Proton Emission from Nuclei Due to Neutron Interactions 247

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PROTON EMISSION FROM NUCLEI. DUE TO NEUTRON INTERACTIONS

BY G. S. MANI AND K. G. NAIR* (Atomic Energy Establishment, Trombay, Bombay) Recr Septcmber 30, 1959 (Communicated by Dr. B. V. Thosar, F.A.SC.)

ABSTRACT

The energy spectra of protons emitted in neutron bombardment of Al ~~ is analysed in terms of the statistical model and volume direct interaction. It is found that the ditfuseness of the nuclear potential or the form of level density used does not alter the shape of the energy spectra very mueh. The available experimental data agree fairly well with theory. The contribution of direct interaction is seen to be small and hence does not alter the general shape of the spectra. The direct interaction cross-section obtained from the measurements of the angular distribution in the case of iron and copper agree reasonably well with theory.

INTRODUCTION IN recent years a large number of experiments 1, 2, s have indicated the inadequacy of the statistical theory in explaining the data. The two assump- tions of the statistical theory are (i) an intermediary compound system is formed which decays after a relatively long time, (ii) the contribution from the several overlapping levels in the highly excited compound system have random phases so that interference effects can be neglected. These assump- tions lead to symmetry about 90 ~ in the angular distribution and linear dependence of the spectra of the emitted particles to the level density of the compound or the residual nucleus. The angular distribution would further be isotropic ir the density of levels of total angular momentum J follows the (2J + I) law. 4 Most of the experimental results indicate that (i) the angular distribution is not symmetric about 90 ~ and is mainly peaked in the forward angles. This anisotropy increases with increasing energy of the emitted particle, corresponding to the low-lying states of the residual nucleus. (ii) The high

* Tata Instituto of Fundamental Research, Apollo Pier Road, Bombay 1. 243 244 G.S. MANI AND K. G. NAIR energy region of the spectra is much larger than predicted by the level density. Also the spectra exhibit peaks in the high energy region which correspond to the discrete low-lying levels of the residual nucleus. There are some indications of the preponderance of low energy protons in (n, p) reactions, s The angular distribution of the emitted particles which leave the residual nucleus in discrete low-lying levels has been explained by the surface direct interaction which closely resembles stripping and pick-up reactions. 6 Unfortunately the theory is at present unable to predict absolute cross- sections. The general increase in the high energy part of the spectra may be due to any one of the following reasons: (i) Modification of level density formula taking into account of the effective mass of nucleons inside the nucleus as shown by the optical model. 7 (ii) Volume direct interaction. 8-1~ (iii) Interactions in which a large number of nucleons take part in the reaction like spot heating of the nucleus and collective excitation. The preponderance of low energy protons in (n, p) reactions may be due to the diffuseness of the nuclear potential, 11 modification of the level density formula taking into account the effect due to pairing energy as suggested by Newton1~ and contributions from reactions like (n, np) due to eompound as well as direct processes. In this paper we shall investigate the effects of these various factors on the shape of the energy spectra of protons emitted from Al ~7 nucleus bom- barded with neutrons. It has been pointed out 13 that volume direct interaction does not contribute appreciably to the total direct interaction cross- section, especially in heavy nuclei, due to surface reflection and attenuation in nuclear volume. The effects of surface reflection and attenuation would be predominant if R >~ A where R is the nuclear radius and ,~ is the mean free path of nucleons in nuclear matter. Since ?t ,~ 3"3 • 10-13 cm. for the energy region under consideration,9,1' the above effects would not appreciably diminish the volume direct interaction cross-section for nuclei with mass number A ,-~ 60. To test this assumption, the experimental value of the direct interaction cross-section for iron and copper has been compared with the theory of volume direct interaction.

