Electron Spin Polarization by Low-Energy Scattering From

REVIEWS OF MODERN PHYSICS VOLUME 4), NUMBER JAN UAR Y 1969

. '..ectron Spin, . o..arization 0y . ow-. 'nergy Scattering ) rom Jnpo, .arizec. ..'argets*

J. KESSLERt' I'hysikulisches Institgt, University of Earlsruhe, West Germany

The status of theory and experiment of electron polarization resulting from spin —orbit interaction in low-energy scattering from unpolarized targets is reviewed. Polarization e8ects in scattering from free atoms, molecules, and solid targets at energies between a few electron volts and a few thousand electron volts are discussed. Apart from a survey of the problems which have been solved, a perspective is given of the work which would be interesting to pursue in this rapidly developing held of research.

CONTENTS field of a nucleus with the atomic number Z. He calculated numerical results for Z=- 79, since the polarization I. Introduction...... ~ ~ ~ ~ ~ ~ ~ 3

II. Theoretical Background...... ~ ~ ~ ~ ~ ~ ~ effects turned out to be significant only at large atomic A. General Relations ...... ~ ~ ~ ~ ~ ~ ~ ~ numbers. . Experiments for checking the theory had to B. Elastic Scattering by Strongly Screened Coulomb F' be made at energies of about 100 keV or higher, since Nld o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 6 C. Resonance Scattering...... 11 only then can the description of the scattering process D. Related Effects 12 as a deflection in the pure Coulomb field of the nucleus III. Experimental Results 12 A. Scattering by Atoms...... 12 be regarded as a good approximation. Besides, it had 1. Mercury. ~ ...... 13 been pointed out by Mott that unless the velocity v of 2. Other Elements...... ~...... 18 3. Resonance Scattering from Neon...... 19 the electrons is comparable with the velocity c of light, B. Scattering by Molecules...... , ...... 19 the polarization effects would become extremely small. C, Scattering by Solid Targets...... 21 The considerable effort of testing Mott's results was IV. Applications and Outlook...... , ...... 22 therefore with V. Addendum...... , ...... 24 made fast electrons. Many experiments Acknowledgments...... , ...... 24 were carried out in the two decades after the theory of "Mott Scattering" had appeared (Tolhoek, 1956), for I. INTRODUCTION this was a very promising way of checking Dirac's theory of the electron on which Mott's calculations A beam or other ensemble of electrons is called were based. However, most of the experiments failed polarized if the spins of the electrons have a preferential to show any polarization effect, and Dirac's theory was orientation. The description of the electron polarization questioned repeatedly for this reason (see, e.g., Sommer- by the expectation value of the spin operator f. d and by feld, 1939). Sound evidence of the effects predicted means of density matrices has been fully discussed could not be given until 1943 when Shull, Chase, and in earlier reviews (Tolhoek, 1956; Farago, 1965). The Myers for the first time detected a genuine scattering present survey will therefore not go into these funda- asymmetry in a double scattering experiment. A great rnental problems, but will be concerned with a dis- number of polarization experiments with fast electrons cussion of the polarization effects in 1ow-energy electron has been made since then )for pertaining papers cf. scattering which have been investigated in recent years. Tolhoek (1956) and Farago (1965) and to date low energies, we mean energies below 10 keV. ), By elaborate results are available for electrons scattered Polarization effects in elastic electron scattering from gold targets by Mikaelyan, Borovoi, and Denisov were predicted for the first time in the famous papers (1963), Apalin et al. (1962), and especially by van of Mott (1929, 1932), where the scattering process is Klinken (1966); these results, for energies above 100 described on the basis of the Dirac equation. The keV and large scattering angles, are in satisfactory results of this theory are too complicated to be repre- agreement with the theoretical findings. sentable in a closed analytic form. Therefore, Mott Looking back, the reason for the failure of the derived approximate formulas for the special case of experiments in the first one or two decades after Mott's elastic electron scattering the unscreened Coulomb by theory can be easily seen. At that time, scattering *Work supported in part by the V.S. Office of Naval Research, experiments with fast electrons constituted a new field Contract Nonr 225(67), by the Deutsche Forschungsgemeinschaft, of research. Frequently one had to use emitters with and the Bundesministerium fur Wissenschaftliche Forschung. P t Part of this paper was prepared during the author's stay as their low intensities (not knowing that one was handling a visiting professor at Stanford Vniversity, Stanford, Calif. polarized electronscm), and there was no experience with )The definition of the polarization as the expectation value of the spin operator in the Lorentz frame in which the electron $ The asymmetry effect in the 90'—270' position of the experi- is at rest gives rise to some difficulties resulting from the fact ment of Chase (1930) was not due to the polarization of the that 6 does not commute with the Hamiltonian of Dirac's equa- electrons emitted by the radium E source, as sometimes assumed. tion. De6nitions which avoid this problem. have also been reviewed It has the wrong sign and thus must be ascribed entirely to an Fradkin and Good, 1961), instrumental asymmetry. 