Vector analyzing power in the 4He(d, p) reaction at 10–11 MeV

Nuclear Physics A264 (1976) 54-62; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

V E C T O R A N A L Y Z I N G P O W E R I N T H E 4He(d, p) R E A C T I O N AT 10-11 MeV * HIROSHI NAKAMURA College of Science and Engineering, Aoyama Gakuin University, Tokyo

and HIROSHI NOYA Department of Physics, Hosei University, Tokyo

Received 22 December 1975 (Revised 6 February 1976) Abstract:The vector analyzing power in the 4He(~l,p) reaction at 10-11 MeV deuteron energy is analyzed by using a semi-phenomenologicalmethod. The values for the free parameters are determined by an analysisof the *He(d, ~p)n reaction at 9 MeV deuteron energy.An excellentfit to the experimental data for the vector analyzing power as well as the absolute differential cross sections is obtained.

I. Introduction

Since we first presented a semi-phenomenological modified impulse approximation (MIA) 1) for analyzing the 2H(~, ~p)n reaction, this approximation has been apl~lied successfully to a n u m b e r of experiments 2-5) involving s-particles with energies in the range 30-165 MeV [ref. 6)]. Most recently, we could reproduce the neutron polarization data by Northcliffe et al. with this method 7). In this article, we will show that the M I A can be used, further, in analyzing the vector analyzing power of the 4He(d, p) reaction involving polarized deuterons with E d = 10-11 MeV. In sect. 2, we extend our previous M I A by including an additional phenomenological term with a single parameter and by making the reducing factor spin dependent. Then, a set o f values for the M I A parameters are determined so as to reproduce the 4He(d, ~p)n reaction data a) of N o g a m i et al. at E d = 9 MeV. Finally in sect. 3, we show that, with this set of values for the parameters, excellent agreement can be obtained with the vector analyzing power data 9) of Keller and Haeberli as well as with the absolute differential cross-section data ~o) of Ohlsen and Young. Some concluding remarks are given in sect. 4.

~" Research sponsored by Nishina Memorial Foundation. 54

4He(d, p)

55

2. The 4He(d, ~p)n reaction at 9 MeV In this article we use the same procedure and notation as in ref. 6). Differing from our previous work, however, we include the spin dependence of the a-2H and a-p (a-n) phase shifts in the calculation o f the reducing factor. This refinement of a reducing factor makes the calculation o f the matrix elements considerably more complicated, as we discuss in the appendix ofref. 6), but there is no essential change. In the analyzing procedures, we neglect the possible spin dependence of the reduction parameters in order to reduce the number of free parameters, and adopt the SOEMW potential 11) for the a-p (a-n) interaction. First, we try the reducing factor which corresponds to the formula (2.5) of ref. 6),

r(Y) = la(Ld)[~p + ~d], ~p = rlp(LpJp) exp {2i(6R(LpJp)+~r(Lp))},

(2.1)

~d = qd(LdJd) exp {2i(6~(LdJd) + ~r(Ld))}, where 6Rp(LpJp) and 6~(LdJd), and qp(LpJp) and r/d(LdJd) represent the real parts of the a-p and a-ZH phase shifts, and the absorption coefficients with orbital and total angular momenta L{p,d ) and J(p,d), respectively. Also in this equation o'(Lp) and ~(Ld) are the Coulomb phase shifts, and Y represents the angular momentum state of the system. In this analysis, we use the experimental data x2) tabulated in table 1. We calculate the absolute differential cross section by means of MIA with reduction parameters a(Ld) = 1 for L d > 3. Since the absorption coefficient for the S-state as given in table 1 suggests that the reaction strength in the S-state is very weak, we set the corresponding reduction parameter ~(0) to 0. The results, however, are not sensitive to the choice of a(0). Although we varied two parameters a(1) and a(2) between 0 and 1 in order to get the best fit to the experimental data, we could not attain any satisfactory reproduction of the data. A comparatively better fit obtained within this approach is shown in TABLE1 The real parts of the ~-2H phase shifts and the absorption coefficientsgivenby Schmelzbachet al. a) (LdJd)

6~(LdJd)(deg)

qd(LdJd)

S1 Po PI Pz D1

60.3 - 27.0 -4.0 1.5 94.3 119.7 157.8 0.0 6.2 7.5

1.00 0.98 0.82 0.78 0.56 0.78 0.62 0.86 0.85 0.85

D2

D3 F2 F3

F4 a) Ref. 12).

