elastic scattering

Nuclear Physicr A282 (1977) 425-434; © North-HoUated Ptrbllahlnp Co., Amtta~tx

Not to be reproduced by photoprint or micro9lm svlthont wrhtm parminion from the pnl>tirher

COMPARISON OF GLAUBER THEORY TO FINAL STATE INTERACTION THEORY FOR pd IIvELASTIC/ELASTIC SCATTERING IAN DUCK, R. D . FELDER and G. S. ML I'CHLER

T. W. Bonne Nuclear Laboratory, Rlee URhxrsity, Houston, Texas 77001, US.! r Received 6 December 1976 (Revised 19 January 1977) Abstract : .The ratio R = o~(B)la~,(8) of pd inelastic/elastic differential cross sections has been calculated using Glauber theory . After the inclusion of D-state efFects following Harrington, and recoil efïects following Gurvitz, Alexander and Rinat, Glauber theory is in close accord with final state interaction analysis .

1. Inbroducdon Inelastic pd scattering in which the target nucleons are left at very low relative energy provides an interesting check on our understanding of the corresponding elastic scattering. An experiment has recently been completed t) in which the inelastic cross section has been resolved by final state interaction (FSn analysis s) into the 3St and tSa contributions. The feature that we are primarily interested in here is the ratio, R, of the inelastic scattering cross section into the (np) 3St continuum state to the purely elastic scattering cross section in which the (np) pair is left in the bound deuteron state. Final state interaction theory provides a simple result. What we report here is an extension to the continuum of Harrington's calculations 3) on elastic pd scattering . Harrington used Glauber theory `) to determine the effect of the deuteron D-state on the elastic cross-section minimum which is due to the interference between single and double scattering amplitudes . In this work we use Glauber theory illustrated in fig. 1 to calculate the ratio Rs - QinIQel for pd scattering for (np) pairs produced with a maximum relative energy of up to 8 MeV. We include D-state and hard core effects using simple analytic deuteron and continuum wave functions s). The results are model dependent but at the level of the present experiments simple FSI analysis suffices to produce the observed continuum bound ratio. Actually the Glauber double scattering amplitude (GDS) is quite different from FSI because of a zero in the inelastic form factor at zero momentum transfer due to the orthogonality of the deuteron and continuum wave functions. A slight modification of the kinematics to include recoil eliminates this feature from the double scattering amplitude and brings it into close agreement with FSI theory . f Worlc supported in part by the LiS Energy Research and Development Administration .

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In a recent paper, Gurvitz, Alexander and Rinat 6) have extended Glauber theory to include a number of corrections. Their double scattering amplitude (GAR) is modified in basically the same way by including recoil in the intermediate states. They go further and estimate offshell corrections to the Glauber amplitude. The calculations reported here do not include off-shell effects but just make on-shell kinematical corrections to Glauber theory . 2. Theory The inelastic cross sections follow by a simple modification of Harrington's work which is briefly reviewed for the sake of completeness. Harrington obtains the elastic differential cross section at momentum transfer d as IF°(d)IZ +IFz(d)I z, n ddz -

(l)

where IFo(d)IZ = I2.f(d~o(~d)+ 2n,~dZq.Î(~d+q)I(id - 4~o(q)Iz, IFz(d)I Z

=

~ 12f(d~z(~d)12+

Iz. 4 12l(d~z(~d)+ 2a J dz q .i(~d +q).Î(~d q~z(q) (3)

So and Sz are the deuteron spherical and quadrupole form factors given by ~ 4mzdr(uz(r)+wz(r))1o(4r),

(4)

~4mzdr2w(rxu(r)-~w(r))Îz(4r),

(5)

so(q) = Sz(q) =

J0

J0

where u(r) and w(r) are the S- and D-state radial wave functions as defined in the appendix . The deuteron wave function is normalised so that So(q = 0) = 1 . S1z(P) is the tensor operator. The cross sections are evaluated using a Gaussian representation of the nucleon-nucleon scattering amplitude .Î(d) = 4~(~p -~RNdzxl-tP) with RN = 0.4 fm, p = -0.3 and Q = 40 mb. The precise values of the parameters R and p are not very well defined') but our results turn out to be very insensitive to them .

