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Off-shell corrections to the glauber theory of nuclear multiple scattering
Nuclear Physics
A240 (1975)
521-532;
@
North-Holland
Publishing
Co., Amsterdam
Not to be reproduced by photoprint or rnicrotilm without written permission from the publisher
OFF-SHELL CORRECTIONS TO THE GLAUBER THEORY OF NUCLEAR MULTIPLE SCATTERING B. SCHURMANN Institut
ftir Theoretische Physik der Universitiit Heidelberg, 69 Heidelberg,
Germany
Received 30 August 1974 (Revised 10 October 1974) Abstract:
We study the first-order off-shell corrections to Glauber’s multiple scattering model. These consist of a direct and a rescattering term, and are obtained from the eikonal corrections studied in a previous paper by a simple substitution. For an independent particle nuclear model the direct term becomes identical to an expression previously derived by Moniz. We show that the rescattering term is in general not small compared to the direct term. For high incident energy a comparison is made with other corrections to the Glauber model. An optical potential is proposed which consists of the Glauber potential and its off-shell, eikonal, Fresnel and spatial correlation corrections.
1. Introduction The scattering of fast particles by nuclei is successfully described by Glauber’s multiple collision model ‘). This model is valid in the high-energy limit k + cc where k is the waye number of the incoming particle. Its basic features are: (i) The target nucleons stay at fixed positions during the time of collision. (ii) The free-space Green function is replaced by a propagator of eikonal type which describes the propagation of the projectile between subsequent collisions by the laws of geometrical optics. (iii) The expression for the projectile-nucleus amplitude depends only on the on-shell values of the elementary interactions. At finite energy, various corrections to the Glauber theory may become significant. To improve on the frozen-nucleus assumption ‘) is quite difficult and the literature on this subject is still scarce t. The effects from deviation of the geometric-optical propagation of the incident wave have been studied extensively in recent years 4- I’), and to first order in k- ’ can be distinguished as Fresnel-type and eikonal-type corrections 16,17). Off-shell effects have been discussed in investigations of the relationship between Watson’s multiple scattering expansion and the Glauber series’8-21), an d in connection with the problem of non-overlapping potentials22-28). In a paper dating several years back, Reading has extracted from a non-local (offshell) optical potential an equivalent local one which allows for an off-shell variation of the elementary interaction 29). To first order, his result is identical to an expression recently derived by Moniz 30). + See, however, the recent paper by Koch 3). 521
B. SCHtjRMANN
522
In the present article we study the first-order off-shell corrections to the Glauber model, using as a starting point a Watson-type multiple scattering series. These corrections are obtained by expanding the elementary amplitudes in a Taylor series around the initial wave number k and retaining only terms to first order. We distinguish between a direct correction (which includes single scattering on the same target nucleon only) and a rescattering correction (which takes account of multiple scattering on the same target nucleon). For an independent particle nuclear model, the direct correction is shown to become identical to an expression derived in ref. 30) by a method different from the one used here. To our knowledge the rescattering correction has not been taken into account in previous work. It is considered here because it might lead to a significant reduction of the [in some cases substantial 3o)] direct off-shell correction. Such a cancellation tendency has previously been noticed for the eikonal corrections to the Glauber model l 7). We shall see that there is in fact a close connection between the eikonal and off-shell corrections which enables us to obtain the latter from the former by a simple substitution. The Watson-type multiple scattering series from which we start is very briefly sketched in sect. 2. The assumptions which lead to the Glauber approximation are discussed in some detail in the same section. In sect. 3 we demonstrate how the offshell corrections can very easily be obtained from the eikonal corrections, and specialize to an inde~ndent particle nuclear model. The effect of the res~ttering off-shell contribution on the direct contribution in the high- and medium-energy region is discussed in sect, 4. The magnitude of the total off-shell correction at high energies is compared to that of other corrections to the Glauber model in sect. 5. We briefly summarize our results in sect. 6. 2. Watson-type multipIe scattering series and Glauber approximation The projectile-nucleus
scattering amplitude is given by 31) F(k’, k) = ; @‘(k’, k)+F,(k’,
k).
(1)
j=l
Here k and k’ are the initial and final projectile wave numbers, A is the nuclear mass number, fi) denotes the direct contribution for j collisions, and I;, is the (in~nite) remainder including all rescattering processes. In the frozen-nucleus approximation, the amplitude (1) can be expressed in terms of the free-particle Green propagators @p, k) = (pz - kZ- ie)- ‘,
(2)
and the projectile-nucleon scattering amplitudes fr(p’, p). Here the index E stands for the incident energy E(k), and p, p’ denote intermediate momenta. The direct terms @ are F$‘#‘, k) = Afdk’, k)(nje-i(A’-k)~rjO), (3)
523
GLAUBER THEORY
x
fE(k),
pj_l).
. . j&,1,
k)e-i(k’-PJ-d.rJ..
pl)fE(pl,
.
e-i(pz-PI)‘r~e-i(P~-k).rllo),
2sjsA.
(4)
The states IO) and In) are the unperturbed initial and final target states, the coordinates ri, . . ., rj are the instantaneous positions of the target nucleons, and the symbol P(l . . . j) means that we have to sum over all permutations of the indices 19. . ., j. The elementary scattering amplitudes are assumed to be equal for protons and neutrons. The single scattering term (3) contains only the on-shell elementary interaction, and is thus uninteresting for the present study. The direct contribution (4) forj collisions contains two half off-shell andj- 2 fully off-shell amplitudes and, together with the remainder F,, is the quantity of interest. The remainder is rather complicated, however, and for this reason will not be given here. The terms needed for this work can be found in ref. l’). The Green propagators z(’ and elementary amplitudes fE are subject to various approximation procedures. In the high energy limit k -+ cc the propagator (2) may be replaced by the eikonal propagator z’,, G”’= (2k)-‘G,,
(5)
with G, = (d,,-is)-‘,
(6)
or in coordinate space G,(b - b’, z-z’)
= $5@)(b - w)@z - z’)eik(z-z’).