56

H. N A K A M U R A

A N D H. N O Y A

fig. 1, where the reduction parameters are chosen to be =(1) = 0.30,

=(2) = 0.30.

(2.2)

In order to get better results, we examine two new possibilities, i.e. (a) the introduction of the n-p final state interaction (n-p FSI) and (b) the modification of the reducing factor (2.1). First, we study the effects of the n-p FSI for the differential cross section by using various methods including several phenomenological parameters. However, if a strong n-p FSI is introduced in order to reproduce the SHe peaks, the calculated curves in fig. 1 are sharply deformed. We obtain a very large discrepancy in the overall features of the spectra between the calculated and observed results. This may be due to the fact that around the SHe peak the phase and magnitude of MIA matrix elements rapidly change and that n-p FSI and MIA matrix elements strongly interfere with each other. Thus, we are led to conclude that the n-p FSI does not play an important role in the experimental geometry given here. sl-I, 1 6C

Op=70" I~-- 30"

2C 1

2

3

Ilk

e.=4o, 2©

2

3

4G

ep: 35" eo=ZS"

2

4 Sl'k ;/~ /f

.,,

3

5

5 Ep (MeV) Fig. 1. Absolute differential cross ~ections (in mb/sr 2 • MeV) for 4He(d, ctp)n at E d = 9 MeV, plotted 4

against the energyof the emitted proton Ep (MeV) for the pairs of proton and ~-particle scatteringangles (0v, 0~).The positions of 5He and 5Li peaks are indicated by arrows. The theoreticalvalues are calculated by MIA with parameter set ~(1)= 0.30, ~(2)= 0.30 for reducing factor (2.1) (dashed lines) and • (1) = 0.20, ~(2) = 0.25, R, = 0.80 for reducing factor (2.3) (solid lines).

"He(d, p)

57

As a second possibility, we try the following form of the modified reducing factor with a new parameter R s for L d < 3, r(Y) = ½~(Ld)[~p + ~d + (-- 1)LRs(~p~d-- 1)/i],

(2.3)

where L = min [Lp, Ld]. The physical implications of the R s term are discussed in the appendix. With this modified reducing factor, the agreement with the data becomes very good, as illustrated in fig. 1. In fig. 1, we used the following set of values for the parameters: ~(1) = 0.20,

a(2) = 0.25,

Rs = 0.80.

(2.4)

The experimental data, especially around the 5He peaks, are excellently reproduced in this calculation. This result suggests that the R term becomes important in the lower energy region. 3. Vector analyzing power at 10-11 MeV

The vector analyzing power i T 11 for vector polarized deuterons is given by iT11 = f Q d ~ h / f w d ~ 2 h ,

(3.1)

where w = 2 [ S I 2 + ~ ( V * V ) + ~ ( A * A ) + 2 ~ IT~j[2, ij

Q = x/~ Im [4(A *N)S - N ( V * x V) - N(A* × A) + 2 ~, ~ A i N J - ~ N ] , ij ~,,. = ~ eikhTk* Thj,

N = d x p/ld x p[,

khj

and where h is the relative momentum between neutron and alpha particle, f2h the solid angle, and d and p the momenta of the incident deuteron and the emitted proton, respectively. The definitions of the amplitudes S, V, A and Tu are given in ref. l a). The differential cross section for unpolarized deuterons is given by dEa

dOpdEp

_

1 14M2ph~wdah ' (2n) s v J

(3.2)

where v is the relative velocity between deuteron and alpha particle in the initial state, and M N is the nucleon mass. The calculated results with the parameter set (2.4) are shown in figs. 2-5. The agreement with the data 9-lo) at E d = 9-11 MeV is quite excellent both for the absolute differential cross sections for unpolarized

58

H. N A K A M U R A

200

I

I

I

A N D H. N O Y A I

I

f

I

Ea -- 9.00 M~'

i

150

100

~

5o

0

I

I

1

2

I

3

4

5

6

7

8

E~, (HEY) Fig. 2. The absolute differential cross sections o f 4He(d, p) for unpolarized deuteron at E d = 9 MeV as a function of proton energy. The arrows refer to the position o f the 5He peak. The theoretical values (solid lines) are calculated by M I A using the reducing factor (2.3) with the same parameter set as in fig. 1, i.e., ~t(1) = 0.20, ~(2) -- 0.25, R s = 0.80.

