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Analyzing power and phase-shift analysis of 3He(p, p)3He between 19 and 30 MeV
~
.L~
Nuclear Physics A311 (1978) 1 - 1 0 ; ~ ) North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher
ANALYZING POWER AND PHASE-SHIFT ANALYSIS
OF 3He(p, p)aHe BETWEEN 19 AND 30 MeV D. M ~ L L E R , R. B E C K M A N N and U. HOLM
I. lnstitut ffir Experirnentalphysik, Universitdt Hambury, D-2000 Hamburg 50 Received 2 August 1978 Abstract: The 3He spin analyzing power of elastically scattered protons has been measured at a lab energy of 25.0 MeV at nine angles from 46 ° to 156 °. The energy dependence is reported at a fixed lab angle of 1'35° for five energies between 19.6 and 26.5 MeV. A phase-shift analysis was performed at 25.0 and 30.5 MeV including all available experimental data. N U C L E A R REACTIONS 3He(p, p), E = 25 MeV; measured A3He(O). Deduced phase shifts. Polarized target.
I. Introduction Proton-3He elastic scattering has been the subject of intense experimental and theoretical work for proton bombarding energies less than 20 MeV [ref. t)], whereas for energies of 20 to 30 MeV, reachable with the Hamburg isochronous cyclotron, experimental data have onlyrecently become available. Vanetsian et aL 2) measured the differential cross section at 19.4 MeV, Hutson et al. 3) at 19.5 MeV, Kim et al. 5) at 31 MeV and more recently Morales et al. 4) at 20.0, 22.5, 25.0, 27.5 and 30.0 MeV. Furthermore, cross-section and polarization data have been published by Darves-Blanc et al. 6) at 19.5 and 30.5 MeV, by Ware et al. 7) at 27.5 MeV, by Harbison et al. at 30.6 MeV [ref. 8)] and 30 MeV [ref. 9)], and by Baker et al. 10) at 19.4 MeV, who additionally measured spin correlation parameters. Phase-shift analyses in the above energy regions were l~erformed in refs. 4.6, 8, 1o). The phase shifts of ref. ~0)at 19.5 MeV may be regarded as very reliable due to the fact that they included cross-section data, proton and aHe polarization data and spin correlation data. The phase shifts of ref. 6) at 19.5 MeV are also very similar. Between 20 and 30 MeV, in 2.5 MeV steps, phase shifts were computed by Morales et al. 4), who fitted their own cross-section measurements and other proton polarization data resulting from an interpolation of experimental data at 19.5 and 30.5 MeV. At 30.5 MeV there are further phase shifts 5,a) deduced from cross-section and proton polarization data only. I November 1978
2
D. MfALLERet al.
The object of the present work was to provide further 3He polarization data at 25.0 MeV proton energy so that the phase-shift analysis could be performed more accurately. The importance of phase-shift solutions in this energy region is given by predictions of the structure of the 4Li nucleus ~~' ~2) and resonating group predictions about phase-shift behaviour versus energy ~3). The present paper gives the results of a phase-shift analysis at 25.0 and 30.5 MeV using all available cross-section, proton and 3He polarization and correlation data in this region, inclu.ding our measurements.
2. Experimental details The experiments were performed with the proton beam of the Hamburg isochronous cyclotron HAIZY and the ~ptically pumped polarized 3He target. The cyclotron beam, with a FWHM of about A E / E = 3 °/o o, was not momentum analyzed because of the large foil straggling. The central lab energy of the incident beam was measured with the analyzing system to _+0.5 °/o o. Two collimating diaphragms of 8 mm diameter in the beam-optics system, 150 mm and 930 mm before the target, defined, together with the detector collimating systems, the reaction volume in the extended gas target and prevented fictitious asymmetries by possible beam-focusing variations. The cyclotron beam made maximum angles of + 0.3 ° with respect to the axis of the target cell. The air-cooled electrically suppressed Faraday cup had a length of 330 mm and a diameter of 130 ram; its distance from the target was 4 m, so that the entire beam could be collected. The 3He target cells consisted of thin-walled (0.3 mm) cylindrical glass bulbs of 40 mm diameter and 32 mm height, filled with 2 Torr ~He. The cleaning and filling procedure is described elsewhere ~4). The glass foil windows for the incoming and outgoing beam and the elastically scattered protons were produced in the following way: glass pipes of 20 mm diameter and 1.2 mm thickness were sealed by fusion and the molten glass was drawn into the pipe so that concave foils were formed. The thicknesses of the foils were determined by the energy loss of passing a-particles from a 2l/pb source. Four foils of thicknesses less than 30 pm and usable diameters of at least 10 mm were produced, with pipes of about 20 mm length to the cylindrical glass body (fig. 1). Each experimental point in the angular distribution required its own target cell. The beam entrance and exit windows were coated with silver (about 0.1 pm) and thermically connected with the holder. Constructed in this way, the target could withstand 3 pA of 19 MeV protons for at least several days without any decrease in polarization. The cells could be stored for more than two years with negligible change in their qualities. The 3He cell was located in the center of 200 mm diameter aluminum vacuum chambers, constructed in the "Korex" 15) way, each with three pairs of symmetrically drilled holes for the insertion of the pipes with the counter telescopes. Each telescope consisted of a silicon surface-barrier and a lithium-drifted silicon detector.
3He(p, p)3He
gloss
beom
"
)
I
~
~,~
l
~cm
Fig. 1. Simplified scale drawing of the glass target cell for a scattering angle 0~ab = 45°.