(6a)
The quantity d ,, is defined as the longitudinal component of the momentum transfer A = p-k, with k in the z-direction. There arises the question whether assumptions in addition to eq. (5) have to be made in order to obtain the finite Glauber on-shell result from the infinite on- and off-shell Watson series (1). Although the answer to this problem is known to a number of people it might nevertheless be worthwhile to repeat it here in order to avoid any misunderstandings. Let us first assume that the elementary interactions can be constructed from twobody potentials via the Lippmann-Schwinger equation. Insertion of the eikonal propagator (6) in the Watson series (1) and in the two-body Lippmann-Schwinger equation then yields the Glauber result. This comes about because the remainder F, cancels against appropriate terms in the direct contributions fi) [ refs. ia* 19)]. On the other hand, the Glauber expression can also be obtained by making the additional assumption that the potentials are of finite range and do not overlap. Then
B. SCHURMANN
524
the remainder F, and the off-shell parts in the direct terms fi’ vanish separately. One should note, however, that in the high-energy limit the no-overlap assumption would not be an additional assumption if the contributions from the overlap region vanish in this limit. It is argued later on that in our approach the overlap contributions in fact vanish like k-i [cf. eq. (30)]. Since at high energies the potential concept probably has little meaning it is desirable to be able to obtain the Glauber series from eq. (1) without resorting to the potential concept. This is possible by assuming the elementary interactions to be given by profile functions yE which reproduce the observed high-energy elementary scattering amplitudes, f&i,
p) w &(p;-pJ
=
E
d2be-‘(P’i-PI)‘by~~), f
(7)
where pi and pL are the transverse components of p’ and p. This pragmatic point of view is made plausible in ref. 16) and we shall not repeat the arguments given there. If one takes the effective interaction yE to be of finite range then relations (5) and (7) amount to the no-overlap assumption in the high-energy limit. This is shown in the next section. The insertion of expressions (5) and (7) in eq. (1) yields the Glauber result, with the remainder F, vanishing 16). This is just what one would expect from the previous discussion for the potential scattering case. 3. First-order off-shell corrections to the Glauber model In ref. l ‘) we have assumed the form (7) for the elementary interactions and have studied corrections to the Glauber theory which arise from improving on the eikonal propagator and which become significant at lower energies, Effects from corrections to ass~ption (7) had been neglected. These we wish to study in this section. Sin& here we are only interested in first-order corrections to Glauber theory, and these do not interfere, we can study them separately and simply add them all up in the end. Thus we retain the eikonal propagator but improve on expression (7). This then completes our systematic studies of first-order corrections to the Glauber model in the framework of the frozen-nucleus approximation. We start with expressing the amplitudes fk in terms of the average momentum Z&r+p’) and the relative moments p’ -p. Since for not too light nuclei the nuclear form factors are sharply peaked around p’ = p and thus determine the moments transfer dependence of the projectile-nucleus scattering amplitude, we may assume + f&P
+
lo P’- PI = fE(PY 0).
7 We can generalize eq. (8) by assuming f&Q+p’), p’--p] ifs@, pi -pJ. Since the momentum transfer dependence is of no importance for the present study, and to keep the discussion as short as possible, we use eq. (8).
525
GLAUBER THEORY
In addition, we have neglected the dependence on the direction of p. This is correct for a wide class of interactions, and in any case is a good approximation near the forward direction. Next we expandf, in a Taylor series aroundp = k. To first order this yields fE(P,O)=
f&O)+
[+-&J))]Pz~,,.
(9)
In eq. (9) we have replaced p-k by p,,-k = A,, since p,, zp(l-$8”) for an intermediate scattering angle 8, and O2 & 1 near the forward direction. It is seen that the zero-order term on the right-hand side of eq. (9) is just the onshell forward scattering amplitude, and that the term of first order in A,, determines the off-shell dependence off*. Using eq. (9) and the eikonal propagator (6) in eq. (l), and keeping only terms of first order in A,, will yield the Glauber amplitude plus its first-order off-shell correction. Next we establish the relationship with the previously studied eikonal corrections i7). We write eq. (9) as &(P, 0) = fE(k, WE1 + ,4k)A
,,I,
(94
with
(10)
cLW= An intermediate projectile-nucleon
collision is described by [cf. eq. (4)]
&(A,,)f#, The propagator
including eikonal-type
(11)
OX1+/#)A,,]. corrections
is given by 17)
G,(A,,)[1-(2k)-‘A,,l,
(12)
or in coordinate representation ;
#2’(b _ b’)eik(Z-Z’) [O(z-z’)+i(2k)-‘6(z-z’)].
(124
Inserting eq. (12) into eq. (1) and approximating the elementary interactions by their on-shell values yields the Glauber result plus the eikonal correction. The intermediate scattering corresponding to eq. (11) is in this case described by &(A,,)_&@,o)t-l-(2k)-1A,,1.
(13)
The preceding discussion and a comparison of eqs. (11) and (13) show that the offshell corrections can be obtained from the eikonal corrections by simply replacing in the latter - (2k)- ’ by p(k). We should mention that the eikonal and thus the offshell contribution as well consist of a direct and a rescattering part 17). The physical nature of the off-shell (and eikonal) corrections is best discussed in
526
B. SC~~RMANN
coordinate space. We show that these corrections come from the region of overlapping interactions. This can be seen in the following way. Assume two target nucleons at positions r = (s, z) and r’ = (s’, z’). The projectile-nucleon interactions are in the eikonal approximation given by the profile functions ~(b-s) and y(b’-s’) [cf. eq. (IS)] where b and b’ are impact parameter vectors. We take y to be of finite range a. For the non-overlapping con~guration Is-s’] > 2a the projectile can hit at most one nucleon. The reason is that because of the delta-function aczt(b - b’) in the eikonal propagator (6a) the projectile passes through the nucleus with constant impact parameter vector b. Single scattering involves only on-shell values, however [cf. eq. (311. Therefore the off-shell corrections arise from a configuration with Is-$1 $2a. It can be seen from expression (12a) that in addition they involve z = z’. Thus the off-shell corrections come from the overlap region Is-s’/ s 2a, 2 = z’. We note that the Glauber (on-shell) result involves non-overlapping interaction regions only, Is--s’1 > 2a for single scattering, and Is-s’l 5 2a, z > z’ for the higher-order terms. This can be seen with the same reasoning as before, by noting that z > z’ because of 6(z -2’) in eqs. (6a) or (12a). The preceding discussion shows that the theorem for non-overlapping interactions 26) is valid also in the eikonal approximation. Moreover, it shows that in case of a profile function y of finite range a the amplitude (9) can be interpreted as the sum of a no-overlap and an overlap part. To conclude this section we consider elastic scattering in an independent particle nuclear model. The projectile-nucleus elastic scattering amplitude including the firstorder off-shell corrections to the Glauber model becomes (14) with the phase shift X(b) = -(hv)-’
O”dz V(r). s -K!
