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Isoscalar spin response functions in the quasielastic scattering region
Nuclear Physics North-Holland
A510 (1990) 106-124
ISOSCALAR
SPIN RESPONSE
FUNCTIONS
SCAmERING
IN THE QUASIELASTIC
REGION
T. SHIGEHARA Computer Centre, University of Tokyo, 2-11-16, Yayoi, Bunkyo-ku, Tokyo 113, Japan K. SHIMIZU
and
A. ARIMA
Department of Physics, Faculty of Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113, Japan Received 27 April 1989 (Revised 18 September 1989) Abstract: The role of particle-hole correlations in isoscalar spin response functions in the quasielastic scattering region is examined within the continuum RPA. Both spin-longitudinal and spin-transverse response functions are appreciably modified from the free response function without correlations, due to the repulsive central force and the exchange term of the tensor force in the particle-hole interaction. The latter force also induces an appreciable difference between the two isoscalar spin response functions. It is shown that the ground-state correlation (the backward-going contribution) due to the tensor force is important in the large momentum transfer region and is not weakened, even at large energy transfer. The experimental ratio between the two spin response functions at 4 = 1.75 fm-’ observed in (p, p’) reactions is not largely modified around the quasielastic scattering peak by including particle-hole correlations in the spin-isoscalar mode. On the other hand, it is somewhat reduced in the large energy transfer region above the quasielastic scattering peak.
1. Introduction A recent
experiment
at LAMPF
I), measuring
a complete
set of polarization-
transfer observables in (p, p’) reactions (E, = 500 MeV) in the quasielastic scattering region at a large momentum transfer (q = 1.75 fm-‘), suggested that in the plane-wave impulse approximation the spin-longitudinal response function is not necessarily enhanced but rather suppressed in comparison to the spin-transverse response function. Considering the attractive (repulsive) nature of the isovector longitudinal (transverse) component in the particle-hole (ph) interaction, the longitudinal response function had been expected to be much enhanced compared to the transverse response function in nuclear matter ‘). In order to explain the discrepancy between the experiment and theoretical prediction, many theoretical calculations were promoted in finite nuclei within the random-phase approximation (RPA) 3-6), relying on the so-called ring approximation. These calculations have shown that the effects of nuclear finite geometry and proton distortion are important to diminish the difference between the two spin response functions, although the difference still remains, especially in the small energy transfer region. 0375-9474/90/$03.50 (North-Holland)
@ Elsevier
Science
Publishers
B.V.
107
T Shigehara et al. / Isoscalar spin response functions
In (p, p’) reactions, be excited. tudinal
Under
however,
isoscalar
the kinematical
and transverse
response
modes besides
condition
functions
the isovector
of the LAMPF
ones can also
experiment,
are given in terms of isovector
the longiand isoscalar
parts as follows, R,(q, w) = (3.62R,T=‘(q,
w)+ R:=‘(q,
w))/4.62,
R,(q, w) = (l.lSR,T=‘(q,
w)+R,T=‘(q,
w))/2.15.
Here the coefficients
in eq. (1) are determined
by a phase-shift
(14
analysis
of nucleon-
nucleon elastic scattering data ‘). Eq. (1) shows that, although the isovector part dominates over the isoscalar part in the longitudinal mode, the isoscalar contribution is comparable to the isovector one in the transverse mode. So, the isoscalar as well as the isovector contribution must be carefully treated, especially for the transverse mode. All theoretical calculations done so far, however, depended on the assumption that the isoscalar spin response functions in eq. (1) are given by the free ones without correlations. This assumption seems to have no firm theoretical foundation. The most persuasive reason which justifies this assumption might be the fact that the Landau parameter (I = 0) in the spin-isoscalar (central) mode of the ph interaction is small compared to that of the spin-isospin mode (so-called g’). Nevertheless, one should be cautious in applying this argument straightforwardly to the case of the quasielastic scattering region, because the kinematical situation is different from that in the Landau limit. In addition, the role of the tensor force in the ph interaction should be carefully examined, because it is the tensor force that gives rise to differences between the longitudinal and transverse response functions. One should keep in mind that, although the direct term of the tensor force in the ph interaction is weak in the isoscalar mode, this is not necessarily the case for the exchange term. In fact, in the resonance region (w s 30-40 MeV), the spin-isoscalar response function
in the long-wavelength
limit is known
to be considerably
influenced
by the
exchange term of the tensor force “). From the practical side, one reason why ph correlations in the isoscalar spin response functions have been ignored so far is that there has been no trustworthy numerical method to assess the effects of ph correlations in the spin-isoscalar mode in the quasielastic scattering tion, which can be handled
region. In the spin-isospin mode, the ring approximanumerically without much effort, works well ‘) except
for some ambiguity in treating the so-called screening effect lo). That is, in the exchange channel of the ph interaction not only the central force is really of short range, but also the tensor force is weak enough to be neglected in the spin-isospin mode. The validity of the ring approximation is, however, not obvious in the spin-isoscalar mode. In other words, we do not have any theoretical prescription to derive the ph interaction to be used in the ring approximation in this mode from an effective interaction based on a microscopic theory. This means that, for a quantitative estimate of the ph correlations in the spin-isoscalar mode, it is necessary
108
T. Shigehara et al. / Isoscalar spin response jiuncrions
to go beyond
the ring approximation
and to include
explicitly
the exchange
term
of the ph interaction. Recently,
we have developed
in which the exchange can be exactly method,
treated
to estimate
a numerical
method
term of the ph interaction “). The main
purpose
based
on the continuum
as well as the continuum of this paper
the role of ph correlations
is, according
in the spin-isoscalar
RPA effects to this
mode
in the
quasielastic scattering region. A special emphasis is put on the role of the exchange term of the tensor force in this kinematical region. Some specific features of the tensor correlation at large momentum and energy transfers will be clarified. Also, we will examine to what extent the ph correlations in the spin-isoscalar mode can resolve
the puzzle
on the ratio between
the longitudinal
and transverse
response
functions. This paper is organized as follows. A summary of the necessary formulae and adopted parameters in this paper is given in sect. 2. Numerical results and discussions are presented in sect. 3. In subsect. 3.1, the isoscalar longitudinal and transverse response functions are calculated in the continuum RPA. It will be shown that an appreciable difference between the two spin response functions is caused by the exchange term of the tensor force. In order to get a qualitative understanding of the role of the exchange term of the tensor force in the quasielastic scattering region, some close arguments are given from a perturbative point of view in subsects. 3.2 and 3.3. In the subsect. 3.2, some formulae for the nuclear response function in nuclear matter are given in first-order perturbation theory, according to which, in subsect. 3.3, the isoscalar spin response functions are calculated, taking into account only the exchange term of the tensor force in the ph interaction. The effects of ph correlations in the spin-isoscalar mode on the ratio between the two spin response functions in eqs. (la) and (lb) are discussed in subsect. 3.4. Finally, a summa~ and conclusion
are given in sect. 4.
2. Formalism In this section, the numerical The nuclear
we will summarize
some formulae
and parameters
calculations in the following section. response function for a one-body local operator R(m)=
--!_ Im dr, dr, O+(r,)ZI(r,, 7T I
necessary
O(r)
r,, ri, r,; o)O(r,)
for
is given by ,
(2)
where w is the energy transfer and fl is the polarization propagator (the ph Green function). In eq. (2), spin and isospin variables are suppressed for simplicity of notation. In terms of the one-body Green function G(+‘( r, , r2;
~I=(4
T. Shigehara et al. / Isoscalar spin response functions
one can express the free polarization
propagator
without
fl’O’(r2, rh, rl, r{; w) =F [+th(rz)@(rL,
4;
109
ph correlations
as “)
w+ dh(rl)
++th(4)G(+)(rl, r2; -0 + ~~hh(rS)I,
(4)
where $,,(r) is an eigenfunction of the single-particle hamiltonian Ho with an eigenvalue &h (h denotes a hole state). The method of getting the RPA polarization propagator from the free polarization propagator and a given local ph interaction has been explained in detail in the previous paper “). Here we just stress that the exchange term of the ph interaction as well as the continuum effect of the particle state are exactly taken into account in the calculated RPA polarization propagator. The isoscalar spin-longitudinal and spin-transverse operators are given by = c . q eiq”,
G,“‘(r)
(54
0,7=0(r)=d$uXqeiq’r,
(5b)
where q is the momentum transferred by the external field. We calculate the response functions for 160. The one-body
potential
and the
residual interaction are exactly the same as in ref. ‘). That is, the one-body potential is a square-well type, the depth and range parameters of which are V, = -51.5 MeV and R. = 3.01 fm, respectively, and the residual interaction consists of the central and tensor components of the G-matrix ‘*) at normal density ( kF = 1.36 fm-‘), which is derived from a version of the Bonn potential i3).
3. Numerical In this section, in the quasielastic
results and discussion
the effects of ph correlations scattering
parts. First, in subsect. functions are calculated
region
on isoscalar
are examined.
spin response
This section
is divided
3.1, the isoscalar longitudinal and transverse according to the continuum RPA. In subsect.
functions into four response 3.2, some
formulae for the nuclear response function necessary for a further investigation are given, based on first-order perturbation theory in nuclear matter. According to these formulae, the role of the exchange term of the tensor force in the quasielastic scattering
region,
which
causes
an appreciable
difference
between
the two spin
response functions, is examined in subsect. 3.3. Finally, in subsect. 3.4, we will discuss the effect of ph correlations in the spin-isoscalar mode on the ratio between the longitudinal and transverse response functions.