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To get the inelastic cross section for production into the 3S t continuum state one must judiciously replace bound state radial wave functions u and w by their continuum counterparts ~s and fin, which are functions of the (np) relative momentum k. We obtain inelastic form factors Sio°(k, 4) = Si(k, 9) =

m4~dr{~s'(k, rh~(r)+~n(k, r)w(r)}1o(9r),

J0

~4nrzdr{~s(k, r)w(r)+ ~n(k, r)tdr) - W e~u~(k, r)w(r)}.lz(gr)

J0

and inelastic Glauber amplitudes F'ô(k, d) and FiZ (k, d) exactly as in eqs. (2) and (3). With the normalization the inelastic differential cross section for (np) 3S 1 pairs of maximum energy Eop is 3

~ déZ = J (2n)3 {I~(k, d)IZ+I~(k, d)IZ}

integrated up to k2/m = Eop. The continuum cross section involves a threefold integration on r, q and k which must be done numerically so it is important to have analytic expressions for the radial wave functions. Many refined representations of the deuteron functions exist but no comparable effort has been put into the continuum functions. In order to have a unified and convenient representation of the bound and continuum wave functions

0

-, kF _p

Fig . 1 . (a) Glauber single scattering amplitude (GSS) for Pd almost elastic scattering with (np) relative momentum k F in the final state . (b) Glauber double scattering amplitude (GDS) for pd scattering. Momen tum transfer q to the intermediate state includes a fixed longitudinal (parallel to p) component in the Gurvitz-Alexander-Rinat (GAR) treatment.

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I. DUCK et al.

we have resorted to the Yamaguchi wave functions including the tensor force effects e). The simple expedient of pushing the S-wave radial functions out to accommodate a hard core does not alter the orthonormality or the D-state probability. As simple as they are, the Yamaguchi wave functions involve some lengthy algebra so they have been relegated to an appendix. Before going onto a discussionofnumerical results, weexamine theGDS amplitude and incorporate some recoil corrections which will have a considerable effect on the inelastic/elastic ratio. The GDS amplitude corresponding to fig. 1 b is 3 3 F(d) 4np (2n)3 ~t (k+4~~(k).i(~d +q).i(id - 4)in~P' 9) J (2~)3

= 2a dZgl.f(~d+gl).T(id - 41) fd3~c*(rk~l~'~~(r), J

(12)

where the integration is on momentum transfer ql which is perpendicular to the incident beam momentum p. We can relax the Glauber approximation by replacing ~II) by the energy shell delta function 8(E - E~,) which restricts the intermediate state to on-shell propagation. Energy conservation requires 2P' 9 = -3P' d -d . 9 - q2 ~ - 3P' d = âd2" so the energy conserving delta function becomes 8(p'[q ll -d~l8p']) withp' = p+~dll . In all of this, terms containing k, q, kF and deuteron Fermi momentum have been dropped . This result is the same as that of Gurvitz, Alexander and Rinat ~ in their on-shell correction to the GDS amplitude. The result in eq . (12) is that the double scattering amplitude involves a form factor S(q 2 ) with qz = q~l + qi for qlI = d s /8p' and 0 < q 1 < ao. Since the form factor is never evaluated at zero momentum transfer, the inelastic form factor is not so sensitive to the orthogonality of the continuum and bound state wave functions. Thus, the ratio of continuum bound scattering cross sections is brought into closer agreement with FSI theory . 3. Results Simple FSI produces a continuum bound ratio __ dQQ~dd z -_ ~~ kZ dk I ~or r = j) 2 R tst dQ dd 2 2n2 tP'~ r=0 '

(13)

which depends upon the magnitudes of the wave functions at the origin. In fig. 2 the ratio obtained from eq. (13) using unmodified Yamaguchi S-wave functions [(r~ = 0, t = 0 in eqs. (A.5) and (A .8)] is compared to the prediction of unmodified Glauber theory for incident proton energy of 800 MeV and a laboratory scattering angle B,,b = 45° where the double scattering amplitude dominates. The ratio calculated using GAR theory is nearly the same as that calculated from FSI theory

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42 9

Ep (MeV)

Fig. 2. Effect of the GAR correction to the GDS amplitude is to circumvent the orthogonality of initial and final state wave functions and increase R = a,,la,, to agree with FSI.

and the two are shown as a single curve in fig. 2. The importance of the GAR correction to the GDS amplitude is evident. Without including longitudinal momentum transfer in the form factors the orthogonality of the wave functions reduces the inelastic cross section so that Ra is calculated to be 0.22 whereas with the GAR correction and q~~ = 105 MeV/c the ratio becomes 0.32 in close accord with FSI which produces 0.31 . Pure GDS produces an (np) relative energy distribution that is more peaked than FSI. Even if the peaks are normalized to coincide at zero relative energy, the GDS peak is 16 ~ below FSI at 8 MeV whereas the GAR peak is within 2 ~ of FSI . Teat,e 1