(15)
Here u = hk/m is the projectile speed. The optical potential V is given by V(r) = V,(r)+ w,(r),
(16)
IL(r) = - (27rfiZl+%#&, O)p(r)
(17)
where
is the Glauber potential with the single-particle density p(r). The amplitude&(k, can be written in terms of the elementary profile function yE as (cf. eq. (7)) ik fAk 0) = zn
d2Wbf-
0)
GLAUBER
527
THEORY
For later use we also note the partial wave representation + &(k, 0) = & g (21+ 1x1 -ezidl).
(fga)
I-O The
function w,(r) =
-(W1~L(k)(K&912W9W9+3[%912}
in eq. (16) is the first-order oQhel1 V(r) =
f,(k
correction, with
-(2nh2/m)AfE(k, 0) =
(19)
;
O)p(r),
(20)
d2b[y,(b)]2,
(21)
s
or &(k, 0) = & E (21+ 1x1 -e2idr)2.
(21a)
I-O
term in eq. (19) involving Vi is the direct correction, the other two make up the rescattering correction. The first of them arises from double scattering on the same nucleon, the second one from two double scatterings on two different nucleons. Double scattering is expressed by the square of the profile function yE (cf. eq. (21)). Neglecting the rescattering contributions, eq. (19) becomes
The
WA) = -(W-‘CL(~)[KA~)]~,
(22)
which is identical to Moniz’s off-shell correction given in his eq. (13) ++. For the study in the next section it is appropriate to write eq. (19) as w,(r) = w&)(1 - CE)
(19a)
with the dimensionless quantity C: = fE(k, 0)/J&
0) -+[f,(k,
0)/j&
O)]“.
(23)
4. Influence of rescattering off-shell corrections In this section we discuss the magnitude of CFL_ in the high- and medium-energy region for some simple models of the elementary interaction.
’ In the high- and medium-energy region the difference between the impact parameter and partial wave representation is negligibly small. +’ The quantities v0 and y m Momz’s paper are related to Vc and p by v0 = (2mjh’)Vo and y = p(k)/k. respectively.
B. SCHURMANN
528 4.1, HIGH-ENERGY
REGION
(1 GeV AND ABOVE)
In this region the elementary interaction can be described by a real profile function of Gaussian type,
y,(b) = g
exp
E
&b2 1,
(24)
where c is the averaged projectile-nucleon total cross section. Insertion into eqs. (18) and (21) yields f,(k, 0) = ika(E)/4lc and 3&k, 0) =&k, O)o(E)/8nB, which are purely absorptive. Thus C;“, = ~(E)/8~~-~~(E)/8~~)’
= 0.5,
(251
for typical values of (r and 3 (Z 40 mb and z 0.2 fm2, respectively). The Moniz correction will therefore be reduced by about 50%. 4.2. PION-NUCLEON
ABSORPTIVE
RESONANCE
(z 800 MeV)
Lower down in energy one encounters highly absorptive pion-nucleon resonances. We assume that the elementary interaction is dominated by a F-wave interaction and use for the F-wave phase shift the simple parametrization 30) (taking the nucleon to be spinless) u(k/k~~2r4 243
=
i(k2_k;)2+p.
with kR = 1 GeV/c, r = 250 MeV, a = 2. At the resonance, St = i. Employing the partial wave representations (18a) and (21 a) offx and FE, the rescattering correction C’Fcat the resonance becomes Cfe =; 1 _e2idaR-+(l
_
e2id3R)2w 0.5.
(27)
Thus the direct off-shell correction will again be reduced by about 50 %. 4.3. PION-NUCLEON
ELASTIC RESONANCE
(A FEW HUNDRED
MeV)
Still lower down in energy there occur elastic resonances such as the pion-nucleon (3,3) resonance. Here the elementary amplitude can be assumed to be given by a P-wave interaction. The P-wave resonant phase shift 67 is real with a value near $t. Since for 6: = +7c, C,“, = l-ein-+(l-eZ”)2
= 0,
(28)
according to our theory the rescattering off-shell corrections will become very small at an elastic resonance. This result should be regarded with some caution, however, since a Watson-type multiple scattering expansion of the projectile-nucleus scattering
GLAUBER
529
THEORY
amplitude might not converge in this case. This is indicated by the fact that in our approximation the two rescattering terms in eq. (28) are of the same size, whereas the second is considerably smaller than the first one in eqs. (25) and (27). 5. Comparison with other corrections A comparison of the off-shell correction w, with other corrections to the Glauber model is quite difficult in general since w, depends on the (unknown) off-shell behaviour of the elementary interaction. Moreover, at lower energies W, seems to be sensitive to the choice of off-shell continuation (see, for instance, the comparison in ref. 30) with results of other authors). In the high-energy regime the situation is more fortunate, however. For this reason, and because our results are most reliable for large k, we confine our comparison to this region. The crucial quantity in the expression for W, is p(k) since it contains the off-shell dependence of the elementary amplitude. At high energies it can, however, be expressed in terms of on-shell quantities only if one makes the reasonable assumption that the elementary interaction varies slowly with energy 30). Then
and thus (30) For the amplitude f&k, 0) = ika(E)/4n this yields p(k) = k- ‘. As discussed in sect. 3, the off-shell correction w, is obtained from the eikonal correction w, by the substitution - (2k)- ’ + p(k) and vice versa. Hence from eq. (19) w, = -2w,,
(31)
with w,(r) = (2khu)-‘[&(r)]z(1
-CF=),
(32)
or
w,+w, =
-we.
(33)
In ref. “) the effect of the eikonal and Fresnel corrections has been studied for the specific case of elastic scattering of 1 GeV protons from 4He, “C and 160, with qualitatively similar results. The Fresnel correction is the dominan; correction, the eikonal correction having only a marginal effect at large scattering angles. The smallness of w, is mainly due to the substantial reduction of the direct term by the rescattering term. Although w, + w, and w, have opposite sign, because of the smallness of w, the results of ref. I’) are virtually unaltered by including the off-shell correction.
