding model “X’6), which folds a projectile-target nucleon
interaction with the transition densities described in the preceding section. The other
is an optical model potential ansatz that is suggested by the folding model.
The typical direct reaction computer program (DWBA or coupled-channels)
(see ‘*) for example) uses the standard radial form factor
G,(r)=-P,RdUldr, (3.2)
where U(r) is an optical potential. This form factor is then to be replaced directly
by the G”‘(r) discussed here in order to calculate the corresponding cross sections.
The contributions from t = 0 and t = 1, as well as the Coulomb excitation term, are
coherent, so they should be added when both are present.
3.1. FOLDING MODEL
We write the central and spin-independent part of the effective hadronic interaction
between the projectile at r and each target nucleon i at r, as
2, = U’(S) + U’(S)7 ’ Ti , (3.3)
where s = Ir - r,/ and T, +ri are twice the isospin vectors of the projectile and target
nucleon, respectively. Then u” corresponds to an isoscalar probe and U’ to an
isovector one. We make a multipole expansion of u<(s),
u’(s) = C u;(r, r,) Yy(gi) Y;“(F)*. Im
Then the folding model 1’,16) for the transition potential form factor is
G:‘(r) = a, g:‘(r,)uf(r, ri)rf dr,.
(3.4)
(3.5)
Here a, = 1 and a, = r3, twice the 3-component of the isospin of the projectile. (Note
that 73 = -1 for protons and 3He, but is positive for other nuclei.)
Clearly, these G:‘(r) have properties similar to those already discussed for the
corresponding g:‘(r). Indeed, if the interaction ~1 has zero range, V’(S) =J’S(s),
222 C.R. Satchter / Isospin and macroscopic models
then we have the simple result
G;l’( I) = J’g:‘( r) : zero range (3.6)
(Note: u’, J’ are negative for an attractive interaction.)
Within this model, the optical potential U is given by folding the same interaction
(3.3) into the target ground-state density,
U= U0(r)+r3U1(r), (3.7)
where the isoscaiar and isovector terms are given by the 1= 0 term of (3.14)
f-Jo(r)= I
u:(P, ri)p(ri)r:dri, (3.8a)
U,(r)= uA(r, ri>[p,(ri)-p,(ri)lr:dri. (3.8b)
Here p = pn + pi,. If pn, p,, have the same radial shape, so pn/pp = iV,/Z, U, becomes
U,(r)==[(N-2)/A] d,(r,ri)p(ri)rfdri. s (3.9)
In the zero-range limit, co~esponding to eq. (3.6),
U,(r) = JOp(r) , U,(r) = J’b,(r) -k+ArH : zero range. (3.10)
Sometimes the effective interaction v is taken to be real 16), so that eqs. (3.7)-(3.10)
apply only to the real part of U. In that case, a phenomenoiogical imaginary part
is introduced for U, and frequently the imaginary part of G is obtained by using
the deformed optical model of sect. 3.4. Sometimes an imaginary part for u is
obtained from other considerations. One phenomenological approach “) is to
demand that it reproduce Im U in the folding model (3.8).
It should also be emphasized that, besides the neglect of any spin dependence
or noncentral forces, the discussion above does not take into account the possibility
of density dependence in the interaction. Such a dependence would modify the
formulae somewhat 16).
3.2. COULOMB EXCITATION
The Coulomb field of the projectile only acts upon the target protons. It is less
confusing to use this fact directly, rather than translate into isospin terms. The
multipole component experienced by each proton is then
ut( r, ri) = 47rZ’e*( r!Jry’)/(21+ 1)
G. R. Satchler / Isospin and macroscopic models 223
where Z’e is the charge on the projectile and r<, r, are the lesser, greater of r, r,.
The corresponding transition potential for Coulomb excitation is then
GTC(r) = (-l)q I gf’(r,)vF(r, ri)rf dr,
The phase (-)” arises from the convention adopted in eq. (2.3).
Now the reduced electric transition probability I*) is just
B(EZ)T = gP(r)r”’ dr 2 e*.
(3.11)
(3.12)
Consequently, at large radii r s ri, (3.11) assumes the form
G:‘(r) = i(-)“4~[B(EZ)t]“‘Z’e/(2Z+ l)r’+’ (3.13)
which is model-independent except for the f choice of sign which corresponds to
the sign of the integral in eq. (3.12) and hence depends upon the model adopted
for g;(r). (The Tassie model (2.4), the BM model (2.7) and the GT model (2.12)
all make this integral positive so that G,R” has the sign (-)“.) However, the same
model determines the sign of the nuclear component, for example via eq. (3.5), and
hence the sign of the Coulomb-nuclear interference.