3.1. RESULTS
OF THE
CONTINUUM
RPA
In this subsection, the role of ph correlations in the spin-isoscalar quasielastic scattering region is examined relying on the continuum
mode in the RPA.
T. Shigehara et al. / Isoscalar spin response functions
110
In fig. 1, the isoscalar response
is common consider
longitudinal
both to the longitudinal a one-body
of the spin-orbit
(dashed)
and
at q = 2 fm-’ are shown, together
functions
spin-orbit
potential
and transverse
potential.
causes
only
operators
It has been a slight
transverse
(dashed-dotted)
with the free one (solid), because
which
we do not
shown ‘) that the inclusion
difference
between
the two free
response functions in the quasielastic scattering region. It is seen from fig. 1 that the ph correlations induce an appreciable difference between the longitudinal and transverse response functions. Around the quasielastic scattering region (w L- q2/2M = 82.9 MeV, with M the nucleon mass), the transverse response function is larger than the longitudinal one, in contrast to the fact that the isovector longitudinal (transverse) response function is enhanced (suppressed) compared to the free one due to the attractive (repulsive) nature of the corresponding component in the ph interaction. In order to see the reason why the difference appears between the two modes, we show in fig. 2 the result obtained by leaving out the exchange term of the tensor force in the residual interaction. One can see that in this case there is no essential difference between the longitudinal and transverse response functions. This means that it is the exchange term of the tensor force which introduces the appreciable difference between the two spin response functions in fig. 1: It has a repulsive (attractive) effect on the longitudinal (transverse) response function. This effect of the tensor force on the isoscalar been pointed out in ref. 14). W e will examine
(non-energy-weighted) sum rules has the role of the exchange term of the
0.6
0 60
100
160
mo
260
o(MeV) Fig. 1. The isoscalar within the continuum
longitudinal (dashed line) and transverse (dashed-dotted line) response functions is also RPA at q = 2 fin-’ in 160. The free response function without correlations shown for comparison (solid line).
T. Shigehara et al. / Isoscalar spin response functions
111
q=Z.Ofm-’ i
0 60
100
160
mo
a60
o(MeV) Fig. 2. The same as in fig. 1, except that the exchange term of the tensor interaction.
tensor force in detail in the following subsections. nature of the central force causes a considerable
force is not included
in the ph
Fig. 2 also shows that the repulsive hardening (i.e. it shifts the position
of the quasielastic scattering peak in the free response function to the higher energy region). The small difference between the longitudinal and transverse response functions in fig. 2 is attributed to the direct term of the tensor force, which is weak in the isoscalar mode, contrary to the case of the isovector mode. From the above consideration, one can understand the global features of the RPA response functions in the quasielastic scattering region in fig. 1. They are controlled by two components in the ph interaction; the repulsive central force and the exchange term of the tensor force. Although the former gives a repulsive contribution both to the longitudinal and transverse response functions, the effects of the latter are quite different in the two modes and lead to a clear difference between the two response functions. It is interesting to examine the role of ph correlations also in the resonance region (w 6 30-40 MeV). In fact, one can observe, comparing fig. 1 with fig. 2, that the exchange
term of the tensor force has a considerable
influence
on the RPA response
functions, as pointed out in ref. “) although the momentum transfer considered there is much smaller than that in the present calculation. Here, however, we do not go into detail on this subject. This is because, in the continuum RPA, some careful numerical treatments are demanded in estimating the strength in this kinematical region: There are in the response function some peak structures with a narrow width and yet an appreciable strength. In fact, there are even zero-width peak structures in the free response function in w s 35 MeV, which are not explicitly shown in figs. 1 and 2. (In the present calculation, the response functions have been calculated at
T. Shigehara et al. / Isoscalar spin response functions
112
all energies
in steps of 1 MeV.) So, one must be careful
in the resonance progress
region
due to an elaborate
The isoscalar
the continuum
numerical
spin response
fig. 3. The qualitative region
within
features
functions mentioned
RPA.
in calculating
the strength
The calculation
is now in
method. at q = 1.5, 1.75 and 2.25 fin-’
are shown in
above are valid in the quasielastic
scattering
also in these cases.
3.2. SOME
FORMULAE
IN
FIRST-ORDER
PERTURBATION
THEORY
IN
NUCLEAR
MATTER
In the previous subsection, we have mentioned the importance of the exchange term of the tensor force for isoscalar spin response functions in the quasielastic
76
50
100
LZS
L-50
o(MeV)
L 0.6
0.6
-
I
./- -,.
q=2.25fm-l
-
7
7
%o., P s -
%0.4 2 B -
5go..?, 0’ :
5
-
go..?. 0” :.
-
e:
02
0
-
0 IO0
160
-
zoo
100
Fig. 3. The isoscalar
spin response
200
300
w(MeV)
w(MeV) functions at different momentum are the same as in fig. 1.
transfers
are shown. Other indications
T. Shigehara et al. / Isoscalar spin response functions
113
scattering region. In order to understand its role in a more intuitive way, we will give some more detailed discussions within first-order perturbation theory in nuclear matter. It is considered that the nuclear matter theory is trustworthy enough for getting gross features of the quasielastic scattering, because the shell effects in finite nuclei are not expected to be large in this kinematical region. In this subsection, we will summarize some necessary formulae for the following. In first-order perturbation theory in nuclear matter, the nuclear response function for a local operator can be written as w) .
R(q, w) = R”‘(q, w)+AR(q,
(6)
The first term on the right-hand side in eq. (6) is the free response function, which can be expressed in terms of the Lindhard function 17”‘(q, w) including the spinisospin factor 4, R”‘(q, OJ)= -L
Im n’“‘( q, 7T
w
).
(7)
The second term in eq. (6) represents the effects of ph correlations and for the real ph interaction is composed of two parts,
AR(q, 0) = ARdq, 0) + ARdq, 0) .
(8)
The first term of the right-hand side in eq. (8) is the forward-going the response function, including only forward ph propagations, AR,(q,
W) = -;
1
1 Im (2m)6
2 ARB(% W) = -;
contribution
q, w),
I
Here, the forward and backward ph propagators factor 4) are given by
(9)
involving the backward ph
1 Im (27T)6 dh, dk, &(h,,
x(&,(q)+ VdQ~))n~(hz,
TT,(h, Q,0) =4
q, w)
q, ~1.
(10)
(also including the spin-isospin
~(l~+ql-WV,+\) w-(h+q)2/2M+h2/2M+i7j
rr,(k q,o) = &(k
4,
to
d& dh, n~(hi, 4, w)
x(Vdq)+ VE(QF))G(~~, and the second term is the backward-going propagator,
contribution
-w) ,
’
(114
(lib)
respectively. In eqs. (9) and (lo), we have assumed that the ph interaction is local (in the direct term). The direct momentum transfer q is common both to the forward-
T. Shigehara et al. / Isoscalar spin response functions
114
and backward-going tum transferred hand,
contributions
by the external
the exchange
different
and, in nuclear
momentum
transfers,
QF and
with the momen-
of hole momenta.
On the other
QB in eqs. (9) and
(lo),
are
from each other and given by
QF=hl-h*, Notice that, although the exchange is not the case for QB. 3.3. ROLE
matter, identical
field, independent
OF THE
SCATTERING
EXCHANGE
QB=h,+h2+q. momentum
TERM
OF THE
transfer
TENSOR
(12) QF is independent
FORCE
IN THE
of q, this
QUASIELASTIC
REGION
In this subsection, the role of the exchange term of the tensor force in the quasielastic scattering region is examined from a perturbative point of view. Especially, the importance of the backward-going term V,( QB) in eq. (lo), in which the time direction of the ph propagation is changed by the ph interaction, is stressed. As mentioned in subsect. 3.1, the difference between the isoscalar longitudinal and transverse response functions is mainly attributed to the exchange term of the tensor force. In order to see how it influences both modes in a different manner, we consider the isoscalar spin response functions obtained by including only the exchange term of the tensor force in the ph interaction. That is, in eqs. (9) and (lo), the direct term V, is discarded and the exchange term V, contains only the tensor force. In fig. 4, the isoscalar longitudinal and transverse response functions at
0
50
100
160
200
o(MeV) Fig. 4. The isoscalar longitudinal (dashed line) and transverse (dashed-dotted line) response functions in first-order perturbation theory in nuclear matter at normal density (k, = 1.36 fm-‘). Only the exchange term of the tensor force is taken into account in the ph interaction. The free Fermi-gas response function is also shown for comparison (solid line).
