Revista Mexicana de Física 40, Suplemento 1 (1994) 123-133

Short range nucleon-nucleon correlations probed byreal and virtual photons

ZISIS PAPANDREOUGenter of Nuclear Studies, Department of l'hysics

The George Washington University, Washington DG 20052Received 6 January 1994; accepted 12 May 1994

ABSTRACT.During the Ia.<;tdecade, a wealth of experimental data ha.<;heen gathered from pionabsorption in a nucleus and the subsequent ejection of two nucleons: evidence has been foundpointing to the existen ce of correlated nucleon pairs in the nucleus, prior to the interaction. Theseresults have been confirmed by the first real and virtual photon absorption experiments. \Viththe recent advent of high intensity, high energy, continuous wave electron accelerators, a new setof experiments utilizing real and virtual photons is now possible. The added features of polarizedbeams, large solid angle detectors and polarimeters allow access to hitherto unexplored observableswhich will help quantify the strength of the one- and two-body nuclear current and simultaneou sIyillucidate the nature of the nucleon-nucleon correlation function. This investigation can be bestperformed via the (-y,NN) and (e, e'NN) reactions, which involve coincident detection of the twoemitted nucleons.

RESUMEN.Durante la última década se ha reunido una riqueza de datos experimentales de ab-sorción de piones en un núcleo y la subsecuente expulsión de dos nucleones. Se ha encontradoevidencia que indica la existencia de pares de nucleones correlacionados en el núcleo, anterior ala interacción. Estos resultados han sido confirmados por el primer experimento de absorción defotones reales y virtuales. Con la llegada reciente de aceleradores de electrones de onda continua,de alta intensidad, y alta energía, ahora es posible hacer un nuevo conjunto de experimentos, uti-lizando fotones reales y fotones virtuales. Los ra.<;gosagregados del haz polarizado, los detectoresde ángulos sólidos grandes y polarimetros permiten el acceso a la.<;observables no esplorada.<;ha$tahoy en dia, las cuales ayudarán modificando la intensidad de la corriente nuclear de uno y doscuerpos, simultaneamente aclararán la naturaleza de la función de correlación nucleón- nucleón.Esta.<;investigaciones se pueden desarrollar vía la.<;reaccciones (-y,NN) y (e,e'NN), la.<;cuales in-volucran detección en coincidencia de emisión de dos nucleones.

PAes: 24.70.+s; 25.1O.+s; 25.20.-x

1. INTRODUCTION

The building blocks of a large fraction of stable matter in the universe in its present formare protons and neutrons, which were once cousidered elementary partides. By now it isaccepted that nudeons are built from even smaller constituent partides (quarks). It isapparent that any theory able to account correctly for the behavior of the fundamentalconstituents of matter, must also be able to describe eventually the more global features ofmatter. In the case of quarks, the theory which describes their behavior is caBed QuantumChromodymanics (QCD). However, there are several features of QCD which either have

123

124 ZISIS PAPANDREOU

not been understood or have not been verified (e.g., the existence of the top quark andthe confinement of quarks). Besides these, there is another shortcoming: QCD works onlyin the regio n of large momentum transfer. In this high-Q region, associated with thequark nature of matter, perturbation theory can be employed in a similar manner toQuantum Electrodynamics (QED). In the low-Q region (Q2 < 2 (GeV /C)2), however,which corresponds to hadronic degrees of freedom (nucleons and mesons), perturbationtheory fails and perturbative QCD breaks down due to the large coupling constant of thestrong interaction. The puzzle behind the transition from the high-Q to low-Q regions isa subject of paramount interest in intermediate energy nuclear physics.Clearly, studies aiming at the solution of this problem must involve the examination

of the structure of the nucleus. This can be carried out by employing different pro bes,each sensitive to a different part of the interaction process. Hadronic probes, such as theproton or pion, "feel" all three of the fundamental interactions (gravity is excluded fromthis discussion): weak, eletromagnetic and strong. This fact makes the interpretation ofthe data very difficult. On the other hand, electrons are much cleaner probes, since theyfeel -to first order- only the electromagnetic interaction.

