- Email: [email protected]
Quasi-free electron-proton scattering (I)
Nuclear Physics 32 (1962) 139--151; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
Q U A S I - F R E E E L E C T R O N - P R O T O N S C A T T E R I N G (I) GERHARD
J A C O B t and T H . A. J. M A R I S *t
Instituto de Fisica and Faculdade de Filoso]ia, Universidade do Rio Grande do Sul, Pdrto A legre, Brasil Received 6 J u l y 1961 A b s t r a c t : I t is s h o w n t h a t , f r o m a n g u l a r and energy correlation m e a s u r e m e n t s on electron-proton
pairs emerging f r o m t h e scattering of high energy (300-1000 MeV) electrons on nuclei, detailed i n f o r m a t i o n on the e n e r g y levels a n d s t r u c t u r e s of the u p p e r a n d lower shells of light and m e d i u m nuclei could be obtained. A calculation in which the distortion of the outgoing p r o t o n w a v e is t a k e n into a c c o u n t has been p e r f o r m e d for C12. As c o m p a r e d to the result for zero distortion, the absolute m a g n i t u d e of the correlation cross section is reduced, b u t the shape of its a n g u l a r d i s t r i b u t i o n is practically unchanged. Consequently the observed energy and a n g u l a r correlations would i m m e d i a t e l y give b o t h the binding energy and t h e m o m e n t u m distribution of the nuclear p r o t o n in the shell model s t a t e o u t of which it h a s been ejected. F r o m an e x t r a p o l a t i o n to o t h e r nuclei of the calculated value of the reduction factor for the cross section, it is expected t h a t this situation prevails at least up to nuclei with A = 50. Finally some corrections are qualitatively discussed.
1. Introduction Assuming the approximate validity of the shell model of the nucleus, it is apparent that low energy experiments are able to give detailed information only on certain aspects of nuclear structure, namely on the properties of the least bound proton and neutron shells. To break up the more strongly bound shells a relatively high energy is needed; for low energy experiments the core of the nucleus much resembles an elementary particle of which only overall properties are relevant and can be determined. In recent years high energy (150-440 MeV) investigations of quasi-free proton-proton scattering have given information on the structure of the more strongly bound nuclear shells. Loosely speaking, in a quasi-free scattering event in a nucleus, a high energy particle knocks out a nucleon of the nucleus without any additional violent interaction of the incoming or the two emerging particles with the nucleus. Experimentally these events are recognizable b y the more or less sharp peaks they cause in the spectrum of the summed energies of the two emerging particles, corresponding to the binding energies of the shell model state of the ejected nucleon. The peaks are superimposed on a background caused b y multiple scattering and b y excitations from rearrangements in the residual nucleus. A short discussion of these processes and references m a y be * N o w on leave at the I n s t i t u t e for Theoretical Physics, University of Copenhagen. tt P r e s e n t address: Physics D e p a r t m e n t , University of Illinois, U r b a n a . 139
140
G.
JACOB
AND
TH.
A.
J.
MARIS
found, for example, in the introduction of ref. l) ; for later work see reis. 2-4). Qualitatively, the experimental results of quasi-free proton-proton scattering agree remarkably well with the shell model, and quantitative information on the energy differences of the shells, on their momentum distributions and on the widths of the hole states, has been obtained. One serious drawback of these (p, 2p) experiments is that the inner shells of only the light nuclei can be expected to be, and have indeed been, observed. This is because the total path length which the incoming and the two outgoing protons have to traject in a medium nucleus amounts in general to several times their mean free path in nuclear matter. For medium and heavy nuclei, processes occurring in certain parts of the nuclear surface for which the total path length is particularly small, will be mainly observed I). Because of the sloping edges of the shell model potential and because of the influence of the binding energy on the exponential tail of the single particle wave functions, the nuclear surface will mainly consist of particles of the upper shells. Consequently the inner shells of medium and heavy nuclei are difficult to observe by this method. It is in this respect that quasi-free (e, e’p) scattering should offer a clear advantage over the (p, 2p) processes. We are thinking of an initial electron energy of 300-1000 MeV and an energy of 100-400 MeV for the recoil proton. In a quasi-free (e, e’p) scattering event only the outgoing proton has an appreciable chance of being absorbed in the nucleus. Therefore surface interactions are much less accentuated than in the (p, 2p) scattering and the contributions of the inner shells relatively to those of the upper shell will be much larger, especially for medium or heavy nuclei. One should, however, not forget that the negligible absorption of a high energy electron in the nucleus is caused by the weakness of the electromagnetic interaction. This same fact results in small absolute cross sections for the quasi-free events, which make the experiments difficult, though not out of question t. There is another point of no less importance in which quasi-free electronproton scattering seems to offer a considerable advantage over quasi-free proton-proton scattering. As we have remarked, the angular and energy correlations in quasi-free proton-proton scattering may not only give rather directly the average momentum distributions for the nucleons in the whole nucleus, as has been recognized ten years ago 5)) but also the momentum distributions for the separate shells out of which the nucleons are ejected l-4, “). In case the scattering of the particles before and after the collision could be neglected, the difference between the total momentum of the two emerging particles and the momentum of the incoming particle would equal the momentum the struck nucleon had in the nucleus. (Of course this difference is always equal and opposite to the recoil momentum of the residual nucleus). Therefore, t R. Hofstadter
and J. A. McIntyre,
private communications.
QUASI-FREE
ELECTRON-PROTON
SCATTERING
(I)
141
would be possible to read off directly the momentum distribution of the struck nucleon from the correlation cross sections. However, the distortion of the incoming and outgoing waves by the nucleus destroys the connection between the momentum and correlation distributions in a degree which strongly increases with the size of the nucleus. In (p, 2p) scattering this effect becomes serious for all except the lightest nuclei. In (e, e’p) scattering one would hope that for light and medium nuclei mainly the magnitude but not the shape of the cross section will be influenced by the relatively small distortion. The following calculation verifies this expectation for a nucleus like C12. The shapes of the correlation curves calculated with distorted waves differ negligibly from those following directly from the momentum distributions, but the absolute magnitudes are reduced; this reduction differs only by 25 O/Ofrom a nai’ve estimate using an average absorption. We expect this situation to be still essentially true for much heavier nuclei than C12. In sect. 2 the necessary formulae of a distorted wave Born approximation for our case have been written down. Calculations for Cl2 are performed in sect. 3. In sect. 4 an extrapolation of the cross sections to heavier nuclei is made; the results are discussed from a physical point of view and the inherent possibilities of quasi-free e-p scattering for the investigation of nuclear structure are outlined. In sect. 5 the conclusions are summarized; some details of the calculation are given in the Appendix. it
2. Cross Section Formula For the sake of simplicity a pure j-j coupling single particle model for the nucleus is assumed. In the case of mixed configurations the single particle wave function of the nuclear proton has everywhere in this paper to be replaced by the overlap integral of the wave functions of the initial and the residualnucleus; the differences in binding energies of the nuclear protons have then the more general meaning of differences in excitation energies of the residual nucleus. We use the Born approximation for the electromagnetic interaction and a distorted plane wave for the outgoing proton. Furthermore it will be assumed that the effective interaction between the incoming electron and the knockedout proton is of such a range that the variation of the distortion of the proton wave over the effective interaction region can be neglected. This assumption is well fulfilled for the considered large momentum transfers to the proton, as may be estimated from the wavelength of the intermediating virtual photon. For the general case of arbitrary angles and energies of the outgoing particles the formulae become complicated and little transparent. We shall therefore choose an example which seems theoretically most simple although experimentally measurements of non-coplanar electron and proton momenta are required. The following nomenclature will be used: the indices e,, p,,, e and p refer to
142
G. JACOB AND TH, A. J . NIARXS
the initial and final electron and proton respectively; (E, k) is an energy momentum four-vector; we have taken the proton mass, ~ and c as units. The plane through k% and k e is called the "scattering plane"; 0 is the angle between the projection of a vector on the scattering plane and k% ; ~ is the angle between a vector and the scattering plane. First we shall consider the scattering on a nuclear proton in rest, bound with a definite binding energy E B. For this case ~p ~ 0 and Ee, Ep, 0p follow from energy-momentum conservation and are given as functions of E B and 0e in the Appendix. The experiment is supposed to be performed by considering E B as a variable and measuring the cross section da/d.QedEed.QpdE p for coincidently emerging electron-proton pairs as a function of E B for fixed Eeo, 0e and ~p ~ 0. The resulting spectrum will contain a peak having its maximum at the value of E B equal to the binding energy of the proton and its width given b y the lifetime of the final state of the residual nucleus and the experimental widths for the proton and electron energies. To measure the momentum distribution of the nuclear proton the measurement is repeated for non-zero values of ~p. Denoting the final proton momentum for the just considered rest case b y kpr, one finds from energy-momentum conservation for the initial momentum of the nuclear proton kpo :
--keo-~-ke-~-k p : kp--kp r,
kpo ~- 2kp sin ½~p ~ kp9~p .
