I Nuclear Physics 83 (1966) 46--64; (~) North-Holland Publishing Co., Amsterdam 2.B I

No t to be reproduced by photoprint or microfilm without written permission f rom the publisher

POLARIZATION IN p-~ S C A T T E R I N G F R O M 17 T O 27 MeV

w. G. WEITKAMP t and W. HAEBERLI

University of Wisconsin, Madsion, Wisconsin tt

Received 7 January 1966

Abstract: The proton polarization in p-~ scattering has been measured for proton energies between 17.5 and 27 MeV at six laboratory scattering angles from 45 ° to 115 °. The 8He(d, p)*He reaction was used as the source of polarized protons. The measured extremes of polarization at 17.5 MeV are --0.68±0.03 at 75 ° and 0.92--0.04 at 115 °. The magnitude of the polarization decreases slowly with increasing energy except near 23 MeV, where an anomaly, attributed to the ,~+ state in 6Li, is observed. A phase shift analysis of all available p-~ scattering data was made at 14.3, 17.5, 20.6, 26.1, 27.7 and 31.0 MeV. These phase shifts together with resonance parameters for the .~+ state obtained by analysing the 3He(d, p)4He total cross-section data were used to cal- culate the energy dependence of the p-~ polarization and the cross section near the resonance. The results are in good agreement with the measurements.

E NUCLEAR REACTIONS SHe(d, p), E = 5.5, 8.0, 10.7 MeV; measured polarization (Ep).

4He (polarized p, p), E = 17.5-27 MeV; measured polarization (E; 0). Deduced scattering phase shifts. 5Li resonance parameters. Enriched 8He target.

1. Introduction

Elastic p-~ scattering has often been used as a p ro ton polar izat ion analyser, prin-

cipally because very large polar izat ions are observed over a wide energy range. F o r

p ro ton energies up to 14.5 MeV the polar izat ion has been measured in a number o f

double-scat ter ing experiments t t t . However , at higher energies there is a p ronounc-

ed lack of detailed informat ion.

In 1958, G a m m e l and Thaler 2) made a phase shift analysis of data at 17.5 and 40

MeV. The p-c~ polar izat ion between these energies was then calculated by interpolat ing

the phase shifts with the use o f a potent ia l model. However , subsequent measurements

o f p-~ polar izat ion at 22, 29 and 38 M e V have shown that the G a m m e l and Thaler

predict ions are not reliable 3-5) , so that addi t ional measurements are necessary be-

fore p-7 scattering can be used as a polar izat ion analyser in this energy region. The

present exper iment was under taken to measure the p-ct polar izat ion in the energy

region f rom 17.5 to 27 MeV. One specific aim of the exper iment was to determine to

what extent the polar izat ion near a p ro ton energy of 23 MeV is affected by the

4He(p, d)3He react ion threshold and the ~+ excited state in 5Li at 16.64 MeV. The

t Now at the University of Washington, Seattle, Washington. tl" Work supported in part by the U.S. Atomic Energy Commission.

ttt For a list of references see table 5.5 ofref. 1).

46

p-~ S C A T T E R I N G 47

threshold and excited state are shown on the energy level diagram of 5Li in fig. 1. The

excited state mentioned has been extensively studied by means of the inverse reaction 3He(d, p)4He but has not previously been observed in p-~ scattering, although an anomaly corresponding to the mirror state in 5He has been observed in the n-~ cross section. For further discussion of measurements and analysis of n-~ scattering, we refer to the following paper by Hoop and Barschall 6).

In the present experiment, the 3He(d, p)4He reaction, initiated with 5.5 to 10.8 MeV deuterons, was used as a source of polarized protons. The protons were scat- tered from 4He into one of two identical detectors located at azimuthal angles of zero

16.385 3He + d

16.64 3 / 2 ~"

j'rr= :5/2- ~ / / / / / / / / / / / / / ~ / / I / / / / / / / / / / / . ~

SLi

27O 18.610

17846 3 H e + n + p

17.5

- I . 9 6 7 4He+p

Fig. 1. Energy levels of ~Li.

and ~r. From the ratio r of the counting rates in the two detectors the product Pres was obtained, where Pr is the polarization of the protons produced in the 3He(d,p)4He reaction and Ps the proton polarization produced in p-e scattering t

Two sets of measurements were performed. First, Pr was determined by slowing protons with polyethylene foils so that the p-e scattering took place at about 12 MeV, where the values of Ps are well-known 7). Making use of these results, Ps was then determined for various proton energies between 17.5 and 27 MeV and several scat- tering angles.

f Subscripts r and s will be used consistently in the experimental part of this paper on quantities which are associated with the SHe(d, p) reaction and p-~ scattering, respectively.

48 W.G. WEITKAMP AND W. HAEBERLI

2. Apparatus

The apparatus used was a modification of the double scattering chamber described by Brown et aL 7). All major dimensions of the apparatus have been given there. A simplified sectional view is shown in fig. 2. The collimated deuteron beam entered through aperture A and penetrated the reaction cell B, which contained 19 a tm of SHe. The deuteron beam stopped in the air-cooled tantalum beam stop C. Protons produced in the 3He(d, p)4He reaction near a laboratory reaction angle ~k r = 30 ° passed through a foil at D to slow down the protons before they entered the 4He- filled scattering cell E. The foils were of polyethylene and as much as 5 m m thick. Scattered protons were detected in one of two counter telescopes F, each consisting

I0 cm I~ I

Fig. 2. Diagram of the apparatus. A---entrance aperture, B---aHe cell, C--beam stop, D----energy ab- sorbing foils, E 4He cell, F----counter telescope, G--proportional counter, H--scintillation detector.

of a proportional counter G and a scintillation detector H. The scattering angle ~s was adjustable f rom 45 ° to 115 ° with an uncertainty of + 0.1%

For measurements taken at high deuteron energies, the reaction cell B was a simple 1 cm diamcylinder of 4.9 mg/cm2 foil t. At low deuteron energies, where beam heating in the cell was a serious problem, a cell which could be cooled was used. The cell con- sisted of a 0.5 cm diam cylinder made of 3.3 mg/cm 2 foil surrounded by an 0.8 cm diam concentric cylinder of 2.2 mg/cm 2 thick foil. Forcing helium between the cylin- ders cooled the foils and increased the maximum tolerable deuteron beam current by more than a factor of five compared to the simple cell.

