Vector-polarization measurements and phase-shift analysis for d-α scattering between 3 and 11 MeV

1 2.B:2.L 1

Nuclear Physics A156 (1970) 465-476;

Not to be

North-Holland

Publishing

Co., Amsterdam

reproduced by photoprint or microfilm without mitten permission from the publisher

VECTOR-POLARIZATION ANALYSIS

@

MEASUREMENTS

FOR d-a SCATTERING

AND PHASE-SHIFT

BETWEEN 3 AND 11 MeV

L. G. KELLER t and W. HAEBERLI University

of Wisconsin,

Madison,

Wisconsin tt

Received 20 July 1970 Abstract: Vector-polarized deuterons from a tandem accelerator were used to measure the vector analysing power in d-cr elastic scattering at deuteron energies from 3 to 11 MeV in steps of 1 MeV, for c.m. angles between 21” and 152”. A phase-shift analysis was carried out of the differential cross section and of tensor and vector polarization data. It was found that f-waves must be included above 8 MeV. The analysis confirms the presence of the two known d-yrave resonances. The p-wave phase shifts were found to be small. An alternate solution with large p-wave phase shifts was found at 8 MeV. E

NUCLEAR REACTIONS 4He(vector-polarized d, d), E = 3-11 MeV; measured vector analyzing power
1. Introduction

Several studies of d-cc elastic scattering have been made to determine the scattering phase shifts in the energy range from 3 to 12 MeV. In an analysis based upon measurements of the differential cross sect& Senhouse and Tombrello ‘) found phase shifts which confirmed to known “) 2+ and l+ levels in 6Li at 4.6 and 6.3 MeV, respectively, and which indicated the presence of three p-wave resonances between 8 and 12 MeV. Later, McIntyre and Haeberli found that these phase shifts were inconsistent with the measured tensor polarization “). A phase-shift analysis “) was carried out which included the polarization data. The 2+ and 1+ levels were confirmed but all p-wave phase shifts were found to be small. Later measurements “) of the vector polarization at two angles in the same energy range were consistent with the phase shifts of McIntyre and Haeberli. However, recent measurements “) of the vector polarization at forward angles at 9 MeV are in disagreement with the predictions of ref. “). In the experiment reported here, angular distributions of the analysing power for vector-polarized deuterons were measured at c.m. angles between 21” and 152” for deuteron energies from 3 to 11 MeV in 1 MeV steps. A phase-shift analysis was carried out of the cross section, the tensor polarization, and the present vectorpolarization measurements. t Now at Purdue University, Department of Physics, Lafayette, Indiana. it Work supported in part by the US Atomic Energy Commission. 465

466

L. G. KELLER

AhJ

W.

HAEBERLI

Vector-polarization measurements are of interest not only because they provide data for a phase-shift analysis, but also because d-a scattering is used as a polarization analyser or as source of polarized deuterons for scattering experiments “). 2. Experimental procedure The vector-polarized deuteron beam ‘) from the University of Wisconsin tandem accelerator was scattered from a 4He gas target placed in the center of the scattering chamber described in ref. “). For scattering angles smaller than 30” a gas cell 5 cm in diameter, pressurized to 1 atm, was used. Another cell, 1.9 cm in diameter and pressurized to 3 atm, was used for larger angles. Windows on each cell consisted of 2.5 pm thick Havar t. The beam current on target was between 0.2 and 2 nA. Scattered deuterons were observed in pairs of solid state detectors placed symmetrically on opposite sides of the beam, 12.5 cm from the center of the target. The scattering angle was defined by a pair of rectangular slits, one 3 mm wide and 25 mm high placed directly in front of the detector, the other 2.4 mm wide and 25 mm high and located 5 cm from the center of the target. The extreme angular spread for this geometry was * 2”. In the present experiment the polarization vector of the beam was always perpendicular to the incident deuteron momentum ki,. The cross section for the scattering of a purely vector-polarized deuteron beam can be written “) as a(& f$) =a@)(1 +2
coscp),