530
B. SCHtjRMANN
There is another correction to the Glauber potential which arises from the spatial correlations between a pair of target nucleons. It is given by ‘) w,(r) = (ihu)_ iEF[v,(Y)]*,
(34)
where & is the so-called correlation length. In a detailed study of the effect of spatial correlations on the differential cross section for elastic scattering of 1 GeV protons from light nuclei 32) it has been found that 1, = 0.74 fm including both short-range and Pauli correlations, and 1, = 0.85 fm for short-range correlations alone. Since the repulsive short-range force tends to keep the target nucleons apart one should expect that the situation of non-overlapping elementary interactions is better satistied if the repulsive correlations are included 23,26). This would mean that there is a cancellation effect between w, and w, which in turn implies that both are of a similar nature. At high energies this is apparently not the case. This can be seen from the fact that for a purely (or mainly) absorptive elementary amplitude the off-shell corrections - as well as the eikonal and Fresnel corrections - are purely (or mainly) real whereas the correlation corrections are purely (or mainly) imaginary. This leads to physically distinct effects on the projectile-nucleus differential cross section. Since the off-shell, eikonal and Fresnel corrections add a real part to the Glauber potential they mainly influence the depth of the diffraction minima but do not change the positions. In contrast the correlation correction does not influence the depth of the minima but changes the widths of the secondary diffraction maxima and thus the positions of the subsequent minima. These qualitative features are reproduced in the numerical results of refs. ’ 7, 32). Including the corrections discussed in this section leads to an optical potential of the form V(r) = V,(r) + w,(p)+ w,(r) + w&) + w,(r),
(35)
where wr(r) = (2khv)-‘b $ [VG(r)12 is the Fresnel correction+. Such a potential probably has a wider range of validity than the Glauber potential. In addition it has the attractive feature that from it the projectile-nuclei elastic scattering can still be calculated via the simple eikonal formulae (14) and (15). 6. Summary We have studied the first-order off-shell corrections to Glauber’s high-energy collision model by making use of their close relationship to the eikonal-type cort This ISthe expression for a spherically symmetrical Glauber potential. In the general case wF is more complicated 17).
GLAUBER
THEORY
531
rections derived in a previous paper 17). Like the eikonal-type corrections the offshell corrections consist of a direct and a rescattering term. For elastic scattering in an independent particle nuclear model the direct term becomes identical to an expression obtained by Moniz 30). We have demonstrated by simple examples that the rescattering term can in general not be neglected compared to the direct term. A comparison with the eikonal and Fresnel corrections “) has shown that at high energies the sum of the off-shell and eikonal contributions is of the same size as the eikonal correction alone, but has opposite sign. This size is small compared to the Fresnel contribution because of cancellations between the direct and rescattering terms. We have pointed out that at high energies the off-shell and short-range correlation corrections are of a distinct physical nature which is somewhat unexpected. Finally we have proposed a form for the optical potential which includes the firstorder corrections to Glauber theory and should thus have a wider range of validity than the conventional Glauber potential. I wish to thank K. Schafer and J. Htifner for helpful discussions.
References 1) R. J. Glauber, Lectures in theoretical physics, vol. 1, ed. W. E. Brittin science, New York, 1959) p. 315 2) L. L. Foldy and J. D. Walecka, Ann. of Phys. 54 (1969) 447 3) J. H. Koch, Nucl. Phys. B66 (1973) 464 4) T. T. Wu, Phys. Rev. 108 (1957) 466 5) D. S. Saxon and L. I. Schiff, Nuovo Cim. 6 (1957) 614 6) J Pumplin. Phys. Rev. 173 (1968) 1651 7) E. A. Remler, Phys. Rev. 176 (1968) 2108 8) R. S. Sugar and R. Blankenbecler, Phys. Rev. 183 (1969) 1387 9) H. D. I. Abarbanel and C. Itzykson, Phys. Rev. Lett. 23 (1969) 53 10) V. S. Bashin and V. S. Varma, Phys. Rev. l&l (1969) 1338 11) S. J. Wallace, Phys. Rev. Lett. 27 (1971) 622; Ann. of Phys. 78 (1973) 190 12) E. Kujawski, Phys. Rev. D4 (1971) 2573; Ann. of Phys. 74 (1972) 567 13) A. Baker, Phys. Rev. LMi (1972) 3462 14) K. Gottfried, Ann. of Phys. 66 (1971) 868 15) G. Faldt, Nucl. Phys. B46 (1972) 460 16) W. E. Frahn and B. Schtirmann, Ann. of Phys. 84 (1974) 147; B. Schtirmann and W. E. Frahn, Nucl. Phys. B62 (1973) 365 17) B. Schtirmann, Nucl. Phys. B67 (1973) 425 18) D. R. Harrington, Phys. Rev. 184 (1969) 1745 19) J. M. Eisenberg, Ann. of Phys. 71 (1972) 542 20) M. Yamazaki, Prog. Theor. Phys. 47 (1972) 1593 21) J. M. Namyslowski, Nuovo Cim. 12A (1972) 321 22) M. A. B. Beg, Ann. of Phys. 13 (1961) 110 23) F. Scheck and C. Wilkin, Nucl. Phys. B49 (1972) 541 24) D. Agassi and A. Gal, Ann. of Phys. 75 (1973) 56 25) E. Kujawski, Phys. Rev. C7 (1973) 18; E. Kujawski and E. Lambert, Ann. of Phys. 81 (1973) 591
and L. G. Dunham
(Inter-
532
26) 27) 28) 29) 30) 31)
B. SCHURMANN
J. Hiifner, Nucl. Phys. B98 (1973) 55 D. R. Harrington, Nucl. Phys. B59 (1973) 305 J. H. K&h and J. D. Walecka, Nucl. Phys. B72 (1974) 283 J. F. Reading, Phys. Rev. 156 (1967) 1116 E. J. Moniz, Phys. Rev. C7 (1973) 1750 K. M. Watson, Phys. Rev. 105 (1957) 1388; M. L. Goldberger and K. M. Watson, Collision theory (Wiley, New York, 1964) p. 749 32) E. J. Moniz and G. D. Nixon, Ann. of Phys. 67 (1971) 58.