* All foils used for the gas targets consisted of a high tensile strength cobalt alloy ("Havar") sold by Hamilton Watch Company, Lancaster, Pennsylvania.

p-~ SCATTERING 49

The type of scattering cell described by Brown et aL 7) was used for measurements at scattering angles of 45 ° to 60 °. The cells were filled with 20 atm of 4He. At larger angles, the cross section for p-~ scattering is so low that it became necessary to in- crease the target density to maintain an acceptable counting rate. This was done by cooling the second target to the temperature of liquid nitrogen. The cell (E of fig. 2) was a 3.2 cm diam cylinder of 8.7 mg/cm 2 wall thickness suspended from the bottom of a liquid nitrogen reservoir and filled with 12.2 arm of 4He. The average temperature in this cell was 89°___ 4 ° K.

Two identical gate circuits and a pulse-height analyser were used to obtain the spectra of those pulses from the scintillation detectors which were coincident with pulses in the proportional counter. The resolving time of the circuit was about 3 ps.

3. Measurement Techniques

The number of counts in each counter was deduced from the pulse-height spectra. Two sets of measurements are shown in fig. 3. Most spectra showed well-defined peaks with little background similar to those in the upper half of fig. 3. In fact, the spectra shown were obtained without the coincidence circuit. However, under certain conditions, the background became large even when the coincidence circuit was used. The worst case is shown in the lower half of fig. 3. The background (open circles) was measured by inserting a thick piece of tantalum between the first and second targets. There are several factors which cause the background to the unusually large in this case. The number of good counts is relatively small since the p-~ cross section has a minimum near this angle. Also, the large scattering angle causes the energy loss to the recoil a-particle to be large. This is undesirable because the number of background pulses increases rapidly with decreasing pulse-height. The background is particularly large for the spectrum on the right because this counter is near the beam stop, and the number of good counts is particularly small because of the large polarization at this angle.

The net number of good counts was obtained by adding the pulses in all channels above a certain cut-off (marked by an arrow in fig. 3) and, when necessary, subtract- ing from this the number of background counts above the cut-off. All points were measured with the reaction angle ~k r both to the left and to the right. Since for a given counter the pulse-height of the good protons is the same in both cases, the same cut- off was used. Thus the arrow on the bottom right hand spectrum of fig. 3 was based on the run in which this same counter was on the other side. It should be emphasized that the choice of this cut-off does not affect the data; the counter efficiency cancels entirely when the ratio r of the counting rates in the two detectors is taken to be the geometric mean of the counting ratios with the reaction angle to the left and to the right. 8)

In order to test the accuracy of the background measurements at those angles and energies where the background was largest, the number of counts in the channels just

50 W. G. WEITKAMP AND W. HAEBERLI

below the cut-offs was compared for background and foreground runs. I f the tech- nique for measuring background is satisfactory, this number of counts (which is presumably all background) should be the same in both kinds of runs. However, the number of counts was found to be approximately 15 ~o lower in background runs than in foreground runs. A 15 ~o error in the background determination can cause at most an error of 0.03 in polarization. Since it seemed difficult to correct accurately for this error, a corresponding uncertainty was included in the final results for those cases where the background was large.

400

bd Z 30C Z

W Q. I00

I--

~ 4O 0

W 3C

I I I I I I I I I I

22.0 MeV 4 5 ° LAB

!

27.0 115" LAB MeV

Z 2G J" o

IO o °o i ° O ~ o e ~ o~ ° o o - ~ I - 4 0 60 SO 40 60 80 I00

CHANNEL NUMBER

Fig. 3. Pulse-height spectra. The upper pair of spectra show the most clearly resolved pro ton peaks. The lower pair show the least clearly resolved proton peaks. Open circles show results of a background

measurement. The arrows indicate the cut-offs described in sect. 3.

The measured ratio r of counting rates was corrected for the effect of finite geometry The calculation was carried out by computing r for the geometry of the actual experi- ment by numerical integration of the intensity over target volumes and detector areas, and comparing the result with the ratio that would have been obtained with point geometry. The calculation took into account the variation of cross sections and polarizations in both targets with angle and with energy as well as the effect of stragg- ling and multiple scattering in the polyethylene foil between reaction and scattering cell. The assumed angular dependence of polarizations and cross sections was ob-

p-~ SCATTERING 51

tained by drawing smooth curves through available data. Since no cross-section data were available near the 23 MeV resonance, the correction was obtained by linear interpolation of the results at other energies. The calculation is described in more detail in ref. 9).

4. Polarization of the Protons from the SHe(d, p)4He Reaction

The polarization of the protons from the aHe(d, p)4He reaction was measured to check the earlier results lo) and to improve their accuracy. The earlier measurement depended on the polarization analysing power of carbon for 14 MeV protons which is relatively uncertain. The availability of accurate, absolute measurements 7) of p-ct polarization at proton energies up to 11.9 MeV made it advantageous to use p-~ scattering as a polarization analyser for the Pr measurements.

In order to measure Pr, foils of polyethylene were used to slow the reaction protons to about 11.9 MeV. Accurate Pr measurements were made at three deuteron energies, E r = 5.55, 7.97 and 10.68 MeV, corresponding to reaction proton energies at the centre of the reaction cell of Ep = 22.3, 24.8 and 27.3 MeV, respectively. All of these measurements were made at a laboratory reaction angle of 30 °, where Pr is a maxi- mum. The p-~ laboratory scattering angle was 45 °.