Frequently, eq. (3.13) is extended to small radii by using the model of a point
charge interacting with a uniform charge distribution of radius R,. Then
G:“(r) = +(-)“4~[B(EZ)~]“‘.Z’e/(2Z+ 1) x l/r'+', rz R,,
r’/ Rz’+’ , (3.14)
r < R, .
In the Tassie model (2.4), eq. (3.12) becomes
B(EZ)T = /(2Z+ l)cqZ(r2’-2),/4~~2 e* , (3.15a)
where (ry)p is taken over the ground-state proton distribution. For the BM model,
(2.7), we have instead
B(EZ)T = ~(Z+2)6,Z(r’~1),/4rr)2 e2. (3.15b)
For a uniform distribution of protons with radius R, these expressions reduce to
B(EZ)T = )3Zc~~R*‘-*/47r1* e* = )3Z&R’-‘/47~)~ e* (3.15c)
which we recognize as the commonly used form if we put 6, = /3,R or (Y, = p,R*-‘.
In this approximation, a single n = 0 or 1 state that exhausts the sum rule for Z 2 2
has
2 21-2
B(EZ)T = Z(2Z+ 1) g 3fnI& e*, x
(3.15)
224 G.R. Sntchler / Isospin and macroscopic models
if we assume the neutrons and protons have the same deformation lengths and the
same radial distributions. For 77 = 1 and Z= 1, the sum rule (2.13) corresponds to
(3.16)
3.3. COULOMB-NUCLEAR INTERFERENCE
Although the folding procedure is only a simple model, it should be sufficient to
determine whether the Coulomb-nuclear inteference is constructive or destructive,
particularly at the larger radii which are most important when strong absorption is
present, and for the macroscopic models being considered here.
It is more transparent to re-express eqs. (2.3), (3.5) into neutron, proton language
since the two hadronic (h) components with 1= 0 and t = 1 (if both are allowed)
are coherent. The form factor for the total transition potential then becomes
G~(r)=G:“(r)+G:‘(r)=G:h(r)+G:C(r) (3.17a)
where the nuclear (hadronic) component is
Gyh(r) = Gr(r)+(-)“GP(r) (3.17b)
with (i=n,p)
G;(r)= gf(r,)uf(r, ri)rfdri I
(3.17c)
and from eq. (3.3) the projectile-neutron and projectile-proton interactions are
u”=v”+73v1) ZIP= no- 7$?. (3.18)
In particular, if the projectile is isoscalar (such as an a-particle), so that only t = 0
enters, then U’ = 0, U” = up = 21’ and
G?(r) = r: dr, UgXri)+ (-)“gP(ri)lvY(r, r0-t (-)“gP(ri)nT(r, ri)) . (3.19)
Some general conclusions follow from eqs. (3.11) and (3.17)-(3.19). We note that
g;(r) and g?(r) are “similar” for the collective oscillations of interest here. Then,
if 7~ = 0 and if both Y” and up are attractive (negative) while vc is repulsive (positive),
it is clear that the Coulomb-nuclear interference is destructive. However, if ~7 = 1,
or if T~V’ is stronger than v” so that U” and up have opposite signs, then the sign
of interference depends sensitively upon the differences between g’; and gp.
We only discuss explicitly excitation of “isovector” (7 = 1) states by isoscalar
probes (t = 0), as given by eq. (3.19); the other cases can be obtained by analogous
G.R. Satchler / Isospin and macroscopic models 225
derivations. The Tassie, eq. (2.4), or BM, eq. (2.7), transition densities apply for
132. For these, pn/pp= N/Z is a reasonable approximation so that g;/gp- N/Z
and so, and
(3.20)
Thus the Coulomb-nuclear interference is constructive if N > Z and the hadronic
interaction v” is attractive.
More care is required for the GDR because the assumption pn/pp = N/Z makes
the isoscalar response vanish. The total transition potential form factor is
G:(r)= [g~“(ri)v~(riy r)-gT(ri)vy(ri, r)]rfdr,. I
(3.21)
Within the GT model, eq. (2.12), gy is positive and the Coulomb contribution is
seen to be negative. A reasonable assumption is that the neutron distribution has a
wider radial distribution than the proton one (if N > Z), as embodied in the ansatz
(2.14). As a consequence, g:‘(r) is positive for radii r beyond the nuclear surface,
as in eq. (2.17), and thus the hadronic contribution to G’(r) is also negative if u”
is attractive. Consequently, the Coulomb-nuclear interference is constructive outside
the nucleus and destructive inside. When strong absorption is present, it is the
outside part that dominates.
