Direct interaction in the p—2p reaction

Nuclear Physics 18 (1960) 1 - - 1 3 , (~) North-Holland Publ, sh,ng Co., Amaerdam Not to be reproduced by photoprmt or microfilm without written pernusston from the pubhsher

DIRECT I N T E R A C T I O N A M GREEN

IN THE p---2p REACTION and G

E

BROWN,

Department o] Mathemat,cal Phys,cs, Umvers,ty o/ B*rm*ngham R e c e i v e d 7 April 1960 Abstract: A n a n a l y s i s of p---2p r e a c t i o n s is carried o u t a n d it is p o i n t e d o u t t h a t w i t h good e n e r g y resolution m d e t e c t i o n of t h e particles, ~t w o u l d b e possible to o b t a m i m p o r t a n t

r e f o r m a t i o n a b o u t t h e energies of shell-model levels T r e a t m e n t of t h e dlrect-mteractaon p a r t of t h e process as a s u r f a c e r e a c t i o n is j u s t l h e d Calculations in t h e d x r e c t - m t e r a c t i o n f o r m a h s m s h o w s t r o n g correlations m angle of t h e two e m i t t e d p r o t o n s

1. Introduction Recent experimental studies have shown the p - - 2 p reaction to be a very interesting and useful one. In particular, it can be used as a tool to investigate shell-model levels down in the nucleus :) and is, at present, almost our sole source of knowledge about these levels, since other methods usually give reformation only about the separation energies of the last particles. In adcht:on, Eisberg 9) has proposed using this reaction to find out what proportion of the events take place in the nuclear surface. Such events would be expected to exhibit a strong correlation in angle, whereas the statistical events would not. Experiments of Elsberg and Griffiths 3) have been interpreted as indicating a strong correlahon in angle for a large proportion of the proton pairs, and that the events take place in the surface A theoretical analysis of these experiments 4) assumes the momenta of the struck proton to have a chstnbution up to a maximum momentum kF; the corresponchng Fermi energy E F is found to be 1--2 MeV which seems unreasonably small. There is no doubt that the direct interaction part of this process takes place m the nuclear surface, because protons from any events occurrmg within the nucleus will almost certainly be absorbed Into compound states before getting out s). However, replacement of the nuclear surface b y a Fermi sea drops the most important feature of chrect interaction theories, namely, the centre of rotation; without it, angular momenta cannot be d e f i e d . Angular momentum considerations are of primary Importance in determining the positlon of the peak m the differential cross section in chrect interactions. Furthermore, the momenta k F obtained b y McCarthy et al cannot signify the actual momenta of the nucleons in the surface since with such small momenta, one Is precluded b y the uncertainty principle from localizing the par1 August (1) 1960

2

A M GREEN AND G E. BROWN

ticles within a region less than several times the extent of the nuclear surface. We shall return later to a discussion of the results of McCarthy et al. and show why t h e y obtaan such a small k F. Our main interest in giving a theoretical description of the p---2p process is to see if it can be used at energaes lower than the 185 MeV employed by Tyr6n et al. to study shell model levels. If, for example, such studies can be carried out for incident energies of ~ 50 MeV, the problem of determining the energies of the final protons becomes much easier, and there is more hope of extending the method to heavy nuclei where the shell model levels are closer together. In giving such a description, we wish also to obtain a rough estimate of cross sections and, m this way, predict in which cases one would expect a large amount of direct interaction m the process. In section 2 we chscuss the physical picture behind the process, why one expects it to proceed by direct interaction, and the justification of the use of procedures invented for stripping reactions. In sect. 3 we develop the necessary formalism, and in sect. 4 compare our results with existing experimental data. In sect. 5 we discuss the results and indicate interesting experiments.