To investigate whether multiple scattering and energy straggling of protons in the polyethylene cause an appreciable uncertainty in these measurements, the point at Er = 5.55 MeV (Ep = 22.3 MeV) was also measured by slowing the protons to 7.8 MeV, thereby increasing both the multiple scattering and the straggling by a factor of about 1.5. The difference between the two values of Pr measured at this point was 0.002___0.021. In addition, the E r = 10.68 MeV (Ep = 27.3 MeV) point was also measured by slowing the protons to 22.1 MeV before scattering them from 4He. The value of the p-~ polarization measured at Es = 22.0 MeV in the present experiment was then used to calculate Pr- The difference between the two results was 0.02___ 0.06.

The results of the Pr measurements are given in table 1. In this table r is the experi- mental (uncorrected) ratio of intensities. The values quoted for Pr have been corrected for finite geometry as mentioned in sect. 3. The listed uncertainty in Pr includes the

TABLE 1

Polar izat ion o f p ro tons f rom the 3He(d, p) react ion for a labora tory angle o f 30 °

Er /rp /r, P, r /'r (MeV) (MeV) (MeV)

5.55 22.3 11.86 --0 .409 0.705 0.439 4-0.014 7.97 24.8 12.08 --0 .406 0.592 0.663 -t-0.021

10.68 27.3 12.28 -- 0.403 0.565 0.727 ±0 .040 10.68 27.3 22.13 --0.308 0.642 0.744 4-0.039 10.68 27.3 0 .736±0.028 s)

a) Average o f the two preceding values.

52 W. G. WEITKAMP AND W. HAEBERLI

0 . 8

0 . 6

Pr 0 . 4

0 .2

o

statistical uncertainty, the uncertainty in the geometrical correction and the uncertain- ty in the analysing power P~. The uncertainty in the deuteron energy Er is +__0.03 MeV.

The values of Pr are plotted together with the results of the previous measurements of Brown and Haeberli 1o) in fig. 4. The agreement between the two sets of results is better than was expected from the uncertainty of the earlier measurements. The curve is a smooth line drawn through the points. The values of P~ used in calculating P~ (see sect. 5) were taken from this curve. The assumption that P~ is a smooth function is supported by the fact that the cross section for the 3I-Ie(d, p)4He reaction [and also the mirror reaction 3H(d, n)4He] shows no evidence of resonances with laboratory widths less than several MeV in the deuteron energy range from 4 to 12 MeV (ref. 11)).

1.0 " I' l I I i l i I J

3He(d,p) 4He ,/,,. = :3o* ~ - - ~ - -

I I I 1 I I I I I 4 6 8 I 0 I£

D E U T E R O N E N E R G Y ( M e V )

I I I t I 2 3 2 5 2 7

PROTON E N E R G Y (MeV)

I I I 21 2 9

Fig. 4. Proton polarization in the SHe(d, p)~He reaction. Both deuteron and proton energies are in the laboratory frame of reference. The open circles are from ref. xe). The present measurements

are shown as solid dots.

5. Measurements of the Polarization in p-= Scattering

The polarization in p-~ scattering was measured at six laboratory scattering angles ~s between 45 ° and 115 ° at each of six proton energies E s from 17.5 to 27 MeV. In addition, a number of measurements were made between 22 and 24.4 MeV to study the structure in this region.

At all points for which Es > 22 MeV, the deuteron beam energy was adjusted to produce the correct proton energy so that no polyethylene foils were needed to slow the protons. However, from fig. 4 it may be seen that Pr decreases quite rapidly below a deuteron energy of 5.5 MeV corresponding to a proton energy of 22.3 MeV. Also, deterioration of the energy and angle resolution of the deuteron beam sets in below 5.5 MeV because of the increasing effects of multiple scattering and energy loss in the reaction cell wall. Therefore, the points at Es = 17.5 MeV and 20 MeV were measured by holding the deuteron energy fixed at about 5.5 MeV and slowing the protons to the correct energy with foils.

p-at SCATTERING 5 3

The p-~ polarization is plotted as a function of energy in fig. 5. The anomaly in polarization near Es = 23 MeV stands out clearly. As pointed out in sect. 3, the cal- culations correcting for finite geometry are not accurate near this anomaly. In par- ticular, the spread in energy was large enough that the peaks may not be fully resolved. At 23 MeV, the rms spread in proton energy was 0.13 MeV and the spread in scattering angle was 1.9 °. The curves shown in fig. 5 will be discussed in sect. 8.

The p-~ polarization measurements of Craddock et aL 5) at 29 MeV are shown in fig. 5 for comparison. Craddock et aL and Conzett et al. 3) have also measured the p-~ polarization at 22 MeV. At the three scattering angles at which direct comparison between their results and our measurements is possible, the measurements of Crad- dock et al. average 0.96_+0.03 times our values while those of Conzett et aL average 1.04_+ 0.05 times our values.

0 2

554 ° 0

- 0 2 . } #

- 0 4 L I J I [ [ I 1 I i I I

Z o ~ -o.2 ~ 2 . 7 ° -

N m -0.4 n ~

._1 0 - 0 . 6 ~ ~ ~ J r 13._

-o.4t . 89.2* _

17 19 21 2 3 2 5 2 7 2 9

Ep (MeW

-0.2

-0.4

-0.6

-0.8

~ ~ 986° .

I [ I I I ] I I r I [ :

I ~z ,9 2f 23 25 27 z9

E.(Mey)

Fig. 5. The polarization in p-g scattering. The curves are calculated f rom phase shifts and resonance parameters described in sect. 8. Crosses at 29 MeV indicate the data o f ref. s).

The measurements of Ps are tabulated in table 2. The values of r given in this table are uncorrected counting ratios. The error quoted for Ps contains the statistical un- certainty, the uncertainty in the finite geometry correction and, where appropriate, the background uncertainty discussed in sect. 3. However, the errors do not include the uncertainty in P, because this is a scale factor uncertainty common to all measure- ments at a given proton energy.

54 W . G . WEITKAMP AND W. HAEBERLI

The uncertainty in the proton energy for measurements at ~k, = 45 ° and 60 ° is -t-0.08 MeV. Since a thin film of condensable material was sometimes observed on the cold foils of the liquid nitrogen cooled scattering cell, the uncertainty in Es at the other angles was __.0.12 MeV.

