dirac coupled channel calculations and nucleon scattering at large momentum transfer

Volume 218, number 4 PHYSICS LETTERS B 2 March 1989

DIRAC C O U P L E D C H A N N E L C A L C U L A T I O N S A N D N U C L E O N S C A T F E R I N G AT L A R G E M O M E N T U M T R A N S F E R ~

Jacques R A Y N A L Service de Physique Thborique ~ de Saclay, F-91191 Gif-sur-Yvette Cedex, France

H.S. S H E R I F Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 2 and Service de Physique Thkorique ~ de Saclay, F~91191 Gif-sur-Yvette Cedex, France

A.M. K O B O S 3, E .D. C O O P E R 4 and J.I. J O H A N S S O N Nuclear Research Centre, University of Alberta, Edmonton, Alberta, Canada T6G 2N5

Received 27 June 1988; revised manuscript received 7 November 1988

Attempts to fit the elastic scattering observables for ~ + 4°Ca at 500 and 800 MeV, using a spherical Dirac optical potential of Woods-Saxon geometry fail to reproduce the data at large angles. Coupled channel calculations are carried out within the frame- work of Dirac phenomenology. It is found that the coupling to the lowest 3- and 2 + states improves the agreement with the data at large momentum transfer.

Many exper imenta l da ta on pro ton elastic scattering at in te rmedia te energies have so far been analysed within the f ramework o f the phenomenologica l relat ivist ic opt ical model [ 1-4 ]. The technique em- ployed is to solve the Dirac equat ion with a complex potent ia l consist ing o f the t ime-l ike componen t o f a vector potent ia l and a Lorentz scalar potential . In virtually every case, with only a few exceptions in light nuclei, good agreement is ob ta ined for all observables, including the spin rotat ion function Q [ 5 ]. These analyses, however, were mainly concerned with re- gions o f small m o m e n t u m transfer q [ 6 ]. Fo r example for pq-a°Ca at 500 MeV the da ta analysed cover only the range q < 3 f m - ~, and at 800 MeV they reach

Work supported in part by the Natural Sciences and Engineer- ing Research Council of Canada. Laboratoire de l'Institut de Recherche Fondamentale du Com- missariat/l l'Energie Atomique.

z Permanent address. 3 Present address: Myrias Research Corporation, Edmonton,

Alberta, Canada T5K 2P7. 4 Present address: Department of Physics, University of Surrey,

Guilford, Surrey, GU2 5HX, UK.

up to q,-~ 4 f m - i . On the other hand there are measurements o f cross sections for both Ca and Pb at 800 MeV which reach up to 5 fro-1 [7,8]. Moreover , recent measurements by Hoffmann et al. [ 9 ] have extended the angular range of the cross section ( d a / d O ) and analyzing power (Ay) da ta at 500 MeV up to 0 = 6 0 ° (corresponding to q = 5.3 f m - l ) . These authors found that both relat ivist ic and non-relat ivis t ic impulse approx imat ion calculations fail to describe their da ta in the region q > 3.5 f m - 1. There are also earl ier reports of difficulties in the large angle region for ~ + 2°8pb at 800 MeV [8 ]. In a recent calculation for the T R I U M F data for pq-a°Ca at 300-500 MeV, it is evident that the best possible fits to the da ta fail to describe the observed cross sections for q> 3 f m - 1 [ 10]. In a reanalysis of the data on p + a ° C a at 800 MeV we found that it is possible to fit the three elastic observables using a s imple Dirac potential , pro- v ided the avai lable cross section da ta are t runcated at 0 = 34 ° (q-~ 4 fm-1 ). When all the measured cross section points are included in the search, it is not possible to fit the three observables s imul taneously with s imple Woods -Saxon potentials . It is possible to ob-

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403


tain fits with more complicated radial form factors, but this may be only an attempt to mimic real physi- cal effects such as those discussed below.