The detectors could be cooled down to - 3 5 °C separately by Peltier cooling. Thin foils (5 /~m AI) in front of the detectors prevented the weak discharge light from reaching the detectors. The collimating system consisted of an entrance slit located 50 mm and a rear slit located 440 mm from the target center. Both slits were 5 mm wide and 10 mm high, so that the horizontal acceptance was + 0.7 ° in the lab system. The accuracy of the detector angle setting was + 0.3 °. Small magnets before the rear slits prevented secondary electrons from reaching the detectors. Chamber vacuum was provided by turbomolecular pumps 4 m before and 2 m behind the chamber, as well as by liquid nitrogen traps in the detector arms, which usually maintained a pressure of 4 × 10 -6 Torr. A magnetic field of 10 G, which defined the alignment direction, was provided by 90 cm diameter Helmholtz coils. The infrared circularly polarized optical pumping light was passed to the 3He cell through a focusing glass lens which was cemented to the bottom of the vacuum chamber with Araldit. The light passing the 3He cell was monitored by a photodetector through a light pipe (2 mm diameter) which was cemented into a vacuum feed-through. The effective pumping light intensity had been increased by a reflector at the back of the target cell by a factor of nearly 2 [ref. ~6)]. The reflector consisted of either a silver film evaporated onto the back of the cell or an aluminum foil. With the reflector, typical target polarizations of 15 ~-18 ~ could be achieved. The 3He target nuclear polarization was determined by optical pumping methods ~7). The measurement uses an equation for the absorption of circularly polarized light by 3He metastable atoms ~8). The parameters of this equation have been determined 19) and furthermore the correctness of the equation could be confirmed ~9), so that we have confidence in this optical method. The polarization can be computed from the ratio I(P)/I( - P), where I(P) is the fraction of pumping light absorbed at the 3He cell at the polarization P. In practice, we produced different relative polarizations + P and ~ P by rotating the linear polarizer by an angle of ½~ and then back to its original position. The whole procedure lasted about 1 sec. The I(P) value was obtained by switching off the weak discharge. This method has
4
D. M O L L E R et al.
the great advantage of not destroying the nuclear polarization, so that the measurement could be repeated every 15 min during the experiment. More measurements were not necessary because, in general, the polarization values were constant to within their accuracy of _+0.3 °/o.The sign of the polarization was reversed by rotating the linear polarizer by an angle of ½g about once every half-hour. The protons were detected in each counter telescope by requiring coincidences between pulses from the AE and E detectors. The infbrmation was handled by a Nuclear Data N D 4420 multichannel analyzer. The pulse-height spectra were corrected for a small constant background (always less than 2 I~o).The spin analyzing power was determined in the usual way 7) from the corrected counting rates, in order to eliminate geometrical differences and different numbers of counts per integrated beam current.
3. The 3He spin analyzing power of protons The measured values of the 3He spin analyzing power of protons at a lab energy of 25.0 MeV are listed in table 1. The errors quoted are standard deviations due to counting statistics: all other possible errors are comparatively small. Our experimental results are nearly identical to those at 19.4 MeV of ref. 1o) and at 26.8 MeV of ref. 7) (fig. 2). There is no strong variation with energy as predicted in ref. 7). To confirm this result we have measured the analyzing power at a fixed lab angle of 135 ° at different energies from 19.6 to 26.5 MeV (table 2). Here, too, no essential energy dependence is to be seen.
4. Phase-shift analysis Our measurement of the 3He analyzing power was made in order to test the phases of Morales et al. 4) at 25 MeV. Fig. 2 shows the analyzing power computed with these phases, together with our experimental results. One can see that the principal trend of the curve is confirmed by our results, but that the predicted analyzing power is too high by a factor of about 1.5. Therefore, a new phase-shift analysis also including our data was made, and new phases were computed for 30.5 MeV using extrapolated 3He polarization data besides the available experimental cross-section and proton polarization data. In our analysis, we took into account waves with angular momentum up to l = 3, both singlet and triplet. The spin coupling parameters 2o) ~ and ~2 describe singlettriplet transitions of the P~ and D 2 states (spectroscopic rotation). The parameter e3, which mixes the F 3 states, was set to zero. This was done because the F-phases turned out to be very small, so that e 3 had no noticeable affect on the analysis. Couplings of different angular momentum are described by the parameters t02 for the transitions 3S~ --* 3D~ and t~3 for 3 P 2 -+ 3 F 2. All phases are taken to be real in order to limit the number of fitting parameters to a reasonable amount. With these limitations we have 14 eigenphases and 4 coupling parameters, i.e. 18 fitting parameters.
3He(p, p)3He
5
• 'th~S work O.5J--- calc. our phases
I
.~x
J-.- calc.phoses re,.z,
05~- Ep: 25MeV !. . . . .
I ....
0
50
~k~.
" I
100
....
!,, lC-~ Oc.m!deg)
Fig. 2. The 3He analyzing power of 3He(p, p)3He at 25.0 MeV proton energy versus the scattering angle. The closed circles are the new data of this work, the dashed curve is calculated with the phase shifts of ref. 4), and the smooth curve is calculated from the derived phase shifts of table 3.