3.3. DEFORMED OPTICAL POTENTIALS FOR TRANSITIONS
The folding model (3.5), especially in the limits (3.6) and (3.10) of zero range
for the interaction, implies that the general behavior of the hadronic transition
potential follows that of the transition density 12). This suggests that the prescriptions
that were applied to the ground-state density distributions in order to generate the
transition densities also be applied to the ground-state optical potentials to generate
the transition potentials. While this relationship is not exact, it has been widely
used to describe the excitation by hadron scattering of low-lying collective states
of the isoscalar type with 12 2 and results in amplitudes (deformation lengths) or
multipole moments that correlate well with those deduced from electric excitation ‘I).
Use of this model has also been extended to giant dipole and monopole
resonance 7,15,‘9).
The optical potential U is complex and the deformation prescriptions apply to
both real and imaginary parts, thus giving complex transition form factors G:‘(r)
and “complex coupling” ‘I).
Again, we only discuss explicitly the excitation by an isoscalar probe, t = 0; the
other cases follow in a similar way. Then U = U. in eq. (3.7). It is usually assumed
that U. depends only upon (r - R), where R is the potential radius; of course, the
226 G.R. Satchler / Isospin and macroscopic models
real and imaginary parts may have different radii. Almost invariably the BM form
(2.7) is used, rather than the Tassie form (2.4), in the deformed potential model for
12 2 excitations. Then we use for 12 2
Gy(r) = -6, dU, (r-R)/dr, 6 = P,R , (3.22a)
to which should be added the Coulomb term (3.13) or (3.14) with the sign +(-)“.
The remaining question is how to relate this PI of the potential to the underlying
density deformation. Frequently it is assumed that the potential and density informa-
tion lengths 6, are equal, although others equate the PI themselves. We believe that
?$ is a more fundamental quantity then /3,. The folding model suggests that the
deformed potential extends a fixed distance beyond the deformed density. The
distinction is particularly marked when the projectile and target nuclei have compar-
able sizes, such as in heavy-ion scattering. Sometimes this is taken into account by
writing the potential radius as R = r,(A:‘3 +A:‘3) and then using as deformation
length the quantity 6, = P,Rx where R, = roAii3 and x refers to the nucleus being
excited. We prefer to use the 6, associated with the transition density. (This also
implies using the same &, rather than the same p,, for the real and imaginary parts
of the potential.) There is even more ambiguity if the Tassie form (2.4) of density
deformation is more appropriate than the BM one, (2.7), as is the case of giant
resonances exhausting most of the sum rule strength. (These questions are avoided
by using the densities directly ‘) in the folding model (3.9, but this procedure has
its own uncertainties.)
The giant monopole excitation can be described in a similar fashion by using
3O,,(r)+ry I
,
Gr( r) .
[This corresponds to “version I” of ref. ‘“).I
The analogous prescription for the GDR may be written
dU,, dU ZF-N--$ , 1
(3.23a)
(3.23b)
(3.24)
where U, = U,,+ Up and the two parts U,, , Up arise conceptually by putting p =
pn + pP in eq. (3.8a). There is some ambiguity over their precise definition, but it is
reasonable to assume that U,, Up behave like pn, pp. Thus pn/pp = N/Z suggests
that U,,/ Up = N/Z also, so that the isoscalar response (3.24) vanishes. (This would
appear to correspond to the “AR = 0” case in ref. *), for which nonzero cross sections
were given.) On the other hand, the assumption of different radii c, # cP for pn and
G.R. Satchler / Isospin and macroscopic models 227
pP, such as in eq. (2.14), suggests a similar prescription for U, and U,; that is,
R,#R, with AR=R,-R,=c,-c,.
It remains to relate the strengths (real and imaginary depths K, Wi) of U, and
U, to those of U,. The form (2.14), with
v,,, = f( * EX) v0, w”,,=;(l*EX)WO, (3.25)
keeps V,+ VP = V,, W,+ W, = W, and maintains (to lowest order) their volume
integrals in the ratio J,,/Jp = N/Z. On the other hand, only the tails of the potentials
are important in the presence of strong absorption, suggesting that one impose the
constraint
Vn ew (R,la)+ VP exp (&,/a) = V. exp (WA), (3.26)
where u is the surface diffuseness of the potential, with the analogous relation
between the Wi. Fortunately, the two choices give very similar results.
