High-energy electron scattering from 6Li

Nuclear Physics

Al62

(1971) 583-592;

@ North-Holland

Publishing

Co, Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

HIGH-ENERGY G. C. LI, High Energy

ELECTRON

SCATTERING

I. SICK t, R. R. WHITNEY

Physics Laboratory,

Stanford

Received

and M. R. YEARIAN

University,

1 December

FROM 6Li

Stanford,

California

94305 ti

1970

Elastic electron scattering cross sections have been measured for 6Li at 500 MeV and values of q’, the momentum transfer squared, up to 13 fm- 2. Calculations including nucleon-nucleon correlations predict a diffraction minimum beyond the range of previous measurements for 6Li. The results for 6Li do show a diffraction minimum at q* = 8 fme2; they are also compared with a phase-shift calculation using an empirical charge distribution.

Abstract:

E

NUCLEAR

REACTIONS 6Li(e, e), (e, e’), E = 200, 500 MeV; measured a(@; deduced form factors. 6Li deduced rms nuclear radius,

1. Introduction In previous lm3) e 1ec t ron scattering experiments from 6Li the elastic form factor has been reported in the range of the momentum transfer squared q2 = 0.2-6.9 fm-‘. The form factor has been shown to correspond to a central Gaussian charge distribution plus a tail which shows that there is slightly more charge around r w 4 fm than is given by a purely Gaussian distribution. On the basis of these data several theoretical calculations 4-6 ) published prior to the start of this experiment predicted relatively strong (although different) effects of short-range correlations on the 6Li form factor. In order to check these predictions we have therefore extended the 6Li elastic data to higher momentum transfers. Taking advantage of a high beam current and using a thicker target than in ref. I), makes it possible to measure the very small cross sections of z lo-’ mb/sr expected at these higher values of q2.

2. Experiment The experiment was performed at the Stanford Mark III electron accelerator. The details of the experimental equipment have been reported in ref. ‘) and we forego giving them there. Some changes relative to ref. ‘) should be mentioned however: a new floating-wire calibration of the energy-defining deflecting magnet gives the value of the incident energy to + 0.1 %; a new angle calibration allows an accuracy of 0.03”; a feedback system allows the stabilization of the beam position (and hence the scattering angle) to +0.5 mm. t Present address: CEN Saclay, B.P. No. 2, 91 Gif-sur-Yvette, France. +t This work was supported in part by the US Office of Naval Research Contract and the National Science Foundation. 583

[Nonr. 225 (67)l

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584

The 6Li targets used were isotopically enriched to 99 % and had thicknesses of 233 mg/cm2 and 745 mg/cm’ (the latter target was only used for the high q2 points); the target thicknesses were determined by measuring the weight and area to +2 %. The clean targets were transferred to the vacuum scattering chamber under benzene in order to reduce the formation of lithium hydroxide and lithium nitride as much as possible. 6Li E, = 499.3

MeV

2.16 MeV

0 = 40.00°

I 356 4.52 535

t

1

i 01

I

I

30

25 EXCITATION

I

20

1

I

I

I

15

IO

5

0

ELECTRON

ENERGY

IN

_Y_

MeV

Fig. 1. Spectrum of scattered electrons for an incident energy of 499.3 MeV, a scattering angle of 48”, and target thickness of 745 mg/cm 2. The spectrum has been corrected for relative channel efficiencies and has been radiatively unfolded.