TABLE 1 The 3He spin analyzing power of 25.0 MeV elastically scattered protons as a function of c.m. scattering angle 0. 0.... (deg)
A3He
46.2 58.9 71.1 94.0 114.7 132.8 141.0 148.8 156.1
- 0.085 _+0.003 -0.140+0.020 -0.165+0.035 -0.216+0.030 0.000+0.040 0.303 + 0.040 0.269+0.035 0.216+0.035 0.130+0.030
TABLE 2 The 3He spin analyzing power of elastically scattered protons at a fixed scattered angle function of the proton energy Ep, lab (MeV) 19.6 21.1 23.4 25.0 26.5
A3He 0.069 ___0.035 0.148+0.040 0.150+0.040 0.216+0.035 0.217+_0.042
01a b =
135° as a
6
D. MOLLER et al.
In the following we give a brief description of the program used for fitting the data. The program which calculates cross sections and analyzing powers as functions of the phase shifts uses the formulae of Lindner 24). These formulae were completed by adding the Coulomb interaction. To test the program we calculated the Z2 for the cross section and proton analyzing power with the phases of Morales et al. 4) at 30 MeV. The good Z2 showed that the program works correctly. The fit program 2 l) searches for the minimum ofxZ(a), where a = ai • . . a, are points in parameter space with the eigenphases and mixing parameters as coordinates. Each step carried out in parameter space is calculated from the gradient o f z 2 and the matrix Mi, k = 62(Z2)/ 6ai3a~ of second derivatives. Near the minimum, x2(a) should be a nearly quadratic function of the fitting parameters a i. For a quadratic function, the errors rri of the parameters a~ are given by rr~ = G,, with G~k = ( 1 M i 0 - ~. These errors are computed by the program after finding the minimum. In a further error analysis, other errors s~ are computed in the following way: the parameter a i is replaced by the fixed value aj + rrj and the minimum z2(a, o-j) is searched by varying the other parameters. Then sj is defined as s~ = rrj(z2(a, r r ) - zZ(a))- ~. For further information about this error analysis and the search program we must refer to ref. 2~). In a first search at 25 MeV we fitted cross-section data and our measured 3He analyzing power. Different sets of phases resulted, which all described the experimental data with good ;(2. To get a unique fit, further data had to be included in our search program. We therefore interpolated, in the same way as Morales et al. 4), between the proton polarization data of Tivol 22) at 19.5 MeV and of Harbison 9) at 30.5 MeV. From fig. 3 it can be seen that the angular distributions at these two energies are very similar. Only the rapid change of polarization from negative to positive values is shifted by about 10°. Therefore, one can suppose that the proton polarization changes only slightly with energy so that interpolation between these two data sets seems to be reasonable. Using these additional data, a new search gave only one set of phases with good ;(2. The results are listed in table 3. The 3He analyzing power and proton polarization calculated with these phases are shown in figs. 2 and 3. At 30.5 MeV, the phases of Darves-Blanc 6) and Morales 4) both describe the experimental cross section and proton polarization, but the predicted 3He analyzing power with Ami n = --0.5 and Areax = 0.4 seems to be too high. Fig. 4 shows the analyzing power data from Baker et al. ~0) at 19.4 MeV, our data at 25.0 and 19.6 MeV, and data from Ware et al. 7) at 26.8 MeV. From this figure it can be seen that there is no noticeable energy dependence between 19.4 and 26.8 MeV. Therefore, we do not think it probable that the analyzing power increases in the strong way predicted by the phase shifts at 30.5 MeV. Our phase-shift analysis at 30.5 MeV included not only cross-section and proton polarization data, but also our 3He analyzing power data at 25 MeV. Because of the energy independence mentioned above, this seemed to us to be reasonable. Nevertheless, we have taken the fact that these are only extrapolated data into account by
3He(p, p)aHe TABLE 3 Phase shifts for 3He(p, p)3He at Ev = 25.0 and 30.5 MeV
iS o 3So IP 1 3P 0 3P 1 3P 2 ID 2 3D 1 3D::
25 MeV
30.5 MeV
- 1 4 2 ___4 -112 +2 29 -+3 30 -+8 41 -+2 65 -+2 4.3-+1.1 4.0-+0.5 - 1.1 -+0.8
-127 +4 - 1 2 3 -+2 32 -+2 25 -+2 42 + 2 58 -+1 0.6_+0.8 1.8+_0.6 12.3± 1.2
I
3D 3 IF 3 3F z 3F 3 3F 4 e1 e,2 t02 t13
25 MeV
30.5 MeV
5.2+0.5 2.2-+1.7 3.3+0.5 3.8-+1.5 3.9-+0.5 - 4 1 -+8 -23 +8 - 4 -+4 -0.5-+4
6.9+0.6 7.6±0.6 3.2-+0.5 6.3±0.9 3.3+__0.4 - 3 1 -+7 19 -+4 0.2-+3 2.2-+3.8
• Ep: 19.5 MeV,ref. 22 , / ~
....
I ....
0
I ....
50
1,,
I(3(3
150 Oc.m.ldeg}
Fig. 3. The proton analyzing power of 3He(p, p)aHe versus the scattering angle. The crosses are data from ref. 22) at Ep = 19.5 MeV, the open circles are data from ref. 9) at E v = 30.0 MeV and the smooth curve is calculated with the phase shifts of table 3 for Ep = 25.0 MeV.
0.4 0.2
o Ep = 19./~ MeV,ret.lO & Ep= 19.6 MeV, this work Ep= 25,0 MeV, this work ~,Ep= 26.8 MeV, ret7
x • x
x*
o
x e &
x
4
.c
x
Iox •
-0
x,*~ o xo,~
io
-O.t
0
20
~0
60
0
10
1
1~0
160 180 Oc.m.(deg )
Fig. 4. The 3He analyzing power o f 3He(p, p)3He versus the scattering angle. The open circles are data at E v = 19.4 MeV from ref. ~o), the triangle is a new data point at Ep = 19.6 MeV from this work, the closed circles are new data from this work at Ep = 25.0 MeV and the crosses are data at Ep = 26.8 M e V from ref. 7).