2. A SIMPLE PICTURE OF THE NUCLEUS

One approách to the investigation of the hadronic structure is via electro n scatteringoff a proton at energies of a few hundred MeV. This interaction involves the exchangeof a virtual photon between the incoming electron and the target proton. The virtualphoton carries an energy w and a momentum Q, which are also known as the energy andmomentum transfer, respectively.Based on the simplicity of this model, it was natural to attempt to extend it to the

electron-nucleus system, by considering the nucleus as being simply a collection of nucle-ons and electron-nucleus scattering as the incoherent sum of electron-nucleon scattering.This model is known as Quasi-Free Scattering (QFS). Here, it is possible to factorize thedifferential cross section of the e-nucleus process into several terms, which separately de-scribe the elementary e-N scattering cross section, the nuclear structure effects (structurefunctions) and the kinematical factors. The QFS model has been very successful since ithas verified the shell model by revealing nuclear shells even in the center of the heaviestnuclei [lj.The successes of the QFS model are accompanied by serious deficiencies which have

proven that the e-nucleus interaction is in fact quite complexo When one explores the w-spectrum aboye 200 MeV, one encounters features not explainable by the QFS mode!. Onesu eh prominent feature is the ll-resonance, which is the excitation of a bound nucleon.This demonstrates directly the subnucleonic degrees of freedom within nuclei and the factthat the nucleon is a not a pointlike object but rather it has a complex structure. Suffice itlo say that the QFS model is inadequate on its own to describe the e-nucleus interaction.To summarize: sorne aspects of nuclei can be described by considering the nucleus as acollection of pointlike nucleons and mesons, whereas others require us to employ quarkdegrees of freedom.

SHORT RANGE NUCLEON-NUCLEON CORRELATIONS... 125

3. PROBING TIIE STRUCTURE OF THE NUCLEUS

In this paper, processes in which two nueleons are emitted will be discussed. Much infor-mation, to date, has been provided by pion scattering and absorption, invoIving energies ofup to a few hundred Me V. Here, the t..-resonance plays an important role, although otherprocesses contribute as well. This resonance, when created on a free nueleon, de-excitesinto a nueleon and a pion. In the presence of a nueleus, an additional channel becomesavailable: the absorptive channel, in which two nueleons are emitted. This occurs sincethe pion originating from the decay of the t.. may remain off-shell and be reabsorbed byan additional nueleon, leading to the ejection of two nueleons.

Essentially, due to the large energy and momentum mismatch between the pion and asingle nueleon in the nueleus, pion absorption takes place predominantly on two nu-eleons but also to sorne extent on three or 1Il0re. Inside a nueleus, pion absorptionfollows very elosely the elementary " + d - NN reaction. This is evident by COIll-paring the angular correlation of the emitted nueleons, as well as the angular distri-bution of the differential cross section, between the ,,-A and ,,-d reactions. In otherwords, the nueleus behaves to a large extent as a collection of deuterons, known asQuasi-Deuteron Absorption (QDA). This model elearly invokes initial state nueleon-nueleon (NN) correlations: the pion is absorbed on a correlated eluster of two (or more)nueleons.

Similar evidence has been obtained from virtual and real photon scattering experiments,as shown in Figures 1 and 2. The narrow peak in Figure 1 corresponds to the ground stateof the residual deuteron nueleus resulting from the 3He( e, e'p) reaction. The continuum ata larger missing energy (broad peak) has been interpreted as evidence of NN-correlationsin the ground state wave function of the target 3He nueleus, based on kinematical anddynamical arguments [21. The broad peak, in other words, corresponds to the absorptionof the virtual photon by a correlated nueleon pair, which results in the subsequent ejectionof these two nueleons.

It is well known that a real photon needs at least two nueleons to absorb its four-momentum due to the large momentum mismatch between the photon and the boundnueleon. The absorption of the photon by a T = O pair of nueleons (QDA) has beenelearly seen experimentally [3]' as illustrated in Figure 2. The energy distribution of pro-tons emitted after the absorption of photons on 9Be is shown as an inelusive spectrumand for the co;ncidence between the proton and a photon, neutron, pion and proton,respectively. Two structures appear in the spectrum: the one at a lower energy COrre-sponds to pinn production (the t.. resonance) whereas the other is related to tWQ-nueleonemission, since it is present only in the n . p and p . p spectra. Additional significant in-formation emerges from this work: the similarity in the structure between 9Be and 2Hconfirms the QDA model, and the cross section ratio of the pp over the pn channel isroughly 0.070.

The complexity nf the aboye mentioned interactions dictates the need to combinethe information from pions and photons (real and virtual). The characteristics of eachprobe establish thcm as complimentary tools in the investigation of nuclear interac-tions. For a more detailed comparison of these pro bes, the reader is directed to refer-ence [41.