T, Shigehara et
Ql. /
~SOSCdUr
Spin
response
11s
fUflCiionS
q = 2 fm-’ in nuclear matter at normal density (kF= 1.36 fm-‘) are shown. It is noticed that, in general, the longitudinal and transverse modes are mixed due to the exchange term of the tensor force. So, it is only within first-order perturbation theory that it is unnecessary to consider the mixing between the two modes. One can see from fig. 4 that the ph correlations quench (enhance) the free response function for the longitudinal (transverse) mode in almost the whole range of energy transfer, which is consistent with the results for a finite system in subsect. 3.1 (compare fig. 1 with fig. 2). This feature is mainly owing to the backward-going contribution in eq. (10). In order to see this, we show the case that only the forwardor backward-going contribution in eq. (8) is taken into account in figs. 5a and 5b, respectively. One can observe from fig. 5a that, compared to the free response function, the longitudinal (transverse) response function is enhanced (suppressed) in the small energy transfer region, and suppressed (enhanced) in the large energy transfer region. It should be noticed that the forward-going contribution does not contribute to the non-energy-weighted sum rule. Fig. 5b shows that, on the other hand, the backward-going contribution suppresses (enhances) the free response function for the longitudinal (transverse) mode in the whole range of energy transfer. Fig. 5 also indicates that the tensor correlation in the forward-going term is weaker than in the backward-going term, and that for both modes the forward-going part tends to cancel the backward-going part at small energy transfer, while at large energy transfer the corresponding contributions are coherent. In order to get a more deep understanding for the role of the tensor correlation, we will rely on a qualitative argument in the following. The tensor force in the nucleon-nucleon interaction in free space is mainly due to the exchange of isovector mesons such as pi and rho mesons. Reflecting this fact, the tensor interaction in the
0.9
03
0.2s
0.26
-
I
0
so
100
150
I
,
,
I
,
,
,
,
,
ON
-
,
,
,
,
)
,
,
,
q=Z.Ofm-i
,
,
,
-
.Eal
o(MeV) Fig. 5. The same as in fig. 4, except that only the forward-going (backward-going) into account in eq. (8) in fig. 5a (5b).
contribution
is taken
116
nuclear
T. Shigehara
medium
et al. / Isoscalar spin response functions
is dominated
So, we may approximately direct tensor interaction,
by the isospin-dependent
neglect
G:(q)
part (in the direct term).
the isospin-independent
part and write for the
* T2,
= H’(q)S,,(~)T,
(13)
where S,,(4)
=
3u, . 4az. 4^--Ul. u2,
(14)
with q the direct momentum transfer. In the ring approximation, where only the direct term of the ph interaction is considered, the tensor force in eq. (13) is effective only for isovector spin response functions. If one takes the exchange term into account, however, it is effective for isoscalar spin response functions the help of the explicit expressions for the exchange operators, P,=+(l+u,*uz),
as well. With
P,=f(l+l,*TJ,
(15)
or the relations P&,(B)
=
S,,(4)
9
(16)
P7T1*T2=+(3-T1*T2),
(17)
one can obtain G:(Q)
= -P,P,G;(Q)
= -H’(Q)S&);(3
-?
. 72))
(18)
where Q is the exchange momentum transfer; Q = QF for the forward-going term and Q = Qe for the backward-going term. Together with the ph interaction in eq. (18), if the quantization axis is taken parallel to q, one obtains for the exchange term in eq. (9) or (lo), V,(Q)
= V;,;“(Q)
= a’=“H’(Q)P2(cos
0,)
(i = L, T) ,
(19)
where ffL
for the isoscalar
longitudinal
T=O= -3,
and transverse
&‘O=$, channels,
(20) respectively.
In eq. (19), 8,
is the angle between q and Q, and P2(x) is the second-order Legendre polynomial (P2(x) =+(3x2- 1)). In order to obtain a rough estimate of eq. (19), we introduce an angle-average of the Legendre polynomial at fixed q as follows, (P2(cos
Q)),
= I
dh, dh2 P,(cos 0,) /I h, ,hzkF
dh, dh2.
(21)
h, .&
In fig. 6, the average (P2(cos f3,)), is shown both forthe forward- and backward-going terms as a function of the magnitude of the momentum transfer q. It is seen that the average for the forward-going term becomes less important than that for the
T. Shigehara et al. / Isoscular spin response functions
0
2
1
3
117
4
q/b Fig. 6. The angle-average ofthe second-order Legendre polynomial in eq. (21) as a function of momentum transfer in units of the Fermi momentum k,. The dashed (dashed-dotted) line corresponds to the case of the forward-going (backward-going) term.
backward-going term if q is larger than typically k,. This can be easily understood from the following: Because the momentum transfer QF in eq. (12) is independent of q and merely the difference between two hole momenta, the average of the second-order Legendre polynomial in eq. (21) becomes small for large q for the forward-going term. On the other hand, the momentum transfer Qe in the backwardgoing term in eq. (12) is essentially if q is large enough. to the integration
Thus,
governed
the integrand
in a coherent
by q for any choice of the hole momenta in the numerator
way. It is noticed
of eq. (21) contributes
that, in fact, the average
for the
forward-going term becomes zero for q > 2k, without the Pauli blocking, and that on the other hand, the average for the backward-going term is nearly equal to one for q %=k,. This is the reason why the backward-going contribution largely modifies the free response function in fig. 5b, although the forward-going contribution does not induce
a strong ph correlation.
Also, together
with the fact that H’(Q)
is negative
(-64.7 MeV . fm3 at Q = Qe- q = 2 fm-i), one can notice from eqs. (19) and (20) that the backward-going term quenches (enhances) the free response function for the longitudinal (transverse) mode, and that the amount of the induced modification for the longitudinal response function is roughly twice as large as for the transverse one. These features are well reproduced in the actual numerical calculation in fig. 5b. One might raise the question, however, why the backward-going contribution in eq. (10) plays an important role in the large energy transfer region in spite of the energy denominator in the backward ph propagation. In order to answer this question, we refine the angle-average of the Legendre polynomial in eq. (21) and
T. Shigehara et al. / Isoscalar spin response functions
118
introduce
another
angle-average,
which is energy-dependent,
(P*(cos@,Nq,,=
dhl dhz 6
(h+qY
0 ---+=
as follows,
h:
2M
Ih,+ql,lh2++kF
X
[Ib,+dh2++kF )I ~-(h,~+* 2M
h,.h,
In eq. (22), we have taken in eq. (10) denominator
into account
2M
the fact that the forward
--I.
(22)
ph propagation
must be on-shell. It is noted that, if both the numerator and the in eq. (22) are integrated with respect to w, we obtain just the average
(Pz(cos 6,)), of eq. (21). In fig. 7, the numerical result of eq. (22) is shown for the backward-going term as a function of w for fixed q = 2 fin-‘. It can be seen that the average (PJcos O,)),, increases with increasing energy transfer, because in the large energy transfer region the on-shell condition favors the kinematical condition that the hole momentum h, aligns to the direction of q. This ensures the importance of the backward-going contribution even in the large energy transfer region, although it is weakened by the energy denominator in the backward ph propagator in eq. (10). This is clear also from a simple estimate. Under the following two assumptions, one can relate the average (P2(cos 6,)),, to the ratio between the backward-going contribution in eq. (10) and the free response function in eq. (7). If one approximates the energy denominator -w-(h + q)*/2M + h2/2M in the backward ph propagator in eq. (lib)
by -w-q2/2M
and neglects
0
50
the Q-dependence
100
160
of H’(Q)
in eq. (19),
200
o(MeV) Fig. 7. The angle-average of the second-order Legendre polynomial in eq. (22) in case of the backwardgoing term as a function of energy transfer w for fixed q = 2 fm-‘.
T Shigehara et al. / Isoscalar spin response funcrions
119
then one gets, AR;,;‘(q, w)lR”‘(q, =-
w)
2
w -@$$+s)
(2wY
I
h,.hzkF 4
x G,YO< Qd
-w-(h2+q)2/2M+h;/2M
o -&k&+$)]
-’
F (i = L, T).
(23)
Using the following numerical values, 4 (2r)3
dhz = 0.154 fmm3
at
q =
2 fm-’ (k, = 1.36 fm-‘),
(24)
I hl
1b,++k,
and H’ = H’( QB = 2 fm-‘) = -64.7 MeV . fm3, the numerical result of the ratio in eq. (23) is shown for the longitudinal channel in fig. 8, compared to the exact ratio obtained from fig. 5b. Although, in the exact calculation, the isospin-independent part of the tensor force is also included in the ph interaction, it is, however, very
0
so
100
160
o(MeV) Fig. 8. The ratio of the backward-contribution in eq. (10) to the free Fermi-gas response function in the longitudinal mode at q = 2 fm-‘, using the approximate formula in eq. (23) (solid line). The exact ratio deduced from fig. 5b is also shown for comparison (dashed line).
T. Shigehara et al. / Isoscalar spin response functions
120
weak compared approximate
the large energy transfer
to the isovector
estimate
somewhat
transfer
QB is larger than
calculation.
