Study of the spectra of fast deuterons produced in the interaction of 670 MeV protons with nuclei

Nuclear Physics A195 (1972) 581 --595; (~) North-Holland Publishin# Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

STUDY OF T H E SPECTRA OF FAST D E U T E R O N S P R O D U C E D IN T H E INTERACTION OF 670 MeV P R O T O N S W I T H NUCLEI L. S. A Z H G I R E [ , Z. CISEK, Z. V. K R U M S T E I N , Yu. P. M E R E K O V , Z. M O R O Z , N G O Q U A N G ZUI, V. I. P E T R U K H I N , A. I. R O N Z H I N and G. A. S H E L K O V

Joint Institute for Nuclear Research, Dubna, USSR O. D. D A L K A R O V

Institute of Theoretical and Experimental Physics, Moscow, USSR Received 10 April 1972 Abstract: The spectra o f deuterons emitted at angles o f 6.5 °, 9.5 °, 13.5 ° and 16 ° (lab system) in the

interaction o f protons with the nuclei tH, 2H, 6Li, 7Li, C, AI, Cu, R h and Pb have been measured in the m o m e n t u m range 700-1700 MeV/c. The secondary particles were analyzed with the help o f magnetic and time-of-flight spectrometers. The differential cross sections for deuteron production in the reactions p ÷ ( 2 N ~ - - ~ N ÷ d and p + ( N ~ - - - > d + ~ r have been measured for selected nuclei and angles. The effective numbers of nucleons and o f two-nucleon clusters for the investigated nuclei were determined. The results obtained are discussed in terms o f the dispersion theory o f direct nuclear reactions. E [

I

N U C L E A R R E A C T I O N S 1"2H, 607Li, C, AI, Cu, Rh, Pb (p, d), E : 670 MeV; measured tr(Ed); analyzed reaction mechanism.

I

I

1. Introduction

The explanation of the direct nuclear reaction mechanism is one of the important problems in nuclear physics. Quasielastic scattering processes such as (p, 2p), (p, dN), (p, tp), etc., play a special part in these problems. The unambiguous kinematics of these processes leads to a sufficiently simple geometry for the experimental arrangement. On the other hand, when the incident proton energies exceed several hundred MeV and the momentum transfer is great enough, we may neglect in the first approximation the interaction of the protons with the rest of the nucleus. The last circumstance allows us to study the mechanism of the interaction between the incident proton and separate nucleons or nuclear clusters. The quasielastic scattering of protons on nuclei, i.e. the (p, 2p) reaction, has been investigated sufficiently well up to an incident energy of 460 MeV [ref. 1)]. This process was found to be an effective tool for studying nuclear shell structure. The more complicated (p, dN) process, i.e. the scattering of a proton by a two-nucleon cluster, gives information on some of the finer features of nuclear structure. However, the experimental data for reactions of this type are considerably poorer 2.3). In 1969, we measured the spectra of deuterons produced in the bombardment of nuclei by 670 MeV protons at 6.5 ° (lab system) with respect to the proton beam direction in 581

582

L.S. AZHGIREI et al.

the wide momentum range from 700 MeV/c up to 1700 MeV/c [refs. 4, 5)]. The preliminary analysis of deuteron spectra showed, that besides the known process p + ( 2 N ) --* d + N there is also a new reaction channel with the deuteron in a final state accompanied by one-pion emission p + ( N ) --, d + n. The corresponding region in the deuteron spectra extends up to 1400 MeV/c. The present experiment is the continuation of earlier investigations 4, 5) which were made for a systematic study of the mechanism of deuteron production in the p + ( 2 N ) --* d + N and p + ( N ) d + n reactions in a wide range of deuteron momenta for 670 MeV incident protons. The deuteron spectra in this work were measured for a number of nuclei from deuterium to lead at lab angles of 9.5 °, 13.5 ° and 16°. The results, together with the spectra at 6.5 ° [ref. s)], are discussed within the framework of the dispersion theory of direct nuclear reactions. Pole and triangular graphs were taken into account.