A240 (1975)
521-532;
@
North-Holland
Publishing
Co., Amsterdam
Not to be reproduced by photoprint or rnicrotilm without written permission from the publisher
OFF-SHELL CORRECTIONS TO THE GLAUBER THEORY OF NUCLEAR MULTIPLE SCATTERING B. SCHURMANN Institut
ftir Theoretische Physik der Universitiit Heidelberg, 69 Heidelberg,
Germany
Received 30 August 1974 (Revised 10 October 1974) Abstract:
We study the first-order off-shell corrections to Glauber’s multiple scattering model. These consist of a direct and a rescattering term, and are obtained from the eikonal corrections studied in a previous paper by a simple substitution. For an independent particle nuclear model the direct term becomes identical to an expression previously derived by Moniz. We show that the rescattering term is in general not small compared to the direct term. For high incident energy a comparison is made with other corrections to the Glauber model. An optical potential is proposed which consists of the Glauber potential and its off-shell, eikonal, Fresnel and spatial correlation corrections.
1. Introduction The scattering of fast particles by nuclei is successfully described by Glauber’s multiple collision model ‘). This model is valid in the high-energy limit k + cc where k is the waye number of the incoming particle. Its basic features are: (i) The target nucleons stay at fixed positions during the time of collision. (ii) The free-space Green function is replaced by a propagator of eikonal type which describes the propagation of the projectile between subsequent collisions by the laws of geometrical optics. (iii) The expression for the projectile-nucleus amplitude depends only on the on-shell values of the elementary interactions. At finite energy, various corrections to the Glauber theory may become significant. To improve on the frozen-nucleus assumption ‘) is quite difficult and the literature on this subject is still scarce t. The effects from deviation of the geometric-optical propagation of the incident wave have been studied extensively in recent years 4- I’), and to first order in k- ’ can be distinguished as Fresnel-type and eikonal-type corrections 16,17). Off-shell effects have been discussed in investigations of the relationship between Watson’s multiple scattering expansion and the Glauber series’8-21), an d in connection with the problem of non-overlapping potentials22-28). In a paper dating several years back, Reading has extracted from a non-local (offshell) optical potential an equivalent local one which allows for an off-shell variation of the elementary interaction 29). To first order, his result is identical to an expression recently derived by Moniz 30). + See, however, the recent paper by Koch 3). 521
B. SCHtjRMANN
522
In the present article we study the first-order off-shell corrections to the Glauber model, using as a starting point a Watson-type multiple scattering series. These corrections are obtained by expanding the elementary amplitudes in a Taylor series around the initial wave number k and retaining only terms to first order. We distinguish between a direct correction (which includes single scattering on the same target nucleon only) and a rescattering correction (which takes account of multiple scattering on the same target nucleon). For an independent particle nuclear model, the direct correction is shown to become identical to an expression derived in ref. 30) by a method different from the one used here. To our knowledge the rescattering correction has not been taken into account in previous work. It is considered here because it might lead to a significant reduction of the [in some cases substantial 3o)] direct off-shell correction. Such a cancellation tendency has previously been noticed for the eikonal corrections to the Glauber model l 7). We shall see that there is in fact a close connection between the eikonal and off-shell corrections which enables us to obtain the latter from the former by a simple substitution. The Watson-type multiple scattering series from which we start is very briefly sketched in sect. 2. The assumptions which lead to the Glauber approximation are discussed in some detail in the same section. In sect. 3 we demonstrate how the offshell corrections can very easily be obtained from the eikonal corrections, and specialize to an inde~ndent particle nuclear model. The effect of the res~ttering off-shell contribution on the direct contribution in the high- and medium-energy region is discussed in sect, 4. The magnitude of the total off-shell correction at high energies is compared to that of other corrections to the Glauber model in sect. 5. We briefly summarize our results in sect. 6. 2. Watson-type multipIe scattering series and Glauber approximation The projectile-nucleus
scattering amplitude is given by 31) F(k’, k) = ; @‘(k’, k)+F,(k’,
k).
(1)
j=l
Here k and k’ are the initial and final projectile wave numbers, A is the nuclear mass number, fi) denotes the direct contribution for j collisions, and I;, is the (in~nite) remainder including all rescattering processes. In the frozen-nucleus approximation, the amplitude (1) can be expressed in terms of the free-particle Green propagators @p, k) = (pz - kZ- ie)- ‘,
(2)
and the projectile-nucleon scattering amplitudes fr(p’, p). Here the index E stands for the incident energy E(k), and p, p’ denote intermediate momenta. The direct terms @ are F$‘#‘, k) = Afdk’, k)(nje-i(A’-k)~rjO), (3)
523
GLAUBER THEORY
x
fE(k),
pj_l).
. . j&,1,
k)e-i(k’-PJ-d.rJ..
pl)fE(pl,
.
e-i(pz-PI)‘r~e-i(P~-k).rllo),
2sjsA.
(4)
The states IO) and In) are the unperturbed initial and final target states, the coordinates ri, . . ., rj are the instantaneous positions of the target nucleons, and the symbol P(l . . . j) means that we have to sum over all permutations of the indices 19. . ., j. The elementary scattering amplitudes are assumed to be equal for protons and neutrons. The single scattering term (3) contains only the on-shell elementary interaction, and is thus uninteresting for the present study. The direct contribution (4) forj collisions contains two half off-shell andj- 2 fully off-shell amplitudes and, together with the remainder F,, is the quantity of interest. The remainder is rather complicated, however, and for this reason will not be given here. The terms needed for this work can be found in ref. l’). The Green propagators z(’ and elementary amplitudes fE are subject to various approximation procedures. In the high energy limit k -+ cc the propagator (2) may be replaced by the eikonal propagator z’,, G”’= (2k)-‘G,,
(5)
with G, = (d,,-is)-‘,
(6)
or in coordinate space G,(b - b’, z-z’)
= $5@)(b - w)@z - z’)eik(z-z’).