6. Phase Shift Analysis

Several authors 2,12-14) have reported the results of phase shift analyses of p-~ data between 11 and 18 MeV. Part of this energy region is reconsidered here because in the earlier analyses no polarization measurements were available. Gammel and Thaler 2), for instance, found five different sets of phase shifts at 17.5 MeV which fit the cross sections equally well; they selected one of these on the basis of a single polar- ization measurement at 15.5 MeV. This section also reports the analysis of the present data and other new polarization and cross-section measurements between 20 and 31 MeV. New analyses for proton energies above 20 MeV have recently also been re- ported by other authors 15).

The relevant expressions for the cross section and polarization in terms of the scat- tering phase shifts have been given repeatedly t. The contribution to the scattering amplitude f rom a particular partial wave is proportional to

f ~ _ e x p ( 2 i r ~ ) - 1, ( I )

2i

where 67 is the complex phase shift of orbital angular momentum l and total angular m o m e n t u m j = l+½. In the following, 6 will be expressed in terms of /t = Re 6 and z = e x p ( - 2 I m 6):

f ~ _ z + exp (E ip~) - 1 (2)

2i

Below 23.02 MeV, which is the threshold of the 4He(p, d)3He reaction, no inelastic processes take place and the inelastic parameter z -- 1. Above the threshold, the total reaction cross section aR is related to • by

0. R ~--. ~ 2 E ( j + ½ ) t l - ( z ~ ) 2 l . (3) I , j

The search for the phase shifts which best fit the polarization, the elastic scattering cross section and the total reaction cross section was carried out with a computer programme which was available f rom previous work x6). The programme followed

t See e.g. refs. ae, 17). The notation of ref. xe) is being used here. The formulas of ref. xT) include first-order relativistic corrections to the Coulomb phase shifts. In our analysis, these corrections were found to be small in comparison to the experimental uncertainties, and were omitted. However, first order relativistic corrections to the Coulomb parameter U and the proton wave number k were used. In addition, exact rather than integer mass values were used throughout.

p-~t SCATTERING

TABLE 2

Measured polar izat ion in p-u scat ter ing

55

~Ps 0, E~ Er / ' r r e~ (degrees) (degrees) (MeV) ( M e V )

45.0 55.4 17.51 5.55 0.439 0.753 - -0 .328--0 .017 20.09 5.55 0.439 0.770 --0.3044-0.013 22.03 5.55 0.439 0.767 --0.308 ±0 .010 22.34 5.84 0.475 0.746 --0.313 4-0.020 22.64 6.14 0.512 0.728 --0.3154-0.020 22.94 6.44 0.546 0.687 --0.347 -4-0.018 22.93 6.36 0.537 0.686 - -0 .354--0 .030 22.94 --0.3494-0.016 a) 23.25 6.75 0.577 0.664 --0.3574-0.019 23.54 7.04 0.602 0.759 --0.235 4-0.019 23.84 7.35 0.624 0.791 --0.1944-0.019 24.19 7.70 0.648 0.770 --0.208 4-0.019 24.44 7.97 0.663 0.787 --0 .186 4-0.018 25.69 9.26 0.716 0.784 --0.1754-0.017 27.06 10.68 0.736 0.805 --0.153 4-0.022

60.0 72.7 17.54 5.55 0.439 0.648 --0.4984-0.023 20.05 5.47 0.428 0.660 --0.489 4-0.020 22.02 5.52 0.433 0.691 --0.433 4-0.020 22.34 5.82 0.472 0.663 --0.441 4-0.021 22.65 6.12 0.510 0.631 - - 0.454 4-0.026 22.92 6.39 0.540 0.621 --0.4444-0.021 23.25 6.72 0.574 0.566 - - 0.494 t 0.019 23.54 7.02 0.600 0.610 --0.4154-0.020 23.74 7.23 0.616 0.654 --0.351 4-0.021 23.86 7.35 0.624 0.626 --0.3804-0.021 24.17 7.66 0.645 0.617 - -0 .379±0.021 24.44 7.95 0.662 0.604 --0.383 4-0.020 25.54 9.07 0.710 0.631 --0.3294-0.020 27.05 10.66 0.736 0.666 --0.282 4-0.021

75.0 89.2 17.47 5.68 0.455 0.539 --0 .682 4-0.021 18.98 5.50 0.432 0.590 --0.6204-0.035 19.96 5.62 0.447 0.560 --0.6544-0.021 20.87 5.50 0.432 0.581 - -0 .636±0 .037 22.00 5.71 0.459 0.584 --0.595 4-0.022 22.32 5.83 0.474 0.582 --0.5824-0.042 22.86 6.36 0.537 0.580 --0.5204-0.034 23.21 6.91 0.591 0.597 - - 0.452 4- 0.025 23.82 7.32 0.622 0.471 --0.6054-0.039 24.43 8.14 0.672 0.461 --0.5775:0.017 25.49 9.24 0.715 0.472 --0.5324-0.021 27.01 10.82 0.737 0.489 --0.4984-0.021

84.0 98.6 17.47 5.68 0.455 0.551 --0.6674-0.021 19.96 5.62 0.447 0.569 - - 0.646 4- 0.024 21.96 5.69 0.456 0.624 --0.539-4-0.040 22.00 5.71 0.459 0.603 --0.571 4-0.030 21.98 --0.5615:0.025 ~) 22.60 6.30 0.530 0.611 --0.4884-0.037 23.01 6.71 0.573 0.676 --0.371 4-0.026

56 W. G. WEITKAMP AND W. HAEBERLI

TABLE 2 (continued)

W8 08 Es Er / 'r r /'8 (degrees) (degrees) (MeV) (MeV) (MET)