8
D. M0'LLER et al.
I • Ap, ref.9 0.51_- Ap, cole - - A3He, cole.
P~'f"k
l
< o i-- , - - + . ~ ~
/j
_o.+ I
I
0
50
I
I00
1
I
]
I
1
~ ec.m.tdeg)
Fig. 5. The 3He or proton analyzing powers of 3He(p, p)3He at Ep = 30 MeV versus the scattering angle. The closed circles are experimental proton analyzing powers of ref. 9), the smooth curve is the calculated proton analyzing power with the phase shifts of table 3 and the dotted curve is the calculated 3He analyzing power with the phase shifts of table 3.
Ep {MeV) -~-
°
40
- 80
I0
2Or
'
iSo
"'.
i
° * ° . ,o
30 -t.0 l '.. - 80
~a
.,2o I
-120
~ -too I
-160[
+o. :1 f
' ' " "...
' 351
~
80 60
g
~0
.. .....
~
:
20 :i
31~
~o
6oi
,~ ,
+
2o
.'"'"'"
"
:
-
60[
°
,b
~
sb
10
20
30
20t""-"".'.. EII-J ~.,
-40
g
+°f I f.., 60
40
40
.
20
........', ,o
~ ~Po
a
~ ~
2b
Jo
40
20
.'"
~PI o
"
°
,o
o x
~
Jo
~' a
Fig. 6. The S and P phase shifts of 3He(p, p)3He as functions of the bombarding energy. The closed circles are the results of Tombrello 23), the horizontal crosses are from Baker et al. ~o), the diagonal crosses from Darves-Blanc et al. 6), the triangles from Morales et al. a), and the open circles are results of this work.
3He(p, p)3He
9
multiplying the experimental errors by a factor of 1.5. The resulting phases are listed in table 3. The 3He analyzing power calculated with these phases (dotted line) and the calculated proton polarization together with the proton polarization data of Harbison 9) are shown in fig. 5. The errors given in table 3 are calculated as described above. Since the calculation of the errors s i is very time consuming for the computer, they were computed only for the S- and P-phases and for the mixing parameter e~. For the other phases and mixing parameters, the errors in table 3 are the above defined ~r~. 5. Discussion of the results
In the following we wish to compare our results with those of other authors and discuss the energy dependence of the phases. When comparing phases of different authors, it must be kept in mind that the definition of the mixing parameters is not unique. For instance, one gets the same scattering matrix by replacing ~l with ex ---90° and interchanging the ~PI phase with the 3P l phase. Here we follow the convention that el goes to zero for small Ep. In fig. 6 the singlet and triplet eigenphases and the mixing parameter ~1 are shown for proton incident energies Ep up to 30 MeV. The D- and F-phases are all small, so no attempt was made to prove their continuous energy dependence. The two S-phases continuously increase with energy, showing the hard sphere behaviour already found at lower energies. At the same energy, the phases of different authors show good agreement. The 3P2 phase also shows a continuous behaviour with energy. For the 3p~ and ~Px phases it is remarkable that the phases of Morales 4) at 20 and 25 MeV widely deviate from continuity. In their own publication, their phases fit the energy dependence with respect to lower energies very well. The reason for this discrepancy lies in the fact that they exchanged the singlet and triplet P-phases of Tombrello et al. 23). But this is not in agreement with their convention about the mixing parameter e~, which is the same as in this paper. For the 3P0 phase, the extrapolation of our phases to lower energies is not so simple. The phase seems to have a maximum at about 15 MeV. The energy dependence of the mixing parameter e x is not very satisfactory. But one must keep in mind that the errors of this parameter are very large compared with those of the eigenphases. The phase-shift analysis was very insensitive with respect to a variation of this parameter. We would like to thank Prof. A. Lindner for helpful discussions and Dr. D. Eppel for placing his subroutines at our disposal. This work has been financially supported by the Bundesministerium fiir Forschung und Technologie.
