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185 MeV proton scattering from 6Li and the impulse approximation
Nuclear Physics Al54 (1970) 261-272;
@ North-Holland
Not to be reproduced by photoprint or microfilm without written
185 MeV PROTON SCATTERING AND THE IMPULSE
Publishing
permission
Co., Amsterdam from the publisher
FROM 6Li
APPROXIMATION
R. M. HUTCHEON? Institut fiir Kernphysik
der Universitc?t Mainz,
0. SUNDBERG The Gustaf
Werner Institute,
West Germany
and G. TIBELL
University
of Uppsala,
Uppsala, Sweden
Received 12 June 1970 Abstract:
Cross sections for the scattering of 185 MeV protons from 6Li have been measured for the elastic case and for inelastic excitation of the first two excited states. These cross sections are compared with plane-wave impulse approximation calculations in which the nuclear matrix elements are taken directly from electron scattering data in order to eliminate model dependence. The experimental to theoretical cross-section ratio is 0.5f15 % for a momentum transfer of 0.7 fm-’ and has a strong momentum-transfer dependence. Previous calculations made in the distorted wave impulse approximation bring the ratio closer to unity but do not lead to complete agreement.
E
NUCLEAR
REACTIONS 6Li(p, p), (p, p’), E = 185 MeV; measured a(Ep,, 0). Enriched target.
I
1. Introduction The plane-wave and distorted-wave impulse approximations (PWIA and DWIA) are used very often in the analysis of medium-energy proton scattering data, and thus it is important to have a test of the degree of validity of these approximations. In such tests the uncertainty in the nuclear structure information usually hinders one from reaching any definite conclusions. For example, the difficulties in light nuclei of theoretically reproducing the experimental values of (p, p’) cross sections ‘) and the (p, 2p) angular distributions and occupation numbers ‘) might be caused either by a breakdown of the impulse approximation or by the assumption of incorrect nuclear wave functions. The general approach in this paper is to perform such a test using the impulse approximation formalism for proton scattering and assuming the nuclear structure information to be given by electron scattering experiments. The latter assumption is reliable because the electromagnetic interaction is well known and because the new linear electron accelerators are producing very accurate, consistent data over a wide _.,_.___ _J? -_ ____ -I-- ~~ . rdnge UI momenrum transfers. The “ii nucieus was chosen for its reiativeiy iownucleon density (the rms radius of the p-shell nucleon distribution is w 3.0 fm), for t Present address: Physics Dept., Queen’s University, Kingston, Ontario, Canada. 261
262
a. M. mrrcmo~
et al.
its reIative ease of handling, and because a large amount of accurate electron scattering data exists for it. An incident energy in the region of 200 NeV was chosen because at this energy the nucleon-nucleon total elastic cross section is at a minimum (a 25 mb) [ref. “)I and thus multiple scattering effects in proton-nucleus scattering should be minimized. It should be noted, however, that the direct application of electron scattering information to the analysis of proton scattering is only justified if, during the proton-nucleus interaction, there are no compound states formed and the excitation does not go through intermediate nuclear states (i.e. fist Born approximation for the --,:r,-L-- -_-~.^-‘_-\ O-CL rf-_^_ ,,,?Z:c:,,, are GAFQLru dyI~S+PS4lcv r\ hUe ..,a11 EP+;&P~ hn CX~ILCLCIUII ~~~~~i~~~~. DULU LU~;~C; CIVUUIL~~~~~ mc11OuLIOMUu “, inelastic proton scattering from 6Li at this energy,
2. Theory t 2.1. PWIA PROTON-NUCLEUS SCATTERING FORMALISM
The PWIA expression for the proton nucleus scattering cross section 4*“) (neglecting antisy~etrization between @e projectile and the target nucleons) in the case where the nuclear ground state has isospin T = 0 is given by do
4 * iti -
-=
d&IL
f M-+1
-
2 kf
)
4n
-ko25,+1
+[E2-#32+C2+F2)]Agl,
~[A~+c~~CN:+IIB~+CZ+F~I~S;, A
,
(; ; ;)((t; ;)P’+W~+l)’ x
(- l)+“-“‘s;, I’S& I , 1
(0
where M is the nuclear mass in units of the proton mass, Ais the multipolarity of the transition, J,, is the ground state spin, k. and kf are the initial and final momenta in the proton nucleus c.m. system and q the momentum transferred to the nucleus in the lab system. The coefficients A’, B2, C2, E2, F2 are functions of k, and q; they have the units of cross section and their numerical vaIues can be extracted from the nucleonnucleon scattering data for a particular energy and momentum transfer. There are two sets of these parameters corresponding to the cases AT = 0 and AT = 1. Hereafter, the subscript CIdenotes AT = 0 and /? denotes AT = 1. The reduced nuclear matrix elements NAand S,, are also isospin dependent and are given for Tr = 0 by
t AI1 formulae given in this section have been derived from basic principles and compared with the ~""*-nlmt.4 -v-_-_y___. rPfPrPncf?P.
6Li
PROTON
263
SCATTERING
and for T = 1 by
A computer program was written to calculate 2’) the complex Kerman-McManusThaler amplitudes A, B, C, E, F, using recently published nucleon-nucleon scattering -
Q5A&;: 0.5
1.0
1.5
qLapd)
Fig. 1. The appropriate combinations of nucleon-nucleon scattering amplitudes plotted as a function of the nucleon-nucleon scattering lab momentum transfer.
phase shifts “) in the formalism of Stapp et al. ‘) with higher phases calculated using the OPEP approximation “). The amplitudes have been calculated so as to include the Coulomb interaction between two protons. The effect of neglecting the Coulomb interaction is shown by the dotted line on fig. 1 where the appropriate squares of the amplitudes are plotted. One can re-write the proton scattering cross section resulting from the AT = 0, non-spin dependent matrix elements (NL) in terms of the electron scattering first Born approximation Coulomb form factors because the nuclear transition operators
R. hi. mmmo~
264
et al.