The failure of the simple optical potential to reproduce the cross section data at larger momentum transfer suggests that other effects, not taken into account in such a restricted model, must play an important role as the momentum transfer increases. One of the possible effects is the strong coupling to higher excited states. The deformation parameter f13 for the lowest octupole state in 4°Ca is close to 0.4 and hence one expects non-negligible effects in the elastic channel due to virtual excitations of this state. It should be pointed out that the possibility of channel coupling effects in elastic scattering has indeed been sug- gested earlier. Amado and Sparrow [ 11 ] have inves- tigated these effects using the non-relativistic eikonal approximation for proton scattering on Pb at 800 MeV. These authors reported large coupling effects for momentum transfer q> 3 fm-1. Later, however, their conclusions were contested by Ray and Hoffmann [12], who pointed out that their Schr6dinger equation based coupled channel (CC) calculations showed effects much smaller than those predicted by Amado and Sparrow. Hoffmann et al. [ 9 ] have also reported that similar CC calculations fail to account for the new ~+4°Ca data at 500 MeV. The purpose of the present note is to present new coupled channel calculations based on the Dirac equation for the scattering of protons on 4°Ca at 500 and 800 MeV. We show that the coupling to the lowest 3- and 2 + states goes a long way towards improv- ing the accord with the data at momentum transfer q>3.5 fm -1. The extension of the CC approach to the Dirac equation is motivated by the success of the latter in dealing with nucleon scattering at low momentum transfer [13]. Moreover, the proper treatment of spin dependence inherent in the Dirac approach makes it possible to study the effect of CC on the spin observables. Spin dependence was not fully treated in the CC calculations of refs. [ 9,11,12 ].

Our treatment relies on the recently developed Dirac coupled channel model for nucleon inelastic scattering (see ref. [ 14] ~' and ref. [ 17] ). The out- line of the calculations is as follows: we start with the

~' Earlier CC calculations for muonic atoms have been done in ref. [ 15 ], and for electron scattering see e.g. ref. [ 16 ].

Dirac equation describing the nucleon motion in a complex potential consisting ofa Lorentz scalar term Us and a time-like component of a four-vector potential Uv. These potentials are allowed to undergo shape deformations and hence can couple different states of the target. The coupled equations for the radial up- per component Fi and lower component Gz of the Dirac wavefunction corresponding to channel i are given by (in writing these equations we leave out the Coulomb interactions, but they are included in the final calculations; we take h = c= 1 )

-~r + F,-(Ez+m+U°-U°)G,

= E g~( U~ - U~)Gs, ( 1 )

= g , s ( U , , + U ~ ) F j , (2) xj

where E~ and m are the total energy of channel i and the rest mass of the nucleon, respectively. The factor x~ is the ith channel expectation value for the opera- tor 1 +tr-L. The superscript 2 refers to the multipole order of the particular potential; thus for example U ° and U ° are the monopole parts of the potentials. The coefficients g,~ are geometrical coefficients per- taining to channels i and j for the multipole order 2. The above equations are solved by iteration using the ECIS method [ 18 ]. The computer code ECIS88 has been developed for this purpose.

The procedure we followed was to first obtain the best possible fit to the data using a spherical potential, i.e. without coupling to excited states. This was done both at 500 and 800 MeV. The results obtained are summarized in terms of their Z 2 per data point in table 1. We also give the volume integrals (per nucleon) for the different parts of the Dirac potentials. The calculated cross section at 500 MeV is shown as the dashed curve in fig. 1. It is seen that the agreement with the data is good out to about 35 °, but de- teriorates beyond this point. Similar results hold at 800 MeV. For theAy at 500 MeV, the agreement with the data is good in the same angular range as for the cross section. However, when one again reaches 40 ° there begins a shift in the position of the minima, which gets worse as the angle increases. At 800 MeV

404


Table 1 Z 2 (per data point) and volume integrals (per nucleon) for the best fit potentials for ~+4°Ca at 500 and 800 MeV. CC (No CC) denote calculations with (without) channel coupling. J~ is the volume integral per nucleon for real vector potential in MeV fm 3, etc.