The direction of kpo is approximately orthogonal to the scattering plane. For the conditions stated one finds to a good approximation for the cross section (see Appendix) f
da da ~r d.C2edE~C2p dEp d E s = ~ ~ - -
1 (1 q-2Eeo sin 2 ½00) ~ Ig'~ (kp, kp r) 12. (1)
In the right hand side of this formula the only dependence on ~opis contained in
Ig' l 2. For the more general case of arbitrary outgoing proton angles eq. (1) can still be used but the free cross section ~) on the right hand side has to be replaced b y a function of the proton scattering angles. Without the distortion of the outgoing proton wave, g'~ would have been the momentum distribution of the proton whose ejection leads to the considered final state. Neglecting the distortion of the electron wave function but taking the distortion for the proton into account b y a WKB-method one has s) g'~ (kp, kp r) = (2a)-]
/E
(
e x p ( i q , r)~0,(r)exp - - i
Vds
)]
dr.
(2)
In eq. (2) q = k%--l%--kp = kpr--kp is the recoil momentum of the residual nucleus; for the single particle model one has - - q = Rp0; V(r) is a suitable optical model potential over which one integrates along the classical path of
143
Q U A S I - F R E E E L E C T R O N - P R O T O N SCATTERING ( I )
the outgoing proton and the single particle state ~0~(r) is normalized to unity. The summation over n in eq. (1) takes into account that some final states, for example those which differ only in the orientation of their spins, cannot be separated experimentally. In case the initial nucleus has a non-zero spin, the cross section has in addition to be averaged over the initial states. It m a y be remarked that, for unpolarized nuclei, ~nig'n(kp, kpr)l 2 and therefore also the cross section (1), as a function of kp for fixed kpr, is rotationally symmetric with respect to the direction of kpr. This is because for fixed kpr all the elements entering into the calculation of the dependence of the distorting function on kp are rotationally symmetric around kp r. It follows that g'n(kp, kpr) for our case is just a function of q = lkp--kpr]. It m a y be noted that the experimental situation for fixed momenta of the incoming and outgoing electrons is only reflection symmetric with respect to the "scattering plane". 3. C a l c u l a t i o n
of the Matrix
Elements
In this section the distorted wave calculation will be performed and applied to C~. The function which distorts the momentum distribution is, from eq. (2), D(r) = exp
(--iEp(°°Vds). \
(3)
kp dr
For an outgoing proton with energy between 150 and 400 MeV the real part of the optical potential can be neglected to a good approximation. Expressing the imaginary part in the nucleon density p and in the mean free nucleonnucleon cross section e, which is practically energy independent and equals s) 29 mb, one gets O0
D ( r , : exp (
lfr
p-eds) -~ exp
[
fr
OO
,Po
exp ( - - ~ ) d , ] .
(,)
In the last step we have taken for the nucleon density p(r) = Po exp(--rZ/bg), where P0 = A~ b3:~t (A being the mass number). Because of the axial symmetry of D(r) it is convenient to take the z-axis of our coordinate system in the direction of the outgoing proton and the origin at the centre of the nucleus. Thus we find D(r) = exp
[
--fiexp
( v,)f -- ~
1
,lb exp (--s'~)ds ' ,
(5)
where fl = ~ A / 2 6 2 ~
a n d v 2 = x 2 + y 2.
In order to be able to perform a closed integration in eq. (2) we have replaced D (r) b y the following function, which for C1~ can be adjusted so as to differ less
144
G. JACOB AND TH. A. J. MARIS
t h a n a b o u t 5 ~ f r o m e x p r e s s i o n (5) in m o r e t h a n 500 p o i n t s s p r e a d e q u i d i s t a n t l y over the nucleus: D ( r ) ~,
1--{c-t-d(z--zo)+/(z--zo)
exp
[--g(z--zo)~]} × {1--
(h+jz)v 2} e x p (--kv~).
(6)
T h e p e r t i n e n t c o n s t a n t s for t h e case of C TM, for w h i c h 7) b = 1.93 fm, h a v e b e e n d e t e r m i n e d g r a p h i c a l l y w i t h t h e result: c = 0.387, d = - - 0.0901 f m -1, ] = - - 0.155 fm -1, g = 0.158 f m -2, h = 0.0539 f m -~, /" = 0.0147 f m -3, k = 0.152 f m -2 a n d z 0 =- 0.579 fm. F o r ~vn (r) we h a v e t a k e n e i g e n f u n c t i o n s of a h a r m o n i c oscillator p o t e n t i a l w h i c h gives a m e a n s q u a r e n u c l e a r r a d i u s e q u a l t o ~b ~.
!
J9.1\ /
-O.2
o.o
O.2
q[X'.]
Fig. 1. ~.[g'~[~ of equation (2) of text for outgoing protons of 150-400 MeV and 0.59 Limes the momentum distribution of the ls-shell in CTM.