2. Measurements and data processing The experimental set-up has been described in ref. 4). The measurements were made by means of magnetic and time-of-flight spectrometers. The momentum resolution of the magnetic spectrometer was A p / p = + 1.8 %, while the total time resolution of the time-of-flight spectrometer was 2z = 0.9 ns. Under the present experimental conditions the divergence angle of the secondary particles was equal to +_0.6 °. An on-line system with a Minsk-22 computer was used in the experiment 6). After preliminary processing by the computer the experimental data were writen on magnetic tape. Corrections were made for the dead-time of the detection apparatus, and for deuteron multiple scattering in the target, air and scintillator. The absolute values of the differential cross sections of deuteron production for the angles of 6.5 ° and 9.5 ° were determined from the ratio of the number of the deuterons detected for the given target to the number of deuterons from the pp ~ dn + reaction. The cross section for this reaction were taken from refs. 7, 8). However, this calibration method looses its precision for the deuteron angle 13.5 °, owing to the strong angular dependence of the pp --* dn + reaction near its limiting angle. At the angle of 16° (beyond the limiting angle) deuterons are not produced any more in the pp ~ dn + reaction. So the cross section for deuteron emission at the angles 13.5 ° and 16° were determined from the ratio of the deuteron number detected for the given target to the number of all the charged particles, detected for the carbon target. The absolute values of the differential cross sections for charged-particle production in carbon were taken from ref. 9). As a check, values of the differential cross section at the angles 6.5 ° and 9.5 ° were determined by both methods and found to be in agreement within the limits of experimental errors.

3. Experimental results 3.1. MOMENTUM SPECTRA The momentum spectra of the deuterons produced at 6.5 ° for the nuclei 1H,2 H, 7Li, C, AI, Cu, Rh and Pb are shown in fig. 1. As seen in fig. 1, the spectrum for the

FAST DEUTERONS

583

pp ~ dTr+ reaction on free protons consists of two peaks corresponding to deuteron emission in the forward and backward directions in the c.m. The peak positions coincide well with the values 1378 MeV/c and 859 MeV/c calculated from kinematics. The widths of peaks, Ap ~ 45 MeV/c, correspond to the momentum resolu-

,~.~;, ;

. i~

~

.oJ

,:

J

,

Q

......

. . . . .

e ~

e~c

~. e ~

0,5

0

oo

oJ['i

Fig. I. M o m e n t u m spectra o f deuterons p r o d u c e d by 670 MeV p r o t o n s at 6.5 °. The curves for all the nuclei except 1H are the results o f the calculation based on the dispersion theory o f direct nuclear reactions. T h e thin c o n t i n u o u s line represents the pole graph; the thick c o n t i n u o u s line, the pole g r a p h + triangle graph; the dotted line, the triangular graph, a n d the dot-dashed line, the interference o f the pole and triangle graphs (fig. lc). T h e 1 -- 1 dashed line is pole graph with taking into a c c o u n t the form factor 1 -- 1.

tion of the magnetic spectrometer. In the deuteron momentum spectrum for deuterium (fig. 1) three peaks atpa ~ 870 MeV/c, 1375 MeV/c and 1625 MeV/c are well seen. The peak atPa = 1625 MeV/c is due to elastic scattering on free deuterons. The width of this peak is equal to 52 MeV/c and corresponds to the momentum resolution of the magnetic spectrometer. The two other peaks correspond kinematically to the p + ( N ) --* d + n reaction on the nucleons of the deuterium. The widths are equal to 150 MeV/c and 85 MeV/c. The characteristic peculiarity of the deuteron momentum spectra for the heavier nuclei (TLi, C, A1, Cu, Rh, Pb) is the presence of two peaks at Pd ~ 1600 MeV/c andpd ~ 1370 MeV/c. The peak atPd ~ 1600 MeV/c is due to the quasielastic proton scattering on two-nucleon clusters, i.e. the reaction A(p, dN)B.