500 MeV 800 MeV

No CC CC No CC CC

Z~ 20.2 7.1 25.4 13,2 Z~ 6.2 3.7 3.5 3,8 z~ 4.4 3.8 2.7 2,9

J~ 1370 1310 1161 1152 J~ -502 -423 -669 -612 J[ -2085 -1964 -2056 -2025 J~ 389 275 559 481

where Ay da ta cover only the region 0 < 30.5 ° the results are in good agreement with the da ta except for the last 2 -3 o. The same good agreement is ob ta ined for the Q data. We note, however, that these cover an even more l imited angular range ( 0 < 22 ° at 500 MeV and 0< 17.1 ° at 800 MeV) .

In the next step in our calculat ions we included coupling to the octupole state at 3.74 MeV. The state is taken to be a one-phonon vibra t ional state. In the search procedure the inelast ic cross section and analyzing power were included in the ;(2 min imiza t ion in such a way so as to contr ibute about half o f the total ;(2. This const ra int ensures an acceptable descr ip t ion

10 ~

10 a

10 a

~_ 10 1 uq

10 o

10_1

I0 -z

lO_a

0 10 -~-

10-5

10-6

10-'7

10-8

+40 V

. . . . CC ~

....... no CC i i

I ] ]L0. z~0. 3~0. 40. s~0. 60. 0 .... (de 9 )

Fig. 1. Best fit results for the elastic cross section of ~ + 4°Ca at 500 MeV. The dashed curve (No CC) is obtained using a simple Dirac optical potential, The solid curve (CC) is the result of coupled channel calculations (see text). The data are from ref. [9].

for both elastic and inelastic scattering. The values adop ted for the deformat ion pa ramete r r3 were 0.39 at 500 MeV and 0.37 at 800 MeV, this lat ter resulting from an actual search. These values correspond to deformat ion lengths fi3 of 1.42 fm and 1.29 fm at 500 and 800 MeV, respectively, in agreement with earl ier results from nucleon inelastic scattering [ 7,19,20 ]. The inclusion o f the coupling to the 3 - state resulted in a considerable reduct ion in the ;(2 for the cross sec- t ions at 500 and 800 MeV for the Ay at 500 MeV, i.e. for the observables that cover a wider range o f mo- men tum transfer. We have made extensive system- atic searches for a large range of values of the imaginary vector potent ia l and found that the CC results are invar iably superior to the best fit No CC results ment ioned above.

The coupling scheme was then further extended to include the 2 + state at 3.9 MeV. For the deformat ion pa ramete r of this state we used the value r2 = 0.14, taken from ref. [ 19 ]. The f13 for the 3 - state were those given in the preceding paragraph. The search procedure on the potent ia l parameters resulted in further improvemen t in the Z 2. The effect o f adding the coupling to the 2 + state was not, however, as dramat ic as that o f the 3 - state. The solid curve in fig. 1 shows the results o f the CC calculat ions for the cross section at 500 MeV. It is seen that the best fit CC results are superior to the best fit spherical potent ia l (No CC) results. Identical results are obtained at 800 MeV. Note that the major improvement in the accord with the da ta takes place in the region of large m o m e n t u m transfer. There are similar, but slightly less dramat ic , improvements in the analyzing power results at 500 MeV.

Table 1 shows the Z 2 and volume integrals o f the resulting potent ia l for both the CC and No CC calculations. The reduct ion in the Z 2 for the da ta that extend to large m o m e n t u m transfer (da/d£2 at both energies and Ay at 500 MeV) is quite evident. There is also some improvement in the ; ( 2 for Q at 500 MeV. The apparent slight increase in the Z z for A and Q at 800 MeV is superficial and is mainly due to the fact that these data contr ibute only marginal ly to the total X 2. We should emphasize that the Z 2 compar ison in table 1 relates to the entire angular dis tr ibut ion. I f we consider only the region o f high m o m e n t u m transfer the improvemen t is much more dramatic . Fo r example table 1 shows that the reduction in Z 2 at 500

405


10-1

: ~ , P +'~°Ca

L 10-2 " LR "" "'""" - -''""*

- ~ 10-3

n~

lo-~ _ _ C 8 ( 0 " , 3 - , 2 " ) ' , ~

lO . . . . . . . O" o n l ~ I I

I I 610, 30. qO. 50. @ .... ( d e 9 )

Fig. 2. Large angle cross sections at 500 MeV for ~+4°Ca. The solid curve is the result of CC calculations. The dashed curve shows the result without coupling. The data are from ref. [9].