10
D. MCILLER et al.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
23) 24)
S. Fiarman and W. E. Meyerhof, Nucl. Phys. A206 (1973) 1 R. A. Vanetsian and D. E. Fedchenko, Sov. J. Atomic Energy 2 (1957) 141 R. L. Hutson, N. Jarmie, J. L. Detch, Jr. and J. H. Jett, Phys. Rev. C4 (1971) 17 J. R. Morales, T. A. Cahill, D. J. Shadoan and H. Willmes, Phys. Rev. CII (1975) 1905 C. C. Kim, S. D. Bunch, D. W. Devins and H. H. Forster, Nucl. Phys. 58 (1964) 32 R. Darves-Blanc, N. Van Sen, J. Arvieux, J. C. Gondrand, A. Fiore and G. Perrin, Nucl. Phys. AI91 (1972) 353 R. H. Ware, W. R. Smythe and P. D. Ingalls, Nucl. Phys. A242 (1975) 309 S. A. Harbison, R. J. Griffiths, N. M. Stewart, A. R. Johnston and G. T. A. Squier, Nucl. Phys. A150 (1970) 570 S. A. Harbison, F. G. Kingston, R. J. Griffiths, A. R. Johnston and J. H. P. C. Megaw, Nucl. Phys. A l l 2 (1968) 137 S. D. Baker, T. A. Cahill, P. Catillon, J. Durand and D. Garreta, Nucl. Phys. AI60 (1971) 428 P. Kramer and M. Moshinsky, Phys. Lett. 23 (1966) 574 P. P. Szydlik, Phys. Rev. CI (1970) 146 I. Reichstein, D. R. Thompson and Y. C. Tang, Phys. Rev. C3 (1971) 2139 R. Beckmann, Ph.D. Thesis, Univ. of Hamburg, 1969 H. Briickmann, private communication R. Beckmann, U. Holm and D. Mtiller, Nucl. Instr. 141 (1977) 525 F. D. Colegrove, L. D. Schearer and G. K. Waiters, Phys. Rev. 132 (1963) 2561 J. M. Daniels and R. S. Timsit, Can. J. Phys. 49 (1971) 525; R. S. Timsit and J. M. Daniels, Can. J. Phys. 49 (1971) 545 R. Beckmann, U. Holm and B. Lindner, Z. Phys. A275 (1975) 319 J. M. Blatt and L. C. Biedenharn, Rev. Mod. Phys. 24 (1952) 258 V. Blobel, DESY report 71/18 (1971) W. F. Tivol, Thesis, Univ. of California, 1968: R. Darves-Blanc et al., Proc. Int. Conf. on Few particle problems in the nuclear interaction, ed. I. Slaus, S. A. Moszkowski, R. P. Haddock and W. T. H. van Oers, (North-Holland, Amsterdam, 1972) p. 617 T. A. Tombrello, Phys. Rev. 138 (1965) B40 A. Lindner, Nucl. Phys. A230 (1974) 477
.L~
Nuclear Physics A311 (1978) 1 - 1 0 ; ~ ) North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher
ANALYZING POWER AND PHASE-SHIFT ANALYSIS
OF 3He(p, p)aHe BETWEEN 19 AND 30 MeV D. M ~ L L E R , R. B E C K M A N N and U. HOLM
I. lnstitut ffir Experirnentalphysik, Universitdt Hambury, D-2000 Hamburg 50 Received 2 August 1978 Abstract: The 3He spin analyzing power of elastically scattered protons has been measured at a lab energy of 25.0 MeV at nine angles from 46 ° to 156 °. The energy dependence is reported at a fixed lab angle of 1'35° for five energies between 19.6 and 26.5 MeV. A phase-shift analysis was performed at 25.0 and 30.5 MeV including all available experimental data. N U C L E A R REACTIONS 3He(p, p), E = 25 MeV; measured A3He(O). Deduced phase shifts. Polarized target.
I. Introduction Proton-3He elastic scattering has been the subject of intense experimental and theoretical work for proton bombarding energies less than 20 MeV [ref. t)], whereas for energies of 20 to 30 MeV, reachable with the Hamburg isochronous cyclotron, experimental data have onlyrecently become available. Vanetsian et aL 2) measured the differential cross section at 19.4 MeV, Hutson et al. 3) at 19.5 MeV, Kim et al. 5) at 31 MeV and more recently Morales et al. 4) at 20.0, 22.5, 25.0, 27.5 and 30.0 MeV. Furthermore, cross-section and polarization data have been published by Darves-Blanc et al. 6) at 19.5 and 30.5 MeV, by Ware et al. 7) at 27.5 MeV, by Harbison et al. at 30.6 MeV [ref. 8)] and 30 MeV [ref. 9)], and by Baker et al. 10) at 19.4 MeV, who additionally measured spin correlation parameters. Phase-shift analyses in the above energy regions were l~erformed in refs. 4.6, 8, 1o). The phase shifts of ref. ~0)at 19.5 MeV may be regarded as very reliable due to the fact that they included cross-section data, proton and aHe polarization data and spin correlation data. The phase shifts of ref. 6) at 19.5 MeV are also very similar. Between 20 and 30 MeV, in 2.5 MeV steps, phase shifts were computed by Morales et al. 4), who fitted their own cross-section measurements and other proton polarization data resulting from an interpolation of experimental data at 19.5 and 30.5 MeV. At 30.5 MeV there are further phase shifts 5,a) deduced from cross-section and proton polarization data only. I November 1978
2
D. MfALLERet al.
The object of the present work was to provide further 3He polarization data at 25.0 MeV proton energy so that the phase-shift analysis could be performed more accurately. The importance of phase-shift solutions in this energy region is given by predictions of the structure of the 4Li nucleus ~~' ~2) and resonating group predictions about phase-shift behaviour versus energy ~3). The present paper gives the results of a phase-shift analysis at 25.0 and 30.5 MeV using all available cross-section, proton and 3He polarization and correlation data in this region, inclu.ding our measurements.