2. The Physical Picture The p---2p reaction would appear to be a very useful tool for investigating the shell-model levels down in the nucleus. Consider the diagram in fig. 1. < >

Eo E,

>

T Fig

1

Schematic

picture

of the p---2p reaction

Here E o is the energy of the incoming proton and E 1 and E-~ are the energies of the two outgoing protons. The q u a n t i t y x ----- E o - E 1 - E 2 represents the energy needed to eject the second proton from the nucleus We neglect here the recoil energy of the residual nucleus; if it is not neghglble, correction for it can easily be made If the nucleus consisted of non-interacting particles in stationary levels in a potential well, the p---2p process would occur only for certain definite values Bl of z corresponding to the binding energies of the particle in the well Although his is clearly not the situation, some effects of the underlying singie-partlcle

DIRECT INTERACTION IN THE p--2p REACTION

structure remain in the process 6). Restricting our considerations to those protons ejected in the direct-interaction process, we would expect giant resonances centred about Bt to occur, of width 2W, where W is the absorption for holes at the relevant excitation energy of the residual nucleus In other words, the hole excitation formed in knocking the particle out of the shell-model state has a finite hfetime, decaying into compound states in a time ~ ti/W. These hole excitations are completely analogous to the single-particle excitations which are observed in low-energy neutron scattering, where they are called giant resonances. The absorption W for holes can be computed as a function of excitation energy 7) in the s~ane way as that for particles is calculated s-10) The calculations r e f e r r ~ ' o : h a v e all been carried out for the case of the infinite nucleus, and it is clear here that, aside from the curvature of the Fermi surface, there is complete symmetry between particles and holes. It is well known that the value of W corresponding to neutron giant resonances at zero bombarding energy is 1--1 5 MeV, and this corresponds to an excitation energy of m 8 MeV in a typical nucleus, if we take this to be the separation energy of the last particle. If, therefore, a value of z equal to 16 MeV is needed to elect a proton in the p---2p process, leaving the nucleus at an excitation energy of m 8 MeV, the half-width of the hole giant resonance will be m 1--1.6 MeV. Thus, for suitable angles, one would expect the cross section to behave as shown in fig. 2, wath a dSa d~ld~,dx

(A

O MeV Fig

X

~,

2 Typical energy spectrum

sharp peak corresponding to the ejection of protons filling the last shell-model level, occurring at z = 8 MeV, and then further peaks, corresponchng to the lower shell-model levels, which become broader with increasing z. Such peaks were, incidentally, not observed m the Minnesota experiments; however, their energy resolution of 15 % would appear to be insufficient when folded in with the width 2W of the excitation. One circumstance makes the observation of hole excitations more favourable

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than that of particle excitations, namely, the former have no escape wadth, whereas at positive bombarding energies the latter have an escape w i d t h / ' , so that the total width of the excitation IS 2W+F, and thls escape width increases rapidly with Increasing energy, so that the particle excitations broaden out very quickly We shall also see that the differential cross sections in the t>--2p process wall peak at different angles for different shell-model levels, and thus will help to distinguish the angular momentum of the hole excitation. In the future we shall carry out the calculations as if the protons were in stationary states in a shell-model well, we must remember that the effect of interactions is simply to give each of these levels a width 2W as far as the direct-interaction part of the reaction is concerned. Since we are mainly interested m using the p - - 2 p process as a tool to observe the hole excitations, our description of the reaction mechanism will be carried out with the purpose of r i d i n g optimum angles of observation and to make sure that a measurable number of protons might be expected. Relatively crude calculations should suffice for this We consider the entire direct-mteractlon process to occur In the surface of the nucleus That this is a reasonable assumption follows from the work of Elton and Gomes 5) who find, in considering the p - - p ' reaction, that only 10 % of the protons making collisions inside the nucleus are able to get out. This results partly from their short mean free path inside the nucleus, and also because their absorption is accentuated by an effect, analogous to total internal reflection of light, at the edge of the nucleus Most of the protons p' strike the edge of the nucleus at an acute angle, and are reflected several times, since only the wave number perpendicular to the surface is relevant to their transmission probability. These effects are much stronger in the p--2p process, where the initial energy is distributed between two protons, which, even if they strike the edge of the nucleus head on, have little chance of being transmitted Therefore, one would expect the p - - 2 p process to be much more completely confined to the surface than the t>--p' one, and to be more similar, in this respect, to deuteron stripping reactions We shall, therefore, employ the cut-off Born approximation as used by Butler li) It Is, of course, clear that this cut-off procedure is a crude way of taking the chstortion of the Incident and emergent waves into account, but it works exceedingly well in describing the angular distributions in stripping reactions It should, however, be remembered that it does not reproduce the absolute values of the cross sections very well

