A model for the production of (p, n) and (p, 2n) reactions at high energy

Nuclear Physics A151 (1970) 449--458; ( ~ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

A MODEL

FOR

THE

PRODUCTION

O F (p, n) A N D (p, 2n) R E A C T I O N S

AT HIGH

ENERGY

D. J. R E U L A N D Indiana State University, Terre Haute, Indiana t and A. A. CARETTO, Jr. Carnegie-Mellon University, Pittsburgh, Pennsylvania t Received 27 October 1969 (Revised 25 March 1970)

Abstract: The ratios of the cross sections for the (p, 2n) reaction to that of the (p, n) reaction at 400 MeV have been calculated for a variety of target systems from a theoretical model and compared with the experimental ratios. The assumed mechanisms were a p-n scattering followed by evaporation of one neutron for the (p, 2n) reaction and a p-n scattering with no subsequent evaporation of particles for the (p, n) reaction. The large variations of the ty(p, 2n)/a(p, n) ratio within small mass regions experimentally observed are well represented by this calculation. For target systems where the isobaric analogue states lie above the separation energies of the least bound neutrons of the p-n products, the magnitudes of these ratios as well as their variations with mass number are predicted reasonably well. For target systems where the isobaric analogue states lie below the separation energies of the least bound neutrons, the variations of the crosssection ratios with mass number are again accounted for; however the magnitudes of the ratios are high by factors of 2 to 4. This could indicate that enhanced population of the isobaric analogue state in the p-n scattering event does indeed enhance the (p, n) cross section for these systems and thereby reduce the a(p, 2n)/cr(p, n) ratio considerably.

1. Introduction T h e (p, n ) a n d (p, 2n) r e a c t i o n s l e n d t h e m s e l v e s well to t h e o r e t i c a l t r e a t m e n t d u e to the l i m i t e d n u m b e r o f m e c h a n i s m s w h i c h c a n l e a d to these r e a c t i o n s . A few studies ~ - s ) h a v e r e c e n t l y b e e n r e p o r t e d o f these r e a c t i o n s in the e n e r g y r a n g e o f 100-400 M e V . T h e results o f these studies i n d i c a t e t h a t f o r b o t h t h e (p, n) a n d (p, 2n) r e a c t i o n s , c o n s i d e r a b l e s c a t t e r i n g is o b s e r v e d in p l o t s o f cross section versus m a s s n u m b e r . G r o v e r a n d C a r e t t o 6) s u g g e s t e d t h a t these large v a r i a t i o n s o f cross section w i t h i n a n a r r o w m a s s r e g i o n m i g h t be due p a r t i a l l y to e n h a n c e d p o p u l a t i o n o f the i s o b a r i c a n a l o g u e state in t h e (p, n) s c a t t e r i n g event. T h u s , w h e n the i s o b a r i c a n a l o g u e state lies at a n e n e r g y a b o v e the g r o u n d state g r e a t e r t h a n the n e u t r o n s e p a r a t i o n e n e r g y o f the r e s i d u a l n u c l e u s f o l l o w i n g the (P, N ) c a s c a d e tt the (p, 2n) cross section w o u l d be e n h a n c e d r e l a t i v e to the (p, n) cross section. t Work sponsored in part by the US Atomic Energy Commission. tt Upper case letters such as (P, N) or (P, P) refer to individual cascade events, the overall reaction is given by the conventional lower case letters. August 1970

449

450

D . J . REULAND AND A. A. CARETTO Jr.