(1)

This process was observed earlier in refs. 2, 3). The small shift of the maximum from

584

L . S . A Z H G I R E I et al.

the point Pd = 1625 MeV/c, corresponding to free pd scattering, is in agreement with the value of the deuteron separation energy (10-20 MeV). The fact that the width of the peak corresponding to reaction (1), Ap ~ 150 MeV/c, evidently exceeds the momentum resolution of the magnetic spectrometer, appears to be connected with the intranuclear motion of two-nucleon clusters. If a Gaussian momentum distribution is assumed for the two-nucleon clusters then one may estimate the average kinetic energy of two-nucleon clusters, T<2N>. Such an estimation leads to the value T<2N> ~ 15 MeV. The peak in the region ofPd = 1370 MeV/c corresponds kinematically to deuteron production in the reactions: p + p ~ d+rc +, (2) p + n --+ d + r r °,

(3)

on the bound nucleons inside the nucleus, with deuteron emission in the forward direction (c.m.), i.e. to the reaction A(p, dTr)B.

(4)

In contrast to the (p, drr) reaction for deuterium, the peaks corresponding to reaction (4) with deuteron emission in the backward direction (c.m.) do not manifest oaJ~

o.e.9 s.

o

/A o o,,

a4 r

"[

oos~

i

'

;'

a o~;

o:s

?* ~

',

** J~e

. . . . "'"' . .

oo~

.~

.

~

~'!"°°~~,%*.;.~

'

~

o/ ' x

~

o~, p Jee

M~/C

ii

oo,i,

i Fig. 2. M o m e n t u m spectra o f the deuterons produced by 670 MeV protons at 9.5 °. The curves for all the nuclei except X H a r e the results o f the calculations on the basis o f the dispersion theory of direct nuclear reactions (notation is the same as in fig. I).

FAST DEUTERONS

585

....

¢c~.-cl

i

i '

~

' -;~-"

~o

~

i~ ~

,,co au4~ ~ / c sz /

oo~[

¢,s.,3 5.

::°ii;

o o0,i

....

o

r! ! ~

Fig. 3. Momentum spectra of the deuterons produced by 670 MeV protons at 13.5°. The curves for all the nuclei except ~H are the results of the calculations based on the dispersion theory of direct nuclear reactions (notation is the same as in fig. 1).

themselves in the deuteron spectra for the more complex nuclei. This may be caused by the broadening of the peak due to the intra-nuclear motion of the nucleons and by the contribution from other reactions, including the processes of deuteron production accompanied by the decay of the final nucleus. Evidence for the latter is provided by the fact that the ratio of the height of the soft part ( ~ 1200 MeV/c) of the momentum spectrum to the peak height at Pd ~ 1360 MeV/c grows with increasing mass number. The momentum spectra of the deuterons produced from 1H, 2H, 6Li, 7Li, C, AI, Cu, Rh and Pb at 9.5 ° are shown in fig. 2. In these spectra, as in the case of0p = 6.5 °, maxima at Pd ~ 1600 MeV/c and Pd ~ 1300 MeV/c corresponding to the deuterons from reaction (1) and the deuterons emitted in the forward direction (c.m.) from reaction (4), are observed. In comparison with the spectra measured at 6.5 °, the broadening of the peak from the reaction (4) and a relative increase of the contribution from the low-momentum deuterons are found at 9.5 ° . This is particularly significant in the case of the heavy nuclei (Cu, Rh, Pb). Its cause is, apparently, the contribution from processes in which the nucleus carries away a considerable portion of the energy of the incident proton, as well as the contribution from the rescattering

586

L.S. AZHGIREI et al.

of the pions accompanying the deuterons in reaction (4). The last mechanism will be discussed in sect. 4. The spectra of deuterons emitted at 13.5 °, i.e. at the limiting angle for deuteron emission from the reaction pp ~ dn + on free protons with 670 MeV incident protons are presented in fig. 3. For 2H, Li, C, deuteron spectra have a wide

.....