The data were taken at two values of the incident energy computed for center target: 200.0 MeV (q2 = 0.31-2.4 fm-‘) and 499.3 MeV (q2 = 1.9-13.4 fmm2). A typical thick target spectrum at q2 = 4.12 fme2 is shown in fig. 1. The elastic and 2.18 MeV level cross sections which were obtained are given in table 1; the errors given do not include a possible systematic error of f3 % due to the uncertainty of the absolute hydrogen cross sections ‘) which were used to calibrate the detection efficiency of our apparatus. From fig. 2, where the elastic 6Li experimental form factors are displayed, it is evident that there is a gap from q 2 = 7 to 8.5 fmm2 in our data. The spectra which we have taken in this region indicate that with the targets we have used, the 6Li cross sections there cannot be obtained. This is due to the contaminants in our targets, which show up at these very low cross sections. The nucleus ‘Li (w 1 %) has a strongly excited level at 4.6 MeV which in the region q2 = 6.5-10 fmW2 overlaps the 6Li elastic peak and contributes significantly to the observed cross section. As a check we have measured cross sections with a ‘Li target from qz = 6.5-13.2 fmT2 and verified that the scattering observed in the region of the 6Li diffraction minimum (q2 x 8

6Li ELECTRON

585

SCATTERING

TABLET Elastic and 2.18 MeV level cross sections for 6Li(e, e) E, = 200.0 MeV Angle (deg.) 32.00 34.00 36.00 40.00 44.00 48.00 52.00 56.00 60.00 64.00 68.00 72.00 76.00 80.00 84.00 88.00 92.00 96.00 100.00

Elastic cross section (mb/sr) (0.967+0.029)101 (0.712f0.021)10-’ (0.501~0.015)10-’ (0.273 &O.OOS)lO- 1 (0.152-&0.005)10-’ (0.920f0.028)10-2 (0.555f0.017)10-2 (0.316&-0.010)10-2 (0.188+0.006)10-2 (0.116~0.~3)10-z (0.705 +o.021)10-3 (0.416~0.013)10-’ (0.268~0.008)10-3 (0.164~0.005)10-3 (0.103~0.003)10-3 (0.671 +0.020)10-4 (0.438f0.013)10-4 (0.292&0.009)104 (0.187+0.006)10-4

2.18 MeV level cross section(mb/sr) (0.214&0.039)10-2 (0.15510.025)10-2 (0.156&-0.024)10-2 (0‘101~0.013)10-~ (0.812f0.086)10-3 (0.682f0.070)10-3 (0.493~0*050)10-3 (0.334&0*033)10-3 (O.246&O.O25)1O-3 (0.208~0.021)10-3 (0.149f0.014)10-3 (0.960&0.075)10-4 (0.7~~0.059~10-4 (0.549f0.043)10-4 (0.415-+0.031)10-4 (0.288&0.025)10-4 (0.231f0.019)10-4 (0.171 f0.013)10-4 (0.140f0.011)10-4

E, = 499.3 MeV Angle (deg.)

Elastic cross section (mbjsr)

2.18 MeV level cross section (mbisr)

32.00 33.00 34.00 35.00 36.00 38.00 40.00 42.00 44.00 46.00 48.00 50.00 54.00 57.00 59.00 62.00 64.00 66.00 68.00 70.00 72.00 74.00 78.00 81.00 83.00 88.00 94.00 98.00

(0.835f0.025)10-” (0.605&0.018)10-3 (0.451 *o.014)10-3 (0.308*0.~9)10-~ (0.233f0.007)10-3 (0.134*0.004)10-3 (0.~3*0.020)10-4 (0.33410.010)10-4 (0.162&0.005)10-” (0.846*0.025)10-5 (0.392&0.011)10-5 (0.193&0.056)10-5 (0.434~0.012)10-6 (0.123~0.004)10-6 (0.659j0.042)10-7 (0.148~0.013)10-’ (0.48 10.10 )lO-’

(0.355+0.033)10-3 (0.290f0.023)10-3 (0.247&0.018)10-3 (0.215f0.016)10-3 (0.17110.014)10-3 (0.116f0.009)10-3 (0.719*0.053)10-4 (0.510~0.038)10-4 (0.322&0.025)10-4 (0.202*0.0t7)10-4 (0.116~0.009)10-4 (O.754&O.O65)1O-J (0.282~0.02i)10-5 (0.123+0.009)10-” (0.825+0.062)10-6 (0.306~0.024~10-6 (0.185~0.015)10-6 (0.725&0.092)10-’ (0.411*0.059)10-7 (0.296f0.049)10-7 (0.148f0.024)10-7 (0.80 &0.15 )lO-+ (0.45 10.11 )10-a (0.17 kO.09 )lO_8 (0.24 30.05 )lO-’ (0.11 10.02 )10-s (0.83 10.17 )lO-’ (0.88 10.16 )1O-g