4 REVIEWS OE MODERN PHYSICS ' JANUARY 1969 the pitfalls caused by plural and multiple scattering tion eRects could be made by means of computers, and. from thin Glxns, one of which is the reQection-trans- such analysis has produced very rehable data. On the mission asynunetry (Tolhoek, 1956). other hand, the general interest in electron polarization On the other hand, the technique of scattering slow phenomena has inspired experimentalists, so that a electrons was well developed, as shown by the detection great number of papers on polarization eRects in low- of the Ramsauer effect (Ramsauer, 1921) and the energy scattering of electrons have appeared. It is the diffraction experiments with crystals (Davisson and purpose of this review to convey an over-all picture of Germer, 1927) and gases (Arnot, 1931;a list of similar this Geld. experiments is given by Kollath, 1958). As we shall see The theoretical background of these polarization later on, the scattered electrons had a considerable phenomena is surveyed in Sec. II, while a systematic polarization in many of the experiments of that kind. discussion of the experimental results is the subject But in spite of the great interest in electron polarization of Sec. III. A few possible applications are discussed in eRects, this was not realized because hardly anybody Sec. IV, which also gives a list of unsolved problems. looked for polarization phenomena at low electron energies. Mott's statement that the electron velocity s II. THEORETICAL BACKGROUND must be comparable with c if any polarization was to be observed was tacitly considered by many authors to be A. General Relations more general than it really was. Therefore the experiments on electron polarization were generally made Though concerned with low electron energies, our with fast electrons, although slow electrons could be considerations cannot be based on the Schrodinger handled much more conveniently and successfully. wave equation. This equation does not include the spin The few experiments on polarization eRects which had of the electron and therefore cannot be used as a been made with slow electrons (Wolf, 1929;Langstroth, basis for describing the polarization eRects. One has to 1932;Davisson and Germer, 1929;Joffe and Arseniewa, use Dirac's equation, which takes account of the elec- 1929) had failed. However, the reason for the failure tron spin. The fact that this equation describes only was not the low energy, as assumed later (Richter, the normal magnetic moment of the electron and 1937).The negative results were due to the undeveloped neglects the anomalous part is irrelevant for the theoretical understanding of the spin eRects in the Grst scattering processes considered here. work (Wolf, 1929), which was before Mott's theory, The physical quantities which can be measured in a while in the papers of Langstroth (1932), Davisson and scattering experiment are determined by the scattering Germer (1929), and Joffe and Arseniewa (1929), amplitudes f and g, where f is analogous to the scat- where thick, solid targets were used as "polarizer" and tering amplitude known from the scattering theory "analyzer, "presumably the overwhelming contribution based on Schrodinger's equation, and g describes the of multiple scattering together with the inadequacy change of spin direction during the scattering process. of the energy analysis was responsible for the fai1ure to Using the method of partial waves, Mott (1929) Lcf. detect polarization. also Mott and Massey (1965)) showed that the Little attention was paid to calculations of Massey scattering amplitudes* are and Mohr (1941) and Mohr (1943), who considered the polarization eRects in electron scattering down to = — — f(8) P I (k+1)[1 exp (2iBI)] energies of 100 eV, taking into account the screening 2& i=o of the nuclear Coulomb Geld by the atomic electrons +tt1—exp (2i8 I I)]}PI(cos8), and thus generalizing Mott's special results. This was Z — the Grst investigation to show that appreciable polariza- g(8) = —Z Eexp (2ibI) exp (2i8 I,) jPII (cos 8), tion effects could even be expected at small energies, 2k )=0 although the results are not in quantitative agreement with those of recent calculations which will be discussed in the next paragraph. But the experiments of the where I'~ and P~' are the Legendre polynomials and next two decades were still concentrated on the range associate Legendre functions, respectively; 8 is the of fast electrons, and Kollath (1949) seems to have scattering angle; and the wave number k is determined been the Grst experimentalist who seriously considered by p= Sk, where p is the relativistic momentum mys. elaborate polarization measurements at low energies. In order to obtain the phase shifts b~, 8 ~~, one has to Lack of conGdence in the earlier results of the laborious Gnd the asymptotic form of the regular solution for the phase-shift analysis, as indicated by Mohr and Tassie radial part of the wave equation. The phase shift b~ (1954), made the theoretical predictions appear not very reliable. *Needless to say, the scattering amplitudes and the quantities from them later on depend not In recent however, considerable progress has which will be composed only years, on 8, as explicitly indicated, but also on the electron energy Q been made. Now the theoretical analysis of the polariza- and the atomjc number Z of &he scattering center, J. KEssrsa Irieciron Spin Polarieaeioe by Lore Zne-rgy ScaNerieg 5 results from the solution of the differential equation cf. Mott and I Massey (1965), Chap. IX, Eq. (22) j FzG. 1. Scattering T. t I+ PS) O' — Z Gg"+ I Pl(i+ 1)/r'j —Ug(r) I G( =0, (2) asymmetry of a trans- versely polanzed beam. where 2W V' l+1a' 3n" 1u" Ug= V— — + $2~2 g2P p ~ 4 ~2 n= (Sc) '(W —V+mc'); (3) vectors of the two beams are perpendicular to the and to each of the beams. 8' is the total energy, and V is the potential energy of scattering plane apply Eq. (2) the electron in the Geld of the scattering center. For Then we obtain for the number of electrons scattered — into a certain direction calculating b & &, one has to replace l by (l+1) in these equations. Q( —I(1+S), X)=I(1—S), All the observable quantities char acteririn the func- scattering process contain products of two wave where now N t and N~ are the numbers of electrons with tions and are therefore described by quadratic terms of spin parallel or opposite to the vector the complex scattering amplitudes f and g. The following quadratic terms with a straightforward physical n= (kg xk2)/(I kr xk2 I), meaning can be formed (Tolhoek, 1956; Farago, 1965; perpendicular to the scattering plane; k~ and k2 are Schopper, 1959; Motz, Olsen, and Koch, 1964): initial and Gnal momentum of the electron. Here we (1) The differential cross section for an unpolarized electron beam have anticipated the result, following from Eq. (11), that the polarization vector does not change its direction d /«=I(~) = Ifl'+ I el'. (4) during the scattering process if it is perpendicular to the scattering plane. From Eqs. (6) and (8) we 6nd (2) The asymmetry function or Sherman function* —— = —I'+ I') P~ SD,, (9) S(0) e(fr* f'g) /(I f I g (5) describing the scattering asymmetry of a polarized where P& is the polarization obtained by scattering an electron beam: Let us assume we have a beam with a unpolarized electron beam (P~= 0) . transverse polarization whose direction is indicated in (3) The terms Fig. i. 0 Nt and N~ are the numbers of electrons with — spin in or opposite to this direction, the degree of T(0) =(lfl' I el')/(Ifl'+ I el') polarization is given by I') U(0) =(fg*+f*a)/(If I'+ I g P= (&~ —&~) . (6) /(&i+&~) describe the rotation of the polarization vector during The scattering cross section of this beam not only the scattering process. The polarization vector after depends on the angle 0 but also on the azimuthal angle the scattering P2 can be expressed in terms of the q and is given by polarization vector before the scattering (P~—P„+P») I(0) (1 PS sin rp). — (2) in the following way: Therefore, the scattered intensity has a left—right (Pg(+S)n+TPgo+ ULn x Pgof P2- 11a asymmetry, maximum asymmetry being observed 1+Ping S when the scattering planes are perpendicular to P. As a result of this scattering asymmetry, an un- where Pz has been decomposed into a transverse polarized beam can be easily seen to become polarized component perpendicular to the scattering plane in the scattering process: An unpolarized beam can be (Pq~ =Pq~ n) and a component parallel to the scattering considered as a mixture (incoherent superposition) of plane (Pq„) (Fig. 2). If UWO, there is a component of two completely polarized beams with opposite directions the polarization Pe perpendicular to the plane (n, P~„) which is identical with the plane U of the polarization. t We assume that the polarization I (P&„P,„)g, i.e., describes the rotation of the polarization out of its *Sherman (1956) was the first to calculate exact numerical values of this function S(8) over a wide angular range for scatter- initial plane. T is seen from Eq. (11a) to describe the ing by the pure Coulomb field of several nuclei (Z= 13, 48, 80) . change of the polarization component parallel to the f Such a beam in which half the spins point in one direction scattering plane. If the scattering amplitude were and the other half in the opposite direction cannot be distinguished g experimentally from a beam in which the electron spins point zero, T would be equal to 1 while 8 and U would be in all directions at random. A measurement of the spin directions zero; thus, there would be no change of the polarization would give the same result in both cases: Half the spins point in the direction defined by this measurement and the other half during the scattering process. point in the opposite direction. For polarization measurements it is more useful to 6 REvIEws oF MQDERN PHYsIcs JANUARY 1969