..... .

oo~

~

t

Fig. 4. Momentum spectra of the deuterons produced by 670 MeV protons at 16°. The curves are the results of calculation based on the dispersion theory of direct nuclear reactions (notation is the same as in fig. 1).

m a x i m u m a t p = 1100 MeV/c, which corresponds to deuterons emitted at the limiting angle in the reaction pp -~ dn +. In the deuteron spectra for heavier nuclei the peak from reaction (4) does not appear, and the probability of deuteron emission fails monotonically with increasing deuteron momentum. For deuterium (fig. 3) a peak is found at 1570 MeV/c corresponding to the elastic pd scattering. For the 6Li and 7Li nuclei a 1570 MeV/c peak corresponding to the quasielastic scattering of protons on the two-nucleon clusters is observed. For heavier nuclei the corresponding peak is not seen against the background of the great contribution from reaction (4) in this m o m e n t u m region. The spectra of deuterons emitted at 16 ° from 6Li, 7Li, C, A1, Cu, Rh and Pb are given in fig. 4. In the Li spectrum a wide peak a t p a = 1000 MeV/c is observed. Although these spectra have been measured beyond the limiting angle for deuteron emission from the p p - * d~r+ reaction, deuterons produced in the

FAST D E U T E R O N S

587

p ~ drc reaction on nucleons moving in backward direction within the nucleus can contribute to the indicated m o m e n t u m region. The deuteron spectra for the heavier nuclei have a monotonic behaviour in the lowm o m e n t u m region. Here, apparently, deuterons produced in multiparticle processes (for example, cascades) contribute. As to the deuterons from reaction (1), their contribution at 16 ° is hardly seen against the background of the other reactions. 3.2. DIFFERENTIAL CROSS SECTION In the determination of the differential cross section of reaction (1) the high-momentum part of the deuteron spectra was approximated by a Gaussian curve. The calculation of the m o m e n t u m distribution in the pole approximation without taking into account the form factor (see sect. 4) leads to values for differential cross sections differing from those found with Gaussian curves by no more than 10 ~o. Values of differential cross sections obtained in that way at angles of 6.5 °, 9.5 °, 13.5 ° and 16 ° are presented in table 1. The differential cross section of the reaction (p, dN) at 6.5 ° for carbon agrees with the results of ref. 2) within the errors. The differential cross sections corresponding to the other part of the deuteron spectra are shown in table 2. The experimental errors given in table 1 and 2 include, in addition to the statistical TABLE 1 Differential cross sections of the p<2N > -~ Nd reaction (mb/sr) Target 2H 6Li 7Li C AI Cu Rh Pb

6.5 o a)

9.5 °

13.5 °

16 °

0.5±0.05

0,47±0.06 1,7 i 0 . 2 1,9 ~ 0 . 2 2,6 ~ 0 . 3 3.7 ±0.5 5.2 ± 0 . 7 6.0 ±0.8 6.9 ±0.9

0.34~0.04 1.2310.16 1.0 ±0.1 1.5 ± 0 . 2 2.3 ±0.3 3.0 ± 0 . 4 3.8 ±0.5 5.7 ±0.7

0.9!0.7 0.8±0.1 1.4±0.2 1.8±0.2 2.4±0.3 3.0±0.4 4.6±0.6

2.1±0.2 2.7±0.3 4.3~0.6 5.6±0.9 5.6±0.9 7.1±1.0

~) Ref. s).

errors, also the errors due to the correction of the deuteron spectra and the experimental errors of the cross sections used for the absolute normalization. Furthermore, some uncertainties appearing in the approximation of the experimental data by the Gaussian distribution have been included into the errors of the cross sections for reaction (1). For light nuclei (2H, Li, C) the reaction (p, drQ gives the main contribution to the cross sections presented in table 2 at the deuteron angles 6.5 °, 9.5 ° and 13.5 °. In other cases the values mentioned in table 2 should be considered as upper limits of the differential cross section for this reaction. The dependence of the differential cross sections de/d[2 (p, dN) and dcr/d[2 (p, dr0 on the target mass number A is shown in fig. 5.