The exact expression (3.24) may then be used, or the analogue ‘) of (2.15),
(3.27)
Coulomb excitation is included by adding the Coulomb transition potentials G:”
of eqs. (3.14), with (3.16) if the sum rule is exhausted. The appropriate sign for the
GT model is +(-)” = -, as discussed in sects. 3.2 and 3.3.
4. Applications
For illustration, we apply the models discussed above to the excitation of the
“isoscalar” (7 = 0) monopole and quadrupole giant resonances and the “isovector”
(7 = 1) giant dipole resonance in r16Sn by the scattering ‘) of cu-particles (t = 0) of
129 MeV. (This particular case was chosen because it has been used “) to contend
that the cross sections for the dipole and monopole excitations are comparable and
may be confused with each other.) These are supplemented by results obtained
using the folding model for similar excitations in 208Pb+ (Y at 172 MeV. There are
more extensive angular distribution data 20) in this case.
We note in passing that the cross sections for hadronic excitation of the “isovector”
(7 = 1) giant monopole and quadrupole resonances by cz-particles would be much
smaller than those calculated here, even if they exhausted their respective isovector
sum rules. First, there would be a factor of approximately [(N - Z)/A12 obtained
by assuming pn/pp = N/Z, as in eqs. (3.22b) and (3.23b). The sum rule limits provide
another factor E,(IS)/E,(IV) from the difference in excitation energies; typically,
this is -$. The product of these means, for example, that the hadronic “isovector”
cross section can be no more than about 2% of the “isoscalar” one for cY-particles
on 208Pb or 1% for ‘?Sn. Of course, the Coulomb excitation of these states would
228 G. R. ~u~c~ler / Isospin and macroscopic models
only be reduced by the excitation energy factor, times any dynamic reduction due
to the change in Q-values. It would then dominate over the hadronic contribution
in the same way that we find below for the GDR excitation.
The calculations were done in the DWBA, using the program PTOLEMY”).
4.1. DEFORMED POTENTIAL MODEL FOR a f”?3n AT 129 MeV
The optical potential LJ,, taken from ref. I), has the same Woods-Saxon shape
for its real and imaginary parts with radii Rv = R, = 1.40 x 1 161’3 fm, and diffuseness
a, = aw = 0.73 fm, and strengths V = 60.8 MeV, W = 40.9 MeV. The Coulomb poten-
tial from a uniform charge of radius 1.35 x 1 161’3 fm was included.
IJsing h2/2m = 20.736 MeV . fm2, the sum rule prescription (2.4~) gives a deforma-
tion length 6, = 0.753 fm for the transition density if all the quadrupole strength is
concentrated in a giant isoscalar quadrupole resonance (GQR) at E, = 13.2 MeV,
andif . 2 “‘= 4 623 fm for the ground state *I). From eq. (3.15c), we get the
corresponding B(E2)f = 0.288 e2b2. The isoscalar form factor (3.22a) was used with
the same deformation length a2. The measured GQR angular distribution ‘) for this
excitation is shifted forward slightly compared to the predicted one (fig. 1). The
data are compatible with 65-80% of the EWSR limit, according to this model.
The monopole sum rule (2.9) predicts an amplitude a0=0.082 for a state at
15.6 MeV that exhausts the sum. The isoscalar potential form factor (3.23a) was
used with this amplitude and fig. 1 shows that the predicted GMR cross sections
are in agreement with the measured ‘) ones.
The giant isovector dipole resonance occurs at E, = 15.6 MeV also. The deforma-
tion length in the GT model is given by eq. (2.13) if the classical sum rule is
exhausted, LY, = 0.383 fm, with the corresponding B(E1) = 0.271 e2b from eq. (3.16).