n
g.I
/ 2
-0.2
0.0
0.2
q [X-p]
Fig. 2. Same as fig. 1 for the lp-shell and with the factor 0.75 instead of 0.59. As h a s b e e n r e m a r k e d in sect. 2, ~ ] g ' ~ l " d e p e n d s in o u r case o n l y o n q. T h i s d e p e n d e n c e is s h o w n in fig. 1 for t h e I s - s t a t e a n d in fig. 2 for t h e l p - s t a t e of C TM (full lines; for d o t t e d lines see b e l o w ) ; t h e abscissae give t h e m o d u l u s of t h e m o m e n t u m q of t h e n u c l e a r p r o t o n in u n i t s of t h e i n v e r s e r e d u c e d p r o t o n
Q U A S I - F R E E ELECTRON-PROTON
SCATTERING (I)
145
Compton wave length ~p-1 ___ Mpc/?i. The angular scale m a y be obtained b y the relation 9p = 2 arc sin (q/2kp) ~ q/kp. To give an impression of the orders of magnitude involved: For 200 MeV outgoing protons the angle between the two maxima in fig. 2 corresponds to 22 °. 4. D i s c u s s i o n
The results of the calculation are very simple to interpret, as is the case for many of the high energy distorted wave calculations. The relative angular correlation distributions are nearly identical to those directly calculated from the momentum distribution which the proton had before its ejection. The influence of the rest of tile nucleus merely consists of a reduction of the absolute magnitude, because tile outgoing proton m a y be absorbed. This situation is illustrated b y tile dotted curves in figs. 1 and 2 for the Is- and lp-states of C12, which correspond to the sum of the absolute squares of the Fourier transforms of the contributing normalized wave functions, multiplied b y reduction factors which equal 0.59 and 0.75 for the Is- and lp-states respectively. The lp-state factor is, as expected, somewhat larger than the Is-state one, because the lp-states are more located at the nuclear surface and are therefore less absorbed. A rough estimate 6) of tile absorption m a y be obtained b y taking the distorting function in eq. (2) independent of r and equal to the average value D, = exp ( - - ~ ) =
exp(--f,~),
(i = ls, lp)
with ~, = X/RZ---}(riZ}, where R denotes the nuclear radius of the corresponding uniform model. The expectation value (r~ z) is to be taken for the shell in question (i = Is, lp). In the case of CIz this estimate leads to values for the reduction factors which are about 25 °/o too small. A somewhat better estimate of the reduction factor for nuclei other than C1~ m a y be obtained b y taking
D ,(A ) = exp[(]~A)~ in/),(12)]. H e r e / ) i ( 1 2 ) (i ---- Is, lp) are the reduction factors, for the Is- and lp-states respectively, following from our calculation of the C12 case and the other factor in the exponential takes into account the dependence ot the linear nuclear dimensions on the mass number A. The function /),(A) is plotted in fig. 3. The estimate shows, for example, that Dls(¢0 ) -----0.45 which is to be compared with the value 0.005 for the corresponding (p, 2p) quasi-free scattering ~). From this curve it is to be expected that the range of possible measurements of energy spectra and also the possibility of a direct interpretation of the angular distribution as momentmn distribution is much more favourable in the (e, e'p) than in the (p, 2p) case. How good actually the agreement between the
14(}
G. JACOB AND TH. A. 1. MARIS
correlation- and momentum-distribution for a case like Ca4° is will be checked by a distorted wave calculation for this nucleus (see next paper 15)).
Q80
I
I0
I
I
20
I
I
30
I
I
40
I
I
50
I
I
A--*"
Fig. 3. R e d u c t i o n factors for the Is- a n d l p - s h e l l s as f u n c t i o n s of t h e m a s s n u m b e r for o u t g o i n g p r o t o n s of 150-400 MeV.
Some points of interest which we have not yet mentioned will now be briefly discussed. As in the (p, 2p) experiments, the widths of the inner shell peaks in the energy spectra may be interpreted either by assuming a short lifetime for the resulting hole-state or by remarking that the state is composed of m a n y quasi-stationary states with sharp energies lying in a broad spectrum, both interpretations being equivalent 9). In case of an odd initial nucleus the spin- and isospincoupling between the residual core and the upper shell will cause an additional splitting. The ejection of a proton out of a non-closed upper shell will in general result in more than one state of the residual nucleus. The magnitude and the momentum distribution of the overlap integrals of the wave functions belonging to the excited states of the residual nucleus and the ground state of the initial one can be obtained from the energy spectrum and the angular correlations. Because of the correlations between the nucleons in the nucleus, a quasi-free event will not always result in the corresponding hole-state, but often in a higher excited state. The average value of the extra excitation energy involved is the so-called rearrangement energy 10), which is believed to be of the order of 15 MeV for an infinite nucleus. We expect that, at least for not too heavy nuclei, the hole-state will still be recognizable in the measured spectrum of the residual nucleus, because it is by far the most abundant, as the rearrangement excitations will be smeared out over a large energy range. The (p, 2p) experiments have shown this to be the case for the light nuclei. One might hope to get
QUASI-FREE E L E C T R O N - P R O T O N
SCATTERING
(I)
I~
a crude impression of the total probability of a rearrangement excitation by comparing the absolute magnitudes of the measured cross sections for the hole-states with the values calculated using a distorted wave calculation. Connected with the rearrangement energies are the high momentum components of the nuclear protons, which are also caused by correlations between the nucleons in the nucleus. It would be interesting to try to observe these components by measuring the cross sections for large deviations of the proton angle from the angle corresponding to zero momentum of the nuclear proton. This will be experimentally difficult because these components are located in a large solid angle and are anyhow expected to be small. Furthermore it is to be expected that a collision with a proton of very high momentum will most often lead to a state which possesses a rearrangement excitation. Such contributions could not be separated from multiple collision processes, with which they really have much in common. Taking in eq. (1) the free electron-proton cross section we have assumed that the nuclear proton has the same structure as a free one. This assumption is certainly not exact but should be nearly true for the high momentum transfers we are considering. For this case only the charge distribution near the proton centre should be of importance and here the influence of the environment should be small. Otherwise stated: the neighbourhood of other nucleons should modify only the soft tail of the proton charge distribution but this soft part is not able to transfer high momenta. Appreciable deviations from the free proton structure would become apparent if they depended on the momentum of the proton in the nucleus. In this case the free cross section factor in formula (1) would become angle dependent and the correlation cross section would be distorted. The influence of the deuteron binding on the proton structure could be investigated by performing the discussed experiment on the deuteron, the wave function of which is rather well known. Confirmation of the expected identity of the structures of the deuteron proton and the free one would strengthen the determination of the neutron form factors 11), as has been already remarked earlier 16).
5. Concluding R e m a r k s We briefly recapitulate what could be achieved with the discussed experiment. 1) The total energy spectra of quasi-free electron-proton pairs may supply the following information: a) The positions of the peaks in the spectrum give the energy intervals between the single particle states. Thi~ would give direct information on m a n y questions of nuclear structure, such as the spin-orbit splittings (including those of inner shells), the value of the effective mass of the nucleons in the nucleus, the building up of the shells for increasing A, the influence of nuclear deformations on the shell structure.
148
G. JACOB AND TH. A. J . MARIS
b) The height of a peak is proportional to the overlap integral of the considered state of the residual nucleus with the ground state of the initial one and makes an estimate of the importance of rearrangement excitations possible. c) From the width of an inner shell peak the lifetime of the corresponding hole-state can be estimated. The distribution of the peaks resulting from the ejection of a proton out of a partially filled upper shell is closely related to the configurations of the initial and final nuclei. 2) The electron-proton angular correlation distributions would, for light and medium nuclei, nearly directly give the momentum distributions of the separate shells. The character of the single particle states (more general: overlap integrals) could be studied in considerable detail. Observed high momentum components might give indications of the deviations from the single particle model. From a theoretical point of view it seems possible to investigate nuclei up to, at least, A = 50 with this method. We have not yet mentioned the eventual effect of the energy losses by bremsstrahhmg on the total energy spectrum of the electron-proton pairs. These processes are of one order higher in the electromagnetic coupling constant than the ones responsible for the scattering. Therefore, without making any quantitative estimate, one would expect the emergence of gamma quanta with an appreciable energy to be seldom. Measurements of the spectra of 850 MeV and 646 MeV electrons scattered on free protons at angles of 145 ° and 135 ° respectively 1~) show that the influence of bremsstrahlung is indeed negligible for energy resolutions of a few MeV. As it is probably very difficult to perform the discussed experiment with the required energy and angular resolution, we would like to remark that, to our knowledge, at present no other experiment is able to give comparable information on the structure of the shells in a nucleus. As has been remarked in the Introduction, a high energy is necessary to obtain this information. Experimentally, at present only electrons and protons allow sufficiently accurate measurements. The (p, 2p) experiments have already given interesting results in this direction. However, due to the strong distortion of the incoming and outgoing proton waves one looses much of the information, except for the very light nuclei. Because the distortion in the case of (e, e'p) scattering is so nmch smaller, here lies a unique possibility for detailed investigations of nuclei. Provided the relevant measurements can be performed, it m a y therefore be hoped that high energy electron scattering will become equally useful for the investigation of nuclear shell structure as it is already for the determination of nuclear charge distributions. Thanks are due to Professors R. Hofstadter and J. A. McIntyre whose comments oll the experimental feasibility of the relevant measurements made
149
Q U A S I - F R E E E L E C T R O N - P R O T O N SCATTERING ( I )
us start this work, to Professor S. W. McDowell for helpful discussions and to Professor D. Dillenburg for his stimulating interest. We also thank our students, especially Mr. M. G. Zwanziger, for performing the numerical calculations. This work has been partially supported by Conselho Nacional de Pesquisas, Comissfio Supervisora dos Planos dos Institutos (COSUPI) and by the Division of Scientific Development of the Pan-American Union.