588

L. S. A Z H G I R E I

et al.

TABLE 2 Differential cross sections o f the p --~ d~ reaction (mb/sr) Target

6.5 ° ")

1H ZH 6Li 7Li C A1 Cu

14.94-0.4 17.1 4-1.7

9.5 ° 1 6 . 4 + 1.0 14.6 4-1.0 28.5 4-2.0 23.0 4-2.0 27 4-2 40 4-3 61 4-5 76 4-6 93 4-7

25.54-4-2.4 27.94-2.5 46 4-4 64 4-6 77 4-7 96 4-7

Rh

Pb

13.5 °

16 °

19.1 ± 0 . 3 10.6 4-0.5 17.04-1.4 14.9 4-1.3 22.44-1.9 33 4-3 52 4-4 67 4-6 82 4-7

7.7 4-0.5 8.5 4-0.6 11.84-0.8 21.64-1.5 32.7-F2.3 43 4-3 58 4-4

a) Ref. 5).

The experimental data for both reactions are well described by the function da/df2 A =. The x-values for processes (1) and (4) obtained by the least-squares method are shown in fig. 6. The value x for the process (1) is close to ½. Apparently, this is an indication that deuterons in this reaction are knocked out from the nuclear peripheral region. For the reaction (4) the quantity x at 6.5 ° has the value ½ and grows with increasing emission angle (at 0a = 16 °, x ~ ½). This growth may be caused by the d(~ I0 -e7 Cm2

t-d"~-

5F

-E'f (p, dN)

50i~ 6

fO"27 c r u z

sz

e~=Z.O"

(P'd'rO

¢

@=16,0 °

o

@d= 13,5"

o

°

e~=95 °

© 0

o

e,y=65"

i

c

~0 _

¢¢

5 ~-

~';-

~d=13.5"

~

8a=9,5"

¢ ~

L [

~

6

ed=6, 5°

50",7

c c

50~-

o

~0 F

¢

~o

I

' Li~TC Ae

°

I

L

10

¢o

100 Cu Rh Pb

O

o

A D

/YC

Ae

Cu Rf~ Pb

Fig. 5. D e p e n d e n c e o f the (p, d N ) a n d (p, dr0 r e a c t i o n differential cross sections o n the m a s s n u m b e r A o f the target nucleus.

FAST DEUTERONS

589

(p, dU) O,5

+ .~-- . . . . . .

0,5

o . . . . . .

(p, dE)

5"

.

fO'

15" ed (ta~)

Fig. 6. D e p e n d e n c e o f the q u a n t i t y x in the a p p r o x i m a t i o n dcr/dg2 ~ A ~ on the angle o f the deuteron emission.

...f0.27 cm 2 S;t (p, dJ¢)

c LL

5"

~

~0"

,¢0-z7 ¢ m 2 SZ

¢5"

8d ( ta6 ) (p, a~)

100 Rh

Cu 60,

Oa (eat~) Fig. 7. A n g u l a r dependence o f the (p, d N ) a n d (p, d~) reaction differential cross sections. T h e curves in the figures join the points for the s a m e nucleus. T h e d a t a for 6Li coincide within the limits o f errors with the VLi data.