90.0 104.7

105.0 119.2

115.0 128.3

23.21 6.91 0.591 0.734 --0.2944-0.028 23.51 7.21 0.615 0.613 --0.425+0.027 23.82 7.52 0.636 0.541 --0.503 :t:0.026 24.40 8.11 0.670 0.530 -- 0.495 -+-0.024 24.99 8.71 0.697 0.493 --0.525 !0.025 25.48 9.23 0.715 0.514 --0.4884-0.028 25.91 9.66 0.726 0.471 --0.5364-0.029 27.02 10.83 0.737 0.528 --0.4624-0.033

21.96 5.47 0.428 0.77 --0.33 4-0.09

17.52 5.72 0.460 2.30 0.8824-0.037 19.96 5.62 0.447 2.04 0.7934-0.032 21.96 5.69 0,456 2.02 0.7664-0.048 22.01 5.72 0.460 2.05 0.775 4-0.044 21.99 0.771 4-0.032 a) 22.26 5.98 0.493 1.97 0.689 4-0.064 22.86 6.58 0.560 2.41 0.765 4-0.061 23.16 6.88 0.588 2.55 0.7694-0.051 23.77 7.49 0.634 2.32 0.654-4-0.046 24.40 8.11 0.670 2.35 0.629 4-0.032 25.49 9.24 0.715 2.88 0.705 4-0.037 27.02 10.83 0.737 2.95 0.6984-0.047

17.52 5.72 0.460 2.38 0.9224-0.033 19.96 5.62 0.447 2.24 0.891 +0.031 21.96 5.69 0.456 2.12 0.8184-0.042 22.01 5.72 0.460 2.17 0.8344-0.030 21.99 0.8284-0.025 a) 22.26 5.98 0.493 2.28 0.8244-0.053 22.86 6.58 0.560 1.97 0.6174-0.053 23.21 6.91 0.591 1.95 0.578 4-0.043 23.77 7.49 0.634 2.77 0.774 4-0.042 24.40 8.11 0.670 2.95 0.770 4-0.028 25.49 9.24 0.715 3.17 0.761 4-0.035 27.02 10.83 0.737 3.48 0.783 4-0.042

~0 s and 0s are the laboratory and centre-of-mass scattering angles, respectively, Es is the energy of incident protons at the centre of the 4He target, Er the energy of incident deuterons at the centre of the 3He target and Pr the value of the polarization of the protons from the SHe(d, p)4He reaction. The values of the p-g polarization Ps are corrected for geometrical effects while the counting ratio r is not corrected.

a) Average of the two preceding values.

to a m i n i m u m the q u a n t i t y

n W{aR_ s] ] M 2 - ' °°" 2N+ L a (4)

w h e r e a++_Aa a n d a ° a re t he m e a s u r e d a n d c a l c u l a t e d d i f fe ren t ia l c r o s s sec t ions ,

P+_ AP a n d p c a re t he m e a s u r e d a n d ca l cu l a t ed p o l a r i z a t i o n s , trR_+AaR a n d a~ are

p-o. SCATTERING 57

the measured and calculated total reaction cross sections and N and Q are the num- ber of angles for which cross-section and polarization measurements were available. The weighting factors N/Q and IV were introduced to weight the three independent sets of measurements (cross section, polarization and total reaction cross section) equally in spite of the disparity in the number of data points in these three sets. The normalization of M 2 is such that M = 1 if the deviation between measured and cal- culated quantities is equal to the experimental uncertainty.

The energies for which the analysis was carried out and the sources of the data are listed in table 3. The lowest and highest energies are outside the range of this experi- ment, but were included to establish the continuity of the phase shifts to neighbouring energies. The proton energies of table 3 are the energies for which cross-section meas- urements had been made. The values of the polarization were obtained by inter- polation. At 31 MeV it was necessary to interpolate polarization measurements 5) between 29 and 40 MeV. This was difficult for the angular range from 104.8 ° to 115.8 ° because of the rapid variation with energy and angle. The data in this angular region were omitted from the analysis. For all other angles an uncertainty of 0.01 was added to the experimental error in the polarization. At energies less than 31 MeV, the un- certainties of the interpolated polarization values were taken to be the uncertainties of the nearest measurement. At 14.32 MeV, the polarization measurements of Rosen and Leland 21) at 14.5 MeV were used without correction.

TABLE 3

Data used in p-~ phase shift analyses

Proton Cross-section data Polarization data Estimated energy a R (MeV) Ref. N Ref. Q (nab)

14.32 12) 40 22) 11 17.45 is) 33 This work 6 20.62 18) 53 This work 6 26.08 18) 49 This work 6 27.68 18) 49 This work 6

and 5) 31.0 20) 49 ~) 22

4 7 ~ 4 66:~6

78=[=10

The number of datapoints and the values of the reaction cross sections are also given in table 3. The reaction cross sections are estimates based on measurements of the 3He(d, p)aHe cross section 22) using the principle of detailed balance and of the 4He(p, 2p)3H and 4He(p, pn)3He reaction cross section 23).

A reasonable fit to the reaction cross section was obtained in the present analysis even though the weight given to the reaction cross section was equal to only one differential cross section measurement (W = 1).

The results of the phase shift searches are given in table 4 and plotted as a function of proton laboratory energy in fig. 6.

In order to discover whether several sets of phase shifts exist which satisfactorily

TA

BL

E 4

Pha

se s

hift

s fo

r p-

u sc

atte

rin

g

Pro

ton

R

eal

par

t o

f p

has

e sh

ifts

(de

gree

s) a

nd

ine

last

ic p

aram

eter

s a)

en

ergy

(M

eV)

S~

P

~

P~

D

~.

D½

F

~

M

.~

9 14

.32

97.2

63

.1

106.

3 5.

6 7.

1 0.

2 99

.4

64.9

10

8.7

6.8

8.0

1.0

101.

5 59

.7

101.

4 1,

4 3.

6 0.

9 17

.45

96.8

57

.2

96.5

3.

6 4.

0 --

0.2

20

.62

89.8

52

.6

91.3

4.

8 5.

9 1.

3 26

.08

84.3

0.

96

53.7

0.

93

92.5

0.

79

7.8

0.91

16

.8

1.0

3.6

84.8

0.

96

52.3

0.

97

89.2

0.