We used the potential form factor (3.24) for the response to an isoscalar probe. For
LJ,, U, we adopted the same diffuseness as for U, = U, + U, but with different radii,
R,,, = R k+AR. The depths were determined so that V, + V,, = V, W, + W, = W and
the volume integrals were in the ratio N/Z. Proton scattering measurements, as
well as Hartree-Fock calculations, suggest 21) that AR = 0.17 fm for the neutron and
proton density distributions. However, we use the much larger value AR = 0.488 fm
for the potential in order to compare with the calculations of ref. “)_ We note from
the potential analogue of the approximate expression (2.15) that the cross section
for hadronic excitation alone is closely proportional to (AR)‘, so that this choice
of AR will give a hadronic cross section about 8 times larger than from a more
reasonable choice. The corresponding depths were If, = 31.582 MeV, VP =
29.218 MeV, W, = 21.245 MeV and W, = 19.655 MeV.
The results are inctuded in fig. 1 for Coulomb excitation alone, hadronic alone,
and their coherent sum. We note that the Coulomb and hadronic terms add construc-
tively and the plus sign is to be chosen in eqs. (3.13) and (3.15). [The angular
distribution shown in ref. “) can only be understood if destructive interference occurs,
G.R. Satchler / Isospin and macroscopic models 229
Ex=13.2 MeV
P=2
E,=i3.2 MeV
1=2
0.04 - 1 ’ i ’ I \ /I \J 0 4 8 12 0 4 8 12
@,,. (deg)
Fig. 1. Inelastic scattering of 129 MeV cr-particles from ‘I’Sn. The upper curves refer to the giant
quadrupole resonance at 13.2 MeV, the lower curves to the combined giant monopole and dipole
resonances at 15.6 MeV. The data are from ref. I). (a) Calculations using the deformed potential model
with 100% depletion of the EWSR. The f = 2 curves show the combined hadronic plus Coulomb excitation
(h+c) and the hadronic alone (h). The f= 1 curves also show the Coulomb excitation alone (c). The
hadronic contribution was calculated for the same unrealistic value AR =0.488 fm that was used in
ref. s). (b) Calculations using the folding model with 100% depletion of the EWSR. The I:= 2 results
were obtained using either the Tassie (T) or BM transition densities; both include Coulomb excitation. The hadronic contribution for I= 1 was calculated for c, - cP = 0.185 fm.
although the text claims it to be constructive. Refs. 4*5) state incorrectly that the
interference is destructive.]
It is clear that Coulomb excitation dominates for the GDR, even when an
unrealistically large AR is used. (The Coulomb-nuclear interference from a more
reasonable value of Al? 2 0.17 fm results in an overall enhancement of the cross
sections by ~20%.) The GDR cross sections are much smaller than those for the
monopole resonance, and simply fill in the deep minima in the angular distribution
for the latter. The GDR accounts for no more than about 10% of the observed peak
230 G.R. Satchler / Isospin and macroscopic models
cross sections and thus does not endanger the conclusion that the excitation of the
monopole resonance approximately exhausts the monopole sum rule.
We also note that, although the cross section magnitudes are in close agreement,
our angular distribution for pure Coulomb excitation is smoother than that shown
by Shlomo et al. ‘) but is in agreement with that obtained by Nakayama and
Bertsch ‘).
A more favorable case for competition from the GDR occurs in ‘24Sn, for which
(N - Z)/A is about 40% larger and for which 2’) AR = 0.28 fm is about 65% larger
than for ‘?Sn. However, the predicted hadronic cross section would still be smaller
than that obtained here for “?Sn when the too large AR was used. Further, the sum
rule value (3.16) for B(E1) is only 5% larger, so again the GDR excitation remains
much weaker than the monopole one.
4.2. FOLDING MODEL FOR a+“%n
Calculations were also made using the folding model (3.5) with the transition
densities (2.4) and (2.7) for the GQR, the breathing model (2.8) for the GMR and
the GT model (2.12) for the GDR. The isoscalar alpha-nucleon interaction was
taken to be the gaussian that is frequently used [see ref. 2), for example],
U’(S) = -(v+iw) exp-(s/p)2. (4.1)
We adopted values obtained ‘) from fits to elastic scattering at 140MeV which yielded
/* = 1.94 fm with average v = 36.4 MeV, w = 22.5 MeV. Woods-Saxon (Fermi) shapes
were used for the density distributions, guided by the analysis of proton measure-
ments *I). A common diffuseness of 0.515 fm was taken, with radii 5.433 fm for the
total density, 5.512 fm for the neutrons and 5.3275 fm for the protons. This choice
results in a difference of 0.13 fm in the RMS radii, as observed *‘) for ‘16Sn.
A folded optical potential, eq. (3.8a), was used, for consistency, to generate the
distorted waves for the DWBA calculations. The corresponding elastic cross sections
were very similar to, but not exactly the same as, those obtained from the Woods-
Saxon potential used in the preceding section. The use of this folded potential gave
inelastic cross sections about 15% larger than the Woods-Saxon one when using
the same transition potential. It is expected that this difference would be removed
if the two potentials were adjusted to give exactly the same elastic scattering.