One
region, q,
tensor force [ H( QB = 2 fm-‘) overestimates
the tensor
where a typical
leading
can see, however,
value of the exchange
to the weaker that
= 5.2 MeV . fm3]. The
correlation
tensor
the w-dependence
correlation
especially
in
momentum in the exact
in the exact
ratio
is
qualitatively well reproduced by the approximate estimate. This is due to the appropriate treatment of the w-dependence included in the average (PJcos eo)),, in eq. (23), which implies that the ph correlation induced by the exchange term of the tensor force is important in the backward-going contribution even at large energy transfer as well as at small energy transfer. Before closing this subsection, we will give a short comment on the role of the exchange term of the tensor force in isovector spin response functions. The tensor force in eq. (13) is also effective in the exchange channel in isovector spin response functions. Eq. (18) indicates, however, that the exchange term of the tensor force is not so important in the spin-isospin mode as in the spin-isoscalar one. In fact, we have shown in fig. 5 of the previous paper ‘) that the inclusion of the exchange term of the tensor force can lead only to a slight modification of the isovector spin response functions. Note that in fig. 5 of ref. 9), the small difference between the response function based on the RPA with the exchange term explicitly included and that within the ring approximation is essentially due to the exchange term of the tensor force. This is because the zero-range approximation for the exchange term of the central force is very good in the spin-isospin mode. Eq. (18) also indicates that, contrary to in the isoscalar mode, the inclusion of the exchange term of the tensor force enhances (suppresses) the longitudinal (transverse) response function in the isovector mode. Furthermore, similar to the isoscalar case, its effect on the longitudinal response function is roughly twice as large as that on the transverse
one. These features
3.4. RATIO BETWEEN
are well reproduced
THE LONGITUDINAL
in fig. 5 of ref. “).
AND TRANSVERSE
RESPONSE
FUNCTIONS
In this subsection, we will discuss the effects of ph correlations in isoscalar spin response functions on the ratio between the longitudinal and transverse response functions in eqs. (la) and (lb), based on the considerations of the previous subsections. Also for isovector spin response functions, we consider the effects of ph correlations due to the same method as described in the preceding section. The isovector spin operators are given by eq. (5) multiplied by r3. In fig. 9, the isovector longitudinal and transverse response functions are shown for the momentum transfer q = 1.75 fm-‘. These results correspond to those shown in fig. 5 of the previous paper 9), except that the momentum transfer was taken as q = 2 fm-’ in the previous calculation. Because the features of the isovector spin response functions in the RPA are
TO Shigehara
et al. / Isoscalar
spin response functions
121
0.8
0
so
100
160
8w
o(MeV) Fig. 9. The isovector within the continuum
longitudinal (dashed line) and transverse (dashed-dotted line) response functions RPA at CJ= 1.75 fm-’ in 160. The free response function is also shown for comparison (solid line).
well known and the characteristics of them in our model have been explained in detail in the previous paper, we restrict ourselves here to recalling some points relevant for the present discussion. First, delta-hole effects are not included in our calculation. Secondly, we do not introduce a phenomenological short-range repulsion (so-called g’ force), but take the G-matrix as the ph interaction. Because, as a well-known fact, the G-matrix alone does not give sufficient repulsion in the central part, the collectivity in the small energy transfer region is prominent in the longitudinal response function. In this sense, our calculation for the isovector spin response functions might be unrealistic. We concentrate, however, on examining how the effects of ph correlations in the spin-isoscalar mode affect the ratio between the longitudinal and transverse response functions. In fig. 10, the ratio between the longitudinal and transverse response functions is shown as a function of w at q = 1.75 fm-’ (solid line). For comparison, we show the result based on the assumption that the isoscalar spin response functions are taken to be the free ones without
ph correlations
(dashed
line). Here, we concentrate
on the quasielastic scattering region of w 340 MeV. (See subsect. 3.1.) It is seen that the ph correlations in the isoscalar spin response functions reduce the ratio appreciably in the large energy transfer region (just above the quasielastic scattering peak of w = q2/2M = 63.5 MeV). In the small energy transfer region, the ratio is not largely modified on the average by including the ph correlations in the isoscalar mode, although the w-dependence reflecting the shell structure is somewhat different in the two calculations in fig. 10. These features can be qualitatively understood as
122
T Shigehara
et al. / Isoscalar spin response functions
q=
20
40
60
1.75fm-’
80
1
loo
120
o(MeV) Fig. 10. The ratio between the longitudinal and transverse response functions in eqs. (la) and (lb), respectively, in the continuum RPA at 9 = 1.75 fm-’ in I60 (solid line). The dashed line corresponds to the case that the isoscalar spin response functions are the free ones. The experimental points are taken from ref. ‘).
follows: As apparent from eq. (l), the ratio is not sensitive to the isoscalar longitudinal response function, but to the transverse one. At large energy transfer above the quasielastic peak, both the central and tensor ph correlations enhance the isoscalar transverse response function, leading to an appreciable reduction of the ratio. On the other hand, at small energy transfer, the effect of the tensor correlation is quite different from that of the central repulsion in the isoscalar transverse mode. The former correlation enhances the response, while the central repulsion suppresses it. As seen from fig. 3b, the former is not strong enough to overcome the latter. (This is partly because the backward-going contribution due to the exchange term of the tensor force which enhances the transverse response function is weakened by the forward-going contribution below the quasielastic scattering peak, as was discussed from the perturbative point of view in the previous subsection.) As a result, the total ph correlations slightly tend to quench the isoscalar transverse response function and to raise the ratio in the small energy transfer region.
4. Summary
In this response with the the effect first-order
and conclusion
paper, we have examined the role of ph correlations in isoscalar spin functions in the quasielastic scattering region, using the continuum RPA exchange term of the ph interaction explicitly taken into account. Also, of the exchange term of the tensor force has been examined in detail in perturbation theory in nuclear matter. Furthermore, the effects of ph
T. Shigehara et al. / Isoscalar spin response functions
correlations transverse
in the spin-isoscalar response
functions
in (p, p’) reactions The repulsive large energy
mode on the ratio between at the momentum
transfer
123
the longitudinal
of q = 1.75 fm-’
and
observed
have been investigated. central
transfer
force in the spin-isoscalar
region
both in the longitudinal
mode shifts the strength and transverse
response
to the func-
tions. Although the tensor force is weak in the direct term, it has, however, considerable effects in the exchange term comparable with the central force, leading to a quenching (enhancement) of the longitudinal (transverse) response function. As a net result, the isoscalar spin response function is appreciably different from the free one without correlations, both for the longitudinal and transverse modes. Below the quasielastic scattering peak, the longitudinal response function is significantly quenched as compared to the free one. On the other hand, the transverse response function is slightly quenched in the small energy transfer region and appreciably enhanced above the quasielastic scattering peak. Concerning the effect of the exchange term of the tensor force, some close examinations based on first-order perturbation theory in nuclear matter have shown that the backward-going term plays an important role in the quenching (enhancement) of the isoscalar longitudinal (transverse) response function. Taking the average of the second-order Legendre polynomial in an appropriate way, we have shown the reason why the importance of the backward-going term is not weakened even in the large energy transfer region. The forward-going term which is recognized not to be important compared to the backward-going term at large momentum transfer gives, roughly speaking, an attractive (repulsive) contribution to the longitudinal (transverse) response function. So, the effect of the backward-going term is somewhat weakened at small energy transfer in both modes. The ratio between the longitudinal and transverse response functions at q = 1.75 fm-’ is insensitive to the isoscalar longitudinal mode. Therefore, in the small energy transfer region around w = 40 MeV, the ph correlations in the isoscalar spin response functions cannot reduce the ratio, in spite of the considerable suppression of the isoscalar longitudinal response function, but even tend to raise the ratio due to the central repulsion in the transverse mode. In the large energy transfer region just above the quasielastic by the central and tensor For a further
scattering peak, however, the ratio is appreciably reduced ph correlations in the isoscalar transverse mode.
investigation,
we have to examine
the role of the 2p2h degrees
of
freedom on the spin response functions in the quasielastic scattering region. It has been indicated r5) that for low-lying collective states the tensor correlation in the spin-isoscalar mode is appreciably weakened by the screening effect. So, the role of this effect on the spin response functions should be carefully examined in the quasielastic scattering region. We are grateful tional
calculation
to W. Bentz for careful for this work
reading
was performed
of the manuscript.
The computa-
at the Computer
Centre
of the
124
T. Shigehara et al. / Isoscalar spin response functions
University
of Tokyo. The figures in this paper were drawn using the KEK-DRAWER
developed
in the National
Laboratory
for High Energy
Physics
(KEK).