Appendix At present we only indicate the way of obtaining formula (1) of the text, realizing that a careful derivation would require more detailed considerations. In Born-approximation the cross section for quasi-free electron proton scattering, with the notation and units as defined in the text, is given by 13) : da d-~dk---~ -----2~l
lgn(kpo)12~(Ei--E,).
(Al)
Here the electron is assumed to have light velocity; the matrix element is the one for free scattering, except for the fact that the energy conservation is somewhat different; gn(kp,)is the Fourier transform of ~v~(r) (more general: of the overlap integral); the other symbols have the usual meaning. Neglecting the variation of the distortion of the outgoing proton wave over the effective interaction region, which should be very small for the high momentum transfer we have chosen, one can, in a WKB-approximation, simply incorporate the effect of the distortion of the outgoing proton wave in eq. (A1) by substituting g~(kpo) by e) g',(kp, kp r) = (2~)-]
exp(iq- r)~o~(r) exp
--i
Vds
dr. (A2)
The argument of the energy 6-function in eq. (A1) is
Et--E t = Eeo--Ee--E~--ER--EB, with E R the recoil energy of the residual nucleus and E B the binding energy of the ejected proton. Integrating both sides of eq. (A1) over EB, one finds da
f dOedE-~-d.C2pdEpdEB ---- 2:~Eo2Ep~/Ep2--1ll2 X Ig'~(k,, kpr)l z.
(A3)
n
In eq. (A3) the energy-momentum vectors are determined by energy-momentum conservation.
1~0
G. JACOB AND TH. A. J. MARIS
We consider first the case in which the nuclear proton is at rest. From energymomentum conservation follows (note that E R = 0)
E e = Ee°+EB(½EB--Ee*--I),
(A4)
1+2E% sin z ~Oe--E B Ep ----- 1+2E% sin 2 -~Oe(E%+I--EB)+EB(~EB--1) 1 + 2Eeo sin z ½0e-- E B
,
(A5)
q~, -= O,
(A6)
E e sin 0e tg 0p = E%_Ee cos 0e"
(A7)
For later use we remark that, for the rest-case, in practice Ee, Ep and in particular 0p are only slightly different from the corresponding values for the free case (E B = 0). If the final momenta are so chosen that the nuclear proton has a non-zero momentum, the recoil energy of the residual nucleus will still be negligibly small as one m a y see from the following estimate: In the single particle model the recoil momentum distribution of the nucleus equals the inversed momentum distribution of the ejected proton. For a nuclear proton with a kinetic energy of say, 20 MeV, the nuclear recoil energy is (l/A) • 20 MeV, which corresponds to only (10/A) % of the absolute momentum of a 100 MeV proton. Therefore, taking Oe, Op, Ee, Ep the same as in the rest case but introducing a non-zero angle 9p one still will obtain energy-momentum conservation but for a non-zero initial momentum of the nuclear proton. The theoretical simplicity of the set of measurements which we have chosen in the text is due to the fact that the T-matrix element occurring in eq. (A3) is practically independent of ~%. This is because kp0 corresponds to a momentum of a non-relativistic proton and is nearly orthogonal to k%, k, andkp. A Lorentz transformation which brings the nuclear proton to rest negligibly affects E o, Oe, Ep and 0p and therefore transforms the arguments of the T-matrix into the ones of the rest case. On the other hand, it can be seen that the energy factor arising from the relativistic transformation properties of the T-matrix (see, for example, ref. 14)) in typical cases deviates less than 1 % from unity and can therefore be neglected. So, indeed the right hand side of eq. (A3) depends on 0p to a good approximation only through ~ [g'~J2. As remarked above, the parameter values for the rest case (E B va 0) differ only slightly from the ones for the free case (E B = 0). Therefore we can express the T-matrix element in (A3) in the usual free cross section for the same values of Ee, and 0e. This approximation is not quite as good as the approximations made up to now, but it affects only the normalization of the angular distribu-
QUASI-FREE ELECTRON-PROTON SCATTERING (I)
151
tion. Integrating eq. (A1) over k e and kp, using some well-known properties of the S-function and putting E e = ke, one finds for the free cross section da tr EI~Ee 2 df2---~~ 2~z[(ke, kplT[keo, 0)l 2 l + 2 E e o sin2 ½0e •
(A8)
From eqs. (AS) and (A3) (taken for kpo = 0) eq. (1) of the text follows. References 1) Gerhard Jacob and Th. A. J. l~Iaris, Nuclear Physics 20 (1960) 440 2) B. Gottschalk and K. Strauch, Phys. Rev. 120 (1960) 1005 3) Peter Hillman, H. Tyr~n and Th. A. J. l~Iaris, Phys. Rev. Letters 5 (1960) 107; H. Tyr~n and P. Isacsson, Proc. Intern. ConL on Nuclear Structure, ed. b y D. A. Bromley and E. W. Vogt (North-Holland Publishing Co., Amsterdam, 1960) p. 429 4) J. P. Garron, J. C. Jacmart, Iv[. Riou et C. Ruhla, preprint (1961) 5) Owen Chamberlain and Emilio Segr~, Phys. Rev. 87 (1952) 81; J. B. Cladis, W. N. Hess and B. J. Moyer, Phys. Rev. 87 (1952) 425 6) Th. A. J. NIaris, Peter Hillman and H. Tyr~n, Nuclear Physics 7 (1958) 1; Th. A. J. Maris, Nuclear Physics 9 (1958/59) 577 7) Robert Hofstadter, Ann. Rev. Nuc. Sc. 7 (1957) 231; Robert H e r m a n and Robert Hofstadter, High-energy electron scattering tables (Stanford University Press, Stanford, 1960) 8) C. J. Batty, Nuclear Physics 23 (1961) 562 9) H. Tyr~n, Th, A. J. Maris and P. Hillman, Nuovo Cim. 6 (1957) 1507 10) V. F. Weisskopf, Nuclear Physics 3 (1957) 423 11) D. N. Olson, H. F. Schopper and R. R. Wilson, Phys. Rev, Letters 5 (1961) 286; R. Hoistadter, C. de Vries and Robert Herman, Phys. Rev. Letters b (1961) 290; Robert Hofstadter and Robert Herman, P h y s . Rev. Letters b (1961) 293 12) F. Bumiller, M. Croissiaux and R. Hofstadter, Phys. Rev. Letters 5 (1960) 261; R. Hofstadter, C. de Vries and Robert Herman, Phys. Rev. Letters 5 (1961) 290 13) Geoffrey F. Chew, Phys. Rev. 80 (1950) 196 14) W. Brenig und R. Haag, Fortschritte der Physik 7 (1959) 183 15) Gerhard Jacob and Th. A. J. Maris, Nuclear Physics 31 (1962) 152 16) Loyal Durand III, Phys. Rev. 115 (1959) 1020
Q U A S I - F R E E E L E C T R O N - P R O T O N S C A T T E R I N G (I) GERHARD
J A C O B t and T H . A. J. M A R I S *t
Instituto de Fisica and Faculdade de Filoso]ia, Universidade do Rio Grande do Sul, Pdrto A legre, Brasil Received 6 J u l y 1961 A b s t r a c t : I t is s h o w n t h a t , f r o m a n g u l a r and energy correlation m e a s u r e m e n t s on electron-proton
pairs emerging f r o m t h e scattering of high energy (300-1000 MeV) electrons on nuclei, detailed i n f o r m a t i o n on the e n e r g y levels a n d s t r u c t u r e s of the u p p e r a n d lower shells of light and m e d i u m nuclei could be obtained. A calculation in which the distortion of the outgoing p r o t o n w a v e is t a k e n into a c c o u n t has been p e r f o r m e d for C12. As c o m p a r e d to the result for zero distortion, the absolute m a g n i t u d e of the correlation cross section is reduced, b u t the shape of its a n g u l a r d i s t r i b u t i o n is practically unchanged. Consequently the observed energy and a n g u l a r correlations would i m m e d i a t e l y give b o t h the binding energy and t h e m o m e n t u m distribution of the nuclear p r o t o n in the shell model s t a t e o u t of which it h a s been ejected. F r o m an e x t r a p o l a t i o n to o t h e r nuclei of the calculated value of the reduction factor for the cross section, it is expected t h a t this situation prevails at least up to nuclei with A = 50. Finally some corrections are qualitatively discussed.