several authors (Holzwarth and Meister, 1964a; 1964b; Schonfelder, 1966; Bunyan, 1963; Bunyan and Schon- Fig, 2. Decomposition of arbi- felder, 1965); these show that the few early calculations trary initial polarization. mentioned in Sec. I (which had to be made without the aid of computers) gave a proper qualitative description of the polarization. While this article was being prepared, numerical evaluations of the polarization I'2 the express by unit vectors n&, n2, and 02 &n: have been published for iodine at 400eV by Vates — Similar which been P2 —In(S+PI n)+n2(LP, nI+RPI. LnI xn]) (1968a). calculations have not yet — published were made for iodine by Buhring* at 200, +Ln, xn](LPI InIxn] EPI nI) I/[1+S(PI n) j, 300, 400, and 600 eV and at 300 eV by Walker, *~ who (11b) also made unpublished calculations for bismuth at 300, 400, 500, 700, 900, and 1200 eV. Coulthard~ made where L= Lj' sin 8+T cos 8, R= U cos 8—T sin 8. unpublished calculations for mercury at 10, 15, 20, 25, The results given here show that the scattering 30, 40, 50, 60, 70, 75, 80, and 90 eV and for argon, amplitudes f and g provide a complete description of krypton, and xenon at 40, 1000, and 300 eV, the intensity distribution and the behavior of the respectively. ~ polarization in the scattering process. The problem is As an example of the polarization e6ects predicted to find reliable values for the complex functions f and g. by the new calculations, Fig. 4 gives the function S which describes the polarization of an electron beam B. Elastic Scattering by Strongly Screened Coulomb resulting from the scattering of an unpolarized beam Field LEq. (9)j. Contours S(8, E) = const for Hg with 100 eV &E&3 keV are shown[ in a logarithmic scale. For scattering in the pure Coulomb field, and g have One can see that there are several areas where f ~ S ~ com- been calculated by Sherman (1956) for certain has high values of more than 0.8. According to the binations of the relevant parameters: Z (atomic numerical tables (Holzwarth and Meister, 1964b) and and 8. 3 reminds number), E (electron energy), Figure a paper by Buhring (1968b) to be discussed later, us of the essential results given by the theory for scat- values close or equal to 1 should be attainable at certain tering in the Coulomb potential. The Sherman function combinations of energy and scattering angle in the S, which has very low values at small Z, has appreciable middle of the peaks of Fig. 4. The potential used in values for heavy nuclei. These larger values (up to calculating these results was the relativistic Hartree about 0.5 for Au) are found preferentially at large potential for mercury computed by Mayers (1957). scattering angles, whereas S is very small at small Several effects which are known to be important in angles. The angular and energy dependences of S are given by curves which are relatively sxnooth. These theoretical values for 5 could be con6rmed experimentally (Mikaelyan, Borovoi, and Denisov, for 1963; Apalin et at., 1962; van Klinken, 1966) gold O targets if, as in Fig. 3, the energies and scattering angles u~ -0.2 were chosen high enough so that the description of the U scattering as a deflection in the pure Coulomb 6eld of the nucleus was a good approximation. At somewhat O lower energies where the screening of the nucleus by fL the atomic electrons begins to play a role, the agreement O between theory (Bonham, 1962; Lin, Sherman, and 0 Percus, 1963; Lin, 1964; Holzwarth and Meister, -0.6 I I I I I I 1964a; 1964b) and experiment is doubtful (van Klinken, IOO I20 I40 l60 I 80 1966), even if screening is taken into account SCATTERING ANGLE, DEGREES theoretically. FIG. 3. Dependence of the asymmetry function S on the As for numerical evaluations of and g at low electron scattering angle for a gold target, at various energies (from Motz . f in where the inhuence of screening is crucial, Olsen, and Koch, 1964). These data are valid for scattering energies, the Coulomb field of a point nucleus with no screening. extensive calculations for elastic scattering have been results for the scat- made in recent years. Numerical The author is indebted to Dr. W. Biihring, Dr. M. A. tering amplitudes and/or their products have been Coulthard, and Dr. D. W. Walker for kindly making available published down to 100 eV cross sections even lower their unpublished results. I f See the addendum. (Schonfelder, 1966)7 for some heavy elements (mainly f The author is indebted to Dr. K. Jost for kindly providing Z=80 and 79, some data also for Z=81 and 83) by this figure. J. KzssLsa Eteetrort Spert Potareeatton by Low E-rtergy Scatterertg