84

6.6

0.86

14

.4

1.0

3.9

78.8

1.

0 46

.9

0.97

86

.2

0.92

4.

9 0.

93

14.7

0.

94

2.8

27.6

8 88

.4

0.87

58

.4

0.90

95

.6

0.69

I0

.0

0.85

19

.2

1.0

5.3

86.1

0.

92

54.3

0.

96

89.4

0.

77

7.6

0.81

16

.0

0.99

4.

9 83

.5

0.96

53

.5

0.93

89

.8

0.78

7.

5 0.

84

17.2

0.

98

4.1

31.0

81

.5

0.96

54

.2

0.96

86

.9

0.77

9,

3 0.

79

18,6

1.

0 5.

6 78

.3

0.99

49

.8

0.96

84

.3

0.86

7.

0 0.

81

17.5

0.

96

4.8

82.0

0.

98

51.4

0.

99

82.9

0.

84

8.2

0.75

16

.3

1.0

5.0

1.0

1.0

1.0

1.0

1.0

1.0

0.97

0.

96

0,99

--0

.4

0.0

1.4

--0

.3

1.5

3.7

3.9

3.3

4,8

4.1

3.3

5.0

4.6

4,7

1.0

1.0

0.99

1.

0 1.

0 1.

0 1.

0 0.

99

0.97

1.5

1.5

- 1.

6 1.

3 1.

0 >

1.

0 1.

1 ~

1.2

" 0.

7 ~

0.8

0.8

1.0

1.1

1.1

s) T

he q

uant

itie

s in

ital

ics

are

inel

asti

c p

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s v.

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ey a

re g

iven

whe

n th

ey d

iffe

r fro

m u

nity

.

p - 0 ¢ S C A T T E R I N G 59

reproduce the experimental data, the search was started from a number of different initial sets of phase shifts. At 17.45 and 20.62 MeV all the phase shift solutions found agreed with those given in the table within one degree. In fact, at 17.45 MeV, 14 sets o f widely differing initial phase shifts were used to try to find other solutions, but the result of the search was always the set of values shown. At the other energies, several sets of phase shifts, differing by two degrees or more in at least one phase shift, were found. In each case table 4 gives the three best fits. As may be seen f rom the listed values of M, in all cases the quality of the fit was about as good as would be expected from the published uncertainties in the data.

The uncertainty in each phase shift at 17.45 MeV was investigated by holding that phase shift fixed at a value differing from the best fit value by a small amount, and adjusting the other phase shifts until a minimum in M 2 was found. The change in the original phase shift necessary to increase the minimum M z value to twice its best fit

I10

I 0 0 -"

~o 90

8O

7O !

- 60

5O

I10 t :

,"I Joo[

22I 15

I .

20 2J5 30

E p (MeV)

t 2O

0 #

2O

; o

IG F3

0

rO

0

J

/

i i

i i

$

15 20 25 30 Ep(MO/)

ro I 0.8J

r. LOI ! o.81

1.01

r,+o.o o.6 I Lol

r~ o.8t

0.6t

I

L

J

25 Ep(MeV)

Fig. 6. Phase shifts for p-ct scattering. Circles, squares and triangles indicate the best, second best and third best fits at each energy, respectively. The curves are discussed in sect. 8.

value was then taken to be the uncertainty in the phase shift. The uncertainties ob- tained in this way, which range f rom three to five degrees, are plotted in fig. 6.

There are four regions in which significant deviations (three times published un- certainties or more) were found between measured and calculated values. First, at •4.32 MeV it was not possible to fit polarization measurements at 114.5 and 165.0 °. The calculated polarization was too large by slightly more than three times the experi- mental uncertainty at both angles. Second, at 17.45 MeV, the cross-section measure- ments at 61.05 ° and 66.82 ° could not be fitted. Third, at 20.62 MeV, three cross sec- tion measurements at 0 = 10.05 ° accounted for about ~ of the value of M. Finally, at 31.0 MeV the 6.3 ° cross-section point contributed about ¼ of the value of M. Except for the 17.45 MeV case, all these discrepancies are located in regions where either the cross section or the polarization is changing rapidly with angle and is con- sequently difficult to measure. I t is therefore assumed that these discrepancies are ex- perimental in nature.

30

6 0 W . G . WEITKAMP AND W. HAEBERLI

7. Resonance Parameters

The broad peak in the 3He(d, p)¢He cross section near 0.4 MeV deuteron energy (see fig. 7) has been interpreted as the effect of a J~ = ½+ state in SLi. The excitation ener- gy of the level corresponds roughly to the observed anomaly in the p-g polarization. The question to be discussed is whether the level can account quantitatively for the energy dependence of the p-g polarization near 23 MeV.

Resonance parameters for this state, i.e. the reduced width for deuteron emission 7d 2, the reduced width for proton emission 72 and the resonance energy E R, have been obtained from an analysis of the 3He(d, p) cross section by a number of authors refs. 2¢-27). One complication which prevents an accurate determination of level

parameters is the poor agreement in the absolute cross-section measurements of various authors. Three sets of data 25, zs, 29) are shown in fig. 7. Measurements sup-

1.0 I i ~ f i i i

--..Q~cn(n(n 0.40"60"8 _ 3 H e ( d ' p ) 4 H e ~

0 o 0.2 ~- /

0 o.z 0.4 0.6 0.8 E d (MeV)

Fig. 7. Total cross section for the SHe(d, p)4He reaction. The points are representative measurements from each of the three groups of data listed in table 5. Closed circles ref. u), open circles ref. =a), triangles ref. so). The curves show the results of a single-level fit to each of the three groups of data.

The deuteron energy is in the laboratory system.

porting the lowest curve have also been reported by Bonner et al. 24) and by Booth et al. 30), while the data of Freier and Holmgren 3x) agree quite well with the upper- most curve. We have analysed the three groups of data shown in fig. 7 with the single- level formula * using a channel radius of 5 fro. The resonance parameters correspond- ing to the curves of fig. 7 are given in table 5. The ratio of the reduced widths

2 2 " o O = 7d/)'p ]s quoted because the ratio can be determined to an accuracy of about 5 Yo, while either width alone is accurate to only about 30 ~o. A given set of data determines the resonance energy E R to about +__0.01 MeV.