THEORY Neutron bombardment of Al ~ would yield the following reactions: 1. (n, p) and (n, pn) due to compound process. 2. (n, n'), (n, n' n) and (n, n' p) due to compound process. Proton Emission from Nuclei Due to Neutron interact/ons 245

3. (n, p) and (n, pn) due to direct process. In the case of (n, pn) the neutron can be directly ejected simultaneously with the proton or can boil off the residual nucleus. 4. (n, n'), (n, n' n) and 01, n' p) due to direct processes. In the case of (n, n' n) and (n, n' p) the second neutron or proton can be simultaneously ejected with the ¡ neutron or can boil off the residual nucleus. 5. 01, a) and (n, cLn) reactions. Reactions like (n, y), (n, d), 01, t), (17, 2.o) are neglected either because their cross-sections for energies of incident neutrons up to 20 MeV are negligible x5 or because they are not energetically possible. The spectra of protons from (n, p) and (n, pn) reactions are given in the statistical theory by Blatt and Weisskopf. TM This expression is valid only when the level density follows the (2J-b-l) law.

n (c) de = ~rIc~e (`) oJ (c 0 -- c) dE ~' F, (l) V where ,;~ is the cross-section for the formation of the compound nucleus due to the interaction of the target nucleus with incident neutron, ,re (c) is the cross-section for the formation of the compound nucleus due to the interaction of the residual nucleus in the excited state with the emitted proton of energy r co is the maximum possible energy for the emitted proton and involves the incident neutron energy and Q value of the reaction. ~o (c*) is the level density of the residual nucleus at the excitation energy, c*= c0--E, S F, is the total probability for the compound system P to decay into the various channels v. Since (n, po) for the energies under consideration is very improbable due to Coulomb barrier, we have assumed that only a neutron can boil off the residual nucleus. For the energies of interest we have assumed that simultaneous evaporation of two or more particles from the compound nucleus is improbable. As will be seen later the experimental data seem to support this approximation.

The various cross-sections have been evaluated using the following two formul~e for the level density: o) (c) = C exp. [2a~c ~] '(2) o~ (c) = C exp. [2a t (c -- AE) ~] (3) where 'a' is a measure of the nuclear temperature and we have assumed it to be the same for all nuclei under consideration. 'C' is a constant whicla 246 G.S. MANI AND K. G. NA1R

may vary widely from nucleus to nucleus and in effect it determines the level density for the particular nucleus. A E is the pairing energy for the nucleus under consideration and has been evaluated by Cameron. 17 Since it is very difficult to estimate the value of the constant C due to our lack of knowledge of level density in nuclei, we have eliminated it from the equations by using the experimental value of the total (n, p) cross-section.

fa cqC 5 CGe (E) exp. (2aiE q) dE [cr (,, p)]~o=,. = ,0-B. (4) Z' F, p where Bn is the binding energy of the last neutron in the residual nucleus and E' = (% -- ~) or E' = (Ea -- E --/1 E) depending on the level density formula used. [cr(n,p)],0=p. is the total cross-section for the (n,p) reaction going through the compound process. From (1) and (4) we obtain n (E) dE -- [o (n, p)]~o=p. ,C,e (~) exp. [2~t, '~] d, (5) lo J" Cae (E) exp [2ate 'j] dE whieh is independant of the constant C. [a (n, p)] ~o,,,. is obtained from the relation [,~ (,7, PI co~~. = [" (n, P)l,x,. - [o (n, p)] d,,. (6) where [cr (n, p)],~p, is the experimental vatue for the total (n, p) cross-section and [cr (n,p)]dt~. is the total (n,p) cross-section due to direct interaction and is evaluated later [see eq. (14)]. The spectra of protons due to (n, np) reaction proceeding through the compound system is obtained as follows: Let N (En) den be the spectra of neutrons from the (n, n') reaction. Let Bn and Bao be the binding energies of the last neutron and proton respectively of the residual nucleus. No further proton emission is possible if n > (%n -- Bp) where %n is the maximum energy with which neutrons can be emitted in the (,, n') reaction. Similarly no neutron emission from the residual nucleus is possible ir en > (%n -- Bn). Let N (en, En') denden" be the number of secondary neutrons emitted in the range en' and En' q-d~n' after the emission of N(en) d~n primary neutrons. Similarly let N (en, era)dendEv be the distribution function for secondary protons. From the statistical theory we have N (En, en') dEndEn' --- Cn' K (en) ~e (en') exp. [2a~en"~) de n" (7) N (en, "la) aenaelJ = CI~ K (en) Ore (Ep) exp. (2a'eŸ delo (8) Proton Emission from Nuclei Due to Neutron Interactions 247