(I)

where rr(e) is the unpolarized cross section and 4 is the azimuthal angle of the detector (4 = 0” when the polarization vector is parallel to ki, x k,,,). The quantity (it,,) is the vector polarization of the incident beam. The vector analysing power (iT, I (O)> is equal lo) to the vector polarization of the outgoing beam when unpolarized incident deuterons are scattered through an angle 8. From expression (1) it is clear that a measurement of the left-right asymmetry, that is, the ratio of yields for 4 = 0” and Cp= 180”, will determine (iTI,) at an angle 8.. Instrumental asymmetries were eliminated from the measurements by reversing the direction of the beam polarization at the polarized ion source. The procedure was the same as described in ref. 11) for protons t+. The vector polarization of the beam was determined from the observed asymmetry in d-a scattering at 9 MeV and a c.m. angle of 65.8”, where the vector analysing power is known from a previous measurement “) by Trier and Haeberh. All present results are therefore relative to that point. t Havar is an alloy with a high tensile strength sold by the Hamilton Watch Company, Lancaster, Pennsylvania. tt The measurements at 9 and 11 MeV were made before the purely vector-polarized beam became available. The effect of tensor polarization of the beam was eliminated by using the procedures outlined in ref. I’). Some points in the 9 MeV angular distribution were later remeasured with the purely vector-polarized beam and no inconsistencies were found.

d-u

SCATTERING

461

As an additional check on the beam polarization the present values of changes rapidly with energy. The discrepancy can be attributed to a systematic error of approximately 50 keV in the deuteron energies quoted in ref. “). The deuteron energy in the present experiment was determined by a 90” momentum analysing magnet which had been recalibrated 13) prior to the present work. No momentum analysis was used in the experiment of ref. “). The results of the comparison show that if all of the data of ref. “) at 65.8” had been used to determine the beam polarization, rather than just the 9 MeV point, the value for the beam polarization would have changed by less than 1 %. There is a systematic uncertainty of approximately 3 % in the normalization of the data of ref. “) caused by the uncertainty of the polarization of the incident beam. The overall normalization of the present measurements is therefore also uncertain by the same amount. 3. Results of measurements The results of the measurements are shown in figs. 1 and 2 together with the crosssection data of refs. ‘3I42’ “). Numerical values of the measured polarization have been deposited with the National Auxiliary Publication Service + under file number NAPS-01168. The errors in the vector polarization, which are typically +O.Ol, are a combination of the statistical error of measurement and the uncertainty in the measurement of the beam polarization. In addition, the errors contain the estimated effects of a possible correlation between the beam direction and the spin direction, which causes a false asymmetry when the cross section changes rapidly with angle (0 < 30”). The systematic error in the beam polarization discussed at the end of sect. 2 has not been included. All data were corrected for the effects of the finite extent of the detector apertures and the target length. Large values of
468

la. G, KEY.LER

AND

W.

HAEBERLI

d-a

469

SCATTERING

4. Phase-shift analysis

The results of the present expe~rn~~t together with previously measured crosssection ‘8’ 4~1“) and tensor polarization data 3y17,18) were used to determine phase shifts for d-a scattering. In d:ct elastic scattering a tensor force can couple two partial waves with the same total angular morn~n~rn j but with orbital angular momenta differing by two units. That such a conping exists between the s- and d-waves for j = 1 was shown first by McIntyre and Haeberli ‘j’. The orbital angular momentum is therefore not a good quantum number and the scattering cannot be described by a phase shift associated with each orbital angular momentum quantum number. A detailed treatment of this problem has been given by Blatt and Biedenha~ 19). Their analysis shows that linearly independent combinations of two coupled partial waves can be formed whose scatt~~~g can be described by phase shifts. When s- and d-waves for jR = 1’ are coupled the combinations formed are YE = cos aY(3S&-sin Y! = -sin aY(3S,)i-cos

EY(~D,), EY(~D,).

(3)

The coupling parameter E is chosen by convention to approach zero in the limit of zero incident energy, where no tensor coupling can occur. The phase shifts associated with the scattering of the partial waves Y, and Y, are labehed 6, and 6, ) respectively. In the present analysis the coupling parameter s between the s- and d-waves for * = 1 was assumed to be real and the coupling between p- and f-waves for j = 2 J was excluded. Partial waves up to 1*= 3 were used. Above 4 MeV the phase shifts were allowed to be complex. Previous work ‘“f indicated that g-waves become important only above 20 MeV. Computations were carried out with the phase-shift program described in ref. 3), modified to permit v~ia~o~s of the coupling parameter in the search for a best fit. The program searches for phase shifts which ~ni~ze the x2 function

for N data points D&, with experimental errors t_d.@,. Here D&, may be a crosssection, vector polarization, or tensor polarization measurement. The quantities -I&i0 are the corresponding valnes calculated from the phase shifts. The phase shifts are shown in fig, 3 and listed in tables 1 and 2. The positive parity phase shifts clearly show the effects of the 2* and 1+ levels in ‘Li at 4.6 and 5.3 MeV deuteron energy respectively. The d-wave phase shift for j = 3, which rises approximately from zero to 180” over the 3+ resonance “) at 1.07 MeV, was found to decrease slowly in the energy region considered here. The p-wave phase shifts are small and show no in~~a~on of resonance behaviour. The present phase shifts agree closely with those of McIntyre except that f-waves were found to be necessary above 8 MeV to describe the behavior of at for-

470

L.G. KELLER

AND

W.HAEBERLI

20"

0 RE d:

I

Fig. 3. The phase shifts and coupling parameter (in degrees) found in the search. The opea symbols refer to the real part of the phase shift 6( and the closed symbols to the imaginary part. The solidfines are a guide to the eye.