1. Introduction Assuming the approximate validity of the shell model of the nucleus, it is apparent that low energy experiments are able to give detailed information only on certain aspects of nuclear structure, namely on the properties of the least bound proton and neutron shells. To break up the more strongly bound shells a relatively high energy is needed; for low energy experiments the core of the nucleus much resembles an elementary particle of which only overall properties are relevant and can be determined. In recent years high energy (150-440 MeV) investigations of quasi-free proton-proton scattering have given information on the structure of the more strongly bound nuclear shells. Loosely speaking, in a quasi-free scattering event in a nucleus, a high energy particle knocks out a nucleon of the nucleus without any additional violent interaction of the incoming or the two emerging particles with the nucleus. Experimentally these events are recognizable b y the more or less sharp peaks they cause in the spectrum of the summed energies of the two emerging particles, corresponding to the binding energies of the shell model state of the ejected nucleon. The peaks are superimposed on a background caused b y multiple scattering and b y excitations from rearrangements in the residual nucleus. A short discussion of these processes and references m a y be * N o w on leave at the I n s t i t u t e for Theoretical Physics, University of Copenhagen. tt P r e s e n t address: Physics D e p a r t m e n t , University of Illinois, U r b a n a . 139
140
G.
JACOB
AND
TH.
A.
J.
MARIS
found, for example, in the introduction of ref. l) ; for later work see reis. 2-4). Qualitatively, the experimental results of quasi-free proton-proton scattering agree remarkably well with the shell model, and quantitative information on the energy differences of the shells, on their momentum distributions and on the widths of the hole states, has been obtained. One serious drawback of these (p, 2p) experiments is that the inner shells of only the light nuclei can be expected to be, and have indeed been, observed. This is because the total path length which the incoming and the two outgoing protons have to traject in a medium nucleus amounts in general to several times their mean free path in nuclear matter. For medium and heavy nuclei, processes occurring in certain parts of the nuclear surface for which the total path length is particularly small, will be mainly observed I). Because of the sloping edges of the shell model potential and because of the influence of the binding energy on the exponential tail of the single particle wave functions, the nuclear surface will mainly consist of particles of the upper shells. Consequently the inner shells of medium and heavy nuclei are difficult to observe by this method. It is in this respect that quasi-free (e, e’p) scattering should offer a clear advantage over the (p, 2p) processes. We are thinking of an initial electron energy of 300-1000 MeV and an energy of 100-400 MeV for the recoil proton. In a quasi-free (e, e’p) scattering event only the outgoing proton has an appreciable chance of being absorbed in the nucleus. Therefore surface interactions are much less accentuated than in the (p, 2p) scattering and the contributions of the inner shells relatively to those of the upper shell will be much larger, especially for medium or heavy nuclei. One should, however, not forget that the negligible absorption of a high energy electron in the nucleus is caused by the weakness of the electromagnetic interaction. This same fact results in small absolute cross sections for the quasi-free events, which make the experiments difficult, though not out of question t. There is another point of no less importance in which quasi-free electronproton scattering seems to offer a considerable advantage over quasi-free proton-proton scattering. As we have remarked, the angular and energy correlations in quasi-free proton-proton scattering may not only give rather directly the average momentum distributions for the nucleons in the whole nucleus, as has been recognized ten years ago 5)) but also the momentum distributions for the separate shells out of which the nucleons are ejected l-4, “). In case the scattering of the particles before and after the collision could be neglected, the difference between the total momentum of the two emerging particles and the momentum of the incoming particle would equal the momentum the struck nucleon had in the nucleus. (Of course this difference is always equal and opposite to the recoil momentum of the residual nucleus). Therefore, t R. Hofstadter
and J. A. McIntyre,
private communications.
QUASI-FREE
ELECTRON-PROTON
SCATTERING
(I)
141
would be possible to read off directly the momentum distribution of the struck nucleon from the correlation cross sections. However, the distortion of the incoming and outgoing waves by the nucleus destroys the connection between the momentum and correlation distributions in a degree which strongly increases with the size of the nucleus. In (p, 2p) scattering this effect becomes serious for all except the lightest nuclei. In (e, e’p) scattering one would hope that for light and medium nuclei mainly the magnitude but not the shape of the cross section will be influenced by the relatively small distortion. The following calculation verifies this expectation for a nucleus like C12. The shapes of the correlation curves calculated with distorted waves differ negligibly from those following directly from the momentum distributions, but the absolute magnitudes are reduced; this reduction differs only by 25 O/Ofrom a nai’ve estimate using an average absorption. We expect this situation to be still essentially true for much heavier nuclei than C12. In sect. 2 the necessary formulae of a distorted wave Born approximation for our case have been written down. Calculations for Cl2 are performed in sect. 3. In sect. 4 an extrapolation of the cross sections to heavier nuclei is made; the results are discussed from a physical point of view and the inherent possibilities of quasi-free e-p scattering for the investigation of nuclear structure are outlined. In sect. 5 the conclusions are summarized; some details of the calculation are given in the Appendix. it
2. Cross Section Formula For the sake of simplicity a pure j-j coupling single particle model for the nucleus is assumed. In the case of mixed configurations the single particle wave function of the nuclear proton has everywhere in this paper to be replaced by the overlap integral of the wave functions of the initial and the residualnucleus; the differences in binding energies of the nuclear protons have then the more general meaning of differences in excitation energies of the residual nucleus. We use the Born approximation for the electromagnetic interaction and a distorted plane wave for the outgoing proton. Furthermore it will be assumed that the effective interaction between the incoming electron and the knockedout proton is of such a range that the variation of the distortion of the proton wave over the effective interaction region can be neglected. This assumption is well fulfilled for the considered large momentum transfers to the proton, as may be estimated from the wavelength of the intermediating virtual photon. For the general case of arbitrary angles and energies of the outgoing particles the formulae become complicated and little transparent. We shall therefore choose an example which seems theoretically most simple although experimentally measurements of non-coplanar electron and proton momenta are required. The following nomenclature will be used: the indices e,, p,,, e and p refer to
142
G. JACOB AND TH, A. J . NIARXS
the initial and final electron and proton respectively; (E, k) is an energy momentum four-vector; we have taken the proton mass, ~ and c as units. The plane through k% and k e is called the "scattering plane"; 0 is the angle between the projection of a vector on the scattering plane and k% ; ~ is the angle between a vector and the scattering plane. First we shall consider the scattering on a nuclear proton in rest, bound with a definite binding energy E B. For this case ~p ~ 0 and Ee, Ep, 0p follow from energy-momentum conservation and are given as functions of E B and 0e in the Appendix. The experiment is supposed to be performed by considering E B as a variable and measuring the cross section da/d.QedEed.QpdE p for coincidently emerging electron-proton pairs as a function of E B for fixed Eeo, 0e and ~p ~ 0. The resulting spectrum will contain a peak having its maximum at the value of E B equal to the binding energy of the proton and its width given b y the lifetime of the final state of the residual nucleus and the experimental widths for the proton and electron energies. To measure the momentum distribution of the nuclear proton the measurement is repeated for non-zero values of ~p. Denoting the final proton momentum for the just considered rest case b y kpr, one finds from energy-momentum conservation for the initial momentum of the nuclear proton kpo :
--keo-~-ke-~-k p : kp--kp r,
kpo ~- 2kp sin ½~p ~ kp9~p .