are identical ‘). Thus, (4) where the F,,(q) are the first Born approximation Coulomb scattering form factors o-.-l f;r in\ 10 ;Dthe nrr\tnn rhorme l”llll fnrm IUVLVI. fgrtnr R~rmce the form frrrtnr ncpd UllU “)$\LJ, C&h” y’vc”‘” vU.urav YVYUYY” *II., (L”I__& I--...& -u-1 in -1 the _--proton scattering formalism must be derived from the distribution of the nucleon centers, the effect of the finite proton charge distribution must be unfolded from the electron scattering form factor when inserting it into the proton scattering expression. This is accomplished through dividing the electron scattering form factor by G,,(g). For transitions in which the spin-dependent matrix elements contribute, the comparison with the electron scattering form factors is model dependent and separate calculations must be made in each case. 2.2. THE STRUCTURE OF 6Li AND PWIA CROSS SECTIONS
The charge form factor of the “Li ground state has been measured by electron scattering and is fitted reasonably well up to q = 1.7 fm-’ by a harmonic oscillator shell model with separate parameters for the s- and p-state (as = 1.632 fm, up = 1.98 fm) [ref. ‘“)I. The ground state has a very small quadrupole moment, and in shellmodel analyses is found to be almost pure (99 ‘A) 3S1 state ll). The magnitude and the shape of the electron scattering form factor for the transition to the 3.56 MeV, Ot, T = 1 level is given to within 5 % assuming a shell model ISo configuration for the excited state, indicating it is an almost pure spin tlip transition 12-14). Any other model preserving this pure spin flip character would yield the same normalization. Thus the reaction mechanism of the transition to this level is well understood, and the mnnnZ+.rrl~ nF tha nnlnrriotnrl epAxz,-enA +ronn;+;r\n r\mh~h;l;t~g ;a prepntilllv ;nApnpndpnt muqjruruuti “1 u.lLI ~alru,Pccu IL‘UU_U L,411c-~LL”ll J,.a”“U”l‘lCJ 1.J ““YY”L’U”_I “‘..-~““u”‘“‘ of the radial wave functions. It has also recently been possible to fit the form factor for the transition to the 5.36 MeV, 2+, T = 1 level using the shell model. ‘I) The shape of the form factor for the transition to the 2.18 MeV, 3+, T = 0 level is also reproduced by the shell model, but the magnitude of the calculated form factor is a factor 2 to 3 too small. This is undoubtedly due to the failure of the simple shell model to describe the deuteron-like correlations which the stronger T = 0 forces induce ’ “). Thus although in this case the nuclear matrix eIements are not easily calculable, the reaction mechanism is fairly well understood. The proton scattering PWIA cross section for elastic scattering from 6Li is given by do -=
da,.,.
105.8{[& -/-C,z]F&,(q)i-&[I+,2
+ C: +E,2 -t-F,Z]F;(q))
G+J
9
(5)
Ep
where F,,(q) is the elastic electron scattering form factor for 6Li and F,(q) is the elastic electron scattering form factor for only the p-shell n&eons. Both form factors are normalized so that F(q) = 1 for q = 0. The second term occurs because ‘Li has
6Li PROTON
265
SCATTERING
a ground state spin 1 and thus the equivalent of electromagnetic Ml scattering is possible. The quantity F,(q)/G,,(q) has not yet been measured directly to sufficient accuracy, but the form factor for the 3.56 MeV transition F,.,,(q) because of its spin-flip nature, should be very nearly the same. The contribution of the second term to the total cross section is in any case less than 2 % for a momentum transfer q 5 2 fm-I. The proton scattering PWIA cross section for excitation of the 3.56 MeV level calculated with a simple LS coupled shell model is given by
(6) A comparison
of the proton
and electron
F32.56k) =
scattering
nuclear
matrix
J’idd 3-22m’
B(ML t)
where of the 0.0586 The
elements
yields (7)
F,,(q) is the inelastic electron scattering first Born form factor for the excitation level 13, 14) and th e reduced transition probability B(M1, t) was taken to be fm2 [ref. ‘“)I. F. .n=l _I cross section for excitation of the L.111Nrev state with the same modei is da d%m.
= 105.8 $ [A,Z+C,2+$E,2++(B;+C:+F,2)]F;.18(q),
(8)
0
where it should be noted that the factors 4 and 3 are peculiar to the shell-model LS coupling scheme assumption “) and are thus model dependent. Again, comparison with electron scattering theory yields %
s(q) = F:,(q)/G&(q),
(9
where Fc2(q) is the electron scattering first Born form factor for excitation of the m 10 _L..L^ L. lb Xl-X, 1VlGv blalc. If in all three cases one replaces the theoretical electron scattering from factors with the measured electron scattering form factors, then one can hope to eliminate most of the model dependence of the calculated proton cross sections, including the troublesome c.m. motion corrections ‘3 “). This is especially true for the elastic and the 3.56 MeV transitions, in which the maximum possible model dependence is less than 5 %. 3. Experimental
results
The experiment was performed on the 185 MeV proton synchrocyclotron at the Gustaf Werner Institute, University of Uppsala. A general description of the apparatus and experimental layout as well as the data analysis methods is given in ref. 16). Cross sections have been measured for lab scattering angles between 8” and 50”
266
a. M. FlUTCHEON
et al.
using 1 to 2 mm thick 6Li targets of 99.3 % isotopic purity. The targets were handled as in ref. ’ “) and the absolute thickness was judged to have an error of f 0.01 mm. Four typical spectra are shown in fig, 2 the notable features of which are the improvement in resoiution over other measurements r ‘* I7 j and the presence of a
E
=185 MeV ‘Lab
Q ~b
li-
I
.s
177
=15O
I
I
178
179
._.
160
181
, -.... .
I
-__ 1UJ
2’; E
I \
=185 MeV
/ /
‘Lab e
:
Lab=l7.65o
I ,
/i
5-
4 X&i i_ \
+/ : 176
177
178
.J 179
180
‘1/ , i, 182 Efdt$eV'
Fig. 2. Spectra of protons scattered from 6Li at various angles. The dotted line through the points ._I_Al_-__._ Q-l__L^__11_^_ ^__.._^-1 1_ &L^___1_._1^ ^E_^^l_^__^_^_^_L_..._ iSi0 giiiuemt: eye. ~nc;oase~moa assuuw m we ana~yxs UI yean mr;as altiSIIUWII.
large continuous cross section similar to that found in electron scattering 18plg) and attributed to ad breakup of 6Li. This continuum does not seem to have the shape predicted theoretically “3 17) by monopole excitation to the ad continuum. Only two strong inelastic peaks are seen in the spectra, and, as in the recent electron scattering
6Li PROTON
\‘i 10
15
‘-_‘r-. 20
261
SCATIl3lilNG
* 25
30
35
40
E
:td
Fig. 3. Angular distributions for the 2.18 MeV and 3.56 MeV peaks in 6Li. The dotted lines are guides to the eye. The 3.56 MeV level point without error bars at f&.=. = 13.3” is the normalization point for all spectra.