Church and Caretto s) studied (p, xn) reactions at incident proton energies between 100 and 400 MeV. Their data indicate that the principle mechanism leading to a (p, n) reaction is a p-n or charge exchange scattering event in which the excitation energy of the residual nucleus is less than the separation energy of the least bound particle of the residual nucleus. This excitation energy is comprised of both the neutron "hole" energy and the proton binding energy. The primary mechanism producing (p, 2n) reactions is thought to be a p-n or charge exchange scattering in which the excitation energy is greater than this neutron separation energy but less than that required to evaporate off any more particles. Vegas Monte Carlo calculations substantiate these as the dominant mechanisms producing (p, n) and (p, 2n) reactions. However, as with other simple nuclear reactions the Monte Carlo calculations fail to predict accurately the magnitude of these reactions. It was the objective of this work to develop a model for (p, n) and (p, 2n) reactions consistent with the previously mentioned mechanisms which better accounts for both the magnitude of these reaction cross sections and their variation with mass number. The (p, n) and (p, 2n) reactions were studied for the following systems: 63Cu, 68Zn, 72Ge ' t 24Te ' 125Te and 126Te" The Te systems were selected because the cross sections for these reactions display large variations within this narrow mass number range of 3 units. Also, the Te systems represent targets in which the isobaric analogue state lies above the neutron separation energy of the residual nucleus following the (P, N) cascade; whereas, the other systems represent targets in which the isobaric analogue state lies below this neutron separation energy. Thus, the importance of the enhanced population of the isobaric analogue state on these reactions could also be investigated.

2. Calculations

Assuming that a clean knockout of a neutron by the incident proton by either p, n or charge exchange scattering is the principle mechanism leading to both (p, n) and (p, 2n) reactions, the ratio of the cross section of the (p, 2n) reaction to that of the (p, n) reaction for a given target system is approximately given by: j=oo

a(p, 2n) a(p, n)

~_, P(j)E.(j)

~,~ j = o

j=s=

(1)

Z P(J)

j=O

where P(j) is the probability of a clean p-n scattering event depositing excitation energyj in the residual nucleus, E,(j) is the probability of evaporating one neutron from the nucleus of excitation energyj and Sn is the neutron separation energy of the residual nucleus following the (P, N) cascade. The excitation energy of the residual nucleusj is the sum of the neutron hole energy

(p, n) (p, 2n) REACTIONS

451

and the p r o t o n binding energy. The p r o t o n binding energy is determined by the energy above the Fermi sea of protons of the level into which the p r o t o n is scattered. The neutron hole energy is determined by the depth below the Fermi sea of neutrons of the level in which the struck neutron resided prior to collision. The probability that a proton will be scattered following the (P, N) cascade into a level with binding energy i is given by: 2rc (0,1 d~r sin 0' dO',

P(i) = 32.3.J,o'2 dQ

(2)

where the limits of the integration 0~ and 0'1 are the c.m. scattering angles associated with a p r o t o n being scattered into a level of binding energy i and reflect the width of this level. The factor 32.3 is the total elastic scattering cross section of protons on neutrons 7,8). Using the expression for the differential cross section in the energy range of interest given by Clements and Winsberg 8) and integrating, one obtains the following equation:

P(i) -

2~z 32.3

x [, - 1.68 cos 0' - 0.80 cos 5 0' - 6.93 × 10- 2 cos 15 0' - 3.96 × 10- 2 cos l o 1 0 ,30'2. 0'1

(2b)

This probability was calculated as a function of the p r o t o n binding energy i in increments of 1 MeV for neutrons in each Fermi level, L, up to im,x where /max

= ( r n p + M zA- M z + Al )+913 1 . 4 + 0 . 7 V c ;

Vc being the C o u l o m b i c barrier seen by the proton. F o r this calculation the p r o t o n level density and level width of the target system must be known as a function of excitation energy. The level densities were assumed to be of the f o r m

W(i) = C exp 2[,a(i-7)] ½,

(2c)

where a is the level density p a r a m e t e r taken as ~ A M e V - a i is the excitation energy in M e V and ~/is the pairing energy term obtained f r o m Newton 9). The total probability of scattering a p r o t o n into levels between binding energy i and ( i + 1) M e V f r o m a neutron in level L is given by

P(i, L)

2re

-

-

k=W(i)

y ' [--- 1.68 cos 0 ' - - 0 . 8 0 cos 5 0 ' - - 6.93 x 10 - 2 c o s 15 0 ' 32.3 k=l --3.96 X 10 .2 COSI°1 v.10,2k, n,lo',~

(3)

where the limits O'lk and 0~k reflect the width of the kth level. Thus, 2re

P(i, L) "~ ~ . 3 [ , - 1 . 6 8 cos 0 ' - 0 . 8 0 cos 5 0 ' - 6 . 9 3 x 10 -2 cos 15 0 ' - 3 . 9 6 x 10 .2 cos 1°10']°o~i,+l,(xC exp 2[,o(1-7)]~),

(3a)

452

O.J.