200

I

I

I

I

I

I

f

Eo : 9.00M,~ 150

1O0 '0

50

I -'~'--'-'-'-'-'-'-~ I

0

I

2

~0°

3

4

5

i

6

I

7

Ep (MeV)

Fig. 3. Same as fig. 2 but the experimental values are shown by dashed lines.

deuterons and the vector analyzing powers for polarized deuterons. This seems to reconfirm the validity and effectiveness of our M I A method. We have also calculated the differential cross sections at E d = 8 and 11 MeV using the same parameter set (2.4). The agreement between data and theoretical values is again excellent and the discrepancy does not exceed 25 ~o at the SHe peaks. It should be remarked that these results strongly depend on the new parameter R s. A special case with R s = 0 is given in figs. 4 and 5 to illustrate the R s dependence o f results.

*He(d, p) 200

I

i

i

i

59 i

i

E~ = lO.O01deV

]

/,~

Op = 14 ° 150

//\ /;i !

100

5O E

d '1o "ID

I

I

1

I

I

I

I

I

I

~'~vLJ

F_==lO.OOte~V

30

#p = 45 °

ZO

---. !

10

0

I

I

I

1

2

3

I

4

5

~

/

I

6

7

Ep (MeV)

Fig. 4. Same as fig. 2 but Ed = 10 MeV. The dashed lines are the theoretical values for parameter set ~(1) = 0.20, ~(2) = 0.25, R~ = 0.

4. Conclusion Using the M I A with a reducing factor (2.3) and a c o m m o n parameter set (2.4), we obtained excellent agreement with the experimental data both for the absolute differential cross sections of the 4He(d, p) reaction for unpolarized deuterons at E d = 8-11 MeV and the vector analyzing powers for polarized deuterons at E d = 10-11 MeV, as well as for the absolute differential cross sections of the 4He(d, ~p)n reaction at E d = 9 MeV. In the experiments we consider in the present article, protons are emitted in the forward direction in the barycentric system, the scattering angle ~p of protons being in the range ~p < 100 ° at the 5He peak. As we have shown in the preceding sections, in this region our M I A reproduces the experiments very well. It m a y be a very interesting future problem to examine whether M I A is valid also for the geometries in which high-energy protons are emitted in the backward direction in the barycentric system. F o r this purpose, the measurement of the absolute differential cross sections of2H(a, ap)n and 2H(a, p)cm reactions at E~ = 18-22 MeV

60

H. N A K A M U R A AND H, NOYA 0.1

i

i

i

&

0

~

-0:1

GX

iT,,

-0.4

-o.!

32.2* lab

45" lob ~=IOMeV I

E,,=.M.V

I

I

3

2

I

I

5

I

I

7

Epe ) i

O.ll 0

-O.1

-O.2

iT,, -o.:

-0.~

-0.5

6~.'1c~ Ed---11MeV I

2

I

I

4

I

I

6

E~MeV) Fig. 5. The vector analyzing power in the "He(d, p) reaction for polarized deuterons and the values calculated by MIA. The same parameter set as in fig. 4 is adopted.

is desirable. Since the reaction mechanism is generally much more complicated in low-energy break-up reactions, significant discrepancy between experiment and the MIA theory could be observed for large ~rp. The effects of the n-p FSI did not play any important role in the present analysis. The n-p FSI induced in the D- and F-waves of the 2H-4He system, however, may

4He(d, p)