The direction of kpo is approximately orthogonal to the scattering plane. For the conditions stated one finds to a good approximation for the cross section (see Appendix) f
da da ~r d.C2edE~C2p dEp d E s = ~ ~ - -
1 (1 q-2Eeo sin 2 ½00) ~ Ig'~ (kp, kp r) 12. (1)
In the right hand side of this formula the only dependence on ~opis contained in
Ig' l 2. For the more general case of arbitrary outgoing proton angles eq. (1) can still be used but the free cross section ~) on the right hand side has to be replaced b y a function of the proton scattering angles. Without the distortion of the outgoing proton wave, g'~ would have been the momentum distribution of the proton whose ejection leads to the considered final state. Neglecting the distortion of the electron wave function but taking the distortion for the proton into account b y a WKB-method one has s) g'~ (kp, kp r) = (2a)-]
/E
(
e x p ( i q , r)~0,(r)exp - - i
Vds
)]
dr.
(2)
In eq. (2) q = k%--l%--kp = kpr--kp is the recoil momentum of the residual nucleus; for the single particle model one has - - q = Rp0; V(r) is a suitable optical model potential over which one integrates along the classical path of
143
Q U A S I - F R E E E L E C T R O N - P R O T O N SCATTERING ( I )
the outgoing proton and the single particle state ~0~(r) is normalized to unity. The summation over n in eq. (1) takes into account that some final states, for example those which differ only in the orientation of their spins, cannot be separated experimentally. In case the initial nucleus has a non-zero spin, the cross section has in addition to be averaged over the initial states. It m a y be remarked that, for unpolarized nuclei, ~nig'n(kp, kpr)l 2 and therefore also the cross section (1), as a function of kp for fixed kpr, is rotationally symmetric with respect to the direction of kpr. This is because for fixed kpr all the elements entering into the calculation of the dependence of the distorting function on kp are rotationally symmetric around kp r. It follows that g'n(kp, kpr) for our case is just a function of q = lkp--kpr]. It m a y be noted that the experimental situation for fixed momenta of the incoming and outgoing electrons is only reflection symmetric with respect to the "scattering plane". 3. C a l c u l a t i o n
of the Matrix
Elements
In this section the distorted wave calculation will be performed and applied to C~. The function which distorts the momentum distribution is, from eq. (2), D(r) = exp
(--iEp(°°Vds). \
(3)
kp dr
For an outgoing proton with energy between 150 and 400 MeV the real part of the optical potential can be neglected to a good approximation. Expressing the imaginary part in the nucleon density p and in the mean free nucleonnucleon cross section e, which is practically energy independent and equals s) 29 mb, one gets O0
D ( r , : exp (
lfr
p-eds) -~ exp
[
fr
OO
,Po
exp ( - - ~ ) d , ] .
(,)
In the last step we have taken for the nucleon density p(r) = Po exp(--rZ/bg), where P0 = A~ b3:~t (A being the mass number). Because of the axial symmetry of D(r) it is convenient to take the z-axis of our coordinate system in the direction of the outgoing proton and the origin at the centre of the nucleus. Thus we find D(r) = exp
[
--fiexp
( v,)f -- ~
1
,lb exp (--s'~)ds ' ,
(5)
where fl = ~ A / 2 6 2 ~
a n d v 2 = x 2 + y 2.
In order to be able to perform a closed integration in eq. (2) we have replaced D (r) b y the following function, which for C1~ can be adjusted so as to differ less
144
G. JACOB AND TH. A. J. MARIS
t h a n a b o u t 5 ~ f r o m e x p r e s s i o n (5) in m o r e t h a n 500 p o i n t s s p r e a d e q u i d i s t a n t l y over the nucleus: D ( r ) ~,
1--{c-t-d(z--zo)+/(z--zo)
exp
[--g(z--zo)~]} × {1--
(h+jz)v 2} e x p (--kv~).
(6)
T h e p e r t i n e n t c o n s t a n t s for t h e case of C TM, for w h i c h 7) b = 1.93 fm, h a v e b e e n d e t e r m i n e d g r a p h i c a l l y w i t h t h e result: c = 0.387, d = - - 0.0901 f m -1, ] = - - 0.155 fm -1, g = 0.158 f m -2, h = 0.0539 f m -~, /" = 0.0147 f m -3, k = 0.152 f m -2 a n d z 0 =- 0.579 fm. F o r ~vn (r) we h a v e t a k e n e i g e n f u n c t i o n s of a h a r m o n i c oscillator p o t e n t i a l w h i c h gives a m e a n s q u a r e n u c l e a r r a d i u s e q u a l t o ~b ~.
!
J9.1\ /
-O.2
o.o
O.2
q[X'.]
Fig. 1. ~.[g'~[~ of equation (2) of text for outgoing protons of 150-400 MeV and 0.59 Limes the momentum distribution of the ls-shell in CTM.