126 ZISIS PAPANDREOU

12

•4

",E O."-1: '-~'ª 2-~c:.",

<.O~ O

1.5

E • 560MeV•••• 200MeVBe'-25-

1.0

0.5

O 40 60 80 100

Em(MeV)

FIGURE1. Missing energy speclra in lhe 'He(e, e'p) reaclion [2J.The broad peak serves as evidenceoí NN-correlations.

4. THEORETICAL FRAMEWORK

Two-nucleon emission is a suitable tool for lhe exploration of NN-correlalions in nu-clei [5-7] and for lhe understanding of the dynamics of nuclear currents. In particular, thisprocess can be studied advantageously by employing real or virtual photons as probes. Thepromise of new data on these reactions has stimulated several theoretical groups [8-131.Calculations from one of these groups [12,131 have presented an approach in extractingthe one- and two-body components of the nuclear current, as well as information on theNN-correlation function. This approach is discussed in this paper.

4.1. Observables without polarization

In nuclear reactions where two nucleons are emitted, information on the two-hole spectraldensity function and the two-body density matrix [11-13] can be obtained. From suchinvestigations one hopes to extract the two-body density in the ground state of the targetnucleus amI thus obtain an understanding of dynamical short-range correlations (SRC)which are essential for a proper description of nuclear structure.

In the case where two nucleons are emitted in (-y,NN) or (e,e'NN) reactions, one candescribe the measured differential cross section in terms of structure functions, which rep-resent the response of the nucleus to the longitudinal (electric) and transverse (magnetic)components of the photon in the interaction. Specifically, the triple-coincidence cross sec-tion induced by an electron is a function of six structure functions [121: /00, /11, /1-1,71-1' /01 and 701' When the two nucleons are emitted by a real unpolarized photon, only/11 contributes.

SUORT RANGE NUCLEON-NUCLEON CORRELATIONS... 127

200ll(roo

100

(f) O1-Z 2001.U

100>1.U O

200

O

10

O

....

.........'.

", ,"',1, ,1,

el,,', e. 'i¡ , " "

" ..,' , , ¡I,II 1, ,11 "".. '," 1- , '", , " ,

" ,,.

"

'" , ell",il"!,1iI"'11 ,

<lOO 600 800 <lOO 600 800PROTON MOMENTUM (MeV/c)

FIGURE 2. Proton spectra aCter absorption oC 327 MeV photons on 2H and 9Be [3J. Top: inclusive,coincidence with photons (A), neutrons (B), pions (e) and protons (D).

The fu.' represent the response of the nucleus to the longitudinal (A = O) and trans-verse (A = 1) components of the electromagnetic interaction. These structure functionsdepend only on the kinematical quantities w, Q and the angles between the momentumtransfer Q and the individual particle momenta of the two emitted nucleons, and are linearcombinations of the hadronic tensors W~v [121:

where

WIW = ~ J J~(if)r(if)b(E¡ - Ec) (1)

(2)

The transition matrix element in equation (2) consists of the one- and two-body partsof the current operatc,r, J(1) and J(2), respectively:

(3)

where T, T¡, T2 are the coordinates of the recoiling nucleus and the two ejected nucleons,respectively. The initial state wavefunction >Ir. can be expressed as

(4)

where cI> is the pair correlation function in the shellmodel and g(Ii'I -(21) is a Jastrow-typecorrelation function [121.

128 ZISIS PAPANDREOU

...:.:...:.::.:>.::.:.:.:.: ~:'::'::.:.:.:.:::....

,.'

b)

~ .'.. ,....../ : '.

... ......................

o)10'

zO¡::ULU(/)

~OO:::U

10-1020 60 80 100

y¡('')20 40 60 80 100

y¡('')

FIGURE3. Crass-seclions for lhe 160(e,e'pp) reaclion in coplanar symmelrical kinematics, plolledversus lhe angle of one of lhe emilled nudeans. The paramelers are E = 700 MeY, w = 150 MeYand Q = 400 MeYfe. The solid line is lhe resull of lhe full calculation induding J(l) + J(2)lerms wilh hard-core polenlial. The dOI-dashed line is similar calculalion with sofl-core polenlial.Dashed and dolled lines represenl lhe resulls of lhe calculalion based only on lhe J(I) lerms wilhhard-core and soft-core polenlials, respeclively, laken fram reference [121.