(0.115&0.038)10-* (0.111~0.032)10-s (0.92 f0.22 )10-9 (0.77 10.21 )10-e (0.56 kO.23 )10-9 (0.30 f0.12 )10-g (0.26 *0.08 )lO+

fm-“) was predominately due to the excitation of the 4.6 MeV level of ‘Li. In addition at q ’ w 8 fm-’ the most strongly excited levels at E* c 19 MeV due to the slight surface contamination of oxygen and nitrogen fall on top of the 6Li elastic peak. The 6Li cross section is therefore only a fraction of the measured cross section of M lO-8 mb/sr, and we are not able to extract it. At momentum transfers higher and lower than the described region the contributions due to these contaminants are smaller (their form factors are a less steep function of q2 than F2(6Li) below q2 = 8 fm-’ and much steeper than F2(6Li) above q2 = 8 fm-“), and they are also separated away from the 6Li elastic peak due to the difference in recoil energies. The nucleus 6Li has a spin of one, and therefore contributions due to elastic scattering from the magnetic dipole and electric quadrupole moment are present. For the quadrupole scattering contribution we have computed the cross section using the undeformed p-shell and deformed oscillator potential model [eqs. (35) and (39) of

586

G. C. LI er al.

ref. “)I. With the possible exception of the diffraction minimum at 4’ = 8 fmb2, we have found the quadrupole contribution to be negligible; this is to a large extent due to the extremely small quadrupole moment of 6Li of 0.08 fm2. There is no method of experimentally separating the effects of the two longitudinal components (charge and I

6Li

CHARGE FORM FACTOR

R,,,

= 2.56fm

10-2 =_

C NS

10-S =

S Naz IL-

10-4z-

10-s =

10-6 ?r

10-7;

’ ’ ’ ’ 123456769

’

’

’

I

IO

’

II

I 12

’

I3

I

q~,,(frn’)

Fig. 2. Charge form factor of ‘jLi. The smooth curve is the best fit from the phase-shift includes the folding angle of *0.93”.

analysis and

quadrupole scattering), since we only observe the scattered electron. We have no method to determine the quadrupole scattering other than by calculating it. Due to the smallness of the calculated quadrupole contribution we have assumed in our analysis that the observed cross sections are due entirely to monopole charge scattering. The magnetic (transverse) contribution to elastic scattering has also been computed on the basis of the harmonic oscillator shell model, using the magnetic scattering data at lower momentum transfer from ref. “) as a constraint. Varying the parameters as determined by ref. ‘) within the range of their quoted errors yields magnetic cross sections smaller than the one standard deviation at any data point and therefore were neglected. This result is consistent with the theoretical calculation in ref. lo). In con-

6Li ELECTRON SCATTERING

587

trast to quadrupole scattering, the magnetic contribution can be obtained experimentally by measuring the form factor at different angles and constant momentum transfer. We have performed this type of measurement near q2 = 2 fmm2, in the region 4’ = 6.0-9.5 fm’, and also in the diffraction maximum, at q2 x 10.5 and 11.5 fm2. Within our statistical errors we have found no magnetic contribution above or below the minimum. In the region of the diffraction minimum the observed cross section is due mainly to magnetic scattering; this is compatible with the character of the levels of the contaminants which dominate the scattering there. After subtraction of the dominant magnetic scattering from the contaminants around q2 = 8 fmm2, the charge scattering is significantly smaller than at larger momentum transfers, which indicates that a diffraction minimum is observed rather than just a leveling out of the form factor. 10-lL t =

-

F2(q2) FOR THE 2.18 MeV LEVEL IN ‘Li - 200 MeV * 499.3 MeV fh

10-2 -_#f

i if I

! f

lo-‘:,

’123456789 ’ ’ ’

’

’

’

’

’

I

IO

i

!