(6a)
The quantity d ,, is defined as the longitudinal component of the momentum transfer A = p-k, with k in the z-direction. There arises the question whether assumptions in addition to eq. (5) have to be made in order to obtain the finite Glauber on-shell result from the infinite on- and off-shell Watson series (1). Although the answer to this problem is known to a number of people it might nevertheless be worthwhile to repeat it here in order to avoid any misunderstandings. Let us first assume that the elementary interactions can be constructed from twobody potentials via the Lippmann-Schwinger equation. Insertion of the eikonal propagator (6) in the Watson series (1) and in the two-body Lippmann-Schwinger equation then yields the Glauber result. This comes about because the remainder F, cancels against appropriate terms in the direct contributions fi) [ refs. ia* 19)]. On the other hand, the Glauber expression can also be obtained by making the additional assumption that the potentials are of finite range and do not overlap. Then
B. SCHURMANN
524
the remainder F, and the off-shell parts in the direct terms fi’ vanish separately. One should note, however, that in the high-energy limit the no-overlap assumption would not be an additional assumption if the contributions from the overlap region vanish in this limit. It is argued later on that in our approach the overlap contributions in fact vanish like k-i [cf. eq. (30)]. Since at high energies the potential concept probably has little meaning it is desirable to be able to obtain the Glauber series from eq. (1) without resorting to the potential concept. This is possible by assuming the elementary interactions to be given by profile functions yE which reproduce the observed high-energy elementary scattering amplitudes, f&i,
p) w &(p;-pJ
=
E
d2be-‘(P’i-PI)‘by~~), f
(7)
where pi and pL are the transverse components of p’ and p. This pragmatic point of view is made plausible in ref. 16) and we shall not repeat the arguments given there. If one takes the effective interaction yE to be of finite range then relations (5) and (7) amount to the no-overlap assumption in the high-energy limit. This is shown in the next section. The insertion of expressions (5) and (7) in eq. (1) yields the Glauber result, with the remainder F, vanishing 16). This is just what one would expect from the previous discussion for the potential scattering case. 3. First-order off-shell corrections to the Glauber model In ref. l ‘) we have assumed the form (7) for the elementary interactions and have studied corrections to the Glauber theory which arise from improving on the eikonal propagator and which become significant at lower energies, Effects from corrections to ass~ption (7) had been neglected. These we wish to study in this section. Sin& here we are only interested in first-order corrections to Glauber theory, and these do not interfere, we can study them separately and simply add them all up in the end. Thus we retain the eikonal propagator but improve on expression (7). This then completes our systematic studies of first-order corrections to the Glauber model in the framework of the frozen-nucleus approximation. We start with expressing the amplitudes fk in terms of the average momentum Z&r+p’) and the relative moments p’ -p. Since for not too light nuclei the nuclear form factors are sharply peaked around p’ = p and thus determine the moments transfer dependence of the projectile-nucleus scattering amplitude, we may assume + f&P
+
lo P’- PI = fE(PY 0).
7 We can generalize eq. (8) by assuming f&Q+p’), p’--p] ifs@, pi -pJ. Since the momentum transfer dependence is of no importance for the present study, and to keep the discussion as short as possible, we use eq. (8).
525
GLAUBER THEORY
In addition, we have neglected the dependence on the direction of p. This is correct for a wide class of interactions, and in any case is a good approximation near the forward direction. Next we expandf, in a Taylor series aroundp = k. To first order this yields fE(P,O)=
f&O)+
[+-&J))]Pz~,,.
(9)
In eq. (9) we have replaced p-k by p,,-k = A,, since p,, zp(l-$8”) for an intermediate scattering angle 8, and O2 & 1 near the forward direction. It is seen that the zero-order term on the right-hand side of eq. (9) is just the onshell forward scattering amplitude, and that the term of first order in A,, determines the off-shell dependence off*. Using eq. (9) and the eikonal propagator (6) in eq. (l), and keeping only terms of first order in A,, will yield the Glauber amplitude plus its first-order off-shell correction. Next we establish the relationship with the previously studied eikonal corrections i7). We write eq. (9) as &(P, 0) = fE(k, WE1 + ,4k)A
,,I,
(94
with
(10)
cLW= An intermediate projectile-nucleon
collision is described by [cf. eq. (4)]
&(A,,)f#, The propagator
including eikonal-type
(11)
OX1+/#)A,,]. corrections
is given by 17)
G,(A,,)[1-(2k)-‘A,,l,
(12)
or in coordinate representation ;
#2’(b _ b’)eik(Z-Z’) [O(z-z’)+i(2k)-‘6(z-z’)].
(124
Inserting eq. (12) into eq. (1) and approximating the elementary interactions by their on-shell values yields the Glauber result plus the eikonal correction. The intermediate scattering corresponding to eq. (11) is in this case described by &(A,,)_&@,o)t-l-(2k)-1A,,1.
(13)
The preceding discussion and a comparison of eqs. (11) and (13) show that the offshell corrections can be obtained from the eikonal corrections by simply replacing in the latter - (2k)- ’ by p(k). We should mention that the eikonal and thus the offshell contribution as well consist of a direct and a rescattering part 17). The physical nature of the off-shell (and eikonal) corrections is best discussed in
526
B. SC~~RMANN
coordinate space. We show that these corrections come from the region of overlapping interactions. This can be seen in the following way. Assume two target nucleons at positions r = (s, z) and r’ = (s’, z’). The projectile-nucleon interactions are in the eikonal approximation given by the profile functions ~(b-s) and y(b’-s’) [cf. eq. (IS)] where b and b’ are impact parameter vectors. We take y to be of finite range a. For the non-overlapping con~guration Is-s’] > 2a the projectile can hit at most one nucleon. The reason is that because of the delta-function aczt(b - b’) in the eikonal propagator (6a) the projectile passes through the nucleus with constant impact parameter vector b. Single scattering involves only on-shell values, however [cf. eq. (311. Therefore the off-shell corrections arise from a configuration with Is-$1 $2a. It can be seen from expression (12a) that in addition they involve z = z’. Thus the off-shell corrections come from the overlap region Is-s’/ s 2a, 2 = z’. We note that the Glauber (on-shell) result involves non-overlapping interaction regions only, Is--s’1 > 2a for single scattering, and Is-s’l 5 2a, z > z’ for the higher-order terms. This can be seen with the same reasoning as before, by noting that z > z’ because of 6(z -2’) in eqs. (6a) or (12a). The preceding discussion shows that the theorem for non-overlapping interactions 26) is valid also in the eikonal approximation. Moreover, it shows that in case of a profile function y of finite range a the amplitude (9) can be interpreted as the sum of a no-overlap and an overlap part. To conclude this section we consider elastic scattering in an independent particle nuclear model. The projectile-nucleus elastic scattering amplitude including the firstorder off-shell corrections to the Glauber model becomes (14) with the phase shift X(b) = -(hv)-’
O”dz V(r). s -K!