3. Development The reaction is described by Butler's chrect nuclear reaction formalism, in which differential cross sections are obtained from perturbation theory i.e. by means of the matrix element of the nucleon-nucleon potential between the

DIRECT INTERACTION IN THE p - - 2 p

REACTION

5

initial and final states of the system. This matrix element is of the form M = f dyer* (v)V~,(v),

(3 1)

where v denotes all the spatial and spin coordinates needed to specdy the system; ~ t is the initial state of the system, and so in the case of the (p, 2p) reaction is the product of the wave functmn of the struck nucleus and that of the incident proton, which is taken to be a plane wave of wave number ko; ~ut is the final state of the system, and in this case, is the product of the wavefunction of the residual nucleus and those of the emergent protons, which are also taken to be plane waves of wave numbers ka and k2; V is the interactmn between the incident and struck proton. The antegratlon is carried out only over the space outside a sphere of radius R, in hne with the use of the cut-off Born approximation. If the zero-range approximation is made for V, the matrix element M reduces to the form M :

V'

fr >R

dar

e--tkt r e-~k~ r 12½

- - e 12½

*ko *A,Yz=(O, ¢)ht(1)(,kr),

(3 2)

where AzY["(O, ¢)h~(1)(,kr) is the wavefunction of the struck proton (hz(a~ as a spherical Hankel function), V' as gaven by V' ---- ~d3rV(r), and 12 is the volume an which the wavefunctmn of the emergent protons as normalised. The rachal hmlt on r was justified in sect. 2. Using the procedure in Butler's artmle, the value of this integral is found to be

V' R2 : A'~[4~(2l+l)]½_-v--7=Wr[h(pr ), ht(a)(~kr)]r=R, M 12 p~+k z

(3 3)

where q = k l + k 2 - - k o is the momentum furmshed by the nucleus for given

kx, k2. Therefore the differential cross section is da = 2n M 0 2M0s ~ / E a ( E . E 1 ) ( ~ M ) 2 , ck(21d~dEI ?~ ?~ko (2~rh)e

(3 4)

where

Ex+E2 = Eo--B = E

(3 5)

(B is the banchng energy of the proton that was formerly bound) and M 0 as the mass of a proton. The above is for spmless particles; but if V is considered to be independent of spin, the only change, when span is introduced, as that the factor 2 / + 1 an M is replaced by 2(l+ 1), when the bound proton was formerly an a/" = l+½ state, and by 2l when in a 7 = l--1 state. Instead of using the zero-range approxxmation for V, the t-matrix 12) can

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be used to give da

a ld-0 del- c

(da)

1

~/EI(E_E1)(ff2M)2 '

(3.6)

where C is a constant, (da/dS91h~b is the differential cross-section for free p - - p scattering in the laboratory system, where a proton of energy E 0 collides with another proton nearly at rest 0.e. p << k0) and the two emerge with energies E1 and E~. From experiment 18) the equivalent differential cross section in the centreof-mass system is lsotropm wath the value 11 mb/sr 2 at 40 MeV. In the p---2p experiments we shall need the differential cross section off the energy shell; we assume it to be isotropic m the c.m.s, of the two colliding nucleons, and to have the same value as found in proton-proton collisions at energy E o. The value obtained for the absolute cross section depends strongly on the latter assumptmn, since the proton-proton cross section is strongly energy dependent in the regmn of E 0 we consider. The cross section in the laboratory system is obtained from that m the c m.s. b y the following relation: (d~)

I I - - y cos 011 (da) l~b = [ l + r 2 - ~ r c o s O d t ~ c.m

=F(EI,0~)(da)

(3.7)

~-~ cm.'

where y is the ratio of the velocity of the centre-of-mass in the laboratory system (which is constant in this case and equals ~[k0I 1 ) to the velocity of the observed partmle in the laboratory system, i.e lkx[; 01 is the angle, in the laboratory, through which the particle is scattered; 0 is the angle in the centre-of-mass system, the two being connected b y tan

0 =

sin 01 . cos 01-- Y

(3.s)