’

II

’

I2

’

13

q,~,(fm*)

Fig. 3. Experimental form factor of the 2.18 MeV level in 6Li.

Shown in fig. 3 are the observed form factors of the 2.18 MeV level of 6Li. The cross sections for the 2.18 MeV level were determined by fitting the radiatively unfolded spectrum with the peak shape of the elastic peak; this was necessary at 499.3 MeV incident energy in the region q2 = 1.9-10 fme2 because the strongly excited 3.56 MeV level was not completely resolved from the 2.18 MeV peak. Beyond q2 = 10 fmS2 the 3.56 MeV level cross section was low and the 2.18 MeV level was

588

G. C. LI et al.

well separated. The threshold for break-up (‘jLi + o!+ d) occurs at an excitation energy of 1.47 MeV; any contribution from this process within our resolution of x 0.25 % FWHM is included in the cross sections of the 2.18 MeV level. In addition to the statistics of counting, the quoted errors for the 2.18 MeV level cross sections include elastic radiative tail subtraction statistics and uncertainties in separation from the 3.56 MeV level. Within statistics the Rosenbluth plots near q2 = 2,6 and 11 fm-’ indicate that the 2.18 MeV level is completely longitudinal. The present data are in excellent agreement with previous low momentum measurements 11-14) (q” = 0.05-3.61 fm-“). For references and discussion of the small contribution of the e + 6Li + e’+ u + d process the reader is referred to ref. ‘I). 3. Analysis with phenomenological p(r) We have compared our cross sections with those obtained by using a phenomenological charge distribution in a phase-shift computer code ’ “). In order to facilitate a comparison we have used the same shape of the charge distribution as has been used in ref. ‘)

-z

/lo(r) = 8~” [$exp

(-

-$)

- C2(6ii;r2)exp

(-

-$)I.

This charge distribution has been chosen because it gives a p similar to a Gaussian distribution, as indicated by the approximate e-azq2 dependence of FZ as well as the dip in F2 relative to e-a3q2 at low q2 (see fig. 5). The dip comes from the increased charge in the tail compared to a purely Gaussian distribution. In Born approximation the corresponding form factor would be: F,,(q2)

= exp(-a2q2)-c2q2

exp(-b2q2).

This three-parameter charge distribution gives a very good fit to our data up to q2 = 6 fme2; it does not reproduce the diffraction feature. In order to fit the data above q ’ = 6 fmm2 as well, we have added a small modification to pO(r) of the form Lip(r) =

sin(q04+ $ cos(q,r)] 23 i ____ ew(--*p2r2), [ qor

which in Born approximation O(q)

would correspond to = dexp

[-

(y)3

.

This oscillatory modification in the charge distribution influences the form factor only above q 2 = 6 fmF2 where the magnitude of F2 due to p,(r) itself becomes small. The two parts of p(r) are therefore determined quite independently by the data above and below the minimum. The only poorly determined parameter isp, the decay con-

6Li ELECTRON

589

SCATTERING

stant of the modification. Our best-fit parameters obtained with the phase-shift program are : a

= 0.928 f0.003 fm,

40

= 3.11 f0.20 fm-I,

b

= 1.26 kO.09

fm,

d

= -(1.24f0.28)10-3,

c

= 0.48 50.04

fm,

P

= 0.7OkO.29 fm-‘.

rrms= 2.56 kO.05 fm, The errors given are due to statistical and systematical sources and include the correlations between the parameters. The error in r,,, is mostly due to the uncertainty 0.08

-.

61.i CH.SRGE DENSITY

-c--I 4

5

RADIUS IN fm

Fig. 4. Best-fit charge distribution,

p(r), where p(r) =

p,,(r)+dp(r).