Effect of parameter choices on ratio Rs of inelastic/elastic cross sellions for E,P < 8 MeV at E,, a 800 MeV, B,,s = 45° Parameter choice RN =0 .4 911

411

¢`0

=0

RN =0 .7 411# 0

Rs

r~ = 0.0

p=-0.3 t=0 1 .784 2.5 t = 1 .784

0.31 0 .32 0.32 0.30

r~=0 .3

t=1 .784

0.22

r~=0.3

p=0.0 t= 1.784

0.32

r~=0.3

GAR corrected longitudinal momentum transfer = 105 MeV/c ; r~ = hard core radius in fm ; 911 = r = strength of Yamaguchi tensor fore ; RN = range parameter in the NN amplitude in fm.

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The effect of other parameters besides q~ ~ on the ratio R is quite small at a scattering angle 9,,b = 45° as is to be expected from Harrington's results where the cross section was only sensitive to the D-state near the interference minimum. Table 1 gives a briefaccount of the effects of various parameter choices on Ra at 9,, = 45°. As a function of lab angle, the GAR ratio Ra deviates somewhat from the FSI result as shown in fig. 3. This is especially true if tensor force effects are left out and p is set

Fig . 3 . Ratio R s = o,,la,, as a function of 9,,e including (np) continuum pairs with relative energy up to 8 MeV . FSI labels the value of the angle-independent final state interaction ratio indicated by a bar on the margin, GAR labels four curves for the Gurvitz-Alexander-Ringt calculated ratio, GDS labels the Glauber double scattering result calculated only at 8 ~ 45°, E,e = 800 MeV. Datapoints are from ref. ') .

equal to zero . In this case a deep minimum in Re occurs where single and double scattering amplitudes are equal inmagnitude. A tensor force providing a4 % deuteron D-state is sufficient to fill in this minimum and bring Ra into near equality with FSI theory. This extreme sensitivity to the tensor force is removed when a non-zero value of p is used. Then, as shown in fig. 3, the minimum in Rs is quite shallow and only weakly dependent on t. A small maximum occurs at B,, b x 15° in the region where Glauber single scattering (GSS) dominates but momentum transfer is still sizeable. The orthogonality of the 3S t continuum and bound state wave functions then takes over and reduces the ratio Ra eventually to zero in the forward direction. One feature ofthecalculationis practically independent ofthe choice of RN , p and t. This feature is the value of Ra in the large angle region where the double scattering terms dominate . For a range of values, GAR .produces a value in agreement with FSI (and experiment) and about 40 ~ larger than GDS. In fig. 4 the inelastic form factor S'ô(k, q) defined in eq . (8) is compared to the magnitude of the final state wave function evaluated just outside the hard core, at a radius of one ferrai. FSI theory would assume that the k-dependence of the form factor is contained in the factor ~t k (r x 0). It is apparent that this is a, very good

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0

OJ

02

03

k

0.4

Oâ

43 1

0.6

Fig. 4. Plot of IS°o(k, q)h evaluated with Yamaguchi wave functions with t = 1 .784 and r~ = 0.3 fm, compared to I~t(r = l fm)l'. All moments measured in fm - ' .

approximation for momentum transfers q greater than ~ 0.7 fm - ' . For smaller momentum transfers the form factor falls off much more rapidly with increasing (np) relative momentum than does the FSI factor. At q = 0, orthogonality requires the inelastic form factor to vanish . At an incident proton energy of 800 MeV, FSI theory begins to fail as an approximation to the GSS amplitude at q ~ 0.5, d ~ 1 and 6,,6 ~ 5°. 4. Conclosions

The ratio R = c,o(~/Qe,(~ of inelastic to elastic differential cross sections for pd scattering to the'S1 continuum (np) state and to the deuteron bound state has been calculated according to Glauber theory following Harrington's work on elastic scattering. The ratio is a function of the maximum (np) relative energy included in the inelastic scattering, and following analysis of the experiment of ref. t) we choose Eop = 8 MeV as a reference point. Except for a zero in the very forward direction, R is in close agreement with the predictions of final state interaction theory, eq. (13). It would be extremely difficult to resolve any of the structure in R and perhaps nothing beyond a FSI analysis is justified. The most interesting features of the results are the refinements that bring Glauber theory into accord with FSI theory. First, at large angles where the GDS amplitude dominates, recoil corrections due to Gurvitz, Alexander, and Rinat must be included.