4
DOE,
~185 MeV Lab.
IO -
.
Elastic
X
2.18 MeV
Level
l-
I.1 -
” 5
Fig. 4. Angular distributions
for the elastic and 2.18 MeV transitions. The dotted lines serve to guide the eye.
268
R. hf. nmcmot4
et al.
work with improved resolution, if a peak at 4.5 MeV excitation with a natural width of 600 keV exists, it is only weakly excited. The peak areas and thus the cross sections have been extracted from the spectra by assuming hand fitted baselines under the peaks as shown in fig. 2. It was, however, required that the shape and FWHM of the inelastic peaks be consistent with the elastic peak. The uncertainty in the baseline produces an error in the peak area depending upon the quality of the spectrum, and this error has been linearly added to the statistical error which was commonly between 0.5 and 3 %. A&“IIC: ^^^ sn1g:lc ,.--1,. /Cl -- 5L”S”IllK ..L__,..L- crobs-becLI”u ___^^ ___LI_- call”raLI”” __l:L__r:-- __.^^ --lr sr&inlSL .._..I_“& Al. (I+& = 11 11.110,, au was IlIaCIt: the proton-proton cross section using a polyethylene target in which the proton thickness uncertainty was ) 1.2 %. The proton-proton c.m. cross section at this angle was calculated “‘) and found to be 3.73 mb/sr f2.5 ‘A, using the nucleon phase shifts of McGregor et al. “). The differential cross section in the lab frame for excitation of the 3.56 MeV level at Blat,= 11.1” was determined to be 1.24 mb/sr with a 6 % error. All the experimental points have been normalized relative to this value and the error in this normalization procedure has been added linearly to the error in the individual peak areas. The measured cross sections and their errors relative to the 3.56 MeV, Oiab= 11.1” mennntpment 9re I-_ shnwn in --~_. fivn _? I--2nd 4. . -Tn nrder must -__-_ one ---_ _..~. ________ c__I_ --..-- _____ * to nhtgin 811&Q!IJ~~ error
add 6 % to all errors shown. A comparison between the present cross sections and the previous values 1“) measured with poorer resolution indicates that the assumptions made in determining the peak areas in the earlier experiment were not correct. 4. Comparison with theory and discussion Using eqs. (5), (6) and (8) for the proton scattering cross sections, one can either compare the measured cross section with the cross section calculated using the electron scattering form factor or one can extract from the proton cross sectionsan effective form factor, which because of rescattering effects is expected to be smaller than the form factor extracted from the electron scattering results. In figs. 5 and 6 are shown both the calculated and effective proton scattering form factors for the 3.56 and 2.18 MeV states, and fig. 7 shows the calculated and measured elastic cross sections. The most notable features of fig. 7 are that the measured cross sections are smaller than the calculated ones by approximately a factor 2 at 4 = 0.7 fm-l and by a factor 3 at 4 = 1.2fm-l, but for higher-momentum transfer (larger angles) the curves cross and the measured cross sections become much larger than the PWIA calculations. This latter effect has been seen before, e.g., in the analysis of proton scattering from 4He at 147 MeV [ref. ‘)I. A general comparison is made more clearly in fig.8 in which the calculated to experimental cross-section ratios are plotted for each of the levels. It should be noted that in the momentum transfer region 0.4 fm-l 5 q 5 1.0 fm-l the PWIA to measured cross-section ratios are within 10 % of each other for all cases
6Li PROTON
269
SCATTERING
studied here, the AT = 1 transition ratio being a few percent higher than the AT = 0 ratios. This, coupled with the fact that the three transitions have quite different transition mechanisms, suggests that in this momentum transfer region the nuclear model dependence has been removed from our calculated proton scattering cross sections, and that the difference between calculated and measured cross sections must
jpq-t.j.i,_
MeV Level (Jrr33’I
2.18
,
Ii
/;
‘L*
,+,_I--#- -*-..
‘f‘f
‘f,\
‘f, ‘\\ ‘\ ‘4 \
I
1ci3
7 n
Eigenbrad , Darmstadt
x o .
Neuhausen,Mainz Hulcheon and ~resent,ex!racted
\
\
\
\. \
\
(e. e’)
Caplan,Sask. I from (p,p’)
\ ‘\ ‘\
\
\
\ \
I
I
0
0.5
\I I\ I
1.0
1.5
2.0
2.5
I
sctlll-l, Fig. 5. The form factor for the 2.18 MeV transition. The proton scattering form factor has been extracted under the assumption that the PWIA formalism is valid. 1.0 3.56
Fi5s(q)
MeV Level (Iv=
0’)
0.1
.
Eigenbrod,
0 Hulcheon X Neuhausen l
0.01 0
Present,
Darmstadt el.al.,Saskatcon and exlracled
Hutcheon,Mainz from
(p,p’)
0.5 q
(f&j
Fig. 6. The form factor for the 3.56 MeV transition. The proton scattering form factor has been extracted under the assumption that the PWIA formalism is valid.
270
et
R. M. HUTCHEON
d.
\ 0.5
10
1.5 q IfIn-‘)
2.0
t
25
Fig. 7. The elastic proton scattering cross sections at 185 MeV for 6Li. The calculation is done in the PWIA using experimental electron scattering form factors.
f
3111 ‘, -
/ /
S? 2” vL 22 20
;e
0
2.0 [
.)