6Li -

PHENOMENOLOGICAL FIT TO ALL 0sYi-A 200 MeV 499.3 MeV

1.2 I.1 -

Fig. 5. Low momentum

experimental charge form factor divided by exp curve is the best fit to all data.

-+(2.3)*qzeff.

The smooth

G. C. LI

590

et al.

in c which comes about equally from statistical errors and the normalization uncertainty. Our parameters are, as are the data, in good agreement with the results of ref. I), which are (in Born approximation): c

0.934&0.004 fm,

a

=

b

= 1.30340.11

fm,

= 0.453 LO.05 fm,

r rms =

2.54 kO.06 fm.

The calculated best-fit cross sections have, after folding with the experimental angular resolution of +0.93’, a x2 of 0.8 per degree of freedom. They are shown as a curve in fig. 2; the best-fit charge distributions p,,(r), dp(r), and p(r) = po(r)-tdp(r) are given in fig. 4. As can be seen, the existence of a diffraction maximum requires a lower central charge density, x 5 %, than given by PO(r). Since in many nuclei the two-parameter Fermi distribution provides a fairly good fit to the data, we have utilized this model as well. A good fit up to q2 = 6 fmm2 is possible, but for the higher momentum transfers a modification would be needed since the two-parameter Fermi distribution again does not give a low enough central charge density. The rms radius obtained with this model is 2.54 fm; the c and z values are c = 1.437 fm, z = 0.610 fm. 4. Conclusion The medium momentum transfer points of the form factor (1 fmd2 5 q2 5 6 fme2) would give a p(r) for 6Li which is close to a Gaussian distribution with a rms radius of x 2.3 fm. The dip at low q in F2, which is shown in fig. 5 where F2(&)/exp(

-3(2.3)2&,)

is plotted as a function of q&, ,

is due to the additional small charge density at r x 4F(dp/p(O) = 10d2). In trying to understand this dip we have computed r6) the tail of the 6Li charge distribution on the basis of the shell model using a Woods-Saxon potential with parameters that were adjusted to roughly produce the correct p(r) and simultaneously fit the experimental binding energy of the p+ proton; the proton size was folded in. These calculations indicate that the tail of p(r) of ‘jLi drops off exponentially like e-r2’1.61 fm; i.e., somewhat slower than would a Gaussian distribution p(r) cc e-r2’3.52fm2. This exponential tail, which is a consequence of using a finite well, gives a quite significant increase in r rms of 0.22 fm; this increase compares favorably with the magnitude of the experimentally observed dip at low q in F2 which in turn corresponds to an increase in r,,,,, of 0.26 fm over the value obtained when a pure Gaussian distribution is used.