Examples of correlations can be found in systems other than the nuclear enviranment. Itis a general phenomenon that correlations are induced among lhe constituents of a many-body syslem due to interconstituent forces. As an example one might look al alomicliquid 4He, whose atoms are kept apart by the strong repulsive core of the interatomicpotentia1. The probability of finding two atoms simultaneously at locations T¡ and T2,can be expressed as p2g(r12), wilh p being the density of the liquid, r12 lhe relativedisplacemenl veclor (f12 = T¡ - T2) and g(r) lhe pair correlation function. The latteris determined by the inleratomic potenlia1. A similar approach is used for lhe nuclearsyslem.

The one-body operators J(1) can contribule to two-nucleon emission only via SRC. TheJ(2) operators include meson exchange (MEC) and isobar (IC) currents (.ó.-resonance)and thus can eject two nucleons even if SRC are not included. As such, J(2) may ormay not include correlations but il is important, nonetheless, because it contributes tothe lwo-body absorption of lhe photon. These features have been calculated, in a specifickinemalical configuration, for the 160(e, e'pp) and ¡60b, pp) reactions [12]' and displayedin Figures 3 and 4, respeclively.

An NN-inleraction wilh a large hard-core potenlial (OMY) [141 greatly enhances lhecross seclion expecled from J(I) while a soft-core NN-inleraclion (RSC) [15,161 producesa smaller cross seclion, as can be seen in bolh Figures 3 and 4. In the (e, e' pp) case, lhemagnilude of lhe differenl components changes in going fram longitudinal (where MEC

5110RT RANGE NUCLEON-NUCLEON CORRELATIONS... 129

I

................. /// .

'.'

n 10"3~V)-- 1O.An -r--.E'<-'-" ~Z 10"5 ..... ......... .."- ..,.'.....,.:.............~~>....:~O•....U 10-6 f-W(f)

(f)(f)

O 10.7 -~U

108 , , I

60 80 100

Yl (ol

I

60 80I

100

FIGURE 4. Cross-sections fOf the 160( i. pp) reaction in caplanar symmctrical kinematics withE, = 200 l\leV, and T{ = T~ = 88 MeV. The same potential as in Figure 3 has been used. Parts(a) and (b) are ca1culated with hard-core [14) and soft-core [15,161 potentia!s, respectivel)', takeufrom reference [121. The solid line is the result of the full calculation includiug J(I) + J(2) !erms.The dashed (dot-dashed) lines represent the results of the calculatiou when onl)' the one-bod)'(two-body) part of the nuclear current is considered.

amI the transverse components of IC are suppressed) to transv~rse kinelllaties (Figur~ 3).In the h,pp) reaetions both eurrents are equally importanl in lhe hard-eare case due tolhe presenee of MEC and both lransverse and longiludinal IC coutributions (Figure 4).When elllployiug a soft-eore ;,olential, however, the one-body eurreut is gr~atly r~duc~dwhieh illlplies lhal the eross seetion is less aff~cted by lhe corre1atiou funelion.

These features may be summarized as follows. Eleelron indtle~d lwo-prolon ~missionprobes 5RC, while the pholon indueed r~aelion access~s the lwo-body curr~nt. So althoughthe queslion of SRC is an imporlant one, lhe question of lhe lotal slrenglh of )(1) + )(2)

terms is also vital as part of our need to expIare the exaet nalure aud dynalllical prop~rli~sof lhe tbe (e, e') reaetion. A systematie study of both (e, ~'pp) and (r. pp) rPaCIions, alcarefully chosen kinematics, is absolutdy esscntial as ouIy sl1ch reartiotls aH' COllstrainingenougb for the task al hand.

4.2. Polarization ob.,ervable.<

So far lhe discussion has not includ~d polarization obs~rvabl~s. In faet. int~r~sting infor-mation can b~ obtain~d with a lin~arly polarized photon bealll. The r~sulting aSYlllm~tryin the cross S('ctiOll is giV(,ll in tite follO\ving ('quatioll (121:

(5)

130 ZISIS PAPANDREOU

L 1.0

0.6

0.2

-0.2

-0.6

,..,• •.' .• •• •.' .

",' "•, •........ .'. .'. .'. .'• ••• •'.