4. C o m p a r i s o n w i t h E x p e r i m e n t a l Data The only experimental data existing at these energies are those of Griffiths and Elsberg, which are for 40 MeV protons incident on N1~8 and Cu. One coincidence counter was fixed at 25 ° to the incident direction and the other at a variable angle 01, as shown in fig. 3 In the data that we use, all coincidences with E 1 and E , greater than 10 MeV were registered.We shall calculate quantities for this case, and also for a theoreticallysimpler case, in which we hope experiments wall be carried out m the future. The cross section is considered to be the sum of the contributions from the top four proton shell model levels, namely the 1F[, 1D~, 2S½ and 1Dt. B y interpolation from the article b y Ross, Mark and Lawson 14), who have calculated energy levels in a velocity-dependent potential, we find that the

DIRECT

INTERACTION

IN

THE

p--2p

REACTION

7

binding energies of particles in the above levels are respectively 8 MeV, 11 6 MeV, 13 MeV and 17 MeV. To reproduce these with a square well, we need depths of 39 MeV, 40.8 MeV, 41.5 MeV and 43.5 MeV respectively, for a radius of 5 × 10-is cm. The value of the cut-off radius is rather empirical, andis usually

/

s,

so Fig

3 E x p e r i m e n t a l s e t - u p used m calculatmn

chosen to be 1 or 2 x 10-13 cm larger than the potential-well radius. In this case, it is taken to be 7 x 10-la cm. We now carry out calculations in both Born approximation and impulse approximation. 4 1 BORN APPROXIMATION

The differential cross section is first calculated using M from equation (3.2) i.e using Born approximation together with the zero-range approximation for the potential. We take V(r) = V o e x p ( - - r / b ) / ( r / b ) , where V0 = 50 MeV and b 1.2X lO-~3 cm to calculate V'. =

O01C

kI

oooe /

~; ooo{

•

~ . / - .'.,DIN ,Fz

~a 0004: t~

,?/

.~o 002

/

aI /

.

t/

,, \

2

•

1

2

I

I

~%

~3

4

s',,r~\

7

,,,~

/9

R

--0002

-0004

/

, i

"l~ "

i

i

,

i

i

Fig 4 Matrix elements for the top four filled shell-model levels m N P s

A M GREEN AND G E

BROWN

gO eo ~ 7o ~ 6c ~ 5c _z 40 _

,3c

2o 10

~1 ~

~

1 Fig

go 80 ~n 70 ,Y eo 50 ~' 40 z30

~ ~1

=

~

2

3

i I "~-'~-

i

4

5

PR

5 01 a s a f u n c t i o n of p R for t h e 1 F t l e v e l

........

....~ ~ e . 4

~-20 1

Fzg

2

3

6 0z a s a f u n c t i o n of

O0I 80; 70 60 50 4O ~ 3O _20 ~' 10

4

pR

5

pR

for t h e I D i l e v e l

El=10 MeV ~ "

~

~

i

1 Fig

2

7 e~ a s a f u n c t i o n of

gc eo 7c

3

pR

4

for t h e 2S t leve]

EI = 175M m V ~

~ ec

,OR

'~'~'j~

4e 3c i~- 2c io

iD{ 1

Fig

"

2

3

'~,

,OR

8 01 a s a f u n c t z o n of p R for t h e 1D t l e v e l

5

DIRECT INTERACTION IN THE p--2p REACTION

9

The differential cross section given b y equation (3.4) (modified for spin) is dependent on the angle 01, through the quantity p as shown in figs. 4---8. In most (p, 2p) experiments at this energy, owing to the scarcity of events, the differential cross section in integrated over energies. This is the procedure used in subsection 4.2, where the contributions to the cross section from the four energy levels are integrated over energy ranges, which ensure E 1 and E~ both greater than 10 MeV. e g the 1FI contribution is integrated from E 1 ---- 10 MeV to 22 MeV. However, we first illustrate the results b y a theoretically simpler case, where only particles in the 1F] level are involved 1 e. when only coincidences with E 1 ---- E 2 = 16 MeV are recorded, giving fig. 9. This illustrates the strong correlation one would expect to fmd in experiments wlth good energy resolution.