where Cn' and Cp are the C-values of the corresponding residual nucleus after secondary emission. K(En) is a constant of proportionality, en" and ~p" denote the excitation energies of the corresponding residual nuclei. Since we assume that no other mode of decay is possible [the (n, n' a) is neglected since the alpha particle, which has a relatively low energy, would have to overcome a much higher Coulomb barrier],

t egl 1,1'1II.x, N (en) den = Cn'K (en) O

~pm~r~ +%K(E.) I Epae (Ep) exp. [2a%p'~ ] dcp. (9~ O Hence K(E.)= N (en) den t ten miLY. t t I I~P7aX" Cn j" en% (En) exp. [2aten "~] den + Cp Ev%(ep) exp. [2a~Ep"j] dEp O O (lO) where

ep ,,,. = Con -- En -- Bp. (11) Hence if N (en) den and Cn'/Cp are known, the (n, np) spectrum is obtained from equation (8). The total (n, 2n) cross-section in Al r~ cannot easily be determined since the ground state of Al 26 has a very long half-life, (8' 105y).x6 Hence we cannot eliminate Cn,/Cp from the above equation. For Al ~7, the (n, 2n) reaction yields Al ~s which is an odd-odd nucleus while the (n, np) reaction yields the even-even nucleus Mg". FoUowing the prescription suggested by Brown and Muirhead, lo we have assumed Cn,/Cp = 10. This assumption will not alter the spectra very much since the (n, 2n) contribution in the energy region of interest is small, the threshold of the reaction being around 13 MeV. The (n, n') spectrum can be obtained in the same way as the (n, p) spectrum and we get N (,n) d,n = [~ (n, n')]co,,,, en~e ('n) exp. (2a~en"t) d, n (12) note (en) exp. [2aten "t] den O where the lower limit of the integral is taken as zero since we include aU reactions of the type (n, n'x). 248 G.S. MANI AND K. G. NAIR

[g (n, n')]c0,,p. = arte -- [~ (n, p) -t- ~ (n, pn)]c~,. -- [~ (n, p) + cr (n, pn)]dt,. -- [e (n, ~) + cr (n, an)].~. - [~ (n, n'h,,.] (13) where Vne is the experimentally measured non-elastie cross-section. [cr (n,p) + cr (n, pn)]~o,,~, is obtained from equation (5). [~ (n,p) + ~ (n, pn)]dir, and [~(n,n')]dl,. from equations (14)-(16). [~(n,~)+o(n, an)]~=p, has been evaluated from the experimental value of cr (n, a) following the same procedure as in the case of the (n,p)+ (n, pn) reaetion. We have chosen the total non-elastie cross-section since the (n, n') cross-section is not known for these energies. Thus from (10), (11), (12) and (8) we can evaluate the proton spectrum due to (n, np) reaction. The contribution of the direct interaction process to these various reactions has been evaluated as follows. The nueleus is assumed to be composed of two non-interacting Fermi gases and the coUisions of the incident neutrons with individual nucleons in the nueleus give rise to these reactions. These collisions are considered to be free nucleon-nucleon collisions except for the restriction imposed on their final momentum states by the Pauli exclusion principle. We shall not give below the complete derivation of the various cross-sections but only quote the final results since these have been derived in detail by Hayakawa et al. 9 and by Brown and Muirhead. lo The neutron-nucleon collision inside the nucleus may yield any one of the following reactions: (1) n--p collisions with both particles simultaneously ejected. (2) n--p collisions in which only the protons are emitted. In case (1) the final nucleus may boil off a neutron, thus yielding the (n, 2np) reaction. For the energy region under consideration this is not energetically possible. In case (2) the final nucleus may similarly boil off a neutron yielding the (n, pn) reaction. Since we disregard the (n, pp) reaction, calculation of the (n, p) wouki automatically yield the (n, pn) reaction also. (3) n--p collisions with only the neutron emitted. This would yield a part of the (n, n') process. The final nucleus may boil off a proton to yield the (n, n'p) ora neutron to give the (n, n'n) reaction. (4) n -- n collisions in which either or both of the neutrons are emitted. The latter would give rise to (n, 2n) reaction while in the former case the residual nucleus can boil off either a proton or a neutron to yield the (n, n'p) or the (n, 2n)reaction respectively. Proton Emission from Nuclei Due to Neutron lnteractions 249