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The result is to avoid a zero in the inelastic form factor at zero momentum transfer, and to increase R8 by z 40 ~ from 0.22 for GDS to 0.32 for GAR, which is in close agrcement with FSI which produces a value 0.31 . A second interesting feature of the results is the extreme parameter independence at angles where the comparison to experiment has been made . We have found R8 at large angles ( Z 45°) to be practically independent of RN and p, the parameters of the high energy NN amplitude, and of t, the strength of the 3S1 tensor force. We are therefore confident that the GurvitzAlexander-Rinat correction to Glauber theory has been observed in the experiment of ref. 1). We have enjoyed helpful discussions with Professor G. C. Phillips . Appenüix A.1 . YAMAGUCHI WAVE FLiNCTIONS

The Yamaguchi wave functions are particularly simple in momentum space where the deuteron S-state radial wave function is ~P) = 4nN and the D-state function is

1 - 1 az +Pz ßz +Pz

(A .1)

zPz -4aN(ßz az)t (A .2) ~P) _ (a z +p xY z +P z ) z The original parameters were chosen to be a = 0.2316 fm -1, ß = 5 .759a, y = 6.771 a and t = 1 .784 to produce a 4 ~ D-state probability. Increasing t to 2.5 increases the D-state probability to 7.8 ~. The radial wave functions in coordinate space are obtained as

~r)

a .'~P)pz(~' P) = 2N(ßz-az)t - J (2~) e+p 3 yh(iyr) yzrh'(iyr) x {4 (Yz- a z ) + ~Yz- az)z

where l

3

3~

+ + Cz zz zs

y3h(ryr) a3h(iar) az)z uYz-az)z} uYz+ dh(iz) dz

(A .4)

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The hard core is inserted into the S-state radial function by

Similar expressions are available for the continuum radial functions erx. - e -a" (A ~s(~ r) =10(x)+ 1 +, ,1J (ßz+kz)z r for the S-wave function and .1

1

~

d3p ej° ''Pz(ß' P) ~ - tPz ~ ~~k, r) -- 1+.1.J ßz+kz (2~)3 z -kz - iE (yz+pz)z p

.6)

A.

for the D-wave. Here ~, is the strength of the Yamaguchi interaction A = 8~ and J is the integral

1

tz Saz +4ary+y z -' a {ß(a+ß)z + 8 Y(a+y)

(A.8)

tzpa 1 1 d3p . (A.9) - ,%(2n) 3 Pz- kz- ~~(ßz +Pz)z + (Yz+pz)4 } The continuum radial function ~s can also be pushed out to accommodate a hard core and - e-a,~ sm k(r-r~) .1 1 e~a.-.o~ - e -ac .- .~~~ (A.10) r __

for r > r~, zero for r < r~. For the sake of completeness we record 1

t

ik3 y 3 yz 1 + ~yrh'(rYr) X{(kz+yz)z h(~)+ Yz+kz Ch(ryr) C2 - Y z + kz] J~

(A .11)

and J = Js +J° with

Js

The function

_ _ 1 1 ( S ~ß 4~ (ßz + kz)z

4n{(kz+.yz)4

kz

l ik } . + 2ß

+2F(G+3GG'+G")~

(A .12)

(A.13)

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W1Lh

References 1) R. D. Felder, T. M. Williams, G. S. Mutchlor, h Duck, J. Hudomalj-GabitTSCh, M. Furi6, D. Mann, N. D. Gabit~sch, J. M. Clement, G. C. Phillips, T. R. Witten, E. V. Hungerford, M. Warneke, B. W. Mayes, L. Y. Lee and J. C. Alfred, Nucl, Phys. A280 (1977) 308 2) M. Goldberger and K. Watson, Collision thoory (Wiley, NY, 1964) 3) D. R. Harrington, Phys. Rev. Lett . El (1968) 1496 4) V. Franco and R. Glauber, Phys. Rev . 14Z (1966) 1155 5) Y. Yamaguchi and Y. Yamaguchi, Phys . Rev. 93 (1954) 1635 6) S. Gurvitz, Y. Alexander and A. Rinnt, Ann. of Phys. 93 (1975) 152 7) S. Gurvitz, Y. Alexander and A. Rinnt, Ann. of Phys. 98 (1976) 346 8) Y. Yamaguchi and Y. Yamaguchi, loc. cit. 9) S. Gurvitz, Y. Alexander and A. Rinnt, Phys . Lett . 59B (1976) 22