2. Experimental details The experiments were performed with the proton beam of the Hamburg isochronous cyclotron HAIZY and the ~ptically pumped polarized 3He target. The cyclotron beam, with a FWHM of about A E / E = 3 °/o o, was not momentum analyzed because of the large foil straggling. The central lab energy of the incident beam was measured with the analyzing system to _+0.5 °/o o. Two collimating diaphragms of 8 mm diameter in the beam-optics system, 150 mm and 930 mm before the target, defined, together with the detector collimating systems, the reaction volume in the extended gas target and prevented fictitious asymmetries by possible beam-focusing variations. The cyclotron beam made maximum angles of + 0.3 ° with respect to the axis of the target cell. The air-cooled electrically suppressed Faraday cup had a length of 330 mm and a diameter of 130 ram; its distance from the target was 4 m, so that the entire beam could be collected. The 3He target cells consisted of thin-walled (0.3 mm) cylindrical glass bulbs of 40 mm diameter and 32 mm height, filled with 2 Torr ~He. The cleaning and filling procedure is described elsewhere ~4). The glass foil windows for the incoming and outgoing beam and the elastically scattered protons were produced in the following way: glass pipes of 20 mm diameter and 1.2 mm thickness were sealed by fusion and the molten glass was drawn into the pipe so that concave foils were formed. The thicknesses of the foils were determined by the energy loss of passing a-particles from a 2l/pb source. Four foils of thicknesses less than 30 pm and usable diameters of at least 10 mm were produced, with pipes of about 20 mm length to the cylindrical glass body (fig. 1). Each experimental point in the angular distribution required its own target cell. The beam entrance and exit windows were coated with silver (about 0.1 pm) and thermically connected with the holder. Constructed in this way, the target could withstand 3 pA of 19 MeV protons for at least several days without any decrease in polarization. The cells could be stored for more than two years with negligible change in their qualities. The 3He cell was located in the center of 200 mm diameter aluminum vacuum chambers, constructed in the "Korex" 15) way, each with three pairs of symmetrically drilled holes for the insertion of the pipes with the counter telescopes. Each telescope consisted of a silicon surface-barrier and a lithium-drifted silicon detector.
3He(p, p)3He
gloss
beom
"
)
I
~
~,~
l
~cm
Fig. 1. Simplified scale drawing of the glass target cell for a scattering angle 0~ab = 45°.
The detectors could be cooled down to - 3 5 °C separately by Peltier cooling. Thin foils (5 /~m AI) in front of the detectors prevented the weak discharge light from reaching the detectors. The collimating system consisted of an entrance slit located 50 mm and a rear slit located 440 mm from the target center. Both slits were 5 mm wide and 10 mm high, so that the horizontal acceptance was + 0.7 ° in the lab system. The accuracy of the detector angle setting was + 0.3 °. Small magnets before the rear slits prevented secondary electrons from reaching the detectors. Chamber vacuum was provided by turbomolecular pumps 4 m before and 2 m behind the chamber, as well as by liquid nitrogen traps in the detector arms, which usually maintained a pressure of 4 × 10 -6 Torr. A magnetic field of 10 G, which defined the alignment direction, was provided by 90 cm diameter Helmholtz coils. The infrared circularly polarized optical pumping light was passed to the 3He cell through a focusing glass lens which was cemented to the bottom of the vacuum chamber with Araldit. The light passing the 3He cell was monitored by a photodetector through a light pipe (2 mm diameter) which was cemented into a vacuum feed-through. The effective pumping light intensity had been increased by a reflector at the back of the target cell by a factor of nearly 2 [ref. ~6)]. The reflector consisted of either a silver film evaporated onto the back of the cell or an aluminum foil. With the reflector, typical target polarizations of 15 ~-18 ~ could be achieved. The 3He target nuclear polarization was determined by optical pumping methods ~7). The measurement uses an equation for the absorption of circularly polarized light by 3He metastable atoms ~8). The parameters of this equation have been determined 19) and furthermore the correctness of the equation could be confirmed ~9), so that we have confidence in this optical method. The polarization can be computed from the ratio I(P)/I( - P), where I(P) is the fraction of pumping light absorbed at the 3He cell at the polarization P. In practice, we produced different relative polarizations + P and ~ P by rotating the linear polarizer by an angle of ½~ and then back to its original position. The whole procedure lasted about 1 sec. The I(P) value was obtained by switching off the weak discharge. This method has
4
D. M O L L E R et al.
the great advantage of not destroying the nuclear polarization, so that the measurement could be repeated every 15 min during the experiment. More measurements were not necessary because, in general, the polarization values were constant to within their accuracy of _+0.3 °/o.The sign of the polarization was reversed by rotating the linear polarizer by an angle of ½g about once every half-hour. The protons were detected in each counter telescope by requiring coincidences between pulses from the AE and E detectors. The infbrmation was handled by a Nuclear Data N D 4420 multichannel analyzer. The pulse-height spectra were corrected for a small constant background (always less than 2 I~o).The spin analyzing power was determined in the usual way 7) from the corrected counting rates, in order to eliminate geometrical differences and different numbers of counts per integrated beam current.
3. The 3He spin analyzing power of protons The measured values of the 3He spin analyzing power of protons at a lab energy of 25.0 MeV are listed in table 1. The errors quoted are standard deviations due to counting statistics: all other possible errors are comparatively small. Our experimental results are nearly identical to those at 19.4 MeV of ref. 1o) and at 26.8 MeV of ref. 7) (fig. 2). There is no strong variation with energy as predicted in ref. 7). To confirm this result we have measured the analyzing power at a fixed lab angle of 135 ° at different energies from 19.6 to 26.5 MeV (table 2). Here, too, no essential energy dependence is to be seen.
4. Phase-shift analysis Our measurement of the 3He analyzing power was made in order to test the phases of Morales et al. 4) at 25 MeV. Fig. 2 shows the analyzing power computed with these phases, together with our experimental results. One can see that the principal trend of the curve is confirmed by our results, but that the predicted analyzing power is too high by a factor of about 1.5. Therefore, a new phase-shift analysis also including our data was made, and new phases were computed for 30.5 MeV using extrapolated 3He polarization data besides the available experimental cross-section and proton polarization data. In our analysis, we took into account waves with angular momentum up to l = 3, both singlet and triplet. The spin coupling parameters 2o) ~ and ~2 describe singlettriplet transitions of the P~ and D 2 states (spectroscopic rotation). The parameter e3, which mixes the F 3 states, was set to zero. This was done because the F-phases turned out to be very small, so that e 3 had no noticeable affect on the analysis. Couplings of different angular momentum are described by the parameters t02 for the transitions 3S~ --* 3D~ and t~3 for 3 P 2 -+ 3 F 2. All phases are taken to be real in order to limit the number of fitting parameters to a reasonable amount. With these limitations we have 14 eigenphases and 4 coupling parameters, i.e. 18 fitting parameters.