If we obtain the effective (n, n') spectrum due to (3) and (4) with one of the particles remaining inside the nucleus, then we can calculate the spectrum of protons due to (n, n'p) reaction with the proton boiling off the residual nucleus as in the case of the compound process [cf. eq. (10)]. These various eross-sections are given by:

(d~p)np = --,Fpp'\dx2)T(')~(')r + Xnn] (14) [ Pp~np PnCtnnXnpJ r [ do "~ P~ ~,axz ) T (*n) ~ ('n) [1 -- T (,p) ~ (,~)] Xnn] (15) k ) nY = f lŸ p LP~anr~ + pnann XnzoJ &r

~'~p +

T (E) is the penetrability and ~ (~) takes into account loss due to attenuation in nuclear matter and due to rettection at the nuclear surface. e-O.75R/x%l (') = [i- -- { 1 -- T (,)} e-X'33n/x%~] (17) Where )t (~o) is the mean free path of nucleons inside the nucleus with energy (E + V), V being the depth of the nuclear potential and E the energy of the nucleon outside the nuclear fiel& Xnp (El) and Xnn (El) are the total cross-sections for free n--p and n- n collisions with incident neutron energy Ex. AII other notations are the same as in ref. I0. The functions (d~r/dxz'), (&r/dx2"), (&r/dx2"") which occur in equations (14)--(16) are given by:

)x2" = F (x~, x0 dx2 (18) where x," is x,', x,", or x,'" are given by x~' = l-(xl+Q'-xp) x2" = 1 - (xi - x,O x2'" = 1 -- (xi" -- xn') (19) A2 250 G.S. MANI AND K. G. NA[p,

where

X1 __ E: + V . Q' - Q xp - Ep + V

~.Fp ' -- EF----p , EFP Xn En + V x ' En + V , EI + V -- › p ; - n -- " X1 -- EFT ~ EF ~

Fp ----- V -- Bp; EF r : V -- Bn. (20) Bn and Bp are the binding energies of the last neutron and proton in the nucleus respectively.

3 I +x~-)tanh-X (2x2' ~/2 (x, +x2)) F (x~, xi) = 4 V'2 (xi xi + 3x~ (21) In (21), xi is to be replaced by xi' for the case xe ---- x2"'. Figure 1 gives values of (d~/dx2") for various values of xx as a function of x,~. We can define a general function ct (x.~) given by 1 a (x3) = f (~), dx~" (22) .l'3 P where, xa = 2 -- (x~ + Q') for (n, p) and (2x - xi') for (n, n'). From equation (22) ann and ,znp can be evaluated. Figure 2 gives a (x3)as a function of x3

601 v3,..uuE O;RECT t),TEq,Zc"r'o~ / &S & ~UNCT~ON~F 1{l l VOLOME: OlRECT .NTs [xi.,- ~~..o' -~,, r z- (x,.d, ,o ~tx;.,_,~ _,., 40@ Xzlu2~ X I l-1) I~ I'1 4"~ i z,,~i

1.4

I-3 I0 I.I)

I.I

0 OI 02 0"3 04 0$ 06 0"7 @11 ~'1 1.0 0 .I "l $ .4 "6 4 ? 4 lO. } ~ XI XI FIG. 1 FIG Proton Emission from Nuclei Due to Neutron Interactions 251

for various values of xa. From these two figures the various direct interaction cross-sections can be evaluated for any nucleus. Neutron and Proton Penetrabilities.--The cross-section oe(e) for the formation of the compound nucleus due to an incident particle of energy E with a target nucleus is given by % (E) = =k a Z' (2l + 1) Ti (E) (23) I where T! (E) are the partial penetrabilities as defined by Blatt and Weisskopf: e The penetrabilities T (E) which occur in the above derivation for the proton spectra are de¡ as oe (E) T (E) -- [= (R -1- ,t) ~] (24) where ,~ = ()t/27r), A being the wavelength of the incident particle and R Ÿ the incident channel radius. We have chosen for R the value of 1"5 A x/s Ferrnis. For square well potential for protons ~re (E) has been calculated from the computations of Tz(E) by Feshbach etal. TM The penetrabilities for charged particles have been calculated by Kikuchi11 using the W.K.B. approximation. We have chosen the same potential for evaluating T (E). The potential is given by v (r)=ve (r) § v,, (r) ZZ' e 2 Ve (r) = r