n
g.I
/ 2
-0.2
0.0
0.2
q [X-p]
Fig. 2. Same as fig. 1 for the lp-shell and with the factor 0.75 instead of 0.59. As h a s b e e n r e m a r k e d in sect. 2, ~ ] g ' ~ l " d e p e n d s in o u r case o n l y o n q. T h i s d e p e n d e n c e is s h o w n in fig. 1 for t h e I s - s t a t e a n d in fig. 2 for t h e l p - s t a t e of C TM (full lines; for d o t t e d lines see b e l o w ) ; t h e abscissae give t h e m o d u l u s of t h e m o m e n t u m q of t h e n u c l e a r p r o t o n in u n i t s of t h e i n v e r s e r e d u c e d p r o t o n
Q U A S I - F R E E ELECTRON-PROTON
SCATTERING (I)
145
Compton wave length ~p-1 ___ Mpc/?i. The angular scale m a y be obtained b y the relation 9p = 2 arc sin (q/2kp) ~ q/kp. To give an impression of the orders of magnitude involved: For 200 MeV outgoing protons the angle between the two maxima in fig. 2 corresponds to 22 °. 4. D i s c u s s i o n
The results of the calculation are very simple to interpret, as is the case for many of the high energy distorted wave calculations. The relative angular correlation distributions are nearly identical to those directly calculated from the momentum distribution which the proton had before its ejection. The influence of the rest of tile nucleus merely consists of a reduction of the absolute magnitude, because tile outgoing proton m a y be absorbed. This situation is illustrated b y tile dotted curves in figs. 1 and 2 for the Is- and lp-states of C12, which correspond to the sum of the absolute squares of the Fourier transforms of the contributing normalized wave functions, multiplied b y reduction factors which equal 0.59 and 0.75 for the Is- and lp-states respectively. The lp-state factor is, as expected, somewhat larger than the Is-state one, because the lp-states are more located at the nuclear surface and are therefore less absorbed. A rough estimate 6) of tile absorption m a y be obtained b y taking the distorting function in eq. (2) independent of r and equal to the average value D, = exp ( - - ~ ) =
exp(--f,~),
(i = ls, lp)
with ~, = X/RZ---}(riZ}, where R denotes the nuclear radius of the corresponding uniform model. The expectation value (r~ z) is to be taken for the shell in question (i = Is, lp). In the case of CIz this estimate leads to values for the reduction factors which are about 25 °/o too small. A somewhat better estimate of the reduction factor for nuclei other than C1~ m a y be obtained b y taking
D ,(A ) = exp[(]~A)~ in/),(12)]. H e r e / ) i ( 1 2 ) (i ---- Is, lp) are the reduction factors, for the Is- and lp-states respectively, following from our calculation of the C12 case and the other factor in the exponential takes into account the dependence ot the linear nuclear dimensions on the mass number A. The function /),(A) is plotted in fig. 3. The estimate shows, for example, that Dls(¢0 ) -----0.45 which is to be compared with the value 0.005 for the corresponding (p, 2p) quasi-free scattering ~). From this curve it is to be expected that the range of possible measurements of energy spectra and also the possibility of a direct interpretation of the angular distribution as momentmn distribution is much more favourable in the (e, e'p) than in the (p, 2p) case. How good actually the agreement between the
14(}
G. JACOB AND TH. A. 1. MARIS
correlation- and momentum-distribution for a case like Ca4° is will be checked by a distorted wave calculation for this nucleus (see next paper 15)).
Q80
I
I0
I
I
20
I
I
30
I
I
40
I
I
50
I
I
A--*"
Fig. 3. R e d u c t i o n factors for the Is- a n d l p - s h e l l s as f u n c t i o n s of t h e m a s s n u m b e r for o u t g o i n g p r o t o n s of 150-400 MeV.
Some points of interest which we have not yet mentioned will now be briefly discussed. As in the (p, 2p) experiments, the widths of the inner shell peaks in the energy spectra may be interpreted either by assuming a short lifetime for the resulting hole-state or by remarking that the state is composed of m a n y quasi-stationary states with sharp energies lying in a broad spectrum, both interpretations being equivalent 9). In case of an odd initial nucleus the spin- and isospincoupling between the residual core and the upper shell will cause an additional splitting. The ejection of a proton out of a non-closed upper shell will in general result in more than one state of the residual nucleus. The magnitude and the momentum distribution of the overlap integrals of the wave functions belonging to the excited states of the residual nucleus and the ground state of the initial one can be obtained from the energy spectrum and the angular correlations. Because of the correlations between the nucleons in the nucleus, a quasi-free event will not always result in the corresponding hole-state, but often in a higher excited state. The average value of the extra excitation energy involved is the so-called rearrangement energy 10), which is believed to be of the order of 15 MeV for an infinite nucleus. We expect that, at least for not too heavy nuclei, the hole-state will still be recognizable in the measured spectrum of the residual nucleus, because it is by far the most abundant, as the rearrangement excitations will be smeared out over a large energy range. The (p, 2p) experiments have shown this to be the case for the light nuclei. One might hope to get
QUASI-FREE E L E C T R O N - P R O T O N
SCATTERING
(I)
I~
a crude impression of the total probability of a rearrangement excitation by comparing the absolute magnitudes of the measured cross sections for the hole-states with the values calculated using a distorted wave calculation. Connected with the rearrangement energies are the high momentum components of the nuclear protons, which are also caused by correlations between the nucleons in the nucleus. It would be interesting to try to observe these components by measuring the cross sections for large deviations of the proton angle from the angle corresponding to zero momentum of the nuclear proton. This will be experimentally difficult because these components are located in a large solid angle and are anyhow expected to be small. Furthermore it is to be expected that a collision with a proton of very high momentum will most often lead to a state which possesses a rearrangement excitation. Such contributions could not be separated from multiple collision processes, with which they really have much in common. Taking in eq. (1) the free electron-proton cross section we have assumed that the nuclear proton has the same structure as a free one. This assumption is certainly not exact but should be nearly true for the high momentum transfers we are considering. For this case only the charge distribution near the proton centre should be of importance and here the influence of the environment should be small. Otherwise stated: the neighbourhood of other nucleons should modify only the soft tail of the proton charge distribution but this soft part is not able to transfer high momenta. Appreciable deviations from the free proton structure would become apparent if they depended on the momentum of the proton in the nucleus. In this case the free cross section factor in formula (1) would become angle dependent and the correlation cross section would be distorted. The influence of the deuteron binding on the proton structure could be investigated by performing the discussed experiment on the deuteron, the wave function of which is rather well known. Confirmation of the expected identity of the structures of the deuteron proton and the free one would strengthen the determination of the neutron form factors 11), as has been already remarked earlier 16).
5. Concluding R e m a r k s We briefly recapitulate what could be achieved with the discussed experiment. 1) The total energy spectra of quasi-free electron-proton pairs may supply the following information: a) The positions of the peaks in the spectrum give the energy intervals between the single particle states. Thi~ would give direct information on m a n y questions of nuclear structure, such as the spin-orbit splittings (including those of inner shells), the value of the effective mass of the nucleons in the nucleus, the building up of the shells for increasing A, the influence of nuclear deformations on the shell structure.
148
G. JACOB AND TH. A. J . MARIS
b) The height of a peak is proportional to the overlap integral of the considered state of the residual nucleus with the ground state of the initial one and makes an estimate of the importance of rearrangement excitations possible. c) From the width of an inner shell peak the lifetime of the corresponding hole-state can be estimated. The distribution of the peaks resulting from the ejection of a proton out of a partially filled upper shell is closely related to the configurations of the initial and final nuclei. 2) The electron-proton angular correlation distributions would, for light and medium nuclei, nearly directly give the momentum distributions of the separate shells. The character of the single particle states (more general: overlap integrals) could be studied in considerable detail. Observed high momentum components might give indications of the deviations from the single particle model. From a theoretical point of view it seems possible to investigate nuclei up to, at least, A = 50 with this method. We have not yet mentioned the eventual effect of the energy losses by bremsstrahhmg on the total energy spectrum of the electron-proton pairs. These processes are of one order higher in the electromagnetic coupling constant than the ones responsible for the scattering. Therefore, without making any quantitative estimate, one would expect the emergence of gamma quanta with an appreciable energy to be seldom. Measurements of the spectra of 850 MeV and 646 MeV electrons scattered on free protons at angles of 145 ° and 135 ° respectively 1~) show that the influence of bremsstrahlung is indeed negligible for energy resolutions of a few MeV. As it is probably very difficult to perform the discussed experiment with the required energy and angular resolution, we would like to remark that, to our knowledge, at present no other experiment is able to give comparable information on the structure of the shells in a nucleus. As has been remarked in the Introduction, a high energy is necessary to obtain this information. Experimentally, at present only electrons and protons allow sufficiently accurate measurements. The (p, 2p) experiments have already given interesting results in this direction. However, due to the strong distortion of the incoming and outgoing proton waves one looses much of the information, except for the very light nuclei. Because the distortion in the case of (e, e'p) scattering is so nmch smaller, here lies a unique possibility for detailed investigations of nuclei. Provided the relevant measurements can be performed, it m a y therefore be hoped that high energy electron scattering will become equally useful for the investigation of nuclear shell structure as it is already for the determination of nuclear charge distributions. Thanks are due to Professors R. Hofstadter and J. A. McIntyre whose comments oll the experimental feasibility of the relevant measurements made
149
Q U A S I - F R E E E L E C T R O N - P R O T O N SCATTERING ( I )
us start this work, to Professor S. W. McDowell for helpful discussions and to Professor D. Dillenburg for his stimulating interest. We also thank our students, especially Mr. M. G. Zwanziger, for performing the numerical calculations. This work has been partially supported by Conselho Nacional de Pesquisas, Comissfio Supervisora dos Planos dos Institutos (COSUPI) and by the Division of Scientific Development of the Pan-American Union.