/ /l

!I\ 30

~O

gO

0 t IN DEC.~REES

Fig 9 Angular correlation in the ldeahzed case of E t = E a = 16 MeV

4 2 IMPULSE APPROXIMATION

We now go over to the impulse approximation b y replacing the potential V b y the t-matrix This modification is important, because the t-matrix depends markedly on angle in the laboratory system, even though the proton-proton differential cross section is isotropic in the centre-of-mass system. Using equations (3.6) and (3.7), the replacement of V IS carned out b y multiplying each differential cross section da/dQ, d~2~dE1 of the method described in subsection 4.1 b y the appropriate factor (de~ C \d~/e.m.

F(E1,_01) / VE1

~9'1~M 0 2 M o 8 "-~ ~ k O

(2~) 6'

i.e. b y C ' F ( E 1, OI)/X/E x where C' is a constant, and then integrating over energies as before. The quantity F ( E x, 01) is plotted in fig. 10. Angles 01 < 20 ° are not considered since F becomes very large in this region. In adchtion there are no experimental data for such angles. We still limi+ the integration over

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energies so that E 1 and E , __--_lO MeV. since further modification is necessary to include protons of lower energy. 5

EI- 22 MeV

4

13

~-3

~ "

%'%.

10 ..

•

1

~..~.

o

3;

~.~.o

do

~o

e I IN DEGREES

F*g 10 F ( E = , e , ) as a function of O, for various values of E , .

ooe %

z 002

01 IN DEGREES Fig

ll

Angular correlation, when protons from the top two shell model levels contribute 01(5

012

E008

!

u

e I IN DEGREES

Fig 12 Angular correlation, when protons from the four shellmodellevelscontribute

D I R E C T I N T E R A C T I O N IN T H E p - - 2 p

REACTION

11

If we assume that only the top two proton levels, namely 1F[ and 1D t, contribute to the total differential cross section, we get fig. 11. The 1D| level contributes only little, so that the main contribution comes from the 1F½ level When all four levels contribute we get fig. 12 The relative Minnesota data are also plotted 5. C o n c l u s i o n s a n d D i s c u s s i o n

It should be noted, first of all, that the agreement between theoretical calculations and the Minnesota experimental data, as shown in figs 11 and 12, is not very good. A number of reasons might be adduced for this. The rise in our curve at small angles in fig 12, which is not observed in the experimental results, is due to the overwhelming contribution here from the 2S½ protons. However, the cut-off theory may give the ratio of 2S½ to I F [ contributions badly; in the case of deuteron stripping a more realistic treatment of the distortion b y Tobocman and Kalos 15) gave a somewhat different ratio of S to P and S to D contribution than the cut off calculation, although the P to D ratio is given well b y the cut off calculations Even if our calculation gives absolute values only roughly, a number of interesting qualitative features are shown. Firstly, in the region of minimum momentum transfer p to the nucleus the F and D contributions are minimal (see figs. 5 and 6 and fig. 9). This is because the iz(pR) in eq. (3.3) goes to zero as p -+ 0 for non-zero l, this illustrates the importance of angular momentum arguments of the type emphasized b y Butler, Austern and Pearson le). Since the nucleus loses angular momentum l in the process, l = 1O^ R[, where R is the radius vector of the region in which the process takes place, in so far as this region can be localized For p = 0 and non-zero l, tins approximate equality cannot be satisfied. In the neighbourhood of p = 0, the contribution from l = 0 Is, however, large; the maximum of i0GoR), which is ]o = 1, occurs for p = 0, and this is, in fact, larger than the mamma of Bessel functions of non-zero order (e g., the maximum value of 18~R) is 0.24 and occurs for pR = 4 5; in general, jz(pR) is maximum for pR slightly greater than l). Hence, in the region of zero momentum transfer to the nucleus, contributions from S states predominate The latter qualitative point probably holds at higher energies, in the 100---200 MeV region, where the strong absorption forces the contributions to come mmuly from the surface region. It is tempting to interpret the large peak in the Uppsala data for Ca as coming from the 2S½ protons, bound b y about 14 MeV (their counters were set so that they selected small values of p). This binding energy would fit in well with our estimate of 13 MeV for the binding energy of the 2S½ proton in Ni. Presumably contributions from the D particles, which should lie on each side of the 2S½ peak, are obscured b y the large