/
/
i
0’ ’
Elastic
---2.18
t&V
--.-3.56
MeV Level
Level
Jz
1.0
ZE z
0.5 10
9 (ftil)
1.0
20
30
40
50
Gt.4
Fig. 8. The ratios of the calculated PWIA proton scattering cross sections to the measured proton scattering cross sections. from an inadequacy in our treatment of the projectile-nucleus interaction mechanism. The most obvious improvement in this respect would be to perform DWIA calculations for the transitions: that is, to allow for distortion and absorption of the incident and outgoing projectile wave functions. DWIA calculations have been made for 155 MeV proton scattering for both the 2.18 and 3.56 MeV transitions ‘9‘), using optical potentials with a real part between 7 and 10 MeV and an imaginary part between 10 and 20 MeV. The (PWIA/DWIA) ratio was found to be 1.4 at q = 0.9 fm -I for the 2.18 MeV transition, and 1.2 at q = 0.7 fm-l for the 3.56 MeV transition. Taking these corrections into account one still finds that the calculated to measured cross-section ratio is about 1.7. Another DWIA calculation was made at 185 MeV tar the 3.56 MeV transition ““) and the (PWIA/DWIA) ratio was again found to be approximately 1.20 for a momentum transfer of 0.7 fm-l.
come
6Li PROTON
271
SCAlTERING
Thus these distorted wave calculations do not bring our calculated and measured cross sections into agreement. Undoubtedly a fit to the data could be obtained by increasing the values of the deforming and absorbing potentials, but then one must provide a physical justification for the increased depth of the potential parameters. As the p-shell particle density is a little less than one nucleon per 50 fm3 it is plausible that almost all of the deformation and rescattering effects should occur through interaction with the rather tightly bound core. However, DWIA calculations for pa particle scattering have also indicated a clear disagreement between theory and experi_^^C.‘^^ e-c:,. ,.c ,.--_,...:-_c^l.. -_-_&. ..a UI. -r 41,ll11u C-d ^ /nxx7r.4 I-___..__31\ ^_^^^ IlKllC. v^_-^^ Re‘lllall cl. a \Y vvlkl,“‘t;ilL”Ll~tu, C;I”bS-btxLI”II lilLl” “I apy’“.“““dLLcxy 1.7 for elastic scattering of 310 MeV protons on cc-particles. Thus, although it is possible that our results may be reproduced by a complete distorted-wave treatment, it does not at present seem likely. 5. Conclusions The present experiment clearly demonstrates the discrepancy both in magnitude and shape between PWIA calculations and measurements of the inelastic proton scattering cross sections for 6Li. Corrections to the PWIA using existing distortedwave calculations tend to improve the agreement but are not sufficient. Obviously more extensive DWIA calculations should be made with careful consideration of absorption and multiple scattering effects in order to see if agreement can be obtained with physically reasonable parameters. However, accepting the evidence of the available calculations, the discrepancy can probably be explained through one or a combination of the two following effects: (i) the impulse approximation is not sufficiently accurate, (ii) the outer two nucleons in 6Li are strongly correlated implying a nucleon density at the point of interaction which is much higher than we indicated. In this second case then, one might expect larger double scattering and absorption effects than assumed in the previous calculations, which were based upon independentparticle shell-model calculations. Recent successes of the ad model of 6Li [ref. 22)] ..1 _x.rr 1 --._..,X _______Ill.,-’ =._r__ -II- ~~ ,~. f-, WOUIUsuggest tndt rurure arremprs to nr our data wlrn u w 1A caicuiations shouid take this possibility This
work
into was
consideration. supported
by the
Swedish
Atomic
Research
Council.
One
of us
(R.M.H.) would like to express his gratitude for the kind hospitality offered at the Gustaf Werner Institute, Uppsala, and is grateful to the National Research Council of Canada for financial support. References 1) D. F. Jackson and J. Mahalanabis, Nucl. Phys. 64 (1965) 97 2) H. Tyr&n, S. Kullander, 0. Sundberg, R. Ramachandran, P. Isacsson and T. Berggren, Nucl. Plnys. 79 (i966j 32i; J. C. Roynette, M. Arditi, J. C. Jacmart, F. Mazloum, M. Riou and C. Ruhla, Nucl. Phys. A95 (1967) 545; B. K. Jain and D. F. Jackson, Nucl. Phys. A99 (1967) 113
272
R.M. HUTCHEON
et cd.
3) W. N. Hess, Rev. Mod. Phys. 30 (1958) 369 4) A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 5.51 5) D. F. Jackson, Nucl. Phys. 35 (1962) 194 6) M. H. MacGregor, R. A. Arndt and R. M. Wright, Phys. Rev. 182 (1969) 1714 7) H. P. Stapp, T. J. Ypsilantis and N. Metropolis, Phys. Rev. 105 (1957) 302 8) P. Cziffra, M. H. MacGregor, M. J. Moravczik and H. P. Stapp, Phys. Rev. 114 (1959) 880 9) T. de Forest, Jr. and J. D. Walecka, Adv. in Phys. 15 (1966) 1 10) L. R. Suelzle, M. R. Yearian and H. Crannell, Phys. Rev. 162 (1967) 992 11) F. C. Baker, Nucl. Phys. 83 (1966) 418 12) F. Eigenbrod, Z. Phys. 228 (1969) 337 13) R. M. Hutcheon, T. E. Drake, V. W. Stobie, G. A. Beer and H. S. Caplan, Nucl. Phys. 107 (1968) 266 14) R. Neuhausen, Z. Phys. 220 (1969) 456 15) A. Aurdal, J. Band and J. M. Hansteen, Nucl. Phys. Al35 (1969) 632 16) S. Dahlgren, D. Hasselgren, B. Hoistad, A. Ingemarsson, A. Johansson, P. -U. Renberg, 0. Sundberg and G. Tibell, Nucl. Phys. A90 (1967) 673 17) B. Beoffrion, N. Marty, M. Morlet, B. Tatischeff and A. Willis, Nucl. Phys. All6 (1968) 209 18) R. M. Hutcheon and H. S. Caplan, Nucl. Phys. Al27 (1969) 417 19) R. Neuhausen and R. M. Hutcheon, to be published 20) T. Erikson, private communication in Nucl. Phys. 69 (1965) 81 21) R. M. Hutcheon and 0. Sundberg, The Gustaf Werner Institute, Uppsala, to appear as an internal report 22) V. G. Neudatchin and Yu. F. Smirnov, Atom. Energ. Rev. 3 (1965) 157; V. G. Neudatchin, Proc. of the Int. Conf. on clustering phenomena in nuclei, Bochum (1969)
@ North-Holland
Not to be reproduced by photoprint or microfilm without written
185 MeV PROTON SCATTERING AND THE IMPULSE
Publishing
permission
Co., Amsterdam from the publisher
FROM 6Li
APPROXIMATION
R. M. HUTCHEON? Institut fiir Kernphysik
der Universitc?t Mainz,
0. SUNDBERG The Gustaf
Werner Institute,
West Germany
and G. TIBELL
University
of Uppsala,
Uppsala, Sweden
Received 12 June 1970 Abstract:
Cross sections for the scattering of 185 MeV protons from 6Li have been measured for the elastic case and for inelastic excitation of the first two excited states. These cross sections are compared with plane-wave impulse approximation calculations in which the nuclear matrix elements are taken directly from electron scattering data in order to eliminate model dependence. The experimental to theoretical cross-section ratio is 0.5f15 % for a momentum transfer of 0.7 fm-’ and has a strong momentum-transfer dependence. Previous calculations made in the distorted wave impulse approximation bring the ratio closer to unity but do not lead to complete agreement.