Vn(r) =- V( 1 + exp'(r-- a)) -i b ' (25) where Ve (r) and Vn (r) are the Coulomb and nuclear potentials respectively. We have chosen for the constants the values: a=4.1f; b=0.5f. Then T (E) is given in the W.K.B. approximation by

T (E) = exp. -- 2 -~- v'V (r) -- E dr (26)

ti where ra and r2 are obtained from the zeros of the function [V (r)- E]. Since the above approximation does not take into account the centrifugal force, it would yield a much higher value for the penetrability of high energy protons than given by the exact solution. To some extent this has been correeted for by smooth extrapolation of the above curve for low energy protons with the asymptotie value of T (E) given by 252 G.S. MANI AND K. G. NA[R

where B is the Coulomb barrier height for the nucleus. Figure 3 gives the penetrability as a function of proton energy for Mg ~7 for square and diffuse potentials.

RESULTS AND DISCUSSlON

The above theory has been applied to compute the energy spectra of protons emitted from Al ~7 nucleus due to neutron interaction. The incident neutron energies chosen are 14.7 MeV and 17.5 MeV. The constants used in the computation are given in Table I. The value of 'a' which occurs in the level density formula has been varied from 1.5 Mev -1 to 3 MeV -x in steps of 0.5 MeV -1. Figure 4 gives a typical spectrum computed for an incident energy of 14.7 MeV together with contributions from the various types of reactions considered above. It is seen from the figure that the contribution due to direct interaction is negligibly small and does not alter the shape of the total spectra appreciably except in the very high energy part. Since the differential cross-section for high energy proton emission is small, experimental data have large statistical errors and hence cannot resolve the effect due to

~;O~SOI.IO&TED SP(CrR& alŸ

~.*t4 7 MeV A(*O

S~ARE W[kL. 0-)0

•RANSMISSK)N i~qO8&611.1TY FOR PROTONS hN Mgt~

""~ - - - t%*.Rpn) COm*POuNO ' ...... (~~iP:- CO~POUNO | ' Dr POT(NTgALi R" I 3StKilO~m |. g,~ ii1'(i.I, ffOT(hT#AI. Ril'~l ~~aIO C~ " ....-~' **V~2 ......

0"6

. | .... O 2 4 $ IO z+" m 16 ~0 t~ 6~ 40 6.O 4,0 t.O $0 0 qO0 I O I1~ Ee (I,e+W I~~OTON (NERGY tN-MeV FIG. 3 FIO. 4 Proton Emission from Nuclei Due to Neutron Interactions 253

direct interactions. The only other method of evaluating the direct interaction cross-section from experimental data is to study the angular distribution. Allen x uses this procedure and assumes that the total non-isotropic part of the distribution is due to direct interaction. It has been shown by Ericson and Strutinski 4 that the angular distribution in the statistical model need not be isotropic and a more realistic assumption would be that it is of the form (A +B cos ~ 0). The experimental angular distributions are not accurate enough to permit such an analysis. Figures 5-8 give loge oJ (E) as a function of E, for the two types of potentials and level density formulae that we have used. Here ~,(E) and E are defined as:

o~ (E) = N (,p) dcp (28) ~ao [,~e ('v)],~,.-,,,

Lo~ W(E) AGAINST RESIDUAL EXCITATION ENEAGY F~ PROTON EMISSION FROM ~r LOq. W(E) AGAINST RESIDUAL EXCITATION INr NEL~TRON ENE.RGY -F4,7 M~ ENERGy FOR PROTON EMIS$1CN FROM ~7 tNCIOI[NT N[UTRON [NERGu -14 7 MCV (] ,,= i s MW" Q - 30 ~.v-'

SQ~LMI~! AE 0 IIt $~aA[~ [ 81s o 0 *9-Z WE 11 Al: ,Z-I / IIIFF 11 Al[ 0 / ]~ t|.ll II[LL Al[ "21