References 1) T.A. Carey et al., Phys. Rev. Lett. 53 (1984) 144; L.B. Rees et al., Phys. Rev. C34 (1986) 627 2) W.M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A379 (1982) 429 3) W.M. Alberico, A. Molinari, A. De Pace, M. Ericson and M.B. Johnson, Phys. Rev. C34 (1986) 977; W.M. Alberico, A. De Pace, M. Ericson, M.B. Johnson and A. Molinari, Phys. Lett. B183 (1987) 135; Phys. Rev. C38 (1988) 109 4) Y. Okuhara, B. Castel, I.P. Johnstone and H. Toki, Phys. Rev. C34 (1986) 2019; Phys. Lett. B186 (1987) 113 5) T. Shigehara, K. Shimizu and A. Arima, Nucl. Phys. A477 (1988) 583 6) M. Ichimura, K. Kawahigashi, T.S. Jorgensen and C. Gaarde, Phys. Rev. C39 (1989) 1446 7) R.A. Arndt et al., Phys. Rev. D28 (1983) 97 8) J. Wambach, A.D. Jackson and J. Speth, Nucl. Phys. A348 (1980) 221 9) T. Shigehara, K. Shimizu and A. Arima, Nucl. Phys. A492 (1989) 388 10) 0. Sjiiberg, Ann. of Phys. 78 (1973) 39; S. Babu and G.E. Brown, Ann. of Phys. 78 (1973) 1 11) S. Shlomo and G. Bertsch, Nucl. Phys. A243 (1975) 507; G.F. Bertsch and S.F. Tsai, Phys. Reports 18 (1975) 125 12) K. Nakayama, S. Krewald, J. Speth and W.G. Love, Nucl. Phys. A431 (1984) 419 13) K. Holinde, Phys. Reports 68 (1981) 121 14) G. Orlandini, M. Traini and M. Ericson, Phys. Lett. Bl79 (1986) 201 15) K. Nakayama, Phys. Lett. B165 (1985) 239; S. Droidi, J.L. Tain and J. Wambach, Phys. Rev. C34 (1986) 345
A510 (1990) 106-124
ISOSCALAR
SPIN RESPONSE
FUNCTIONS
SCAmERING
IN THE QUASIELASTIC
REGION
T. SHIGEHARA Computer Centre, University of Tokyo, 2-11-16, Yayoi, Bunkyo-ku, Tokyo 113, Japan K. SHIMIZU
and
A. ARIMA
Department of Physics, Faculty of Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113, Japan Received 27 April 1989 (Revised 18 September 1989) Abstract: The role of particle-hole correlations in isoscalar spin response functions in the quasielastic scattering region is examined within the continuum RPA. Both spin-longitudinal and spin-transverse response functions are appreciably modified from the free response function without correlations, due to the repulsive central force and the exchange term of the tensor force in the particle-hole interaction. The latter force also induces an appreciable difference between the two isoscalar spin response functions. It is shown that the ground-state correlation (the backward-going contribution) due to the tensor force is important in the large momentum transfer region and is not weakened, even at large energy transfer. The experimental ratio between the two spin response functions at 4 = 1.75 fm-’ observed in (p, p’) reactions is not largely modified around the quasielastic scattering peak by including particle-hole correlations in the spin-isoscalar mode. On the other hand, it is somewhat reduced in the large energy transfer region above the quasielastic scattering peak.
1. Introduction A recent
experiment
at LAMPF
I), measuring
a complete
set of polarization-
transfer observables in (p, p’) reactions (E, = 500 MeV) in the quasielastic scattering region at a large momentum transfer (q = 1.75 fm-‘), suggested that in the plane-wave impulse approximation the spin-longitudinal response function is not necessarily enhanced but rather suppressed in comparison to the spin-transverse response function. Considering the attractive (repulsive) nature of the isovector longitudinal (transverse) component in the particle-hole (ph) interaction, the longitudinal response function had been expected to be much enhanced compared to the transverse response function in nuclear matter ‘). In order to explain the discrepancy between the experiment and theoretical prediction, many theoretical calculations were promoted in finite nuclei within the random-phase approximation (RPA) 3-6), relying on the so-called ring approximation. These calculations have shown that the effects of nuclear finite geometry and proton distortion are important to diminish the difference between the two spin response functions, although the difference still remains, especially in the small energy transfer region. 0375-9474/90/$03.50 (North-Holland)
@ Elsevier
Science
Publishers
B.V.
107
T Shigehara et al. / Isoscalar spin response functions
In (p, p’) reactions, be excited. tudinal
Under
however,
isoscalar
the kinematical
and transverse
response
modes besides
condition
functions
the isovector
of the LAMPF
ones can also
experiment,
are given in terms of isovector
the longiand isoscalar
parts as follows, R,(q, w) = (3.62R,T=‘(q,
w)+ R:=‘(q,
w))/4.62,
R,(q, w) = (l.lSR,T=‘(q,
w)+R,T=‘(q,
w))/2.15.
Here the coefficients
in eq. (1) are determined
by a phase-shift
(14
analysis
of nucleon-
nucleon elastic scattering data ‘). Eq. (1) shows that, although the isovector part dominates over the isoscalar part in the longitudinal mode, the isoscalar contribution is comparable to the isovector one in the transverse mode. So, the isoscalar as well as the isovector contribution must be carefully treated, especially for the transverse mode. All theoretical calculations done so far, however, depended on the assumption that the isoscalar spin response functions in eq. (1) are given by the free ones without correlations. This assumption seems to have no firm theoretical foundation. The most persuasive reason which justifies this assumption might be the fact that the Landau parameter (I = 0) in the spin-isoscalar (central) mode of the ph interaction is small compared to that of the spin-isospin mode (so-called g’). Nevertheless, one should be cautious in applying this argument straightforwardly to the case of the quasielastic scattering region, because the kinematical situation is different from that in the Landau limit. In addition, the role of the tensor force in the ph interaction should be carefully examined, because it is the tensor force that gives rise to differences between the longitudinal and transverse response functions. One should keep in mind that, although the direct term of the tensor force in the ph interaction is weak in the isoscalar mode, this is not necessarily the case for the exchange term. In fact, in the resonance region (w s 30-40 MeV), the spin-isoscalar response function
in the long-wavelength
limit is known
to be considerably
influenced
by the
exchange term of the tensor force “). From the practical side, one reason why ph correlations in the isoscalar spin response functions have been ignored so far is that there has been no trustworthy numerical method to assess the effects of ph correlations in the spin-isoscalar mode in the quasielastic scattering tion, which can be handled
region. In the spin-isospin mode, the ring approximanumerically without much effort, works well ‘) except
for some ambiguity in treating the so-called screening effect lo). That is, in the exchange channel of the ph interaction not only the central force is really of short range, but also the tensor force is weak enough to be neglected in the spin-isospin mode. The validity of the ring approximation is, however, not obvious in the spin-isoscalar mode. In other words, we do not have any theoretical prescription to derive the ph interaction to be used in the ring approximation in this mode from an effective interaction based on a microscopic theory. This means that, for a quantitative estimate of the ph correlations in the spin-isoscalar mode, it is necessary
108
T. Shigehara et al. / Isoscalar spin response jiuncrions
to go beyond
the ring approximation
and to include
explicitly
the exchange
term
of the ph interaction. Recently,
we have developed
in which the exchange can be exactly method,
treated
to estimate
a numerical
method
term of the ph interaction “). The main
purpose
based
on the continuum
as well as the continuum of this paper
the role of ph correlations
is, according
in the spin-isoscalar
RPA effects to this
mode
in the
quasielastic scattering region. A special emphasis is put on the role of the exchange term of the tensor force in this kinematical region. Some specific features of the tensor correlation at large momentum and energy transfers will be clarified. Also, we will examine to what extent the ph correlations in the spin-isoscalar mode can resolve
the puzzle
on the ratio between
the longitudinal
and transverse
response
functions. This paper is organized as follows. A summary of the necessary formulae and adopted parameters in this paper is given in sect. 2. Numerical results and discussions are presented in sect. 3. In subsect. 3.1, the isoscalar longitudinal and transverse response functions are calculated in the continuum RPA. It will be shown that an appreciable difference between the two spin response functions is caused by the exchange term of the tensor force. In order to get a qualitative understanding of the role of the exchange term of the tensor force in the quasielastic scattering region, some close arguments are given from a perturbative point of view in subsects. 3.2 and 3.3. In the subsect. 3.2, some formulae for the nuclear response function in nuclear matter are given in first-order perturbation theory, according to which, in subsect. 3.3, the isoscalar spin response functions are calculated, taking into account only the exchange term of the tensor force in the ph interaction. The effects of ph correlations in the spin-isoscalar mode on the ratio between the two spin response functions in eqs. (la) and (lb) are discussed in subsect. 3.4. Finally, a summa~ and conclusion
are given in sect. 4.
2. Formalism In this section, the numerical The nuclear
we will summarize
some formulae
and parameters
calculations in the following section. response function for a one-body local operator R(m)=
--!_ Im dr, dr, O+(r,)ZI(r,, 7T I
necessary
O(r)
r,, ri, r,; o)O(r,)
for
is given by ,
(2)
where w is the energy transfer and fl is the polarization propagator (the ph Green function). In eq. (2), spin and isospin variables are suppressed for simplicity of notation. In terms of the one-body Green function G(+‘( r, , r2;
~I=(4
T. Shigehara et al. / Isoscalar spin response functions
one can express the free polarization
propagator
without
fl’O’(r2, rh, rl, r{; w) =F [+th(rz)@(rL,
4;
109
ph correlations
as “)
w+ dh(rl)
++th(4)G(+)(rl, r2; -0 + ~~hh(rS)I,
(4)
where $,,(r) is an eigenfunction of the single-particle hamiltonian Ho with an eigenvalue &h (h denotes a hole state). The method of getting the RPA polarization propagator from the free polarization propagator and a given local ph interaction has been explained in detail in the previous paper “). Here we just stress that the exchange term of the ph interaction as well as the continuum effect of the particle state are exactly taken into account in the calculated RPA polarization propagator. The isoscalar spin-longitudinal and spin-transverse operators are given by = c . q eiq”,
G,“‘(r)
(54
0,7=0(r)=d$uXqeiq’r,
(5b)
where q is the momentum transferred by the external field. We calculate the response functions for 160. The one-body
potential
and the
residual interaction are exactly the same as in ref. ‘). That is, the one-body potential is a square-well type, the depth and range parameters of which are V, = -51.5 MeV and R. = 3.01 fm, respectively, and the residual interaction consists of the central and tensor components of the G-matrix ‘*) at normal density ( kF = 1.36 fm-‘), which is derived from a version of the Bonn potential i3).