3He(p, p)3He
5
• 'th~S work O.5J--- calc. our phases
I
.~x
J-.- calc.phoses re,.z,
05~- Ep: 25MeV !. . . . .
I ....
0
50
~k~.
" I
100
....
!,, lC-~ Oc.m!deg)
Fig. 2. The 3He analyzing power of 3He(p, p)3He at 25.0 MeV proton energy versus the scattering angle. The closed circles are the new data of this work, the dashed curve is calculated with the phase shifts of ref. 4), and the smooth curve is calculated from the derived phase shifts of table 3.
TABLE 1 The 3He spin analyzing power of 25.0 MeV elastically scattered protons as a function of c.m. scattering angle 0. 0.... (deg)
A3He
46.2 58.9 71.1 94.0 114.7 132.8 141.0 148.8 156.1
- 0.085 _+0.003 -0.140+0.020 -0.165+0.035 -0.216+0.030 0.000+0.040 0.303 + 0.040 0.269+0.035 0.216+0.035 0.130+0.030
TABLE 2 The 3He spin analyzing power of elastically scattered protons at a fixed scattered angle function of the proton energy Ep, lab (MeV) 19.6 21.1 23.4 25.0 26.5
A3He 0.069 ___0.035 0.148+0.040 0.150+0.040 0.216+0.035 0.217+_0.042
01a b =
135° as a
6
D. MOLLER et al.
In the following we give a brief description of the program used for fitting the data. The program which calculates cross sections and analyzing powers as functions of the phase shifts uses the formulae of Lindner 24). These formulae were completed by adding the Coulomb interaction. To test the program we calculated the Z2 for the cross section and proton analyzing power with the phases of Morales et al. 4) at 30 MeV. The good Z2 showed that the program works correctly. The fit program 2 l) searches for the minimum ofxZ(a), where a = ai • . . a, are points in parameter space with the eigenphases and mixing parameters as coordinates. Each step carried out in parameter space is calculated from the gradient o f z 2 and the matrix Mi, k = 62(Z2)/ 6ai3a~ of second derivatives. Near the minimum, x2(a) should be a nearly quadratic function of the fitting parameters a i. For a quadratic function, the errors rri of the parameters a~ are given by rr~ = G,, with G~k = ( 1 M i 0 - ~. These errors are computed by the program after finding the minimum. In a further error analysis, other errors s~ are computed in the following way: the parameter a i is replaced by the fixed value aj + rrj and the minimum z2(a, o-j) is searched by varying the other parameters. Then sj is defined as s~ = rrj(z2(a, r r ) - zZ(a))- ~. For further information about this error analysis and the search program we must refer to ref. 2~). In a first search at 25 MeV we fitted cross-section data and our measured 3He analyzing power. Different sets of phases resulted, which all described the experimental data with good ;(2. To get a unique fit, further data had to be included in our search program. We therefore interpolated, in the same way as Morales et al. 4), between the proton polarization data of Tivol 22) at 19.5 MeV and of Harbison 9) at 30.5 MeV. From fig. 3 it can be seen that the angular distributions at these two energies are very similar. Only the rapid change of polarization from negative to positive values is shifted by about 10°. Therefore, one can suppose that the proton polarization changes only slightly with energy so that interpolation between these two data sets seems to be reasonable. Using these additional data, a new search gave only one set of phases with good ;(2. The results are listed in table 3. The 3He analyzing power and proton polarization calculated with these phases are shown in figs. 2 and 3. At 30.5 MeV, the phases of Darves-Blanc 6) and Morales 4) both describe the experimental cross section and proton polarization, but the predicted 3He analyzing power with Ami n = --0.5 and Areax = 0.4 seems to be too high. Fig. 4 shows the analyzing power data from Baker et al. ~0) at 19.4 MeV, our data at 25.0 and 19.6 MeV, and data from Ware et al. 7) at 26.8 MeV. From this figure it can be seen that there is no noticeable energy dependence between 19.4 and 26.8 MeV. Therefore, we do not think it probable that the analyzing power increases in the strong way predicted by the phase shifts at 30.5 MeV. Our phase-shift analysis at 30.5 MeV included not only cross-section and proton polarization data, but also our 3He analyzing power data at 25 MeV. Because of the energy independence mentioned above, this seemed to us to be reasonable. Nevertheless, we have taken the fact that these are only extrapolated data into account by
3He(p, p)aHe TABLE 3 Phase shifts for 3He(p, p)3He at Ev = 25.0 and 30.5 MeV
iS o 3So IP 1 3P 0 3P 1 3P 2 ID 2 3D 1 3D::
25 MeV
30.5 MeV
- 1 4 2 ___4 -112 +2 29 -+3 30 -+8 41 -+2 65 -+2 4.3-+1.1 4.0-+0.5 - 1.1 -+0.8
-127 +4 - 1 2 3 -+2 32 -+2 25 -+2 42 + 2 58 -+1 0.6_+0.8 1.8+_0.6 12.3± 1.2
I
3D 3 IF 3 3F z 3F 3 3F 4 e1 e,2 t02 t13
25 MeV
30.5 MeV
5.2+0.5 2.2-+1.7 3.3+0.5 3.8-+1.5 3.9-+0.5 - 4 1 -+8 -23 +8 - 4 -+4 -0.5-+4
6.9+0.6 7.6±0.6 3.2-+0.5 6.3±0.9 3.3+__0.4 - 3 1 -+7 19 -+4 0.2-+3 2.2-+3.8
• Ep: 19.5 MeV,ref. 22 , / ~
....