There is reason to prefer the third set of resonance parameters of table 5. According to the principle of charge symmetry of nuclear forces one would expect the resonance

t See ref. s=), eq. (XII, 1.13). Specifically, the boundary value was chosen as suggested there (p. 323) so that at the resonance energy the level shift ,4 vanishes.

p-ct SCATTERING 61

parameters for the corresponding states in SLi and 5He to be similar, except of course for a small difference in resonance energy. Using the same formalism as above, Hoop and Barschall 6) found that the 3H(d, n)*He total cross section requires the para- meters given in the last line of table 5. In this case there is no significant disagreement in the magnitude of the peak cross section determined in different experiments 3s). Of the three sets of resonance parameters for *Li, only the third (g = 45) agrees with the resonance parameters of the state in 5He. However, for comparison, the p-~ polarization corresponding to the first set will also be shown.

TABLE 5

Resonance parameters for the {+ state in 5Li and 5He

Peak total 7a 2 Eli Data used Reaction cross section ?a 2 # = ~ (MeV) Ref.

(mb) (MeV) ?p (c.m.)

8He(d, p)4He

SH(d, n)4He

700 3.0 70 18.73 24, ~5, so) 800 1.4 53 18.71 2s) 900 1.5 45 18.70 ~9, sl)

2.0 40 17.67 6)

8. Calculation of the p-~ Polarization near the Resonance

In order to calculate the expected p-~ polarization and cross sections near the reso- nance, one must superimpose the resonance amplitude upon the smoothly varying non-resonant or "potential" phase shift. Since the compound state has total angular momentum J~ =~+, only the amplitude f 2 is affected. Most conveniently f 2 (eq. (1)) is expressed directly in terms of the resonance parameters a,):

where

f [ = sin@d°+ F_~_p sin/~d (#+2.), (5) Fp + F n

fl = tg- 1 ½(rp + Fa) (6) ER+ A - E '

and where ~b is the non-resonant phase shift. For the boundary condition chosen, the level shift A vanishes at the resonance energy ER, so that fl = ½g at E = ER. The quantity E is the proton bombarding energy in the centre-of-mass system.

The phase shifts used for the calculation of the energy dependence of the p-~ polarization are shown by the curves in fig. 6. With exception of the 62 phase shift, the energy dependence was approximated by a straight line drawn to best follow the points. The curve shown for the real part P2 of the 62 phase-shift results from eqs. (2) and (5), where ~b was again assumed to vary linearly with energy. When the ab- sorption parameter x 2 was calculated from eqs. (2) and (5), it was found that the resonance does not adequately describe the absorption at 26 MeV and above, where

62 W. G. WEITKAMP AND W. HAEBERLI

the observed values of ~2 are smaller than the calculated values. A correction, which was a linear function of energy, was subtracted from the calculated values of ~2. The correction was zero at the 4He(p, d)3He threshold and increased to 0.16 at 29 MeV.

Eq. (5) has a simple geometric interpretation. If the resonance is narrow, (i.e., tk, Fp and F d approximately constant) the amplitude f 2 describes a circle of diameter Fp/(Fp'-I-Fd) as the bombarding energy sweeps over the resonance, as shown in the left hand part of fig. 8. In the present situation the resonance is very near the (p, d) threshold, and the deuteron width I'd increases rapidly with energy while the proton width Fp stays constant. As a consequence the diameter of the resonance circle decreases with increasing energy. The curve which f 2 actually follows for the third set of parameters of table 5 is shown in the right-hand part of fig. 8.

l lrn f

~ ~0.5

/ \ / / \ , \ /

f' / rp /

I 23.4 23.3

/, 23.5

-o15 o o o.'5 Re f Re f~

Fig. 8. Behaviour of the amplitude f in the vicinity of a resonance. If the partial widths for decay and the non-resonant phase shift remain constant, the trajectory which f follows is the small circle (left-hand. part of the figure). The actual path which f2- follows is shown on the right, where the num-

bers refer to the proton bombarding energy in MeV. The large circle is the unitary circle.

9. Comparison between Calculated and Measured Polarization and Cross Section

The calculated polarization is shown in fig. 5. The dashed and solid curves are for the first and third set of resonance parameters of table 5, respectively. To provide a better comparison with the experimental data, the calculated polarization, weighted by the calculated differential cross section, was averaged over an energy interval of 0.4 MeV width, which corresponds to a root-mean-square energy spread of 0.13 MeV as used in the experiment.

In general the agreement between calculated and measured polarization is reason- ably good. In particular the resonance has the right shape for all scattering angles. A further test of the phase shifts proposed here can be obtained from the recent work of Allison and Smythe 19) who measured the differential cross sections at two angles over the resonance. The results are shown in fig. 9. The calculated curve for the larger

p-~x SCATTERING 6 3

deuteron width (dashed curve, # = 73) agrees better with the experimental cross section than the curve for the smaller width (solid curve, g = 45) unless one assumes that the energy spread of the beam was such that the resonance was not fully resolved. The energy spread in the experiment is not known accurately, but is definitely less than 0.2 MeV (ref. 35)). Outside the resonance region, the calculated cross section is about 10 % too large for both scattering angles. This difficulty is due to the necessary compromise in drawing smooth phase shift curves through the phase shift solutions obtained at the various proton energies. It clearly would be much better to correlate the analysis at several energies, assuring a smooth energy dependence of the phase shifts. Work along these lines is presently in progress 36, 37).

220

2 0 0

180

C/)

E

Z 0 160

I-- I0 ¢..) LI.I

t I I I "

8 -- . ~ 114.6 o _

t.O (/) o

4 I ""~" I I 21 22 23 24 25 26

Ep(MeV) Fig. 9. Differential cross section for p-~ scattering. Da t a po in t s are f r om ref. 19). The curves are

f r om the calculat ion described in sect. 8.