The GQR cross sections predicted by using transition densities that exhaust all
of the EWSR (2.5a) and (2.5b) are shown in fig. 1 for both the BM model (2.7)
with a2 = 0.763 fm and the (probabily more realistic) Tassie model (2.4) with a2 =
0.126. The BM folding model gives cross sections very close to those from the
deformed optical model (3.22a), while the Tassie ones are about 20% larger.
Consequently, one would conclude that the percentage of the sum rule exhausted
by the measured cross sections is similar for the BM model and about 20% smaller
for the Tassie model. (We remark again that differences in the optical potentials
G.R. Satchler ,t’ Isospin and macroscopic models 231
alone change the inelastic cross sections by 15% .) These numbers give some measure of the model dependence of the EWSR percentage deduced from the data. However, the angular distributions for all three models are very similar.
The monopole transition density (2.8) was used for the GMR. The predicted cross sections for 100% of the EWSR (2.9) are included in fig. 1; they are only 55% the magnitude of those given by the deformed optical model (3.23a), although the angular distributions are the same. Consequently, one would conclude from the data that the observed GMR requires about 180% of the EWSR.
This kind of discrepancy has been noted before *) for measurements on severai nuclei at 152 MeV. It may be an indication that we need to include a density dependence 22) in the elective interaction uO(s) which reduces its strength as the tr-particle penetrates the target nucleus. The integrand of the folding integral (3.5) includes considerable cancellation between interior and exterior contributions because of the change in sign of the monopole transition density (2.8). Density dependence would reduce this cancellation and enhance the transition potential near the surface. On the other hand, the folded transition potential for the GQR could be reduced by density depenence, hence increasing the EWSR percentage deduced from the data. Calculations are under way to test this suggestion.
The GT model (2.12) was used for the GDR, and the results shown in fig, 1, Again the cross sections are almost entirely due to CouIomb excitation. If the purely hadronic results for the deformed potential model, shown in fig. 1 for AR = 0.488 fm, are scaled as (AR)* to the same AR = 0.185 fm used here for the transition density (a reduction by a factor of 7’1, we find almost the same cross sections as obtained from the folding model. The presence of density dependence in the interaction o’(s) would be expected to enhance the hadronic dipole cross section somewhat because, like that for the monopole, the isoscalar GT transition density also changes sign near the nuclear surface.
4.3. FOLDING MODEL FOR m +*“Pb
Calculations were made for excitation of the lowest 3- state at E, = 2.615 MeV, as well as for the giant monopole, dipole and quadrupole resonances, by a-parti- cles 20) of 172 MeV. The 3- was also studied at 139 MeV, where more extensive data *3> provide a more stringent test. The distorted waves were generated from folded potentials using the same interaction (4.1) with v = 36.4 MeV, w = 22.5 MeV. (The “best fit”*) to the 139 MeV elastic data for “*Pb led to u =40.5 MeV, w = 26.0 MeV. Use of those values here results in inelastic cross sections just a few percent larger; the somewhat greater absorption in the optical potential is off-set by the larger coupling strength.)
Again we used Woods-Saxon (Fermi) shapes for the ground-state density distribu- tions. We adopted a radius c = 6.67 fm and di~useness a = 0,545 fm, corresponding to a measured 2t) RMS radius of 5.55 fm for the total nucleon distribution p(r). The
232 G.R. Satchler / fsospin and macroscopic models
neutron RMS radius 2’) is about 0.18 fm larger than the proton one, so we chose
c, = 6.80 fm, cP= 6.54 fm with a,= up = a. This gives y = 0.276 in eq. (2.4), and
pn/pp = 1.374 at small radii instead of N/Z = 1.537.
Although excitation of the lowest 3- state only exhausts about 20% of the isoscalar
octupole sum rule, its transition density is very surface peaked 24). It is well represen-
ted, overall, by a collective model form factor, although we have not made a detailed
comparison. We normalize to the measured 24) B(E3)T = 0.612 e2b3. Then, using the
above proton distribution in eqs. (3.15a) and (3.15b), we obtain ty3 = 0.0150 fin-’
for the Tassie form (2.4) or S3 = 0.806 fm for the BM form (2.7). This corresponds
to 19% of the sum rule (2Sa), or 18% of the sum rule (2.5b). The inelastic cross
sections were calculated assuming the same a3 or S3 for the total density. The Tassie
form provides excellent agreement with the 139 MeV measurements 23), comparable
to that shown previously *), while the BM form gives cross sections only about 3 as
large. This lends further support to the use of the Tassie prescription. The same
difference between BM and Tassie was observed at 172 MeV, although in this case
the less extensive measured cross sections reported “1 for this energy are about 30%
smaller than the Tassie predictions. This discrepancy is not understood. Such a
strong energy dependence of the effective interaction is not expected; for example,
the same Tassie form also provides good agreement ‘) with other measurements 25)
made at 96 MeV.