R E U L A N D A N D A. A. C A R E T T O Jr.

where 0~ is the c.m. scattering angle associated with the scattering of protons from neutrons in level L, into proton levels of binding energy i, and 0'i + 1 is this angle for scattering of protons into a level of binding energy i + 1 MeV. The average level width in MeV is represented by x. In the computation whenever x C e x p 2 [ a ( i - 7 ) ] ~ is greater than unity, it is set equal to 1 and eq. (3a) becomes equal to P(i, L ) rather than approximate. The value of x C used throughout was taken to be 2.0 x 10-a. The average energy of the scattered protons following the p-n scattering event associated with a given c.m. scattering angle O' was calculated from:

E = ½(Ep.-l.-En)+½(Ep-En) cos 0'-4-

PpPn [2(B + 1)3e

sin ~o sin 0 sin 4~,

(4)

where Ep and Pp are the energy and m o m e n t u m of the incident proton inside the nucleus, E n and P , are the energy and m o m e n t u m of the struck neutron, O' is the c.m. scattering angle, ~b' is the c.m. azimuthal angle and co is the lab angle between the m o m e n t a vectors of the incident and struck particles 10); B is given by the expression B = 7pTn--PpPn COS ~0, where 7p and ?n are given by 1

v being the velocity of the incident proton or the struck neutron in the lab system. In these equations momenta are expressed in units of m oc, energies in units of m oc2, and thus the total energy of a particle in these units is 7 = 1 + E . Both the neutron and proton is assumed to have the same rest mass, m o. A Fermi gas distribution of neutrons in the target system was assumed. The energies of the scattered protons were calculated by systematically varying the angle O' in increments of 5 °, the angle qS' in increments of 5 °, the angle co taken at - 9 0 °, 0 ° and 90 ° and the Fermi level of the struck neutron L, from the bottom of the well, L = 1, to the top of the well, L = ½N. In this manner, the average energy of the scattered protons for a given scattering angle 0' and Fermi level of struck neutronL, was calculated. Energy distributions for perpendicular collisions, e) = - 9 0 ° or 90 °, were weighed twice as heavily as head on collisions, ~o -- 0 °, reflecting the belief that (P, N) cascades occur predominately at the nuclear surface where perpendicular collisions would be expected to be more likely than head on collisions. The hole energy, EHL(L), associated with the abrupt removal of a neutron from a level L was calculated by subtracting the energy of this level from the maximum Fermi energy of the neutron distribution. The total excitation energyj of the nucleus following the p-n scattering event is given by j = E H L ( L ) + 1.

(p, n) (p, 2n) REACTIONS

453

The m a x i m u m excitation energy would result from a neutron in the lowest Fermi level, L = 1, scattering a proton into the highest energy bound proton level, Jmax ---- EHL(L) +/max" The probability o f the residual nucleus having excitation energyj was then calculated at intervals o f 1 MeV to Jmax by summing all scattering probabilities P(i, L) which could lead to this excitation energy. 7,0

I

I

I

I

z

9_

sl,.,;b v

I.i.i.. (,.3 o,~ i.iJ

6.0

5.C

4,0

w

g~

,o,_q 60

U.I

3.0--

of

,.if,p:, 2.0 -t--2= i.u I'O0

o ~1.0-I1::

0.0 122

124 MASS

126 NUMBER,

A

Fig. 1. The a(p, 2n)/cr(p, n) ratio versus mass number for target systems whose isobaric analogue state lies above the separation energy of the least bound neutron of the (p, n) product. Black dots - experimental points; triangles-calculated points. The ratio o f the (p, 2n) cross section to the (p, n) cross section was calculated from eq. (I). The evaporation probabilities En(j') were obtained from an evaporation program o f Kiely it). The Fortran IV language was used to write the programs for these calculations, and the programs were run on the Indiana State University IBM 360/30 or IBM 1130 computers. 3. Results The results o f this calculation are summarized in table 1 and figs. 1 and 2. Table 1 lists the experimental values and uncertainties o f the ratios of the (p, 2n) reaction