0 EXPERIMENTAL POINT$ ~P . ( COL LI"~! Of) .fil J (COLLI et 111) /~* *4e ~

'*t.~ "2.1

O O

-IJI -z5 ~,

-411 i i I i ; I jI4 , - Z 4 6 lO 2 -4 E 2* 4i 6I I lOi ~i ~1I 18 R[$1OUAL EXCITATION ENFRGY E (MeV) RFstr)UAL EXCIT&TION ENERGY E (MeV)

FIG. 6 FIG. 6 and E = Enp~ -- Ep where E is the residual excitation energy if all the protons were emitted through the (n,p) and (n, pn) reactions only. Enp~ is the maximum proton energy possible for these reactions and ~p is the enei'gy of the emitted proton. N (~lo)d~p is the spectrum of protons emitted due to aU the various types of reactions considered above. The reason for this 254 G.S. MANI AND K. G. NA1R

TABLE I

Binding energy ~o of protons in: Al 18 ...... 9.5 MeV.

Al r~ ...... 8.3 MeV. of neutrons in: AlS' ...... 7.7 MeV.

Al~ ...... 13.0 MeV. Mg r~ ...... 6.4 MeV.

Cross-sections 14.7 MeV. 17.5 MeV.

cr (n, p)16 53 mb.* 38 mb.*

o (n, a) 15 120 mb. 80 mb.

oo, ~~ 960 mb. 900 mb.

o, 860 mb. 830 mb.t

Paring Energy1~ Mg u ...... 2-1 MeV

Mg ~ ...... 4.3 MeV

Al ~ ...... 2.2 MeV

V = Nuclear potential = 42 MeV:[: * This valuo corresponds to 60 mb. at 14 MeV and is lower than tho ono used in refcrr 15. This lower value seems to agree better with r and is the valuo obtained by Paul and Clarke sI and by Allen? t The ~z has been calculated from the formula r = r (R + ~)~. :~ This is the value obtained from the real part of the optical potential and sectas to fit the total as wr as the elastic nr cross-section. Proton scattering data also yir a similar value. choice of definition is because the experimental data of Colli et al) is also represented in the same way. It is seen from Figs. 5-8 that the experimental points agree better for the case a----3 MeV -1 than for a = 1.5 MeV -1. The function loge ~ (E) is very insensitive to the type of potential used of to the level density formula assumed. The experimental points seem to be in better agreement with the square well potential for ,dE----2.1 MeV Proton Emission from Nuclei Due to Neutron lnteractions 255

LoO, w(E) AGttINST RESlOUAL EXCITA'r~ON L(~I, W(E) AGAtNST RESIDUAL s ENERGY FOR PROTON EMrSSION FROM Ad r FNERGY FOR PROTON [MISSION rROM A~ r INClDENT NrUTRON ENERGY = 17 5 MeV INcInENT NEUTRON ENs - i~ 5 MIV (3 " I"5 MIV "1 Q - 30 M,v"

*D'2 WIrLL I DIFFUSE t: W(LL "2 I tu W.~ OIFF~ISIE II 0 *lis

g (COLLI ito;~ "J *4 6 / o E'7::L%'; 'o"" ,Ÿ

*Z$

0 ll:z ~~ ""

-Z 3 -23

-4,6 ." 4 I 0 I 14 16 0 Z a 8 lO 12 P4 li RESlOUAL FXClTATION ENERGY E (MIV) RESIDUAL EXCITATION [NERGY E (IVeVI

FIG. 7 FIG. 8 but no definite conclusion can be reached since the experimental errors are fairly large. The discrepancy between the 14.7 MeV data as pointed out by Colli et al. 5 is only due to the 01, np) reaction. The total cross-section for the proton production is more sensitive to the potential used and to the level density formula. Unfortunately we are unable to compare our results with the experimental data because of the large error introduced in the computation of the (n,n') cross-section from the total non-elastic cross-section. This can be avoided by a direct measurement of the (n, n') cross-section. There seems to be some discrepancy in the total cross-sections for the various reactions but unless more accurate cross-section measurements are available nothing more conclusive can be derived from the present data. The deviation of the experimental data from the theoretical curves in Figs. 5-8 for large value of E may be due to errors in our estimation of the (n, n') cross-section and our assumption of constant 'a' for all these nuclei. The experimental data for the differential cross-section for direct interaction obtained by Allen ] from angular distribution is compared with the theory in Figs. 9 and 10. The experimental points have been evaluated on the assumption that the statistical model yields isotropic angular 7~56 G.S. MA_~I AND K. G. NAIR