Both Tassie and BM transition densities were also used for excitation of the giant
quadrupole resonance at E, = 10.9 MeV. The sum rule (2.5a) gives LYE = 0.0864 for
the Tassie form, while (2.5b) gives SZ = 0.625 fm for the BM form (or pz = 6,/6.67 =
0.0937, somewhat larger than Q). From eq. (3.12), the corresponding B(E2)T are
0.704 and 0.728 e2b2, respectively. The calculated cross sections are shown in fig. 2.
The BM prescription gives cross sections about 15% smaller than the Tassie one.
(This difference is less marked than for the octupole because the transition densities
differ by a factor of r2 for I= 3 but only by P for 1 = 2.)
The data for this excitation ““) have an angular distribution that is in good
agreement with the theoretical one for f =2. Comparison of the peaks of the
theoretical curves with the measured crosss sections implies between 75 and 100%
of the sum rule for Tassie and about 100% for BM. However, it has been argued **)
that there are significant amounts of Z=4 contributing in this same region of
excitation energy. If these are subtracted, the quadrupole sum rule percentages are
reduced to about 60 and 70, respectively. (Under the same circumstances, we find
that use of the deformed potential model (3.22) implies about 75% depletion of
the sum rule.)
The sum rule limit (2.9) for the monopole gives an amplitude LYE = 0.0543. The
cross sections obtained from this and the transition density (2.8) are included in
fig. 2. Just as for *16Sn, these cross sections predicted by the folding model are only
about 60% those obtained 4*9) using the deformed potential model (3.23a). They
also deviate from the measured “) cross sections by about the same amount. As we
G. R. Safchler / Isos~in and ~ae~osc~pjc models 233
I I I I =
“*Pb+ a
472 MeV 1
EX=i0.9 MeV 1=2 -T
-- BM
- h+c
0 5 (0 45 20
‘&.,,. (%l
Fig. 2. Inelastic scattering of 172 MeV cu-particles from ‘**Pb. The data are from the revised analysis of the second of refs. ‘O). The cross sections were calculated assuming 100% depletion of the corresponding
EWSR. The upper curves refer to the giant quadrupole resonance at 10.9 MeV and compare the predictions
using the Tassie (T) and BM transition densities. The lower curves refer to the combined giant monopole
and dipole resonances near 13.8 MeV. Curves for l= 1 show the hadronic (h) and Coulomb (c) excitation
contributions separately, as well as their coherent sum (h-kc). The hadronic term was calculated using
cn -. cP = 0.26 fm.
234 G.R. Satchler / Isospin and macroscopic models
shall see, only a few percent of this latter discrepancy can be attributed to excitation
of the GDR. The theoretical angular distribution is in good agreement with the
measured one, except that the minima are filled in for the latter; this is mainly due
to excitation of the GDR. Our angular distribution also agrees well with those
generated from the deformed potential model 4*9).
The GT model (2.12) was used for the dipole resonance. Our choice of density
parameters has c,-c,=O.26 fm, corresponding to y =0.276 in eq. (2.14). (The
hadronic cross section is closely proportional to y*.) The sum rule limit (2.13) gives
(or = 0.0950 fm and then B(El)T = 0.535 e*b. Figure 2 shows the individual hadronic
and Coulomb excitation contributions, as well as their coherent sum. The Coulomb
excitation dominates over the hadronic one by one to two orders of magnitude.
(Note that distortion of the trajectories by the nuclear optical potential, as well as
the finite size of the charge distribution, causes this Coulomb excitation to deviate
very strongly from the “point charge” values, both in magnitude and in angular
distribution.) The constructive interference between the two terms modulates the
angular distribution slightly. The resulting dipole cross sections are much smaller
than those for the monopole. When the two are added, the peak cross sections are
enhanced by a few percent, and the dipole fills in the minima of the monopole
angular distribution. Again, we conclude that excitation of the GDR cannot obscure
that of the GMR or confuse its interpretation “).