I ....
0
I ....
50
1,,
I(3(3
150 Oc.m.ldeg}
Fig. 3. The proton analyzing power of 3He(p, p)aHe versus the scattering angle. The crosses are data from ref. 22) at Ep = 19.5 MeV, the open circles are data from ref. 9) at E v = 30.0 MeV and the smooth curve is calculated with the phase shifts of table 3 for Ep = 25.0 MeV.
0.4 0.2
o Ep = 19./~ MeV,ret.lO & Ep= 19.6 MeV, this work Ep= 25,0 MeV, this work ~,Ep= 26.8 MeV, ret7
x • x
x*
o
x e &
x
4
.c
x
Iox •
-0
x,*~ o xo,~
io
-O.t
0
20
~0
60
0
10
1
1~0
160 180 Oc.m.(deg )
Fig. 4. The 3He analyzing power o f 3He(p, p)3He versus the scattering angle. The open circles are data at E v = 19.4 MeV from ref. ~o), the triangle is a new data point at Ep = 19.6 MeV from this work, the closed circles are new data from this work at Ep = 25.0 MeV and the crosses are data at Ep = 26.8 M e V from ref. 7).
8
D. M0'LLER et al.
I • Ap, ref.9 0.51_- Ap, cole - - A3He, cole.
P~'f"k
l
< o i-- , - - + . ~ ~
/j
_o.+ I
I
0
50
I
I00
1
I
]
I
1
~ ec.m.tdeg)
Fig. 5. The 3He or proton analyzing powers of 3He(p, p)3He at Ep = 30 MeV versus the scattering angle. The closed circles are experimental proton analyzing powers of ref. 9), the smooth curve is the calculated proton analyzing power with the phase shifts of table 3 and the dotted curve is the calculated 3He analyzing power with the phase shifts of table 3.
Ep {MeV) -~-
°
40
- 80
I0
2Or
'
iSo
"'.
i
° * ° . ,o
30 -t.0 l '.. - 80
~a
.,2o I
-120
~ -too I
-160[
+o. :1 f
' ' " "...
' 351
~
80 60
g
~0
.. .....
~
:
20 :i
31~
~o
6oi
,~ ,
+
2o
.'"'"'"
"
:
-
60[
°
,b
~
sb
10
20
30
20t""-"".'.. EII-J ~.,
-40
g
+°f I f.., 60
40
40
.
20
........', ,o
~ ~Po
a
~ ~
2b
Jo
40
20
.'"
~PI o
"
°
,o
o x
~
Jo
~' a
Fig. 6. The S and P phase shifts of 3He(p, p)3He as functions of the bombarding energy. The closed circles are the results of Tombrello 23), the horizontal crosses are from Baker et al. ~o), the diagonal crosses from Darves-Blanc et al. 6), the triangles from Morales et al. a), and the open circles are results of this work.
3He(p, p)3He
9
multiplying the experimental errors by a factor of 1.5. The resulting phases are listed in table 3. The 3He analyzing power calculated with these phases (dotted line) and the calculated proton polarization together with the proton polarization data of Harbison 9) are shown in fig. 5. The errors given in table 3 are calculated as described above. Since the calculation of the errors s i is very time consuming for the computer, they were computed only for the S- and P-phases and for the mixing parameter e~. For the other phases and mixing parameters, the errors in table 3 are the above defined ~r~. 5. Discussion of the results
In the following we wish to compare our results with those of other authors and discuss the energy dependence of the phases. When comparing phases of different authors, it must be kept in mind that the definition of the mixing parameters is not unique. For instance, one gets the same scattering matrix by replacing ~l with ex ---90° and interchanging the ~PI phase with the 3P l phase. Here we follow the convention that el goes to zero for small Ep. In fig. 6 the singlet and triplet eigenphases and the mixing parameter ~1 are shown for proton incident energies Ep up to 30 MeV. The D- and F-phases are all small, so no attempt was made to prove their continuous energy dependence. The two S-phases continuously increase with energy, showing the hard sphere behaviour already found at lower energies. At the same energy, the phases of different authors show good agreement. The 3P2 phase also shows a continuous behaviour with energy. For the 3p~ and ~Px phases it is remarkable that the phases of Morales 4) at 20 and 25 MeV widely deviate from continuity. In their own publication, their phases fit the energy dependence with respect to lower energies very well. The reason for this discrepancy lies in the fact that they exchanged the singlet and triplet P-phases of Tombrello et al. 23). But this is not in agreement with their convention about the mixing parameter e~, which is the same as in this paper. For the 3P0 phase, the extrapolation of our phases to lower energies is not so simple. The phase seems to have a maximum at about 15 MeV. The energy dependence of the mixing parameter e x is not very satisfactory. But one must keep in mind that the errors of this parameter are very large compared with those of the eigenphases. The phase-shift analysis was very insensitive with respect to a variation of this parameter. We would like to thank Prof. A. Lindner for helpful discussions and Dr. D. Eppel for placing his subroutines at our disposal. This work has been financially supported by the Bundesministerium fiir Forschung und Technologie.
10
D. MCILLER et al.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)
23) 24)
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