471

d-m SCATTERING TABLE

1

The positive parity phase shifts, in degrees, for laboratory

6%

(ZeV)

Re 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 8.0 “)

SD Im

Re

0.1 0.1 0.8 6.4 5.3 4.2 2.9 69.3

18.4 29.9 54.3 62.3 57.2 47.3 34.1 25.1 25.1

Re

Im

1.5 2.0 0.2 0.0 0.0 0.0 0.2 0.0

6:

G

E

.2.4

112.5 99.3 86.9 74.9 73.2 76.9 77.6 77.2 68.3 45.1

energies between 3 and 11 MeV

171.6 172.9 166.5 165.6 164.8 162.0 157.3 155.0 146.9 155.3

0.3 -4.4 -3.4 -19.1 -71.8 -86.8 --89.9 -90.0 -90.0 -99.6

Im

0.0

4.2 9.3 11.0 15.7 19.0 23.3 0.0

Re

Im

14.7 53.2 94.0 109.3 120.4 125.9 125.6 126.9 131.9 125.2

1.1 3.7 4.6 2.1 2.1 0.9 0.6 3.7

The quantities a,, a8 are the eigen phase shifts for J = 1, E is the coupling parameter for J = 1, and Sg, 8: are the d-wave phase shifts for J = 3 and J = 2; respectively. “) Alternate set of phase shifts. TABLE2 The negative parity phase shifts 8:, in degrees, for laboratory

6:

El Re 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 8.0 “)

1.7 6.6 6.7 6.1 4.0 2.0 0.3 -3.3 -9.8 52.4

a:

6: Im

Re

1.5 1.4 7.5 10.7 18.5 24.9 35.5 14.4

-0.3 1.2 -1.8 -4.5 -6.4 -6.7 -8.6 -9.4 -10.7 33.3

Im

Re

0.0 4.1 6.0 5.1 6.4 6.6 0.1 16.3

-2.3 -0.5 -1.9 -3.6 -5.9 -7.3 -8.4 -10.0 -8.3 112.4

energies between 3 and 11 MeV

62

s4 3

Im

8:

Re

Im

Re

Im

Re

Im

2.2 3.1 3.7 4.1 4.1 7.2 4.5

0.0 0.6 0.6 2.3 3.0 3.9 3.7

3.5 3.6 2.2 2.9 3.2 2.7 2.9

0.0 0.0 0.0 0.0 0.0 1.4 0.0

-0.5 -3.8 -5.6 -6.5 -7.1 -6.1 -5.4

0.0 0.2 0.2 0.0 0.0 2.7 0.5

0.0

2.0 2.9 2.4 2.0 2.5 3.6 28.9

“) Alternate set of phase ~shifts.

ward angles. At 7 and 8 MeV good fits to the data could be obtained without f-waves, but they were included to provide continuity of the f-wave phase shifts with energy. The energy dependence of the phase shifts 6, and 6, and the coupling parameter E is similar to that predicted from the single level analysis “) of the l+ level. The coupling parameter is found to be small below the resonance and to rise rapidly to -90” over the resonance. An examination of eqs. (3) shows that below resonance 6, is therefore predominantly an s-wave phase shift and 6, is largely a d-wave phase shift. The coupling is largest in the region of the resonance where E becomes -45”.

472

L. G. KELLER

AND

W.

HAEBERLI

d-cLSCAT’BRXNG

473

Above the resonance where E reaches - 90”, eqs. (3) show that the two partial waves are again uncoupled, with 6, largely a d-wave phase shifts and 6, largely an s-wave phase shift. The calculated values of the tensor polarization parameters ( T2 J, ( T2 1> and (T,,) at four c.m. angles as a function of energy are shown together with the available data in fig. 4. Only data at the energies of the present experiment were included in the search. The lines shown are smooth curves through the points c~culated in 1 MeV intervals. The tensor moments describe the pola~~tion of the outgoing beam when unpolarized deuterons are scattered from 4He, where the z-axis of the coordinate system is along the direction of the outgoing momentum in the c.m. system and the y-axis normal to k, x k,,,. The normalization of the tensor parameters is that of Lakin ‘I). 5. Uncertainties of the phase shifts