6Li ELECTRON

SCATTERING

591

The diffraction feature in F* is essentially due to the lowering of the central charge density. Recently there have been several published theoretical calculations I’* ‘*) which deal with the significance of this fact; in particular the influence of repulsive cores (short-range correlations) in independent-particle model calculations of the nuclear wave function are considered, also some of the earlier theoretical disagreements are clarified [see also refs. 1g*23*24)].Qualitatively the introduction of repulsive cores clearly will influence the high q2 part of F2 because their influence is seen most strongly as a reduction of p(O), the place where the nuclear density is at its maximum value. The CI-I-d clustering is another possible model for the ground state of 6Li as well as its excited states ‘O), and at small momentum transfer good agreement has been obtained for the elastic and 2.18 MeV level form factors. However, for the description of the high momentum transfer data it may be necessary to introduce repulsive cores in the clusters. In attempting to isolate the in~uence of repulsive cores, the very light nuclei appear to be of more immediate help than the heavier nuclei. For high A, in 40Ca for instance, reasonable amounts of 2p-2h excitations in the 2s shell have been shown to produce a significant effect on the high momentum transfer data “). In the lighter nuclei 2p-2h excitations appear to give much less distinct effects 22). To a large extent this is due to the strong binding of the IS nucleons, which makes a reduction of p(r) in the center through 2p-2h excitations much more unlikely. The application of theoretical calculations on the light nuclei suffers somewhat from the complication which arises through the greater influence on F* of the nuclear recoil correction in the single-particle shell model. It has been shown that the center of mass correction can differ quite appreciably from the one usually applied in the harmonic oscillator shell model *“). In ref. *‘) it has been pointed out that, in con trast to the harmonic oscillator case, this c.m. correction can also change the position of the minimum; the shift in momentum transfer is, at most, a factor (A - 1)/A. Therefore, for a light nucleus the application of a correct c.m. correction becomes imperative. The general conclusion of refs. “* ’ “) is that by a proper choice of the well parameters a diffraction feature in F2 for 6Li can be obtained, but that only inclusion of repulsive cores correctly reproduces the height of the diffraction maximum. This same conclusion has been drawn from attempts to fit, with and without correlations, the second diffraction maximum in 160 [refs. 17**8,24)]. However, we note that conclusions drawn from elastic electron scattering about short-range correlations are strongly model-dependent z 5*22). We wish to thank Professor Robert Hofstadter for his support during this experiment, Drs. T. W. Donnelly and G. E. Walker for their many helpful discussions and Dr. C. Ciofi degli Atti for his correspondence on the theoretical aspects of this problem, and the operating crew of the Stanford Mark III linear accelerator and staff of the High Energy Physics Laboratory.

592

G. C. LI et al.

References 1) L. R. Suelzle, M. R. Yearian and H. Crannell, Phys. Rev. 162 (1967) 992 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

G. R. Burleson and R. Hostadter, Phys. Rev. 112 (1958) 1282 M. Bernheim, Ph. D. thesis (Paris, Orsay, 1965) unpublished S. S. M. Wong and D. L. Lin, Nucl. Phys. Al01 (1967) 663 C. Ciofi degli Atti, Phys. Rev. 175 (1968) 1256 A. Male&i and P. Picchi, Nuovo Cim. Lett. 1 (1969) 81 T. Janssens, R. Hofstadter, E. B. Hughes and M. R. Yearian, Phys. Rev. 142 (1966) 922 U. Meyer-Berkhout, Ann. of Phys. 8 (1959) 119 R. E. Rand, R. Frosch and M. R. Yearian, Phys. Rev. 144 (1966) 859 M. Bouten, M. C. Bouten and P. Van Leuven, Phys. Lett. 26B (1968) 191 R. Neuhausen, Z. Phys. 220 (1969) 456 F. Eigenbrod, Z. Naturf. 23A (1968) 1671 R. M. Hutcheon, T. E. Drake, V. W. Stobie, G. A. Beer and H. S. Caplan, Nucl. Phys. A107 (1968) 266 R. M. Hutcheon and H. S. Caplan, Nucl. Phys. Al27 (1969) 417 J. Heisenberg, private communication The computer code is due to T. W. Donnelly and G. E. Walker C. Ciofi degli Atti and N. M. Kabachnik, Phys. Rev. Cl (1970) 809 D. A. Sparrow and W. J. Gerace, Nucl. Phys. Al45 (1970) 289 C. Ciofi degli Atti and M. E. Grypeos, Nuovo Cim. Lett. 11 (1969) 587 V. G. Neudatchin and Yu. F. Smirnov, Prog. Nucl. Phys. 10 (Pergamon Press, 1969) p. 273 L. R. B. Elton, S. J. Webb and R. C. Barrett, Proc. Third Int. Conf. on high energy physics and nuclear structure, ed. S. Devons (Plenum Press, 1970) T. W. Donnelly and G. E. Walker, Phys. Rev. Lett. 22 (1969) 1121 A. Malecki and P. Picchi, Nuovo Cim. Lett. 1 (1969) 823 C. Ciofi degli Atti and N. M. Kabachnik, Proc. Third Int. Conf. on high energy physics and nuclear structure, ed. S. Devons (Plenum Press, 1970) J. D. Walecka, private communication