2.0-

l.0—

0.5

FIG. 4. Contours of S(8, E) =const for Hg. For the special case of scattering of an unpolarized electron beam, S=P, the polarization of the scattered electrons. Logarithmic energy scale. 0.5

0.2

90' )50

—i.0» S» -0.8 0.0» S» 0.2

T' —0.8» S» -0.6 ~ m ~ T ~ 0.2» S»OA W////A -0.6 —S- -o.e PENNA 0.4 —S —0.6 — 6»S «0.8 -04» S» 02 IIIII IIII II& I II 0.

! 0.8 —S —[.0 slow electron scattering, like exchange effects and agreement of theory and experiment in thee energyenerg ~ ~ electrical polarizability of the atom by the scattered range shown in Fig. 4 turns out to be very good despite electron Lcf. Mott and Massey (1965), Chap. XVIII', these approximations. have not been considered in the theories (Holzwarth Marked polarization effects at low energies, like and Meister, 1964a; 1964b; Schonfelder, 1966; Bunyan, those shown in Fig. 4, could not be anticipated by a 1963; Bunyan and Schonfelder, 1965), so that there simple extrapolation of the facts known from scattering has been some concern about the reliability of the in the pure Coulomb field, for in that case the polariza- theoretical results. As we shall see in Sec. III, the tion of the scattered electrons decreases when the 8 REVIEWS OF MODERN PHYSICS ~ JANUARY 1969