5. Conclusions and discussion
The purpose of this paper was to describe a framework within which to discuss
the excitation of collective motions in nuclei, and in particular to clarify and to
distinguish the use of the terms isoscalar and isovector as applied to the oscillation
itself and to the probe which is producing the excitation. As an aid to this, we
introduced a quantum number IJ which distinguishes oscillations in which the
neutrons and protons move in phase (n = 0, so-called “isoscalar”) and those in
which they move out of phase (7 = 1, so-called “isovector”). This n is to be
distinguished from a quantum number t which labels the probe (t = 0, isoscalar, or
t = 1, isovector). While we expect the strongest response when t = 7, it is possible
to have responses when t # v, especially for nuclei with N # 2. These were discussed
carefully.
We used simple models for the collective motions to illustrate these matters; these
are models frequently invoked in analyses of inelastic scattering data. Except for
the BM model (2.7), the corresponding transition densities are those obtained lo) if
the oscillation exhausts the linearly energy-weighted sum rule for the multipole
operator r’Y;“( 13,4) (or r* for 1 = 0). These multipole fields play an important role
in our understanding of nuclear collective motions 12), but they are not unique, and
the sum rules alone cannot tell us which modes, if any, are eigenstates of actual
nuclei. Further, hadronic probes tend to emphasize the nuclear surface even more
G.R. Sarchler / Isospin and macroscopic models 235
strongly than the radial factor r’. Nonetheless, the use of these models has been
very successful in correlating data from a variety of probes, both electromagnetic
and hadronic. In addition, we do observe excitations in nuclei, the giant resonances,
each of which appear to exhaust the greater part of one of these EWSR.
Particular attention was paid to the excitation of “isovector” (n = 1) states by
isoscalar (t = 0) probes, especially the giant dipole resonance. Once a model is
chosen for such a state, it determines both the t = 0 and the f = 1 response; it is not
necessary to introduce another model for the t f 7 response 20,26). These excitations
will usually be dominated by Coulomb excitation.
We also examined carefully the conditions under which the Coulomb and hadronic
contributions would interfere constructively or destructively; there has been some
confusion over this in the literature 4,5,7*8).
The deformed potential model and the folding model were compared in a few
cases. The predicted cross sections agree to within 20 or 30% for 132 shape
oscillations. The results of using the BM or Tassie transition densities agree to about
the same accuracy. Analysis of measurements for the 3- state at 2.615 MeV in “‘Pb
suggest that the Tassie form is the better choice. On the other hand, the folding
model predicts cross sections for the 1 = 0 breathing mode that nearly a factor of 2
smaller than given by the deformed potential ansatz. They are smaller than measured
cross sections by a similar amount. We suggest that this may be due to the neglect
of density dependence in the effective interaction. Of course, an alternative explana-
tion may be that the transition density (2.8) itself is not entirely appropriate; the
predicted cross sections are sensitive to quite small changes ‘). For example, a more
general transition density is obtained ‘07’2,‘3) for a state which exhausts the EWSR
for the operator j,(qr) rather than the r2 used to generate eqs. (2.8) and (2.9); these
two equations are obtained in the limit of small q.
Uncertainties in the optical potentials and unce~ainties in interpretation of the
sum rule limits can also lead to variations -10 to 20% in cross sections. These
results give some indication of the uncertainties that can be present in the percentage
depletions of the EWSR that are obtained from analyses of the inelastic scattering
data.
I am indebted to J.R. Beene and H.P. Morsch for helpful conversations, and to
K. Nakayama and S. Shlomo for communicating the results of their work 9).
Note added in proof: It has been customary 1-4,8,9*19) to equate LYE in eq. (3.23a)
for the deformed potential model of GMR excitation with the (Ye of the transition
density (2.8), as we did to obtain the results presented here, fig. la, for cy + ‘16Sn.
However, this choice is not consistent (J. Barrette, private communication) with the
philosophy advocated for other multipoles that it is the displacement of the nuclear
surface, i.e. the deformation length, that remains invariant. In accord with this, it
would be a reasonable approximation to replace LY* in the transition potential (3.23a)
by ao(c/R), where c is the density radius and R is the potential radius. In DWBA,
236 G.R. Sarchler / Isospin and macroscopic models
this would reduce the cross section predicted for the GMR by the factor (c/I?)“. This factor is 0.63 for CY + “%I with the parameter values used here, and reduces the cross sections from the deformed potential model to be close to those obtained from the folding model; both are then smaller than the measured values.
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