590

L.S. AZHGIREI

e t al.

growing contribution from other channels in the low-momentum region of a deuteron spectrum. The angular dependence of the differential cross sections for processes (1) and (4) is shown in fig. 7. As it is seen from the figures, the values of da/df2 for both processes monotonically fall when the angle is increased. At the same time the angular dependence of the quasielastic pd scattering cross section is similar to that of the free pd scattering cross section. On the other hand the behavior o f p N ~ d~ differential cross sections for free and bound nucleons in the considered angular region strongly differs. In particular, a strong increase of the angular distribution near the limiting angle for reaction (4) is not obtained. The ratios of the differential cross sections for processes (1) and (4) to the differential cross sections for the corresponding reactions on free deuterons and nucleons, i.e. the quantities da Y/DA ~- ~ (P, a N )

da

/ dda~ (pd ~ /

pd),

(5)

(pp

(6)

for different nuclei are presented in table 3. The quantities nDA and nNA are sometimes called the effective numbers of mass-two clusters and of nucleons, respectively, in TABLE 3 T h e quantities n o . a n d ?tNA le

IH 2H 6Li 7Li C AI Cu Rh Pb

6.5 °

9.5 °

HDA

I1NA

1

1 I. 15 ± 0.05

4.22±0.17 5.404-0.22 8.6 4-0.4 11.1 4-0.5 12.1 4-0.5 14.0 4-0.5

HDA

1 3.62±0.14 1.714-__0.07 4.044-0.16 1.774-0.08 5.534-0.22 3.094-0.13 7.9 4-0.3 4.27_40.19 11.1 4-0.4 5.11-b0.22 12.8 4-0.5 6.5 ± 0 . 3 14.7 ± 0 . 6

13.5 ° /7NA

nDA

nNA

1 0.69 ± 0.04 1.744-0.07 1.414-0.06 1.634-0.07 2.43±0.10 3.70+0.15 4.634-0.19 5.674-0.23

1 3.624-0.15 2.94±0.12 4.504-0.18 6.8 4-0.3 8.8 ± 0 . 4 11.2 4-0.4 16.8 ± 0 . 7

1 0.56 ± 0.02 0.80~0.04 0.78±0.04 1.17±0.05 1.73__40.07 2.714-0.11 3.514-0.14 4.30±0.17

the nucleus A. It is seen from table 3, that the nDA values for a given nucleus do not vary strongly when the angle varies from 6.5 ° to 13.5 °. This justifies, to a certain extent, the use of the concept of an effective number as a notion describing some properties of the internal nuclear structure. In contrast to noA, the quantity nNA for a given nucleus falls with increasing the deuteron emission angle as expected from the difference between the angular dependences of the p N --* dn reaction cross sections on free and bound nucleons. It is, probably, a reflection of the fact that the energy dependence of the p N ~ dn reaction

FAST DEUTERONS

591

cross section near 670 MeV has a resonant character. When the p ( N ) ~ dn reaction proceeds on nucleons moving inside the nuclei, the different deuteron angles correspond to different parts of the momentum spectrum of the intranuclear motion. In view of this, the (p, dn) reaction cross sections at different angles should be effectively compared with values of the pN ~ d~ reaction cross sections taken at different energies of the incident protons and averaged in the proper way. Therefore the presented nNA values may be considered as effective nucleon numbers only approximately. 4. Discussion of the results

The calculations of the deuteron momentum spectra at 6.5 ° for carbon based on the independent particle model have been described in our previous work 4). Qualitative agreement in the shapes of the experimental and theoretical distributions has been observed. It may be of interest to describe the experimental data by means of the dispersion theory of direct processes 1o). Here the reaction mechanism may be represented by a finite number of those Feynman graphs, whose singularities are the nearest to the physical region. The calculations from both the pole and the triangle graphs are considered in this work. The results of the calculation for the processes p + A -~ d + N + B ,

(la)

p+A ~ d+zr+C,

(4a)

are given in subsects. 4.1 and 4.2 where A is the initial nucleus, while B and C are the final ones. 4.1. THE p--A---> dq-Nq-B REACTION

The results of the calculations for the pole diagram (fig. 8) performed without taking into account a form factor and under the assumption that the final nucleus is in its

(2N)

N

B

Fig. 8. Pole diagram for pq-A ~ d + N q - B .

ground state, are shown in figs. 1 - 4 . The calculated curves are normalized with respect to the experimental spectrum at 1600 MeV/c. It is clear from figs. 1-4 that the theoretical curves describe fairly well the experimental data for all the nuclei investigated. Similar calculations were also carried out taking into account the form factor.