Appendix At present we only indicate the way of obtaining formula (1) of the text, realizing that a careful derivation would require more detailed considerations. In Born-approximation the cross section for quasi-free electron proton scattering, with the notation and units as defined in the text, is given by 13) : da d-~dk---~ -----2~l
lgn(kpo)12~(Ei--E,).
(Al)
Here the electron is assumed to have light velocity; the matrix element is the one for free scattering, except for the fact that the energy conservation is somewhat different; gn(kp,)is the Fourier transform of ~v~(r) (more general: of the overlap integral); the other symbols have the usual meaning. Neglecting the variation of the distortion of the outgoing proton wave over the effective interaction region, which should be very small for the high momentum transfer we have chosen, one can, in a WKB-approximation, simply incorporate the effect of the distortion of the outgoing proton wave in eq. (A1) by substituting g~(kpo) by e) g',(kp, kp r) = (2~)-]
exp(iq- r)~o~(r) exp
--i
Vds
dr. (A2)
The argument of the energy 6-function in eq. (A1) is
Et--E t = Eeo--Ee--E~--ER--EB, with E R the recoil energy of the residual nucleus and E B the binding energy of the ejected proton. Integrating both sides of eq. (A1) over EB, one finds da
f dOedE-~-d.C2pdEpdEB ---- 2:~Eo2Ep~/Ep2--1l
(A3)
n
In eq. (A3) the energy-momentum vectors are determined by energy-momentum conservation.
1~0
G. JACOB AND TH. A. J. MARIS
We consider first the case in which the nuclear proton is at rest. From energymomentum conservation follows (note that E R = 0)
E e = Ee°+EB(½EB--Ee*--I),
(A4)
1+2E% sin z ~Oe--E B Ep ----- 1+2E% sin 2 -~Oe(E%+I--EB)+EB(~EB--1) 1 + 2Eeo sin z ½0e-- E B
,
(A5)
q~, -= O,
(A6)
E e sin 0e tg 0p = E%_Ee cos 0e"
(A7)
For later use we remark that, for the rest-case, in practice Ee, Ep and in particular 0p are only slightly different from the corresponding values for the free case (E B = 0). If the final momenta are so chosen that the nuclear proton has a non-zero momentum, the recoil energy of the residual nucleus will still be negligibly small as one m a y see from the following estimate: In the single particle model the recoil momentum distribution of the nucleus equals the inversed momentum distribution of the ejected proton. For a nuclear proton with a kinetic energy of say, 20 MeV, the nuclear recoil energy is (l/A) • 20 MeV, which corresponds to only (10/A) % of the absolute momentum of a 100 MeV proton. Therefore, taking Oe, Op, Ee, Ep the same as in the rest case but introducing a non-zero angle 9p one still will obtain energy-momentum conservation but for a non-zero initial momentum of the nuclear proton. The theoretical simplicity of the set of measurements which we have chosen in the text is due to the fact that the T-matrix element occurring in eq. (A3) is practically independent of ~%. This is because kp0 corresponds to a momentum of a non-relativistic proton and is nearly orthogonal to k%, k, andkp. A Lorentz transformation which brings the nuclear proton to rest negligibly affects E o, Oe, Ep and 0p and therefore transforms the arguments of the T-matrix into the ones of the rest case. On the other hand, it can be seen that the energy factor arising from the relativistic transformation properties of the T-matrix (see, for example, ref. 14)) in typical cases deviates less than 1 % from unity and can therefore be neglected. So, indeed the right hand side of eq. (A3) depends on 0p to a good approximation only through ~ [g'~J2. As remarked above, the parameter values for the rest case (E B va 0) differ only slightly from the ones for the free case (E B = 0). Therefore we can express the T-matrix element in (A3) in the usual free cross section for the same values of Ee, and 0e. This approximation is not quite as good as the approximations made up to now, but it affects only the normalization of the angular distribu-
QUASI-FREE ELECTRON-PROTON SCATTERING (I)
151
tion. Integrating eq. (A1) over k e and kp, using some well-known properties of the S-function and putting E e = ke, one finds for the free cross section da tr EI~Ee 2 df2---~~ 2~z[(ke, kplT[keo, 0)l 2 l + 2 E e o sin2 ½0e •
(A8)
From eqs. (AS) and (A3) (taken for kpo = 0) eq. (1) of the text follows. References 1) Gerhard Jacob and Th. A. J. l~Iaris, Nuclear Physics 20 (1960) 440 2) B. Gottschalk and K. Strauch, Phys. Rev. 120 (1960) 1005 3) Peter Hillman, H. Tyr~n and Th. A. J. l~Iaris, Phys. Rev. Letters 5 (1960) 107; H. Tyr~n and P. Isacsson, Proc. Intern. ConL on Nuclear Structure, ed. b y D. A. Bromley and E. W. Vogt (North-Holland Publishing Co., Amsterdam, 1960) p. 429 4) J. P. Garron, J. C. Jacmart, Iv[. Riou et C. Ruhla, preprint (1961) 5) Owen Chamberlain and Emilio Segr~, Phys. Rev. 87 (1952) 81; J. B. Cladis, W. N. Hess and B. J. Moyer, Phys. Rev. 87 (1952) 425 6) Th. A. J. NIaris, Peter Hillman and H. Tyr~n, Nuclear Physics 7 (1958) 1; Th. A. J. Maris, Nuclear Physics 9 (1958/59) 577 7) Robert Hofstadter, Ann. Rev. Nuc. Sc. 7 (1957) 231; Robert H e r m a n and Robert Hofstadter, High-energy electron scattering tables (Stanford University Press, Stanford, 1960) 8) C. J. Batty, Nuclear Physics 23 (1961) 562 9) H. Tyr~n, Th, A. J. Maris and P. Hillman, Nuovo Cim. 6 (1957) 1507 10) V. F. Weisskopf, Nuclear Physics 3 (1957) 423 11) D. N. Olson, H. F. Schopper and R. R. Wilson, Phys. Rev, Letters 5 (1961) 286; R. Hoistadter, C. de Vries and Robert Herman, Phys. Rev. Letters b (1961) 290; Robert Hofstadter and Robert Herman, P h y s . Rev. Letters b (1961) 293 12) F. Bumiller, M. Croissiaux and R. Hofstadter, Phys. Rev. Letters 5 (1960) 261; R. Hofstadter, C. de Vries and Robert Herman, Phys. Rev. Letters 5 (1961) 290 13) Geoffrey F. Chew, Phys. Rev. 80 (1950) 196 14) W. Brenig und R. Haag, Fortschritte der Physik 7 (1959) 183 15) Gerhard Jacob and Th. A. J. Maris, Nuclear Physics 31 (1962) 152 16) Loyal Durand III, Phys. Rev. 115 (1959) 1020