DtRECT IkTERAr DIFIr[R[NTIAL CROSS-S[CTION f,S*

DrR1ECT INT[RACTION IN Cu INCIO[NT NEUTRON IrN[RGy*kllMeV

INClDENT NEUTRON ENERGY 14 MeV

~ ALL[NS (XPTI,, POINTS ~ ALLEN'$ Exp'r~, POINT,~ m J 3 i ~1~ ~.o v 5-C

40 4.G 30 \\ 30 ZO \ ~0 *0 'O

2 C 8 O 12 14 ~6 2 4 II I0 12 14 E: Mt~ FIG. 9 FIG. 10

distribution. Since the data have large experimental errors, it is not possible to analyse it in terms of the A + B cos a 0 distribution. In the theoretical curve we have used the value of 0.25 for Xnn/Xnp, the ratio of free n -- n to n--p cross-section. The actual value for this ratio lies between 0.25 and 0.5019 and a higher value for this ratio would give a better fit to the experimental data. It seems from these figures that at least for nuclei in this region the major contribution to direct interaction is from volume effects.

The above analysis indicates that a combined measurement of the various cross-sections as well as the energy and angular distribution of the emitted protons is needed for detailed analysis of these reactions. Also the present data are not accurate enough to decide between the types of potential or level density formula. We should like to thank Dr. A. M. Lane, Dr. T. Ericson and Dr. E. B. Paul for reading the manuscript and supplying us with valuable comments.

We gratefully acknowledge the valuable help of Shri M. R. Hiranandani in the above computations. Proton Emission from Nuclei Due to Neutron b~teractions 257

REFERENCES 1. Allcn, D. L. Proc. Phys. Soc., 1957, 70 A, 195; Nuclear Phys., 1959, 10, 348. 2. Gugr P.C. .. Phys. Rev., 1954, 93, 425. 3. Badoni, C., Colli, L. and Nuvo Cimento, 1956, 4, 1618. Facchini, U. Colli, L. and Facchini, U. lbid., 1957, 5, 309. and lbid., 1957, 5, 502. Michelctti, S. 4. Wolfenstein, L. .. Physics Rev., 1951, 82, 690. Ericson, T. and Nuclear Phys., 1958, 8, 284. Strutinski, V. 5. Colli, L., Pignanclli, M., Nuvo Cimento, 1958, 9, 280. Rytz, A. and Zurmuhle, R. 6. Austero, N., Butler, S. T. Phys. Rev., 1953, 92, 350. and MacManus, H. 7. Clementel, E. and Villi, C. Nuvo Cimento, 1958, 9, 950. 8. Goldberger, M.L. .. Phys. Rey., 1948, 74, 1269. 9. Hayakawa, S., Kawai, M. Prog. of Theor. Phys., 1955, 13, 415. and Kikuchi, K. I0. Brown, G. and .. Phil. Mag., 1957, 2, 473. Muirhead, H. I 1. Kikuchi, K. .. Prog. of Theor. Phys., 1957, 17, 643. 12. Newton, T.D. .. Can. J. Phys., 1956, 34, 804. 13. Elton, L. R. B. and Phys. Rey., 1957, 105, 1027. Gomes, L. C. 14. Lanr A. M. and lbid., 1955, 98, 1524. Wandal, C. F. 15. Mani, G. S., McCallum, Nuclear Physics (to be publishr G. J. and Fergusson, A. T. G. 16. Blatt, J. M. and Theoretical Nuclear Physics, John Wiley and Sons, 1952. Weisskopf, V. F. 17. Cameron, A. G. W. Can. J. Phys., 1957, 35, 1021; 1958, 36, 1040. 18. Feshbaeh, H., Shapiro, Tables for the Penetrabilities of Charged Particles, M. M. and Weisskopf, NYO 3077 (1953). V.F. 19. Hess, W.N. .. Rev. Mod. Phys., 1958, 30, 368. 20. Wapstra, A.H. .. Physica, 1955, 21, 367. 21. Paul, E. B. and Clarke, R. L. Can. J. Phys., 1953, 31, 267. 22. Hughes, D. J. and Neutron Cross-sections, BNL 325 (1958). Schwartz, B.