(15)
Here u = hk/m is the projectile speed. The optical potential V is given by V(r) = V,(r)+ w,(r),
(16)
IL(r) = - (27rfiZl+%#&, O)p(r)
(17)
where
is the Glauber potential with the single-particle density p(r). The amplitude&(k, can be written in terms of the elementary profile function yE as (cf. eq. (7)) ik fAk 0) = zn
d2Wbf-
0)
GLAUBER
527
THEORY
For later use we also note the partial wave representation + &(k, 0) = & g (21+ 1x1 -ezidl).
(fga)
I-O The
function w,(r) =
-(W1~L(k)(K&912W9W9+3[%912}
in eq. (16) is the first-order oQhel1 V(r) =
f,(k
correction, with
-(2nh2/m)AfE(k, 0) =
(19)
;
O)p(r),
(20)
d2b[y,(b)]2,
(21)
s
or &(k, 0) = & E (21+ 1x1 -e2idr)2.
(21a)
I-O
term in eq. (19) involving Vi is the direct correction, the other two make up the rescattering correction. The first of them arises from double scattering on the same nucleon, the second one from two double scatterings on two different nucleons. Double scattering is expressed by the square of the profile function yE (cf. eq. (21)). Neglecting the rescattering contributions, eq. (19) becomes
The
WA) = -(W-‘CL(~)[KA~)]~,
(22)
which is identical to Moniz’s off-shell correction given in his eq. (13) ++. For the study in the next section it is appropriate to write eq. (19) as w,(r) = w&)(1 - CE)
(19a)
with the dimensionless quantity C: = fE(k, 0)/J&
0) -+[f,(k,
0)/j&
O)]“.
(23)
4. Influence of rescattering off-shell corrections In this section we discuss the magnitude of CFL_ in the high- and medium-energy region for some simple models of the elementary interaction.
’ In the high- and medium-energy region the difference between the impact parameter and partial wave representation is negligibly small. +’ The quantities v0 and y m Momz’s paper are related to Vc and p by v0 = (2mjh’)Vo and y = p(k)/k. respectively.
B. SCHURMANN
528 4.1, HIGH-ENERGY
REGION
(1 GeV AND ABOVE)
In this region the elementary interaction can be described by a real profile function of Gaussian type,
y,(b) = g
exp
E
&b2 1,
(24)
where c is the averaged projectile-nucleon total cross section. Insertion into eqs. (18) and (21) yields f,(k, 0) = ika(E)/4lc and 3&k, 0) =&k, O)o(E)/8nB, which are purely absorptive. Thus C;“, = ~(E)/8~~-~~(E)/8~~)’
= 0.5,
(251
for typical values of (r and 3 (Z 40 mb and z 0.2 fm2, respectively). The Moniz correction will therefore be reduced by about 50%. 4.2. PION-NUCLEON
ABSORPTIVE
RESONANCE
(z 800 MeV)
Lower down in energy one encounters highly absorptive pion-nucleon resonances. We assume that the elementary interaction is dominated by a F-wave interaction and use for the F-wave phase shift the simple parametrization 30) (taking the nucleon to be spinless) u(k/k~~2r4 243
=
i(k2_k;)2+p.
with kR = 1 GeV/c, r = 250 MeV, a = 2. At the resonance, St = i. Employing the partial wave representations (18a) and (21 a) offx and FE, the rescattering correction C’Fcat the resonance becomes Cfe =; 1 _e2idaR-+(l
_
e2id3R)2w 0.5.
(27)
Thus the direct off-shell correction will again be reduced by about 50 %. 4.3. PION-NUCLEON
ELASTIC RESONANCE
(A FEW HUNDRED
MeV)
Still lower down in energy there occur elastic resonances such as the pion-nucleon (3,3) resonance. Here the elementary amplitude can be assumed to be given by a P-wave interaction. The P-wave resonant phase shift 67 is real with a value near $t. Since for 6: = +7c, C,“, = l-ein-+(l-eZ”)2
= 0,
(28)
according to our theory the rescattering off-shell corrections will become very small at an elastic resonance. This result should be regarded with some caution, however, since a Watson-type multiple scattering expansion of the projectile-nucleus scattering
GLAUBER
529
THEORY
amplitude might not converge in this case. This is indicated by the fact that in our approximation the two rescattering terms in eq. (28) are of the same size, whereas the second is considerably smaller than the first one in eqs. (25) and (27). 5. Comparison with other corrections A comparison of the off-shell correction w, with other corrections to the Glauber model is quite difficult in general since w, depends on the (unknown) off-shell behaviour of the elementary interaction. Moreover, at lower energies W, seems to be sensitive to the choice of off-shell continuation (see, for instance, the comparison in ref. 30) with results of other authors). In the high-energy regime the situation is more fortunate, however. For this reason, and because our results are most reliable for large k, we confine our comparison to this region. The crucial quantity in the expression for W, is p(k) since it contains the off-shell dependence of the elementary amplitude. At high energies it can, however, be expressed in terms of on-shell quantities only if one makes the reasonable assumption that the elementary interaction varies slowly with energy 30). Then
and thus (30) For the amplitude f&k, 0) = ika(E)/4n this yields p(k) = k- ‘. As discussed in sect. 3, the off-shell correction w, is obtained from the eikonal correction w, by the substitution - (2k)- ’ + p(k) and vice versa. Hence from eq. (19) w, = -2w,,
(31)
with w,(r) = (2khu)-‘[&(r)]z(1
-CF=),
(32)
or
w,+w, =
-we.
(33)
In ref. “) the effect of the eikonal and Fresnel corrections has been studied for the specific case of elastic scattering of 1 GeV protons from 4He, “C and 160, with qualitatively similar results. The Fresnel correction is the dominan; correction, the eikonal correction having only a marginal effect at large scattering angles. The smallness of w, is mainly due to the substantial reduction of the direct term by the rescattering term. Although w, + w, and w, have opposite sign, because of the smallness of w, the results of ref. I’) are virtually unaltered by including the off-shell correction.