Because the present phase-soft analysis contains a large number of parameters which are varied to obtain a fit to the data, large uncertainties may be present in the results. There are two procedures often used to estimate the uncertainties of phase shifts. One method estimates the uncertainty in a given phase shift 6i by observing the change in x2 as Sj is varied in steps from the value found for the best fit to the data, keeping ah other phase shifts constant. The error is determined by noting the range of values of 6 i which produce acceptable fits to the data. The un~erta~n~es found in this way ignore possible correlations between phase shifts. An alternate procedure attempts to include correlations between the phase shifts by expanding the x2 function in the region near the best-fit solution, assuming that x2 is a quadratic function of the phase shifts. The objection to this method is that the assumption that x2 is parabolic over the region of interest may not be valid. The method used in the present analysis permitted correlated changes in all of the phase shifts by a step-and-search procedure. fn this method a given phase shift 6, is incremented by several degrees from the value found for the best fit to the data. A search on the other phase shifts is carried out until a best fit is found. For each additional increment of the phase shift 6, the search on the other phase shifts begins with the results of the previous- step. The range of values of 6 i for which one obtains acceptable fits to the data determines the uncertainty of ai. TABLE 3 The uncertainties of the real parts of the phase shifts at 9 MeV

474

L. G. KELLER

AND

W.

RAEBERLI

This procedure has the disadvantage that it requires a large amount of computer time. For this reason the uncertainties in the phase shifts were determined only for the 9 MeV solution. In the present analysis an increase in x2 of more than 40 % above its original value was chosen as unacceptable because for such an increase the deterioration of the fit was obvious by inspection. The uncertainties of the real parts of the 9 MeV phase shifts are presented in table 3. The uncertainties of the imaginary parts of the phase shifts were all approximately 5 IO”, with the restriction, of course, that these phase shifts remain positive. I

I

T

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--s,-5”_

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160

Fig. 5. The sensitivity of the angular distributions of and
The uncertainty of 6, required special consideration. When 6, was increased by 12 from its tabulated value, another minimum in x2 was found which was similar in magnitude to that of the best-fit solution. The minimum was producsd by an improvement in the fit to the polarization data at the expense of the fit to the cross section. The fit to the cross section near 90” was so poor that the solution was considered unacceptable. It should be pointed out that x2 was not a quadratic function of most of the phase shifts near the best-fit solution. The uncertainties of some of the phase shifts are so large that it would be worthwhile to know what further experimental information is required to reduce them. Judging from the changes in the predicted values for
d-a

SCATTERING

475

would be more useful in subsequent phase-shift analyses than more accurate information about either the differential cross section or the vector polarization. Increasing 6, by as much as 30” produced Ghanges of less than 5% in the predictions for the cross section, except at angles smaller than 20” where the cross section changes rapidly. Similarly, changes in were small. The sensitivity of the angular distributions of and reduces the prominence of the peaks in (T,,). Variations of the phase shifts Sf and S: affect only (T,,). The changes in the angular distributions as a function of6: are approximately the opposite of the changes shown for St. Predictions for (Tza) proved insensitive to the above variations in the phase shifts. Since the completion of the present analysis, measurements of the angular dependence of (T,,) [ref. 22)] and of (T2,> and (T,,) [ref. 23)j have become available between 3 and 11.5 MeV. These results suggest that the detailed behavior of the phase shifts above 8 MeV may not be correct. The predicted negative peak shown in fig. 5 for {T,,) near 135” is not con~med. Increasing the imaginary parts of 6, and S(: by 5” and lo”, respectively, and decreasing the imaginary part of6: by 5” improves the fit to these data. The forward angle measurements of ref.22) indicate that the real part of 6, above 8 MeV should be increased by approximately 10” to 15” from the tabulated value. 6. Aiternate sets of phase shifts Near 10 MeV another set of phase shifts was found, which differed from that in tables 1 and 2 p~rna~ly in that the real part of 6, and the imaginary part of Sf were 10” to 15” higher. The two sets gave equally good agreement with the data but predicted different values for (T,,). The tabulated set was chosen because it gave better agreement with the combination of (T,,) and (T,,) measured in a recent double scattering experiment of Ohlsen et al. 24) at 11.5 MeV. In the analysis of cross section and tensor polarization data at 8 MeV, McIntyre and Haeberli ‘> found three additional sets of phase shifts which were compatible with their data. One may raise the question whether these alternate sets can be eliminated on the basis of the vector-pola~zation meas~ements. Pose-soft searches were carried out, using as starting sets the three alternate 8 MeV solutions of table 1 in ref. “). One of the three converged to a solution which gave as good an overall fit to the data as the solution discussed in sect. 5. Also the fit to the recently measured tensor polarization parameters 22fi23) is of similar quality for either solution. The phase shifts for the alternate set at 8 MeV are given on the last line of tables 1 and 2. The solution is characterized by positive p-wave phase shifts. In particular Sy is large. It may be interesting to note that at all energies (iT,,) crosses zero near 90” (figs. 1,2), This behavior is expected if there is no spotting between the three p-wave phase shifts as long as the f-wave phase shifts are small. If the p-waves are split, the vector