3. Numerical In this section, in the quasielastic
results and discussion
the effects of ph correlations scattering
parts. First, in subsect. functions are calculated
region
on isoscalar
are examined.
spin response
This section
is divided
3.1, the isoscalar longitudinal and transverse according to the continuum RPA. In subsect.
functions into four response 3.2, some
formulae for the nuclear response function necessary for a further investigation are given, based on first-order perturbation theory in nuclear matter. According to these formulae, the role of the exchange term of the tensor force in the quasielastic scattering
region,
which
causes
an appreciable
difference
between
the two spin
response functions, is examined in subsect. 3.3. Finally, in subsect. 3.4, we will discuss the effect of ph correlations in the spin-isoscalar mode on the ratio between the longitudinal and transverse response functions.
3.1. RESULTS
OF THE
CONTINUUM
RPA
In this subsection, the role of ph correlations in the spin-isoscalar quasielastic scattering region is examined relying on the continuum
mode in the RPA.
T. Shigehara et al. / Isoscalar spin response functions
110
In fig. 1, the isoscalar response
is common consider
longitudinal
both to the longitudinal a one-body
of the spin-orbit
(dashed)
and
at q = 2 fm-’ are shown, together
functions
spin-orbit
potential
and transverse
potential.
causes
only
operators
It has been a slight
transverse
(dashed-dotted)
with the free one (solid), because
which
we do not
shown ‘) that the inclusion
difference
between
the two free
response functions in the quasielastic scattering region. It is seen from fig. 1 that the ph correlations induce an appreciable difference between the longitudinal and transverse response functions. Around the quasielastic scattering region (w L- q2/2M = 82.9 MeV, with M the nucleon mass), the transverse response function is larger than the longitudinal one, in contrast to the fact that the isovector longitudinal (transverse) response function is enhanced (suppressed) compared to the free one due to the attractive (repulsive) nature of the corresponding component in the ph interaction. In order to see the reason why the difference appears between the two modes, we show in fig. 2 the result obtained by leaving out the exchange term of the tensor force in the residual interaction. One can see that in this case there is no essential difference between the longitudinal and transverse response functions. This means that it is the exchange term of the tensor force which introduces the appreciable difference between the two spin response functions in fig. 1: It has a repulsive (attractive) effect on the longitudinal (transverse) response function. This effect of the tensor force on the isoscalar been pointed out in ref. 14). W e will examine
(non-energy-weighted) sum rules has the role of the exchange term of the
0.6
0 60
100
160
mo
260
o(MeV) Fig. 1. The isoscalar within the continuum
longitudinal (dashed line) and transverse (dashed-dotted line) response functions is also RPA at q = 2 fin-’ in 160. The free response function without correlations shown for comparison (solid line).
T. Shigehara et al. / Isoscalar spin response functions
111
q=Z.Ofm-’ i
0 60
100
160
mo
a60
o(MeV) Fig. 2. The same as in fig. 1, except that the exchange term of the tensor interaction.
tensor force in detail in the following subsections. nature of the central force causes a considerable
force is not included
in the ph
Fig. 2 also shows that the repulsive hardening (i.e. it shifts the position
of the quasielastic scattering peak in the free response function to the higher energy region). The small difference between the longitudinal and transverse response functions in fig. 2 is attributed to the direct term of the tensor force, which is weak in the isoscalar mode, contrary to the case of the isovector mode. From the above consideration, one can understand the global features of the RPA response functions in the quasielastic scattering region in fig. 1. They are controlled by two components in the ph interaction; the repulsive central force and the exchange term of the tensor force. Although the former gives a repulsive contribution both to the longitudinal and transverse response functions, the effects of the latter are quite different in the two modes and lead to a clear difference between the two response functions. It is interesting to examine the role of ph correlations also in the resonance region (w 6 30-40 MeV). In fact, one can observe, comparing fig. 1 with fig. 2, that the exchange
term of the tensor force has a considerable
influence
on the RPA response
functions, as pointed out in ref. “) although the momentum transfer considered there is much smaller than that in the present calculation. Here, however, we do not go into detail on this subject. This is because, in the continuum RPA, some careful numerical treatments are demanded in estimating the strength in this kinematical region: There are in the response function some peak structures with a narrow width and yet an appreciable strength. In fact, there are even zero-width peak structures in the free response function in w s 35 MeV, which are not explicitly shown in figs. 1 and 2. (In the present calculation, the response functions have been calculated at
T. Shigehara et al. / Isoscalar spin response functions
112
all energies
in steps of 1 MeV.) So, one must be careful
in the resonance progress
region
due to an elaborate
The isoscalar
the continuum
numerical
spin response
fig. 3. The qualitative region
within
features
functions mentioned
RPA.
in calculating
the strength
The calculation
is now in
method. at q = 1.5, 1.75 and 2.25 fin-’
are shown in
above are valid in the quasielastic
scattering
also in these cases.
3.2. SOME
FORMULAE
IN
FIRST-ORDER
PERTURBATION
THEORY
IN
NUCLEAR
MATTER
In the previous subsection, we have mentioned the importance of the exchange term of the tensor force for isoscalar spin response functions in the quasielastic
76
50
100
LZS
L-50
o(MeV)
L 0.6
0.6
-
I
./- -,.
q=2.25fm-l
-
7
7
%o., P s -
%0.4 2 B -
5go..?, 0’ :
5
-
go..?. 0” :.
-
e:
02
0
-
0 IO0
160
-
zoo
100
Fig. 3. The isoscalar
spin response
200
300
w(MeV)
w(MeV) functions at different momentum are the same as in fig. 1.
transfers
are shown. Other indications
T. Shigehara et al. / Isoscalar spin response functions
113
scattering region. In order to understand its role in a more intuitive way, we will give some more detailed discussions within first-order perturbation theory in nuclear matter. It is considered that the nuclear matter theory is trustworthy enough for getting gross features of the quasielastic scattering, because the shell effects in finite nuclei are not expected to be large in this kinematical region. In this subsection, we will summarize some necessary formulae for the following. In first-order perturbation theory in nuclear matter, the nuclear response function for a local operator can be written as w) .
R(q, w) = R”‘(q, w)+AR(q,
(6)
The first term on the right-hand side in eq. (6) is the free response function, which can be expressed in terms of the Lindhard function 17”‘(q, w) including the spinisospin factor 4, R”‘(q, OJ)= -L
Im n’“‘( q, 7T
w
).
(7)
The second term in eq. (6) represents the effects of ph correlations and for the real ph interaction is composed of two parts,
AR(q, 0) = ARdq, 0) + ARdq, 0) .
(8)
The first term of the right-hand side in eq. (8) is the forward-going the response function, including only forward ph propagations, AR,(q,
W) = -;
1
1 Im (2m)6
2 ARB(% W) = -;
contribution
q, w),
I
Here, the forward and backward ph propagators factor 4) are given by
(9)
involving the backward ph
1 Im (27T)6 dh, dk, &(h,,
x(&,(q)+ VdQ~))n~(hz,
TT,(h, Q,0) =4
q, w)
q, ~1.
(10)
(also including the spin-isospin
~(l~+ql-WV,+\) w-(h+q)2/2M+h2/2M+i7j
rr,(k q,o) = &(k
4,
to
d& dh, n~(hi, 4, w)
x(Vdq)+ VE(QF))G(~~, and the second term is the backward-going propagator,
contribution
-w) ,
’
(114
(lib)
respectively. In eqs. (9) and (lo), we have assumed that the ph interaction is local (in the direct term). The direct momentum transfer q is common both to the forward-
T. Shigehara et al. / Isoscalar spin response functions
114
and backward-going tum transferred hand,
contributions
by the external
the exchange
different
and, in nuclear
momentum
transfers,
QF and
with the momen-
of hole momenta.
On the other
QB in eqs. (9) and
(lo),
are
from each other and given by
QF=hl-h*, Notice that, although the exchange is not the case for QB. 3.3. ROLE
matter, identical
field, independent
OF THE
SCATTERING
EXCHANGE
QB=h,+h2+q. momentum
TERM
OF THE
transfer
TENSOR
(12) QF is independent
FORCE
IN THE
of q, this
QUASIELASTIC
REGION
In this subsection, the role of the exchange term of the tensor force in the quasielastic scattering region is examined from a perturbative point of view. Especially, the importance of the backward-going term V,( QB) in eq. (lo), in which the time direction of the ph propagation is changed by the ph interaction, is stressed. As mentioned in subsect. 3.1, the difference between the isoscalar longitudinal and transverse response functions is mainly attributed to the exchange term of the tensor force. In order to see how it influences both modes in a different manner, we consider the isoscalar spin response functions obtained by including only the exchange term of the tensor force in the ph interaction. That is, in eqs. (9) and (lo), the direct term V, is discarded and the exchange term V, contains only the tensor force. In fig. 4, the isoscalar longitudinal and transverse response functions at
0
50
100
160
200
o(MeV) Fig. 4. The isoscalar longitudinal (dashed line) and transverse (dashed-dotted line) response functions in first-order perturbation theory in nuclear matter at normal density (k, = 1.36 fm-‘). Only the exchange term of the tensor force is taken into account in the ph interaction. The free Fermi-gas response function is also shown for comparison (solid line).