61

become important in the low-energy region, as we emphasized previously 6). It was also indicated 6) that the n-p FSI in the F-wave may interfere destructively with the MIA matrix element. These n-p FSI effects may become distinct in the 2H(~, ~p)n reaction at E~ = 18-22 MeV for appropriate scattering angles. It is required to measure precisely the absolute differential cross sections around the 5He peak (5Li peak) as well as at Enp ~ 0, because the n-p FSI is observed as the discrepancy between the experimental results and MIA in the differential cross section. The measurement must be done within the range Up ~ < 100° where M I A is considered to be quite reliable. The experimental data reported by Rausch et al. 14), for instance, are not suitable for this purpose, since in their experiment the absolute values of the differential cross sections around the 5He peak and at E,p ~ 0 are not fully measured. According to their data, the spin-triplet n-p FSI does not induce a prominent peak in the spectra. Therefore, it is very difficult to estimate the strength of the n-p FSI using only the shape of the spectra. Finally, we would like to point out that the semi-phenomenological method used in this work may be applied to the deuteron break-up reactions induced by 3H or 3He. The analysis, however, turns out to be considerably more complicated because nucleon transfer processes are expected to be involved in these reactions. For this purpose, the correct values of the matrix elements of elastic 3H-n, 3H-p, 3He-n, 3He-p, 3H-2H and 3He-ZH scatterings including the charge exchange reactions are required. Unfortunately, at present, not all of these data are available. The study of these reactions based on the present idea may be an interesting future problem, because the deuteron break-up reactions induced by 3H or 3He are related directly to nuclear fusion reactions. The authors are very grateful to Prof. Y. Nogami and his collaborators for communicating their results prior to publication. They would also like to thank the computer center at Hosei University and Miss Y. Sakamoto for their assistance in performing these calculations.

Appendix Formula (2.3) is obtained straightforward, if the following approximations are adopted in the formula (3.6) of ref. 6), 0C11 ~___ 0~22 =

0~(Zd) ,

c02 = ~21 = (_ 1)LRs~(Ld).

(A.1)

In order to examine the validity of the simplifications in (A. 1), we carried out a model calculation with a point o~-n interaction and an appropriate cut-off of the reaction strength in the central region as in ref. 6). The result shows that these approximations are quite reasonable, though the parameter R s rather strongly depends on the

62

H. N A K A M U R A AND H. NOYA

momenta of the incident deuterons and emitted protons. Although R s depends also on the angular momentum L d and the cut-off radius of the reaction strength, in the present work we neglect them in order to reduce the number of free parameters.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)

H. Nakamura, Nucl. Phys. A208 (1973) 207 R. E. Warner and R. W. Bercaw, Phys. Lett. 24B (1967) 517; Nucl. Phys. A109 (1968) 21 T. Tanabe, J. Phys. Soc. Jap. 25 (1968) 21 H. G. Pugh, D. I. Bonbright, D. A. Goldberg, P. G. Roose, R. A. J. Riddle and J. W. Watson, Bull. Am. Phys. Soc. 13 0968) 1367 E. Hourany, H. Nakamura, F. Takeutchi and T. Yuasa, Int. Conf. on few particle problems in the nuclear interaction, Los Angeles, 1972; Nucl. Phys. A222 (1974) 537 H. Nakamura, Nucl. Phys. A223 (1974) 599 H. D. Knox, R. G. Graves, F. N. Rad, M. L. Evans, L. C. Northcliffe, H. Nakamura and H. Noya, Phys. Lett. 56B (1975) 33 K. Sagara, M. Hara, T. Motobayashi, N. Nakahashi, F. Soga, F. Takeutchi and Y. Nogami, Int. Conf. on few body problems in nuclear and particle physics, Quebec, 1974 L. G. Keller and W. Haeberli, Nucl. Phys. A172 (1971) 625 G. G. Ohlsen and P. G. Young, Phys. Rev. 136 (1964) B1632 G. R. Satchler, I. W. Owen, A. J. Elwyn, G. L. Morgan and R. L. Walter, Nucl. Phys. Al12 (1968) 1 P. A. Schmelzbach, W. Griiebler, V. K6nig and P. Marmier, Nucl. Phys. A184 (1972) 193 H. Nakamura and K. Nagatani, Nucl. Phys. A101 (1967) 557 T. Rausch, H. Zell, D. Wallenwein and W. von Witsch, Nucl. Phys. A222 (1974) 429