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magnitude of the latter Although the probability of 2S½ particles being m the surface of the nucleus is relatively small compared with t h a t of particles of high angular momentum, they are strongly favoured in processes requiring very small momentum transfers p by the above angular-momentum rule. The observation of any angular correlation m the Minnesota experiments depended on the bias of the counters being such that only particles with E 1 and E 2 greater than 10 MeV were detected, so as to cut out most of the statlstlcal protons We can see this by making a rough estimate of the contribution from the latter. In N1 the binding energy of neutrons exceeds that of protons by about the Coulomb barrier for the latter. Consequently, let us take the cross section for statistical p - - 2 p processes to be about ½ of geometrical, or ~ 400 mb. If this is assumed to be lsotroplc, then d~ptat

400

d,Oldf2-----~ "~ (4~)-----~mb/sr* or about 3 mb/sr* Tins is much larger than the cross sections for direct interaction computed here. However, the statistical distribution will peak for E I + E ~ of the order of twice the Coulomb barrier, so that for E I + E * >= 20 MeV the contribution from statistical protons will be down by a factor of about 100 from the value if would have if protons of all energies were accepted. It does not appear, therefore, that direct emission makes a large contribution to the overall process. But it accounts for essentially all of the high-energy end of the spectrum. We do not beheve the description by McCarthy et al. *) which involves a replacement of the nuclear surface by a Fermi gas, to be a useful one, since in going over to the infinite nucleus, it drops the centre of rotation It was lust the angular momentum considerations that we found to be most important Within our formalism we can, however, understand why they obtain such a low value for their Fermi energy E F In our theory, m which the mare features of the angular distribution are determined by the Bessel function h(pR), the maximum cross section not only occurs for p R ~ l, but the width of the m a n m u m is given by 8pR m 1, i.e., by a change in the argument of the Bessel function of about one unit. If one were to fit the experimental data by assuming the second proton to be picked up out of a Fermi sea, it is clear that the Fermi momentum k F would have to be chosen so that k F ~ ~p ~ l / R , since the vectorial addition of this momentum to the final momenta must then produce the width of the peak in the angular correlation One would, therefore, obtain a Fermi energy E F m ~S/2MR*, where M is the mass of the proton, and this is approximately equal to 1 MeV However, we see from the above chscusslon that k F does not really have the significance of the Fermi momentum of particles in the surface Finally, we hope that the treatment here given will encourage further

DIRECT INTERACTION IN THE p--2p REACTION

13

experimental work with improved energy resolution This should be possible now that cyclotrons in this energy range are becoming available. We beheve t h a t very interesting information can be obtained from these with respect to the shell-model levels, as illustrated in fig. 2. This will allow one to determine empirically the energy dependence of the shell-model potential for energies below the separation energy, where almost no experimental information is available. One of us (A M G ) is grateful to the Department of Scientific and Industrial Research for the award of a maintenance grant. References

i) 2) 3) 4) 5) 6) 7) 8) 9) 1o)

H Tyr6n, P Hillman and Th A J Marts, Nuclear Physics 7 (1958) I0 R M Elsberg, UCRL-2240 (1953) R J Grlfflths and R N Emberg, Nuclear Physlcs 12 (1959) 225 I E McCarthy, E V Jezak and A J Krommmga, Nuclear Physlcs 12 (1959) 274 L R B Elton and L C Gomes, Phys Rev 105 (1957) 1027 G E Brown, Rev Mod Phys 31 (1959)893 E P Pendlebury, private commumcation M Cim and S Fubmt, Nuovo C1m 2 (1955) 75 K A Brueckner, R J Eden and N C Francis, Phys Rev 100 (1955)891 A M Lane and C F Wandel, Phys Rev 98 (1955)1524

11) S T Butler, Phys Rev 106 (1957)272 12) N Austern, S T Butler and H McManus, Phys Rev 92 (1953)350 :3) L H Johnson and D A Swenson, Phys Rev I I I (1958)221 14) A A Ross, R D Lawson and H Mark, Phys Rev 104 (1956)408 15) W Tobocman and M H Kalos, Phys Rev 97 (1955)132 lO) s T Butler, N Austern and C Pearson, Phys Rev I12 (1958)1227