E
NUCLEAR
REACTIONS 6Li(p, p), (p, p’), E = 185 MeV; measured a(Ep,, 0). Enriched target.
I
1. Introduction The plane-wave and distorted-wave impulse approximations (PWIA and DWIA) are used very often in the analysis of medium-energy proton scattering data, and thus it is important to have a test of the degree of validity of these approximations. In such tests the uncertainty in the nuclear structure information usually hinders one from reaching any definite conclusions. For example, the difficulties in light nuclei of theoretically reproducing the experimental values of (p, p’) cross sections ‘) and the (p, 2p) angular distributions and occupation numbers ‘) might be caused either by a breakdown of the impulse approximation or by the assumption of incorrect nuclear wave functions. The general approach in this paper is to perform such a test using the impulse approximation formalism for proton scattering and assuming the nuclear structure information to be given by electron scattering experiments. The latter assumption is reliable because the electromagnetic interaction is well known and because the new linear electron accelerators are producing very accurate, consistent data over a wide _.,_.___ _J? -_ ____ -I-- ~~ . rdnge UI momenrum transfers. The “ii nucieus was chosen for its reiativeiy iownucleon density (the rms radius of the p-shell nucleon distribution is w 3.0 fm), for t Present address: Physics Dept., Queen’s University, Kingston, Ontario, Canada. 261
262
a. M. mrrcmo~
et al.
its reIative ease of handling, and because a large amount of accurate electron scattering data exists for it. An incident energy in the region of 200 NeV was chosen because at this energy the nucleon-nucleon total elastic cross section is at a minimum (a 25 mb) [ref. “)I and thus multiple scattering effects in proton-nucleus scattering should be minimized. It should be noted, however, that the direct application of electron scattering information to the analysis of proton scattering is only justified if, during the proton-nucleus interaction, there are no compound states formed and the excitation does not go through intermediate nuclear states (i.e. fist Born approximation for the --,:r,-L-- -_-~.^-‘_-\ O-CL rf-_^_ ,,,?Z:c:,,, are GAFQLru dyI~S+PS4lcv r\ hUe ..,a11 EP+;&P~ hn CX~ILCLCIUII ~~~~~i~~~~. DULU LU~;~C; CIVUUIL~~~~~ mc11OuLIOMUu “, inelastic proton scattering from 6Li at this energy,
2. Theory t 2.1. PWIA PROTON-NUCLEUS SCATTERING FORMALISM
The PWIA expression for the proton nucleus scattering cross section 4*“) (neglecting antisy~etrization between @e projectile and the target nucleons) in the case where the nuclear ground state has isospin T = 0 is given by do
4 * iti -
-=
d&IL
f M-+1
-
2 kf
)
4n
-ko25,+1
+[E2-#32+C2+F2)]Agl,
~[A~+c~~CN:+IIB~+CZ+F~I~S;, A
,
(; ; ;)((t; ;)P’+W~+l)’ x
(- l)+“-“‘s;, I’S& I , 1
(0
where M is the nuclear mass in units of the proton mass, Ais the multipolarity of the transition, J,, is the ground state spin, k. and kf are the initial and final momenta in the proton nucleus c.m. system and q the momentum transferred to the nucleus in the lab system. The coefficients A’, B2, C2, E2, F2 are functions of k, and q; they have the units of cross section and their numerical vaIues can be extracted from the nucleonnucleon scattering data for a particular energy and momentum transfer. There are two sets of these parameters corresponding to the cases AT = 0 and AT = 1. Hereafter, the subscript CIdenotes AT = 0 and /? denotes AT = 1. The reduced nuclear matrix elements NAand S,, are also isospin dependent and are given for Tr = 0 by
t AI1 formulae given in this section have been derived from basic principles and compared with the ~""*-nlmt.4 -v-_-_y___. rPfPrPncf?P.
6Li
PROTON
263
SCATTERING
and for T = 1 by
A computer program was written to calculate 2’) the complex Kerman-McManusThaler amplitudes A, B, C, E, F, using recently published nucleon-nucleon scattering -
Q5A&;: 0.5
1.0
1.5
qLapd)
Fig. 1. The appropriate combinations of nucleon-nucleon scattering amplitudes plotted as a function of the nucleon-nucleon scattering lab momentum transfer.
phase shifts “) in the formalism of Stapp et al. ‘) with higher phases calculated using the OPEP approximation “). The amplitudes have been calculated so as to include the Coulomb interaction between two protons. The effect of neglecting the Coulomb interaction is shown by the dotted line on fig. 1 where the appropriate squares of the amplitudes are plotted. One can re-write the proton scattering cross section resulting from the AT = 0, non-spin dependent matrix elements (NL) in terms of the electron scattering first Born approximation Coulomb form factors because the nuclear transition operators
R. hi. mmmo~
264
et al.