530
B. SCHtjRMANN
There is another correction to the Glauber potential which arises from the spatial correlations between a pair of target nucleons. It is given by ‘) w,(r) = (ihu)_ iEF[v,(Y)]*,
(34)
where & is the so-called correlation length. In a detailed study of the effect of spatial correlations on the differential cross section for elastic scattering of 1 GeV protons from light nuclei 32) it has been found that 1, = 0.74 fm including both short-range and Pauli correlations, and 1, = 0.85 fm for short-range correlations alone. Since the repulsive short-range force tends to keep the target nucleons apart one should expect that the situation of non-overlapping elementary interactions is better satistied if the repulsive correlations are included 23,26). This would mean that there is a cancellation effect between w, and w, which in turn implies that both are of a similar nature. At high energies this is apparently not the case. This can be seen from the fact that for a purely (or mainly) absorptive elementary amplitude the off-shell corrections - as well as the eikonal and Fresnel corrections - are purely (or mainly) real whereas the correlation corrections are purely (or mainly) imaginary. This leads to physically distinct effects on the projectile-nucleus differential cross section. Since the off-shell, eikonal and Fresnel corrections add a real part to the Glauber potential they mainly influence the depth of the diffraction minima but do not change the positions. In contrast the correlation correction does not influence the depth of the minima but changes the widths of the secondary diffraction maxima and thus the positions of the subsequent minima. These qualitative features are reproduced in the numerical results of refs. ’ 7, 32). Including the corrections discussed in this section leads to an optical potential of the form V(r) = V,(r) + w,(p)+ w,(r) + w&) + w,(r),
(35)
where wr(r) = (2khv)-‘b $ [VG(r)12 is the Fresnel correction+. Such a potential probably has a wider range of validity than the Glauber potential. In addition it has the attractive feature that from it the projectile-nuclei elastic scattering can still be calculated via the simple eikonal formulae (14) and (15). 6. Summary We have studied the first-order off-shell corrections to Glauber’s high-energy collision model by making use of their close relationship to the eikonal-type cort This ISthe expression for a spherically symmetrical Glauber potential. In the general case wF is more complicated 17).
GLAUBER
THEORY
531
rections derived in a previous paper 17). Like the eikonal-type corrections the offshell corrections consist of a direct and a rescattering term. For elastic scattering in an independent particle nuclear model the direct term becomes identical to an expression obtained by Moniz 30). We have demonstrated by simple examples that the rescattering term can in general not be neglected compared to the direct term. A comparison with the eikonal and Fresnel corrections “) has shown that at high energies the sum of the off-shell and eikonal contributions is of the same size as the eikonal correction alone, but has opposite sign. This size is small compared to the Fresnel contribution because of cancellations between the direct and rescattering terms. We have pointed out that at high energies the off-shell and short-range correlation corrections are of a distinct physical nature which is somewhat unexpected. Finally we have proposed a form for the optical potential which includes the firstorder corrections to Glauber theory and should thus have a wider range of validity than the conventional Glauber potential. I wish to thank K. Schafer and J. Htifner for helpful discussions.
References 1) R. J. Glauber, Lectures in theoretical physics, vol. 1, ed. W. E. Brittin science, New York, 1959) p. 315 2) L. L. Foldy and J. D. Walecka, Ann. of Phys. 54 (1969) 447 3) J. H. Koch, Nucl. Phys. B66 (1973) 464 4) T. T. Wu, Phys. Rev. 108 (1957) 466 5) D. S. Saxon and L. I. Schiff, Nuovo Cim. 6 (1957) 614 6) J Pumplin. Phys. Rev. 173 (1968) 1651 7) E. A. Remler, Phys. Rev. 176 (1968) 2108 8) R. S. Sugar and R. Blankenbecler, Phys. Rev. 183 (1969) 1387 9) H. D. I. Abarbanel and C. Itzykson, Phys. Rev. Lett. 23 (1969) 53 10) V. S. Bashin and V. S. Varma, Phys. Rev. l&l (1969) 1338 11) S. J. Wallace, Phys. Rev. Lett. 27 (1971) 622; Ann. of Phys. 78 (1973) 190 12) E. Kujawski, Phys. Rev. D4 (1971) 2573; Ann. of Phys. 74 (1972) 567 13) A. Baker, Phys. Rev. LMi (1972) 3462 14) K. Gottfried, Ann. of Phys. 66 (1971) 868 15) G. Faldt, Nucl. Phys. B46 (1972) 460 16) W. E. Frahn and B. Schtirmann, Ann. of Phys. 84 (1974) 147; B. Schtirmann and W. E. Frahn, Nucl. Phys. B62 (1973) 365 17) B. Schtirmann, Nucl. Phys. B67 (1973) 425 18) D. R. Harrington, Phys. Rev. 184 (1969) 1745 19) J. M. Eisenberg, Ann. of Phys. 71 (1972) 542 20) M. Yamazaki, Prog. Theor. Phys. 47 (1972) 1593 21) J. M. Namyslowski, Nuovo Cim. 12A (1972) 321 22) M. A. B. Beg, Ann. of Phys. 13 (1961) 110 23) F. Scheck and C. Wilkin, Nucl. Phys. B49 (1972) 541 24) D. Agassi and A. Gal, Ann. of Phys. 75 (1973) 56 25) E. Kujawski, Phys. Rev. C7 (1973) 18; E. Kujawski and E. Lambert, Ann. of Phys. 81 (1973) 591
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B. SCHURMANN
J. Hiifner, Nucl. Phys. B98 (1973) 55 D. R. Harrington, Nucl. Phys. B59 (1973) 305 J. H. K&h and J. D. Walecka, Nucl. Phys. B72 (1974) 283 J. F. Reading, Phys. Rev. 156 (1967) 1116 E. J. Moniz, Phys. Rev. C7 (1973) 1750 K. M. Watson, Phys. Rev. 105 (1957) 1388; M. L. Goldberger and K. M. Watson, Collision theory (Wiley, New York, 1964) p. 749 32) E. J. Moniz and G. D. Nixon, Ann. of Phys. 67 (1971) 58.