592

L . S . A Z H G I R E I et al.

Here Butler's form of the vertex part 11) has been chosen, where the orbital m o m e n t u m of the relative motion of virtual deuteron and the residual nucleus B was taken equal to zero. The comparison of the deuteron spectra calculated with the form factor, with the experimental ones allows one to obtain the deuteron reduced with tg~ of the decay A ~ B + d. The channel decay radius was taken to be equal to the nuclear radius. As the state of the final nucleus was not fixed in the experiment, the reduced widths obtained have the meaning of total reduced widths.The reduced widths calculated for the escape angle of 6.5 ° are presented in table 4. For the other angles the O2d values agree, within the errors, with the data presented in table 4. TABLE 4 The data for @dz Nucleus Od2

7Li 2.4~0.5

C

A1

Cu

Rh

Pb

4.4--0.8

7.44-1.4

8.6±1.6

10.3_2.0

12.6±2.5

The result obtained for carbon is in agreement, within the experimental error, with the values 3.8___ 1.0 and 2.74-0.7 obtained in ref. lz) for the deuteron escape angle of 13 ° at initial proton energies of 1260 MeV and 730 MeV, respectively. It should be noted that the values 0 2 depend on the nuclear mass number as A ~. 4.2. T H E p + A ~ d + T t + B R E A C T I O N

The results of the calculation based on the pole diagram of fig. 9a, performed without taking into account a form factor, are given in figs. 1-4. The final nucleus was assumed to remain in the ground state.


~

A

B (a)

A

(B> (b)

B

Fig. 9. Pole diagram for p + A ~ d + ~ - l - B .

The calculated curves are normalized with respect to the experimental spectrum for 6.5 ° at about 1375 MeV/c. As seen from fig. 1, the calculated curves for 2H and 7Li at 0d = 6.5 ° describe the experimental data satisfactorily. For the heavier nuclei, agreement with the experimental data is found only in the region of the peak at 1375 MeV/c. In the low-momentum region the calculated curves are lower than the experimental points. The difference grows with increasing mass number. The calculated curves for the angles 9.5 ° and 13.5 ° (figs. 2 and 3)are systematically lower than the experimental points. This is, apparently, due to the fact that more complicated diagrams,

FAST DEUTERONS

593

including the rescattering of the secondary particles on the residual nucleus, contribute to the p + A ~ d + r c + B process at large angles. As follows f r o m the kinematics o f the pp ~ drr ÷ process, the diagrams with pion rescattering on the residual nucleus must essentially contribute at 0d ~ 10 °, since in this case the pion energy corresponds to the m a x i m u m in the cross section o f the pion-nucleus interaction (fig. 10). ed

(ta6)

I

10'

-27

~t~ncfe),lO cnz

80O

6O0

5~

"~

,/

400

// 2O0

100

200

300

r~rla6

Mev"

Fig. 10. Solid curve-angular dependence of the deuteron emission of its kinetic energy (lab system) in the pp --~ dzr+ reaction. Dashed curve -- dependence of the zrC total cross section on the meson kinetic energy. The calculations carried out with the Butler form factor scarcely vary the shape o f the spectrum. As an example, the result o f the calculation with the form factor (1 = 1) for carbon at 0d = 6.5 ° is shown in fig. lc. Such calculations enable one to estimate the total p r o t o n reduced widths for the investigated nuclei. The calculations were made under the assumptions that the final nucleus is in its g r o u n d state and that the reduced p r o t o n and neutron widths are equal. Here the radius o f the channel was taken either from well-known data, or was considered to be equal to the nuclear radius according to the formula R = 1.35 A ~. The total p r o t o n reduced widths obtained for an angle of 6.5 ° are given in the fourth column of table 5. The p r o t o n reduced widths for the virtual transitions 7Li ~ 6 H e + p and 1ZC --* I~B + p are presented in the fifth column of table 5. In their calculation it has been assumed that the ratio o f the probability o f the transition to the g r o u n d state of the residual nucleus to the probability o f the transition to other states for the vertex 7 Li ~ 6He + p is equal to 0.5 [ref. ~6)], while for the transition tZC ~ 11B+ p it has been considered to be equal to the corresponding ratio for the vertex a2C -~ 11C + n, which is equal to 0.6 [ref. 17)], according to the data from the 12C(p, d ) ~ t c pick-up reaction.