are identical ‘). Thus, (4) where the F,,(q) are the first Born approximation Coulomb scattering form factors o-.-l f;r in\ 10 ;Dthe nrr\tnn rhorme l”llll fnrm IUVLVI. fgrtnr R~rmce the form frrrtnr ncpd UllU “)$\LJ, C&h” y’vc”‘” vU.urav YVYUYY” *II., (L”I__& I--...& -u-1 in -1 the _--proton scattering formalism must be derived from the distribution of the nucleon centers, the effect of the finite proton charge distribution must be unfolded from the electron scattering form factor when inserting it into the proton scattering expression. This is accomplished through dividing the electron scattering form factor by G,,(g). For transitions in which the spin-dependent matrix elements contribute, the comparison with the electron scattering form factors is model dependent and separate calculations must be made in each case. 2.2. THE STRUCTURE OF 6Li AND PWIA CROSS SECTIONS
The charge form factor of the “Li ground state has been measured by electron scattering and is fitted reasonably well up to q = 1.7 fm-’ by a harmonic oscillator shell model with separate parameters for the s- and p-state (as = 1.632 fm, up = 1.98 fm) [ref. ‘“)I. The ground state has a very small quadrupole moment, and in shellmodel analyses is found to be almost pure (99 ‘A) 3S1 state ll). The magnitude and the shape of the electron scattering form factor for the transition to the 3.56 MeV, Ot, T = 1 level is given to within 5 % assuming a shell model ISo configuration for the excited state, indicating it is an almost pure spin tlip transition 12-14). Any other model preserving this pure spin flip character would yield the same normalization. Thus the reaction mechanism of the transition to this level is well understood, and the mnnnZ+.rrl~ nF tha nnlnrriotnrl epAxz,-enA +ronn;+;r\n r\mh~h;l;t~g ;a prepntilllv ;nApnpndpnt muqjruruuti “1 u.lLI ~alru,Pccu IL‘UU_U L,411c-~LL”ll J,.a”“U”l‘lCJ 1.J ““YY”L’U”_I “‘..-~““u”‘“‘ of the radial wave functions. It has also recently been possible to fit the form factor for the transition to the 5.36 MeV, 2+, T = 1 level using the shell model. ‘I) The shape of the form factor for the transition to the 2.18 MeV, 3+, T = 0 level is also reproduced by the shell model, but the magnitude of the calculated form factor is a factor 2 to 3 too small. This is undoubtedly due to the failure of the simple shell model to describe the deuteron-like correlations which the stronger T = 0 forces induce ’ “). Thus although in this case the nuclear matrix eIements are not easily calculable, the reaction mechanism is fairly well understood. The proton scattering PWIA cross section for elastic scattering from 6Li is given by do -=
da,.,.
105.8{[& -/-C,z]F&,(q)i-&[I+,2
+ C: +E,2 -t-F,Z]F;(q))
G+J
9
(5)
Ep
where F,,(q) is the elastic electron scattering form factor for 6Li and F,(q) is the elastic electron scattering form factor for only the p-shell n&eons. Both form factors are normalized so that F(q) = 1 for q = 0. The second term occurs because ‘Li has
6Li PROTON
265
SCATTERING
a ground state spin 1 and thus the equivalent of electromagnetic Ml scattering is possible. The quantity F,(q)/G,,(q) has not yet been measured directly to sufficient accuracy, but the form factor for the 3.56 MeV transition F,.,,(q) because of its spin-flip nature, should be very nearly the same. The contribution of the second term to the total cross section is in any case less than 2 % for a momentum transfer q 5 2 fm-I. The proton scattering PWIA cross section for excitation of the 3.56 MeV level calculated with a simple LS coupled shell model is given by
(6) A comparison
of the proton
and electron
F32.56k) =
scattering
nuclear
matrix
J’idd 3-22m’
B(ML t)
where of the 0.0586 The
elements
yields (7)
F,,(q) is the inelastic electron scattering first Born form factor for the excitation level 13, 14) and th e reduced transition probability B(M1, t) was taken to be fm2 [ref. ‘“)I. F. .n=l _I cross section for excitation of the L.111Nrev state with the same modei is da d%m.
= 105.8 $ [A,Z+C,2+$E,2++(B;+C:+F,2)]F;.18(q),
(8)
0
where it should be noted that the factors 4 and 3 are peculiar to the shell-model LS coupling scheme assumption “) and are thus model dependent. Again, comparison with electron scattering theory yields %
s(q) = F:,(q)/G&(q),
(9
where Fc2(q) is the electron scattering first Born form factor for excitation of the m 10 _L..L^ L. lb Xl-X, 1VlGv blalc. If in all three cases one replaces the theoretical electron scattering from factors with the measured electron scattering form factors, then one can hope to eliminate most of the model dependence of the calculated proton cross sections, including the troublesome c.m. motion corrections ‘3 “). This is especially true for the elastic and the 3.56 MeV transitions, in which the maximum possible model dependence is less than 5 %. 3. Experimental
results
The experiment was performed on the 185 MeV proton synchrocyclotron at the Gustaf Werner Institute, University of Uppsala. A general description of the apparatus and experimental layout as well as the data analysis methods is given in ref. 16). Cross sections have been measured for lab scattering angles between 8” and 50”
266
a. M. FlUTCHEON
et al.
using 1 to 2 mm thick 6Li targets of 99.3 % isotopic purity. The targets were handled as in ref. ’ “) and the absolute thickness was judged to have an error of f 0.01 mm. Four typical spectra are shown in fig, 2 the notable features of which are the improvement in resoiution over other measurements r ‘* I7 j and the presence of a
E
=185 MeV ‘Lab
Q ~b
li-
I
.s
177
=15O
I
I
178
179
._.
160
181
, -.... .
I
-__ 1UJ
2’; E
I \
=185 MeV
/ /
‘Lab e
:
Lab=l7.65o
I ,
/i
5-
4 X&i i_ \
+/ : 176
177
178
.J 179
180
‘1/ , i, 182 Efdt$eV'
Fig. 2. Spectra of protons scattered from 6Li at various angles. The dotted line through the points ._I_Al_-__._ Q-l__L^__11_^_ ^__.._^-1 1_ &L^___1_._1^ ^E_^^l_^__^_^_^_L_..._ iSi0 giiiuemt: eye. ~nc;oase~moa assuuw m we ana~yxs UI yean mr;as altiSIIUWII.
large continuous cross section similar to that found in electron scattering 18plg) and attributed to ad breakup of 6Li. This continuum does not seem to have the shape predicted theoretically “3 17) by monopole excitation to the ad continuum. Only two strong inelastic peaks are seen in the spectra, and, as in the recent electron scattering
6Li PROTON
\‘i 10
15
‘-_‘r-. 20
261
SCATIl3lilNG
* 25
30
35
40
E
:td
Fig. 3. Angular distributions for the 2.18 MeV and 3.56 MeV peaks in 6Li. The dotted lines are guides to the eye. The 3.56 MeV level point without error bars at f&.=. = 13.3” is the normalization point for all spectra.