T, Shigehara et
Ql. /
~SOSCdUr
Spin
response
11s
fUflCiionS
q = 2 fm-’ in nuclear matter at normal density (kF= 1.36 fm-‘) are shown. It is noticed that, in general, the longitudinal and transverse modes are mixed due to the exchange term of the tensor force. So, it is only within first-order perturbation theory that it is unnecessary to consider the mixing between the two modes. One can see from fig. 4 that the ph correlations quench (enhance) the free response function for the longitudinal (transverse) mode in almost the whole range of energy transfer, which is consistent with the results for a finite system in subsect. 3.1 (compare fig. 1 with fig. 2). This feature is mainly owing to the backward-going contribution in eq. (10). In order to see this, we show the case that only the forwardor backward-going contribution in eq. (8) is taken into account in figs. 5a and 5b, respectively. One can observe from fig. 5a that, compared to the free response function, the longitudinal (transverse) response function is enhanced (suppressed) in the small energy transfer region, and suppressed (enhanced) in the large energy transfer region. It should be noticed that the forward-going contribution does not contribute to the non-energy-weighted sum rule. Fig. 5b shows that, on the other hand, the backward-going contribution suppresses (enhances) the free response function for the longitudinal (transverse) mode in the whole range of energy transfer. Fig. 5 also indicates that the tensor correlation in the forward-going term is weaker than in the backward-going term, and that for both modes the forward-going part tends to cancel the backward-going part at small energy transfer, while at large energy transfer the corresponding contributions are coherent. In order to get a more deep understanding for the role of the tensor correlation, we will rely on a qualitative argument in the following. The tensor force in the nucleon-nucleon interaction in free space is mainly due to the exchange of isovector mesons such as pi and rho mesons. Reflecting this fact, the tensor interaction in the
0.9
03
0.2s
0.26
-
I
0
so
100
150
I
,
,
I
,
,
,
,
,
ON
-
,
,
,
,
)
,
,
,
q=Z.Ofm-i
,
,
,
-
.Eal
o(MeV) Fig. 5. The same as in fig. 4, except that only the forward-going (backward-going) into account in eq. (8) in fig. 5a (5b).
contribution
is taken
116
nuclear
T. Shigehara
medium
et al. / Isoscalar spin response functions
is dominated
So, we may approximately direct tensor interaction,
by the isospin-dependent
neglect
G:(q)
part (in the direct term).
the isospin-independent
part and write for the
* T2,
= H’(q)S,,(~)T,
(13)
where S,,(4)
=
3u, . 4az. 4^--Ul. u2,
(14)
with q the direct momentum transfer. In the ring approximation, where only the direct term of the ph interaction is considered, the tensor force in eq. (13) is effective only for isovector spin response functions. If one takes the exchange term into account, however, it is effective for isoscalar spin response functions the help of the explicit expressions for the exchange operators, P,=+(l+u,*uz),
as well. With
P,=f(l+l,*TJ,
(15)
or the relations P&,(B)
=
S,,(4)
9
(16)
P7T1*T2=+(3-T1*T2),
(17)
one can obtain G:(Q)
= -P,P,G;(Q)
= -H’(Q)S&);(3
-?
. 72))
(18)
where Q is the exchange momentum transfer; Q = QF for the forward-going term and Q = Qe for the backward-going term. Together with the ph interaction in eq. (18), if the quantization axis is taken parallel to q, one obtains for the exchange term in eq. (9) or (lo), V,(Q)
= V;,;“(Q)
= a’=“H’(Q)P2(cos
0,)
(i = L, T) ,
(19)
where ffL
for the isoscalar
longitudinal
T=O= -3,
and transverse
&‘O=$, channels,
(20) respectively.
In eq. (19), 8,
is the angle between q and Q, and P2(x) is the second-order Legendre polynomial (P2(x) =+(3x2- 1)). In order to obtain a rough estimate of eq. (19), we introduce an angle-average of the Legendre polynomial at fixed q as follows, (P2(cos
Q)),
= I
dh, dh2 P,(cos 0,) /I h, ,hz
dh, dh2.
(21)
h, .&
In fig. 6, the average (P2(cos f3,)), is shown both forthe forward- and backward-going terms as a function of the magnitude of the momentum transfer q. It is seen that the average for the forward-going term becomes less important than that for the
T. Shigehara et al. / Isoscular spin response functions
0
2
1
3
117
4
q/b Fig. 6. The angle-average ofthe second-order Legendre polynomial in eq. (21) as a function of momentum transfer in units of the Fermi momentum k,. The dashed (dashed-dotted) line corresponds to the case of the forward-going (backward-going) term.
backward-going term if q is larger than typically k,. This can be easily understood from the following: Because the momentum transfer QF in eq. (12) is independent of q and merely the difference between two hole momenta, the average of the second-order Legendre polynomial in eq. (21) becomes small for large q for the forward-going term. On the other hand, the momentum transfer Qe in the backwardgoing term in eq. (12) is essentially if q is large enough. to the integration
Thus,
governed
the integrand
in a coherent
by q for any choice of the hole momenta in the numerator
way. It is noticed
of eq. (21) contributes
that, in fact, the average
for the
forward-going term becomes zero for q > 2k, without the Pauli blocking, and that on the other hand, the average for the backward-going term is nearly equal to one for q %=k,. This is the reason why the backward-going contribution largely modifies the free response function in fig. 5b, although the forward-going contribution does not induce
a strong ph correlation.
Also, together
with the fact that H’(Q)
is negative
(-64.7 MeV . fm3 at Q = Qe- q = 2 fm-i), one can notice from eqs. (19) and (20) that the backward-going term quenches (enhances) the free response function for the longitudinal (transverse) mode, and that the amount of the induced modification for the longitudinal response function is roughly twice as large as for the transverse one. These features are well reproduced in the actual numerical calculation in fig. 5b. One might raise the question, however, why the backward-going contribution in eq. (10) plays an important role in the large energy transfer region in spite of the energy denominator in the backward ph propagation. In order to answer this question, we refine the angle-average of the Legendre polynomial in eq. (21) and
T. Shigehara et al. / Isoscalar spin response functions
118
introduce
another
angle-average,
which is energy-dependent,
(P*(cos@,Nq,,=
dhl dhz 6
(h+qY
0 ---+=
as follows,
h:
2M
Ih,+ql,lh2++kF
X
[Ib,+dh2++kF )I ~-(h,~+* 2M
h,.h,
In eq. (22), we have taken in eq. (10) denominator
into account
2M
the fact that the forward
--I.
(22)
ph propagation
must be on-shell. It is noted that, if both the numerator and the in eq. (22) are integrated with respect to w, we obtain just the average
(Pz(cos 6,)), of eq. (21). In fig. 7, the numerical result of eq. (22) is shown for the backward-going term as a function of w for fixed q = 2 fin-‘. It can be seen that the average (PJcos O,)),, increases with increasing energy transfer, because in the large energy transfer region the on-shell condition favors the kinematical condition that the hole momentum h, aligns to the direction of q. This ensures the importance of the backward-going contribution even in the large energy transfer region, although it is weakened by the energy denominator in the backward ph propagator in eq. (10). This is clear also from a simple estimate. Under the following two assumptions, one can relate the average (P2(cos 6,)),, to the ratio between the backward-going contribution in eq. (10) and the free response function in eq. (7). If one approximates the energy denominator -w-(h + q)*/2M + h2/2M in the backward ph propagator in eq. (lib)
by -w-q2/2M
and neglects
0
50
the Q-dependence
100
160
of H’(Q)
in eq. (19),
200
o(MeV) Fig. 7. The angle-average of the second-order Legendre polynomial in eq. (22) in case of the backwardgoing term as a function of energy transfer w for fixed q = 2 fm-‘.
T Shigehara et al. / Isoscalar spin response funcrions
119
then one gets, AR;,;‘(q, w)lR”‘(q, =-
w)
2
w -@$$+s)
(2wY
I
h,.hz
x G,YO< Qd
-w-(h2+q)2/2M+h;/2M
o -&k&+$)]
-’
F (i = L, T).
(23)
Using the following numerical values, 4 (2r)3
dhz = 0.154 fmm3
at
q =
2 fm-’ (k, = 1.36 fm-‘),
(24)
I hl
1b,++k,
and H’ = H’( QB = 2 fm-‘) = -64.7 MeV . fm3, the numerical result of the ratio in eq. (23) is shown for the longitudinal channel in fig. 8, compared to the exact ratio obtained from fig. 5b. Although, in the exact calculation, the isospin-independent part of the tensor force is also included in the ph interaction, it is, however, very
0
so
100
160
o(MeV) Fig. 8. The ratio of the backward-contribution in eq. (10) to the free Fermi-gas response function in the longitudinal mode at q = 2 fm-‘, using the approximate formula in eq. (23) (solid line). The exact ratio deduced from fig. 5b is also shown for comparison (dashed line).