476

L. G. KELLER

AND

W.

HAEBERLI

polarization at 90” is not zero, unless there is accidental cancellation of different terms in the phase-shift expansion, Such a cancellation apparently occurs for the alternate set of phase shifts at 8 MeV. The continuation of this phase-shift set to neighboring energies was not investigated. The matter was not pursued further since a new analysis is being carried out already by the group at Zurich 22a““). 7. Conclusion Large values of the vector polarization were observed at all energies. The angular distributions are determined largely by the two d-wave resonances in the energy region considered. No new states were found. The phase shifts of McIntyre and Haeberli failed to reproduce the vector polarization data at forward angles above 9 MeV but it has been found that the inclusion of f-waves is the only substantial modification required to reproduce the vector-polarization data. Angular distributions of the tensor polarization parameters would be extremely useful in any subsequent phase-shift analysis. The uncertainty of the phase shifts is still large. The calculations of sect. 5 indicate that much improved phase shifts will be obtained when the recent measurements of the tensor polarization parameters are included in the analysis. It will be interesting to investigate whether the new tensor parameters eliminate the alternate phase-shift solution found in the present work at 8 MeV. References 1) L. S. Senhouse, Jr. and T. A. Tombrello, Nucl. Phys. 57 (1964) 624 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

A. Galonsky and M. T. McEllistrem, Phys. Rev. 98 (1955) 590 L. C. McIntyre and W. Haeberli, Nucl. Phys. A91 (1967) 369 L. C. McIntyre and W. Haeberli, Nucl. Phys. A91 (1967) 382 A. Trier and W. Haeberli, Phys. Rev. Lett. 18 (1967) 915 E. M. Bernstein, G. G. Qhlsen, V. S. Starkovich and W. G. Simon, Phys. Rev. Lett. 18 (1967) 966 T. B. Clegg, G. R. Plattner and W. Haeberli, Nucl. Instr. 62 (1968) 343; T. B. Clegg, G. R. Plattner, L. G. Keller and W. Haeberli, Nucl. Instr. 57 (1967) 167 R. Morrow, Ph.D. Thesis, University of Wisconsin (1967) P. Extermann, Nucl. Phys. A95 (1967) 615 G. R. Satchler, Nucl. Phys. 8 (1958) 65; L. C. Biedenharn, Nucl. Phys. 10 (1959) 620 G. R. Plattner, T. B. Clegg and L. G. Keller, Nucl. Phys. All1 (1968) 481 P. Schwandt and W. Haeberli, Nucl. Phys. All0 (1968) 585 J. C. Davis and F. T. Noda, Null. Phys. Al34 (1969) 361 G. G. Ohlsen and P. G. Young, Nucl. Phys. 52 (1964) 134 L. Stewart, J. E. Brolley, Jr. and L. Rosen, Phys. Rev. 128 (1962) 708 W. Grilebler, V. Kbnig, P. A. Schmelzbach and P. Marmier, Nucl. Phys. Al34 (1969) 686 A. Trier, Ph.D. Thesis, University of Wisconsin (1966) available from Univ. Microfilms, Ann Arbor, Michigan, USA P. 6. Young, G. G. Ohlsen and M. Ivanovich, Nucl. Phys. A90 (1967) 41 J. M. Blatt and L. C. Biedenharn, Rev. Mod. Phys. 24 (1952) 258; Phys. Rev. 86 (1952) 399 P. Darriulat, D. Garreta, A. Tarrats and J. Arvieux, Nucl. Phys. A94 (1967) 653 W. Lakin, Phys. Rev. 98 (1955) 139 V. K&rig, W. Griiebler, P. A. Schmelzbach and P. Marmier, Nucl. Phys. A148 (1970) 380 W. Grtiebler, V. KBnig, P. A. Schmelzbach and P. Marmier, Nucl. Phys. A148 (1970) 391