4
DOE,
~185 MeV Lab.
IO -
.
Elastic
X
2.18 MeV
Level
l-
I.1 -
” 5
Fig. 4. Angular distributions
for the elastic and 2.18 MeV transitions. The dotted lines serve to guide the eye.
268
R. hf. nmcmot4
et al.
work with improved resolution, if a peak at 4.5 MeV excitation with a natural width of 600 keV exists, it is only weakly excited. The peak areas and thus the cross sections have been extracted from the spectra by assuming hand fitted baselines under the peaks as shown in fig. 2. It was, however, required that the shape and FWHM of the inelastic peaks be consistent with the elastic peak. The uncertainty in the baseline produces an error in the peak area depending upon the quality of the spectrum, and this error has been linearly added to the statistical error which was commonly between 0.5 and 3 %. A&“IIC: ^^^ sn1g:lc ,.--1,. /Cl -- 5L”S”IllK ..L__,..L- crobs-becLI”u ___^^ ___LI_- call”raLI”” __l:L__r:-- __.^^ --lr sr&inlSL .._..I_“& Al. (I+& = 11 11.110,, au was IlIaCIt: the proton-proton cross section using a polyethylene target in which the proton thickness uncertainty was ) 1.2 %. The proton-proton c.m. cross section at this angle was calculated “‘) and found to be 3.73 mb/sr f2.5 ‘A, using the nucleon phase shifts of McGregor et al. “). The differential cross section in the lab frame for excitation of the 3.56 MeV level at Blat,= 11.1” was determined to be 1.24 mb/sr with a 6 % error. All the experimental points have been normalized relative to this value and the error in this normalization procedure has been added linearly to the error in the individual peak areas. The measured cross sections and their errors relative to the 3.56 MeV, Oiab= 11.1” mennntpment 9re I-_ shnwn in --~_. fivn _? I--2nd 4. . -Tn nrder must -__-_ one ---_ _..~. ________ c__I_ --..-- _____ * to nhtgin 811&Q!IJ~~ error
add 6 % to all errors shown. A comparison between the present cross sections and the previous values 1“) measured with poorer resolution indicates that the assumptions made in determining the peak areas in the earlier experiment were not correct. 4. Comparison with theory and discussion Using eqs. (5), (6) and (8) for the proton scattering cross sections, one can either compare the measured cross section with the cross section calculated using the electron scattering form factor or one can extract from the proton cross sectionsan effective form factor, which because of rescattering effects is expected to be smaller than the form factor extracted from the electron scattering results. In figs. 5 and 6 are shown both the calculated and effective proton scattering form factors for the 3.56 and 2.18 MeV states, and fig. 7 shows the calculated and measured elastic cross sections. The most notable features of fig. 7 are that the measured cross sections are smaller than the calculated ones by approximately a factor 2 at 4 = 0.7 fm-l and by a factor 3 at 4 = 1.2fm-l, but for higher-momentum transfer (larger angles) the curves cross and the measured cross sections become much larger than the PWIA calculations. This latter effect has been seen before, e.g., in the analysis of proton scattering from 4He at 147 MeV [ref. ‘)I. A general comparison is made more clearly in fig.8 in which the calculated to experimental cross-section ratios are plotted for each of the levels. It should be noted that in the momentum transfer region 0.4 fm-l 5 q 5 1.0 fm-l the PWIA to measured cross-section ratios are within 10 % of each other for all cases
6Li PROTON
269
SCATTERING
studied here, the AT = 1 transition ratio being a few percent higher than the AT = 0 ratios. This, coupled with the fact that the three transitions have quite different transition mechanisms, suggests that in this momentum transfer region the nuclear model dependence has been removed from our calculated proton scattering cross sections, and that the difference between calculated and measured cross sections must
jpq-t.j.i,_
MeV Level (Jrr33’I
2.18
,
Ii
/;
‘L*
,+,_I--#- -*-..
‘f‘f
‘f,\
‘f, ‘\\ ‘\ ‘4 \
I
1ci3
7 n
Eigenbrad , Darmstadt
x o .
Neuhausen,Mainz Hulcheon and ~resent,ex!racted
\
\
\
\. \
\
(e. e’)
Caplan,Sask. I from (p,p’)
\ ‘\ ‘\
\
\
\ \
I
I
0
0.5
\I I\ I
1.0
1.5
2.0
2.5
I
sctlll-l, Fig. 5. The form factor for the 2.18 MeV transition. The proton scattering form factor has been extracted under the assumption that the PWIA formalism is valid. 1.0 3.56
Fi5s(q)
MeV Level (Iv=
0’)
0.1
.
Eigenbrod,
0 Hulcheon X Neuhausen l
0.01 0
Present,
Darmstadt el.al.,Saskatcon and exlracled
Hutcheon,Mainz from
(p,p’)
0.5 q
(f&j
Fig. 6. The form factor for the 3.56 MeV transition. The proton scattering form factor has been extracted under the assumption that the PWIA formalism is valid.
270
et
R. M. HUTCHEON
d.
\ 0.5
10
1.5 q IfIn-‘)
2.0
t
25
Fig. 7. The elastic proton scattering cross sections at 185 MeV for 6Li. The calculation is done in the PWIA using experimental electron scattering form factors.
f
3111 ‘, -
/ /
S? 2” vL 22 20
;e
0
2.0 [
.)