The total inelastic cross section calculated by eq. (3) from the curves for z (fig. 6) reaches a value 55 mb at 31 MeV which is slightly smaller than the value estimated from available experimental information (table 3) because absorption has been neg- lected in phase shifts other than 6~ and 6~. Near the resonance the calculated values of the reaction cross section are slightly larger than the calculated cross sections for the inverse reaction shown in fig. 7 because the curves for z include inelastic processes other than the absorption associated with the resonance. The difference is not large, however. The 3He(d, p)4He cross section calculated from the values of z by detailed balance for a deuteron energy of 0.5 MeV is 860 mb compared to 820 mb obtained from the single level formula. A correction for this difference in the determination of the level parameters is not justified because, except for the resonant part of ~-, the

64 W. G. WEITKAMP AND W. HAEBERLI

va lues o f z a re qu i t e unce r t a in . I n fact , because o f the b e h a v i o u r o f t he ba r r i e r pene -

t rabi l i t ies j u s t a b o v e t h r e s h o l d i t is l ike ly tha t c lose to the r e s o n a n c e z+ is n e a r e r to

un i t y t h a n was a s s u m e d here .

W e w o u l d l ike to t h a n k D r . L. C. M c I n t y r e , L. M o r r o w a n d L. E. P o r t e r fo r

ass i s tance in the expe r imen t . W e also wish to express o u r a p p r e c i a t i o n to D r . P. W.

A l l i son a n d P r o f e s s o r R . S m y t h e fo r p e r m i t t i n g us to see the i r p-~ d i f ferent ia l c ross

sec t ion d a t a in a d v a n c e o f pub l i ca t i on .

References 1) T. Lauritsen and F. Ajzenberg-Selove, Nuclear Physics, to be published 2) J. L. Gammel ~ind R. M. Thaler, Phys. Rev. 109 (1958) 2041 3) H. E. Conzett, G. Igo and A. Nir, University of California Radiation Laboratory Report UCRL

9566 (1961) 4) C. F. Hwang, D. H. Nordby, S. Suwa and J. H. Williams, Phys. Rev. Lett. 9 (1962) 104 5) M. K. Craddock, R. C. Hanna, L. P. Robertson and B. W. Davies, Phys. Lett. 5 (1963) 335;

Nat. Inst. Res. Nucl. Sci. (UK) P.L.A. Progress Report (1963) 6) B. Hoop, Jr. and H. H. Barschall, Nuclear Physics 83 (1966) 65 7) R. I. Brown, W. Haeberli and J. X. Saladin, Nuclear Physics 47 (1963) 212 8) I. Alexeff and W. Haeberli, Nuclear Physics 15 (1960) 609 9) W. G. Weitkamp, Ph.D. Thesis, University of Wisconsin (1965); available from University Mi-

crofilms, Ann Arbor, Michigan 10) R. I. Brown and W. Haeberli, Phys. Rev. 130 (1963) 1163 11) M. D. Goldberg and J. M. LeBlanc, Phys. Rev. 122 (1961) 164 12) J. Sanada, J. Phys. Soc. Japan 14 (1959) 1463 13) K. W. Brockman, Jr., Phys. Rev. 108 (1957) 1000 14) P. D. Miller and G. C. Phillips, Phys. Rev. 112 (1958) 2043 15) B. W. Davies et al., Nat. Inst. Res. Nucl. Sci. (UK) P.L.A. Progress Report (1964) p. 69:

N. Horikawa and H. Kanada, J. Phys. Soc. Japan 20 (1965) 1758; E. T. Boschitz, private communication

16) S. J. Moss and W. Haeberli, Nuclear Physics 72 (1965) 417 17) J. H. Foote, O. Chamberlain, E. H. Rogers and H. M. Steiner, Phys. Rev. 122 (1961) 959 18) K. W. Brockman, Jr., Phys. Rev. 102 (1956) 391 19) P. W. Allison and R. Smythe, Bull. Am. Phys. Soc. 9 (1964) 544;

P. W. Allison, Ph.D. Thesis, University of Colorado (1965) 20) S. M. Bunch, H. H. Forster and C. C. Kim, Nuclear Physics 53 (1964) 241 21) L. Rosen and W. T. Leland, Phys. Rev. Lett. 8 (1962) 379 and private communication 22) L. Stewart, J. E. Brolley, Jr. and L. Rosen, Phys. Rev. 119 (1960) 1649 23) A. F. Wickersham, Jr., Phys. Rev. 107 (1957) 1050 24) T. W. Bonner, J. P. Conner and A. B. Lillie, Phys. Rev. 88 (1952) 473 25) W. E. Kunz, Phys. Rev. 97 (1955) 456 26) W. S. Porter, B. Roth and J. L. Johnson, Phys. Rev. 111 (1958) 1578 27) I. G. Balashko and I. I. Barit, in Nuclear forces and the few nucleon problem, Vol. 2, ed. by

T. C. Griffith and E. A. Power (Pergamon Press, London, 1960) p. 615 28) A. P. Kliucharev, B. N. Esel'son and A. K. Val'ter, Doklady (Soviet Physics) 1 (1956) 475 29) J. L. Yarnell, R. H. Lovberg and W. R. Stratton, Phys. Rev. 90 (1953) 292 30) D. L. Booth, R. S. Hill, F. V. Price and D. Roar, Proc. Phys. Soc. A70 (1957) 863 31) G. Freier and H. Holmgren, Phys. Rev. 93 (1954) 825 32) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 33) V. A. Davidenko, I. S. Pogrebov and A. I. Saukov, Atomn. Energ. 2 (1957) 386 34) J. Vorona, J. W. Olness, W. Haeberli and H. W. Lewis, Phys. Rev. 116 (1959) 1563 35) P. W. Allison, private communication 36) W. G. Weitkamp and W. Haeberli, Karlsruhe Conference on Polarization Phenomena of

Nucleons (1965), to be published 37) C. C. Giamati, H. C. Volkin and R. M. Thaler, Bull. Am. Phys. Soc. 10 (1965) 601