594

L. S. AZHGIREI et al. TABLE 5 The proton reduced widths

Nucleus

R fm

1

Ptot

Pgr

Ref. ~4)

2H 7Li C

4 a) 5 a) 4.3 a)

0 1 1

0.135:0.02 1.1 :~0.2 1.5 5:0.3

0.555:0.10 0.9 :L0.2

0.39 0.21 0.23 0.34 0.26 0.98

A1 Cu Rh Pb

1.36 • A~1.35 • A~1.35.Ak 1.35 • A~

l 1 1 1

0.8 1.6 2.5 2.5

:~0.2 5:0.3 ~0.5 :]:0.5

~) Ref. la).

T h e vertex p a r t o f the d ~ n + p t r a n s i t i o n m a y be f o u n d from np scattering theory at small energies. Its value, I~al 2, in the linear a p p r o x i m a t i o n with respect to the effective r a d i u s rd, is equal to 17dl2 = 4(1 + k r a ) , where k is the nucleon wave n u m b e r o f d e u t e r i u m a n d rd = 4 fm. T h e value o f the r e d u c e d width, 0 2 = 0.19 o b t a i n e d from the r e l a t i o n s h i p between Ivd[2 a n d 0 2 [ref. 15)] is in agreement with the value 0.13 _ 0.02 o b t a i n e d in this work. The p r o t o n r e d u c e d widths o b t a i n e d in o t h e r w o r k s are given in the sixth c o l u m n o f table 5. It m a y be seen t h a t the 0 2 g, values are consistent with those results. One m a y examine this a g r e e m e n t as a n evidence for the d o m i n a t i n g c o n t r i b u t i o n o f the p o l e g r a p h in the p + A -~ d + rt + B process at the angle 6.5 °. To i m p r o v e the description o f the d e u t e r o n m o m e n t u m s p e c t r a at 9.5 ° a n d 13.5 °, the calculations have been c a r r i e d out t a k i n g into a c c o u n t the effect o f p i o n rescattering on the residual nucleus. T h e calculations were based on the triangle graph, fig. 9b. In figs. 2c a n d 3c the calculated curves for c a r b o n at 0d = 9.5 ° a n d 13.5 ° are shown. It is clear that the c o n s i d e r a t i o n o f the p i o n rescattering on the residual nucleus leads to a better description o f e x p e r i m e n t a l data. In b o t h cases the c o n t r i b u t i o n s caused by the rescattering effect are c o n s i d e r a b l y large (from 10 ~o u p to 50 ~o o f the total d e u t e r o n yield in the l o w - m o m e n t u m region), while the rescattering effect at 6.5 ° does not p l a y any essential part. The d e u t e r o n m o m e n t u m spectra for d e u t e r i u m a n d 7Li (figs. 1-3) are also well r e p r o d u c e d within this model. As regards the heavier nuclei (AI, Cu, Rh, Pb), even with c o n s i d e r a t i o n o f rescattering, the calculated curves are lower t h a n the e x p e r i m e n t a l points. A s m e n t i o n e d above, c o n t r i b u t i o n s f r o m multiparticle processes should be taken into account. Thus, the p i o n rescattering effect on the residual nucleus a p p e a r s to p l a y an essential p a r t in the m e c h a n i s m o f the p + A --, d + zr + B r e a c t i o n at large d e u t e r o n angles.

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