T. Shigehara et al. / Isoscalar spin response functions
120
weak compared approximate
the large energy transfer
to the isovector
estimate
somewhat
transfer
QB is larger than
calculation.
One
region, q,
tensor force [ H( QB = 2 fm-‘) overestimates
the tensor
where a typical
leading
can see, however,
value of the exchange
to the weaker that
= 5.2 MeV . fm3]. The
correlation
tensor
the w-dependence
correlation
especially
in
momentum in the exact
in the exact
ratio
is
qualitatively well reproduced by the approximate estimate. This is due to the appropriate treatment of the w-dependence included in the average (PJcos eo)),, in eq. (23), which implies that the ph correlation induced by the exchange term of the tensor force is important in the backward-going contribution even at large energy transfer as well as at small energy transfer. Before closing this subsection, we will give a short comment on the role of the exchange term of the tensor force in isovector spin response functions. The tensor force in eq. (13) is also effective in the exchange channel in isovector spin response functions. Eq. (18) indicates, however, that the exchange term of the tensor force is not so important in the spin-isospin mode as in the spin-isoscalar one. In fact, we have shown in fig. 5 of the previous paper ‘) that the inclusion of the exchange term of the tensor force can lead only to a slight modification of the isovector spin response functions. Note that in fig. 5 of ref. 9), the small difference between the response function based on the RPA with the exchange term explicitly included and that within the ring approximation is essentially due to the exchange term of the tensor force. This is because the zero-range approximation for the exchange term of the central force is very good in the spin-isospin mode. Eq. (18) also indicates that, contrary to in the isoscalar mode, the inclusion of the exchange term of the tensor force enhances (suppresses) the longitudinal (transverse) response function in the isovector mode. Furthermore, similar to the isoscalar case, its effect on the longitudinal response function is roughly twice as large as that on the transverse
one. These features
3.4. RATIO BETWEEN
are well reproduced
THE LONGITUDINAL
in fig. 5 of ref. “).
AND TRANSVERSE
RESPONSE
FUNCTIONS
In this subsection, we will discuss the effects of ph correlations in isoscalar spin response functions on the ratio between the longitudinal and transverse response functions in eqs. (la) and (lb), based on the considerations of the previous subsections. Also for isovector spin response functions, we consider the effects of ph correlations due to the same method as described in the preceding section. The isovector spin operators are given by eq. (5) multiplied by r3. In fig. 9, the isovector longitudinal and transverse response functions are shown for the momentum transfer q = 1.75 fm-‘. These results correspond to those shown in fig. 5 of the previous paper 9), except that the momentum transfer was taken as q = 2 fm-’ in the previous calculation. Because the features of the isovector spin response functions in the RPA are
TO Shigehara
et al. / Isoscalar
spin response functions
121
0.8
0
so
100
160
8w
o(MeV) Fig. 9. The isovector within the continuum
longitudinal (dashed line) and transverse (dashed-dotted line) response functions RPA at CJ= 1.75 fm-’ in 160. The free response function is also shown for comparison (solid line).
well known and the characteristics of them in our model have been explained in detail in the previous paper, we restrict ourselves here to recalling some points relevant for the present discussion. First, delta-hole effects are not included in our calculation. Secondly, we do not introduce a phenomenological short-range repulsion (so-called g’ force), but take the G-matrix as the ph interaction. Because, as a well-known fact, the G-matrix alone does not give sufficient repulsion in the central part, the collectivity in the small energy transfer region is prominent in the longitudinal response function. In this sense, our calculation for the isovector spin response functions might be unrealistic. We concentrate, however, on examining how the effects of ph correlations in the spin-isoscalar mode affect the ratio between the longitudinal and transverse response functions. In fig. 10, the ratio between the longitudinal and transverse response functions is shown as a function of w at q = 1.75 fm-’ (solid line). For comparison, we show the result based on the assumption that the isoscalar spin response functions are taken to be the free ones without
ph correlations
(dashed
line). Here, we concentrate
on the quasielastic scattering region of w 340 MeV. (See subsect. 3.1.) It is seen that the ph correlations in the isoscalar spin response functions reduce the ratio appreciably in the large energy transfer region (just above the quasielastic scattering peak of w = q2/2M = 63.5 MeV). In the small energy transfer region, the ratio is not largely modified on the average by including the ph correlations in the isoscalar mode, although the w-dependence reflecting the shell structure is somewhat different in the two calculations in fig. 10. These features can be qualitatively understood as
122
T Shigehara
et al. / Isoscalar spin response functions
q=
20
40
60
1.75fm-’
80
1
loo
120
o(MeV) Fig. 10. The ratio between the longitudinal and transverse response functions in eqs. (la) and (lb), respectively, in the continuum RPA at 9 = 1.75 fm-’ in I60 (solid line). The dashed line corresponds to the case that the isoscalar spin response functions are the free ones. The experimental points are taken from ref. ‘).
follows: As apparent from eq. (l), the ratio is not sensitive to the isoscalar longitudinal response function, but to the transverse one. At large energy transfer above the quasielastic peak, both the central and tensor ph correlations enhance the isoscalar transverse response function, leading to an appreciable reduction of the ratio. On the other hand, at small energy transfer, the effect of the tensor correlation is quite different from that of the central repulsion in the isoscalar transverse mode. The former correlation enhances the response, while the central repulsion suppresses it. As seen from fig. 3b, the former is not strong enough to overcome the latter. (This is partly because the backward-going contribution due to the exchange term of the tensor force which enhances the transverse response function is weakened by the forward-going contribution below the quasielastic scattering peak, as was discussed from the perturbative point of view in the previous subsection.) As a result, the total ph correlations slightly tend to quench the isoscalar transverse response function and to raise the ratio in the small energy transfer region.
4. Summary
In this response with the the effect first-order
and conclusion
paper, we have examined the role of ph correlations in isoscalar spin functions in the quasielastic scattering region, using the continuum RPA exchange term of the ph interaction explicitly taken into account. Also, of the exchange term of the tensor force has been examined in detail in perturbation theory in nuclear matter. Furthermore, the effects of ph
T. Shigehara et al. / Isoscalar spin response functions
correlations transverse
in the spin-isoscalar response
functions
in (p, p’) reactions The repulsive large energy
mode on the ratio between at the momentum
transfer
123
the longitudinal
of q = 1.75 fm-’
and
observed
have been investigated. central
transfer
force in the spin-isoscalar
region
both in the longitudinal
mode shifts the strength and transverse
response
to the func-
tions. Although the tensor force is weak in the direct term, it has, however, considerable effects in the exchange term comparable with the central force, leading to a quenching (enhancement) of the longitudinal (transverse) response function. As a net result, the isoscalar spin response function is appreciably different from the free one without correlations, both for the longitudinal and transverse modes. Below the quasielastic scattering peak, the longitudinal response function is significantly quenched as compared to the free one. On the other hand, the transverse response function is slightly quenched in the small energy transfer region and appreciably enhanced above the quasielastic scattering peak. Concerning the effect of the exchange term of the tensor force, some close examinations based on first-order perturbation theory in nuclear matter have shown that the backward-going term plays an important role in the quenching (enhancement) of the isoscalar longitudinal (transverse) response function. Taking the average of the second-order Legendre polynomial in an appropriate way, we have shown the reason why the importance of the backward-going term is not weakened even in the large energy transfer region. The forward-going term which is recognized not to be important compared to the backward-going term at large momentum transfer gives, roughly speaking, an attractive (repulsive) contribution to the longitudinal (transverse) response function. So, the effect of the backward-going term is somewhat weakened at small energy transfer in both modes. The ratio between the longitudinal and transverse response functions at q = 1.75 fm-’ is insensitive to the isoscalar longitudinal mode. Therefore, in the small energy transfer region around w = 40 MeV, the ph correlations in the isoscalar spin response functions cannot reduce the ratio, in spite of the considerable suppression of the isoscalar longitudinal response function, but even tend to raise the ratio due to the central repulsion in the transverse mode. In the large energy transfer region just above the quasielastic by the central and tensor For a further
scattering peak, however, the ratio is appreciably reduced ph correlations in the isoscalar transverse mode.
investigation,
we have to examine
the role of the 2p2h degrees
of
freedom on the spin response functions in the quasielastic scattering region. It has been indicated r5) that for low-lying collective states the tensor correlation in the spin-isoscalar mode is appreciably weakened by the screening effect. So, the role of this effect on the spin response functions should be carefully examined in the quasielastic scattering region. We are grateful tional
calculation
to W. Bentz for careful for this work
reading
was performed
of the manuscript.
The computa-
at the Computer
Centre
of the
124
T. Shigehara et al. / Isoscalar spin response functions
University
of Tokyo. The figures in this paper were drawn using the KEK-DRAWER
developed
in the National
Laboratory
for High Energy
Physics
(KEK).
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