-1.0 60 70 80 90 100

FIGURE 5. Asymmetry for the 160(')', pp) reaction in coplanar syrnmetrical kinematics. The curvesfollow the notation of Figure 4.

where q, is the angle between the photon's polarization vector and the reaction planeo Theasymmetry is defined as the difference between the cross sections with the linear photonpolarization parallel (along the x-axis) and perpendicular (parallel to the y-axis) to thereaction plane:

dax - daY~=----

daX + daYf¡-lfl1 .

(6)

A measurement of the cross section in the (-y, pp) reaction determines directly the fl1

structure function. The asymmetry, on the other hand, yields the ratio of the transversestructure functions f¡-l and fl1. This ratio does not depend on the particular choice ofthe correlation function, but is remarkably sensitive to the nuclear current, as shown inFigure 5.At low energies for thc (-y, pn) reaction, hard-core and soft-core correlations give a

similar asymmetry, as shown in Figure 6. At higher energies, while the OMY curve re-mains positive, the RSC curve beco mes negative. Thus, this asymmetry is sensitive to thecorrelation function between the two emitted nucleons.

5. ELECTRON ACCELERATORS

Due to the exact nature of the electromagnetic probes, it is desirable to carry on theinvestigations of the hadronic structure with these as opposed to hadronic probes, which

SIIORTRANGENUCLEON-NUCLEONCORRELATIONS... 131

L 1.0

0.5

O

-0.5

-1.0

-o

"o.

80 120 160 200 240 280 320 360

Ey (MeV)

FIGURE 6. Asyrnrnetry in the l60(-r, np) reaction in coplanar kinernatics as a Cunction oC thephoton energy. The solid (dashed) line has been obtained with the OMY (RSC) correlation 113J.

are subject to the strong interaction that complicates matters. This and additional ad-vantageous features of em probes, have led to the construction of several high energy,high current and continuous wave (100% duty factor) machines, among which we men-tion NIKHEF-K (Amsterdam), ELSA/Bonn and MAMI-B/Mainz (Germany) and CEBAF(USA). The lirst three machines are in operation, and their respective features make themcomplimentary to each other. CEBAF will come onlinein 1994, and employs all neces-sary features (high current, 100% d.f., polarized beams, 411' detector, spectrometers, etc.)including the highest energy, reaching 4 GeV initially and 6-8 GeV at a later stage.

6. TIIE REACTIONSOF INTEREST

6.1. Two-proton emission

The (-y, pp) reaction contains MEC contributions and IC components, due to the couplingnature of the virtual photon. As such it is a suitable tool for investigating the two-bodycurrent which can be accessed via the asymmetry in this reaction when a linearly polarizedphoton beam is used. In different kinematic regions, the IC and MEC strengths can beenhanced or diminished. A complete angular distribution in coplanar kinematics is neces-sary to separate these effects, and to determine the strength of the two-body versus theone-body current (Figure 5). Due to its large angular acceptance, the CLAS spectrometerin Hall-B/CEBAF is most suited for these investigations.The (e,e'pp) reaction is, to lirst order, free of MEC contributions and thus suitable

for investigating SRC. Rednction of the IC contributions is critical because their presence

132 ZISIS PAPANDREOU

greatly affects the choice of NN-interaction assumed, hard-core or soft-core. In order tominimize IC effects one has to choose suitable kinematics. Confining the ejectiles in-plane reduces the six response functions to four. The further requirement of anti-parallelkinematics (nueleons emitted back-to-back) [111, allows only two functions to survive: thelongitudinal (Joo) and the transverse (J1I). An L!T (Rosenbluth, [17]) separation must beperformed to disentangle these. The statistical requirements, due to the low cross sections,demand experiments with high luminosity and the use of three detectors which can alloperate in such an environment. This can be best accomplished in Hall-A!CEBAF, withits two superconducting spectrometers and a third detector armo

6.2. Neutron-proton emission

The (-y, pn) reaction is dominated by MEC and IC, which are intertwined with short-range correlations thus making the extraction of J(1) terms more calculation dependentwhen looking only at the unpolarized cross section. When the asymmetry of the reactionis examined in the case of a linearly polarized photon beam, then information can beextracted on the nature and strength of the correlation function of the initial nueleonpairo Specifically, an energy distribution of the asymmetry reveals a positive sign for ahard core OMY correlation as opposed to a negative sign fór a soft core RSC correlation.According to the calculations of Guisti et al. [13) this is an unmistakable signature for theinvestigation of SRC. Finally, it is expected that due to its strength compared to (" pp)reaction channel, the pn-channel will dominate the multi-nueleon absorption channels andhence merits a thorough study. In addition, it should be accurately measured to determineany interference with the pp-channel via final state interactions.The absorption of a virtual photon on a pn-pair is similarly dominated by MEC and