MeV due to the coupling is about a factor of 3. On the other hand for the region q> 3.3 f ro- ~, this factor increases to 6.5.

As a result of the coupling to excited states there are changes in the optical potential, as can be seen from a comparison of the volume integrals listed in table 1. We note that all volume integrals are reduced when channel coupling is taken into account but that the major reduction is in the imaginary potentials. Similar behavior is known to take place in non-relativistic CC calculations [ 21 ].

The above comparisons between the best fit No CC calculations with the best fit CC calculations, al- though they show the superiority of the latter may not, however, show the true extent of the coupling effects. This is because the flexibility o f free parameters in the No CC calculations allows the potentials to emu- late, among other things, these coupling effects. To get a more realistic picture o f CC effects, we make a comparison between the full CC results and those calculated with the coupling turned off but using the same optical potential. The results for such calculations are displayed for the large angle segment of the cross section data at 500 MeV (fig. 2) and 800 MeV (fig. 3 ), and for the Ay data at 500 MeV (fig. 4). The solid curve represents the full CC calculations with the coupling scheme including the 0 +, 3 - and 2 + states. The dashed curve (labeled 0 ÷ only), is the one obtained without coupling. In all the three cases shown there is hardly any difference between the two types of calculations for CM angles forward of 25 °.

L kn

" 0

ID "O

10-1

10-2

10 -a

10 -o,

10 - s

10-~

N . p+~'°Ca 8 0 0 MeV

i,,ii,[lll ", 12,,

...... 0 ÷ on l y

I I I I 2 5 . 3 0 . 3 5 . q 0 . qS .

O .... ( de 9 )

Fig. 3. Same as fig. 2, but at 800 MeV. The data are from ref. [7 ].

Comparing figs. 1 and 2, it is evident that the coupling effects are more important than may be con- cluded from fig. 1 (note that the solid curve in the two figures is the same). A characteristic feature of the dashed curve in fig. 2, which is also present in the RIA calculations o f ref. [ 9 ], is the failure to reproduce the correct oscillatory pattern of the data. This is present at both energies and is rectified to large extent by the CC calculations. Fig. 4 shows that quali- tatively the same behavior holds for the analyzing power. Similar but weaker effects are found in the case o f ~ + 2°Spb at 800 MeV; the coupling of the 3 - state improves the results for the cross section somewhat. The effect, however,is not as strong as for the 4°Ca case and certainly much less dramatic than those calculated by Amado and Sparrow [ 11 ] using the non- relativistic eikonal approximation.

In conclusion, we have shown by explicit calcula-

O)

o ca

c

kn D'3

m C

( E

1 . 0

0 . 5

0 . 0

- 0 . 5

- 1 . 0 3 0 . qO. 5 0 . 6 0 ,

e .... ( d e 9 )

Fig. 4. Same as fig. 2 but for the analyzing power.

406


t ions the i m p o r t a n c e o f coup l ed channe l effects for

the elast ic sca t ter ing o f p ro tons on 4°Ca at 500 and

800 MeV. T h e ca lcu la t ions are ca r r i ed ou t in the

f r a m e w o r k o f D i r ac p h e n o m e n o l o g y and hence s p i n -

orb i t effects are au toma t i ca l l y inc luded . Th is im-

p roves the non- re la t iv i s t i c C C ca lcu la t ions r epo r t ed

in refs. [ 9,11 ], in wh ich these effects were neg lec ted

in the coupl ing potent ia ls . In a d d i t i o n ou r t r e a t m e n t

also capi ta l ises on the success o f D i r a c p h e n o m e n o l -

ogy for the elast ic sca t ter ing o f nucleons .

O n e o f the au thors (H.S .S . ) acknowledges the

w a r m hosp i ta l i ty ex t ended to h i m by the Serv ice de

Phys ique T h 6 o r i q u e and the Serv ice de Phys ique

Nuc l6a i re ME, C E N , SACLAY.

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dirac coupled channel calculations and nucleon scattering at large momentum transfer

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