/
/
i
0’ ’
Elastic
---2.18
t&V
--.-3.56
MeV Level
Level
Jz
1.0
ZE z
0.5 10
9 (ftil)
1.0
20
30
40
50
Gt.4
Fig. 8. The ratios of the calculated PWIA proton scattering cross sections to the measured proton scattering cross sections. from an inadequacy in our treatment of the projectile-nucleus interaction mechanism. The most obvious improvement in this respect would be to perform DWIA calculations for the transitions: that is, to allow for distortion and absorption of the incident and outgoing projectile wave functions. DWIA calculations have been made for 155 MeV proton scattering for both the 2.18 and 3.56 MeV transitions ‘9‘), using optical potentials with a real part between 7 and 10 MeV and an imaginary part between 10 and 20 MeV. The (PWIA/DWIA) ratio was found to be 1.4 at q = 0.9 fm -I for the 2.18 MeV transition, and 1.2 at q = 0.7 fm-l for the 3.56 MeV transition. Taking these corrections into account one still finds that the calculated to measured cross-section ratio is about 1.7. Another DWIA calculation was made at 185 MeV tar the 3.56 MeV transition ““) and the (PWIA/DWIA) ratio was again found to be approximately 1.20 for a momentum transfer of 0.7 fm-l.
come
6Li PROTON
271
SCAlTERING
Thus these distorted wave calculations do not bring our calculated and measured cross sections into agreement. Undoubtedly a fit to the data could be obtained by increasing the values of the deforming and absorbing potentials, but then one must provide a physical justification for the increased depth of the potential parameters. As the p-shell particle density is a little less than one nucleon per 50 fm3 it is plausible that almost all of the deformation and rescattering effects should occur through interaction with the rather tightly bound core. However, DWIA calculations for pa particle scattering have also indicated a clear disagreement between theory and experi_^^C.‘^^ e-c:,. ,.c ,.--_,...:-_c^l.. -_-_&. ..a UI. -r 41,ll11u C-d ^ /nxx7r.4 I-___..__31\ ^_^^^ IlKllC. v^_-^^ Re‘lllall cl. a \Y vvlkl,“‘t;ilL”Ll~tu, C;I”bS-btxLI”II lilLl” “I apy’“.“““dLLcxy 1.7 for elastic scattering of 310 MeV protons on cc-particles. Thus, although it is possible that our results may be reproduced by a complete distorted-wave treatment, it does not at present seem likely. 5. Conclusions The present experiment clearly demonstrates the discrepancy both in magnitude and shape between PWIA calculations and measurements of the inelastic proton scattering cross sections for 6Li. Corrections to the PWIA using existing distortedwave calculations tend to improve the agreement but are not sufficient. Obviously more extensive DWIA calculations should be made with careful consideration of absorption and multiple scattering effects in order to see if agreement can be obtained with physically reasonable parameters. However, accepting the evidence of the available calculations, the discrepancy can probably be explained through one or a combination of the two following effects: (i) the impulse approximation is not sufficiently accurate, (ii) the outer two nucleons in 6Li are strongly correlated implying a nucleon density at the point of interaction which is much higher than we indicated. In this second case then, one might expect larger double scattering and absorption effects than assumed in the previous calculations, which were based upon independentparticle shell-model calculations. Recent successes of the ad model of 6Li [ref. 22)] ..1 _x.rr 1 --._..,X _______Ill.,-’ =._r__ -II- ~~ ,~. f-, WOUIUsuggest tndt rurure arremprs to nr our data wlrn u w 1A caicuiations shouid take this possibility This
work
into was
consideration. supported
by the
Swedish
Atomic
Research
Council.
One
of us
(R.M.H.) would like to express his gratitude for the kind hospitality offered at the Gustaf Werner Institute, Uppsala, and is grateful to the National Research Council of Canada for financial support. References 1) D. F. Jackson and J. Mahalanabis, Nucl. Phys. 64 (1965) 97 2) H. Tyr&n, S. Kullander, 0. Sundberg, R. Ramachandran, P. Isacsson and T. Berggren, Nucl. Plnys. 79 (i966j 32i; J. C. Roynette, M. Arditi, J. C. Jacmart, F. Mazloum, M. Riou and C. Ruhla, Nucl. Phys. A95 (1967) 545; B. K. Jain and D. F. Jackson, Nucl. Phys. A99 (1967) 113
272
R.M. HUTCHEON
et cd.
3) W. N. Hess, Rev. Mod. Phys. 30 (1958) 369 4) A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 5.51 5) D. F. Jackson, Nucl. Phys. 35 (1962) 194 6) M. H. MacGregor, R. A. Arndt and R. M. Wright, Phys. Rev. 182 (1969) 1714 7) H. P. Stapp, T. J. Ypsilantis and N. Metropolis, Phys. Rev. 105 (1957) 302 8) P. Cziffra, M. H. MacGregor, M. J. Moravczik and H. P. Stapp, Phys. Rev. 114 (1959) 880 9) T. de Forest, Jr. and J. D. Walecka, Adv. in Phys. 15 (1966) 1 10) L. R. Suelzle, M. R. Yearian and H. Crannell, Phys. Rev. 162 (1967) 992 11) F. C. Baker, Nucl. Phys. 83 (1966) 418 12) F. Eigenbrod, Z. Phys. 228 (1969) 337 13) R. M. Hutcheon, T. E. Drake, V. W. Stobie, G. A. Beer and H. S. Caplan, Nucl. Phys. 107 (1968) 266 14) R. Neuhausen, Z. Phys. 220 (1969) 456 15) A. Aurdal, J. Band and J. M. Hansteen, Nucl. Phys. Al35 (1969) 632 16) S. Dahlgren, D. Hasselgren, B. Hoistad, A. Ingemarsson, A. Johansson, P. -U. Renberg, 0. Sundberg and G. Tibell, Nucl. Phys. A90 (1967) 673 17) B. Beoffrion, N. Marty, M. Morlet, B. Tatischeff and A. Willis, Nucl. Phys. All6 (1968) 209 18) R. M. Hutcheon and H. S. Caplan, Nucl. Phys. Al27 (1969) 417 19) R. Neuhausen and R. M. Hutcheon, to be published 20) T. Erikson, private communication in Nucl. Phys. 69 (1965) 81 21) R. M. Hutcheon and 0. Sundberg, The Gustaf Werner Institute, Uppsala, to appear as an internal report 22) V. G. Neudatchin and Yu. F. Smirnov, Atom. Energ. Rev. 3 (1965) 157; V. G. Neudatchin, Proc. of the Int. Conf. on clustering phenomena in nuclei, Bochum (1969)