IC components. It can be used to determine the two- and three-body absorption crosssections. Furthermore, at selected kinematics one has an excellent opportunity to searchfor effects such as three-body forces. Several proposals have been approved at CEBAF toinvestigate such issues.These and other investigations (e.g., precision measurements of the electromagnetic

form factors of the neutron) require the detection of neutrons and sometimes the deter-mination of its polarization. The experimental difliculty associated with neutron detec-tion in an electromagnetic enivironment is being overcome, and several neutro n detectorshave been constructed at various facilities. Finally, a neutron polarimeter employing anovel technique, promises to yield accurate results in experiments at NIKHEF and CE-BAF [181.

6.3. The pp/np ratio

From pion physics we have learned that the ratio of the cross section of the pp to npchannels is roughly 0.050. A similar behavior has been observed with real photons, asmentioned in section 2 (see Figure 2), where a ratio of 0.070 was found on 913e. Thecorresponding theoretical values [131 for the pp!np cross section ratio are 0.058 (OMY)and 0.011 (RSC), showing that the OMY (strong correlation) is elearly favored within theframework of these calculations.

SlIORT RANGE NUCLEON-NUCLEON CORRELATIONS... 133

Similar results have been obtained in recent 12C(e, e'pp) measurements at NIKHEF [19,201 when examined with this mode!. These triple coincidence experiments were the firstof their kind. Although carried out at a low duty factor machine -and thus hamperedstatistically- they established the experimental procedure of detecting two protons amidsta large electromagnetic background. Experiments on the (e, e'pn) reaction are only nowbeing pursued due to the additional technical complexity of detecting neutrons.

7. SUMMARY

By using electrons and linearly polarized photons, both the (-y, NN) and (e, e'NN) reactionscan be deconvoluted. The combined study of pp- and np-emission channels is necessaryin order to disentangle the reaction mechanism. In the (e, e'NN) reactions an LIT separa-tion is required to separate unambiguously the contribution due to the one-body current(correlations) and two-body current. Other kinematical domains (symmetric kinematics,out-of-plane) permit the extraction of additional structure functions. As for the (-y, NN)reactions, the photon asymmetry appears to be a sensitive parameter in accessing thetwo-body current and probing the type of the correlation function. The systematic pro-grams on these topics, planned at the various electron accelerator facilities, will contri bu tevaluable information towards unraveling the dynamics of the nueleus.

REFERENCEs

1. B. Frois and C. Papanieolas, Ann. Rev. Nud. Parto Se;. 37 (1987) 133.2. C. Marehand el al., Phys. Rev. Let!. 60 (1988) 1703.3. M. Kanazawa el al., Phys. Rev. C35 (1987) 1828.4. Th.S. Bauer, Nud. Phys. A546 (1992) 181c.5. J.S. Levinger, Phys. Lell. B82 (1979) 181; Phys. Rev. 84 (1951) 43.6. K. Gottfried, Ann. o/ Phys. 21 (1963) 47; Nud. Phys. 5 (1958) 557.7. W. Weise el al., Z. Phys. 236 (1970) 176.8. W. Kratsehmer, Nud. Phys. A298 (1978) 477.9. J.M. Laget, Phys. Rev. C35 (1987) 832.10. S. Bofli and M.M. Giannini, Nud. Phys. A533 (1991) 441.11. C. Giusti and F.D. Pacati, Nud. Phys. A535 (1991) 573.12. C. Giusti el al., Nud. Phys. A546 (1992) 607.13. C. Giusti el al., Nud. Phys. A, to be published.14. T. Omhura, N. Morita and M. Yamada, Prog. Theor. Phys. 15 (1956) 395.15. O. Benhar el al., Phys. Lelt. B64 (1976) 395.16. R.V. Reid Jr., Ann. Phys. 50 (1968) 411.17. M.N. Rosenbluth, Phys. Rev. 79, (1950) 615.18. Z. Papandreou, High Aeeeplanee Recoil Polanmeler, Conceptual Design Report, Utrecht Uni-

versity (1992).19. A. Zondervan, Ph.D. Thesis, Vrije Universiteit (1992).20. L.J.H.M. Kester, Ph.D. Thesis, Vrije Universiteit (1993).