sign of the spin-orbit potential Lsee Eq. (13)j and thus to the resulting scattering potential is diferent for spin-up and spin-down electrons of the same L. In other words, the "effective radius" of the atom (which can be deGned as that radius where the potential has FIG. 5. Relation between the to a certain is different P I dropped value) for spin-up and +)— l cross sections for spin-up and spin-down electrons of the same L. I spin-down electrons and the I polarization of an electron beam The elastic scattering can be regarded as a diffraction I obtained by scattering an un- of the incident electron wave by the atom. At the I polarized beam. energies discussed here, the electron wavelength X is comparable with the atomic radius E, and one gets a typical interference pattern with distinct maxima and minima of the scattered intensity (quantitatively shown in Fig. 17). Since the positions of the minima and maxima are determined by X/R and the effective radius R is di6erent for spin-up and spin-down elec- electron energy and the atomic number of the scattering trons, there is a relative shift of the diffraction patterns nucleus becomes smaller. In the case under discussion, for the two spin directions as illustrated in Fig. 5. the electrons are very slow and the "effective atomic Apart from this shift, there will also be some diGerence number" is very low because of the strong inQuence of in the shape of the cross sections because of the some- screening at these energies. Therefore, from a naive what different scattering potential. For these reasons, point of view, it seemed obvious that little polarization the cross sections for spin-up and spin-down electrons was to be expected here. are usually different from one another at a certain In order to get a clear understanding of the origin of angle 8, i.e., there are di8'erent numbers of spin-up and the polarization phenomena, it is expedient to use an spin-down electrons in the scattered beam so that this optical model, although nearly all the quantitative beam is polarized. Recalling the deGnition of the calculations have been made by the method of partial polarization LEq. (6)j and the fact that the numbers waves. A simpliGed qualitative explanation of the Et and E~ in the scattered beam are proportional to polarization e6ects can be given as follows: the corresponding cross sections, we 6nd Although the electron beam is scattered by an — electrostatic field (magnetic moments of the atom have P= (d~/da) t (d /da), not been considered by the aforementioned theories), (d~/da) t+(d~/da), the magnetic moment of the electron p plays a role in the scattering process because in the rest frame of the and can readily construct the polarization curve from electron there is a magnetic Geld the cross section curves as indicated in Fig. 5.%henever one of the two cross sections has such a deep minimum H= — IcE (v/c) that its value is very small compared to that of the other cross section at the same angle, I' is close to 1 interacting with p. The interaction energy is —p H ~ ~ because then there are virtually electrons of one which, after substituting H from Eq. (12) and only spin direction in the scattered beam. y= (s/mc) 8, is seen to be the spin-orbit energy Figure 6 shows a quantitative example for these (2m'c')-Ir I(dV/dr) 8 L relations between the cross sections and the polarizations. One can see that the mutual shift and the change except for a factor of 2 caused by the Thomas precession in the shape of the cross section are not very great even (Thomas, 1926; Farago, 1967). Qualitatively, the for a heavy atom like mercury. resulting scattering potential is the sum of the screened The angular dependences of the cross sections for — Coulomb potential and the spin orbit potential and is fast electron scattering (energies as given in Fig. 3) thus dependent on the relative direction of spin S and have no minima and maxima any more, because X is orbital angular momentum L of the incident electron. now much smaller than the atomic dimensions. The Ke can regard the incident unpolarized beam as cross sections for spin-up and spin-down electrons drop being made up of equal numbers of electrons with spins monotonically with increasing 8, so that the construction parallel and antiparallel to the normal of the scattering of the polarization from these cross sections (analogous plane* ("spin-up" and "spin-down" electrons). The to Fig. 5) results in the smooth curves of Fig. 3. There are no high polarization peaks because there are no *This special decomposition of the unpolarized beam has the deep minima in the cross sections. the scatter- advantage that the spin directions are not changed in The optical analogy used here was very suitable for ing process because the magnetic Geld t Eq. (12)g is perpendicular to the scattering plane, too. our heuristic discussion. However, for obtaining J. KEBSLRR Etectrom Spirt Polarisatiort by I.oro Em-ergy Scatterirtg

l6 quantitative results on the polarization effects, an I I I I I I SPIN POLARIZATION optical model has been used only in the investigation l2— Ee= 4eV +45 made by Goldberg (1963). He calculated cross sections and spin polarizations of the scattered electrons for Rp = I.75~. Rp = I.60 /r Z=18 (argon) and energies between 0.01 and 100 eV. w~ ~ The optical model was based on a Thomas-Fermi- Dirac potential obtained from the screening function I-z —4— tables of Abrahamson plus the standard spin- W poI (1961) CJ orbit term and an additional term tag 8 CL —45 ~o l2 I- I N 8— I I I which takes into account the electrical polarizability a Ee 20 eV 0 Rp= l.60 of the atom; Rz is an adjustable parameter. Figure 7 " 0 ~ Rp -l.75' gives some results for two electron energies (4 eV, 20 eV) and three values of Ro, including the case R„= or V„=O.One can see that the results are similar to those obtained for mercury by other methods. S(8), I I I I I I I the polarization I'(8) of the scattered electrons for the 4 0' 80 l20' 160 8 special case of scattering of an unpolarized electron beam, is an oscillating curve whose structure is very FIG. 7. S(tt) for argon at 4 and 20 eV calculated with an optical model sensitive to the exact form of the scattering potential, (reproduced from Goldberg, 1962). (For the meaning of Sd. the caption of Fig. 4 or the text. ) especially in the case of the lower electron energy, where the electrical polarizability of the atom an plays much about the polarization peaks for argon from the important role. The polarization curve does not to go up few curves calculated so far.* gite. l values close to 1 (or in Fig. also in Fig. 4 only 100%) 7; For the other elements, which we did not mention, a small fraction of the possible curves S(II, E= const) no theoretical polarization data are available at present. go through high peaks. Thus, one cannot say very In view of their importance (see, e.g., Sec. III), these data should be calculated at least for such a variety of atomic numbers that the remaining data can be I.O ~ ~ t I.OxlO Io~o 11 Ij interpolated. ] do It is interesting to note that there is a close analogy dQ 1 between the polarization 11 effects in electron scattering and those of nucleon scattering which have been in- l1 vestigated for a much longer period and more exten- l1 sively. At energies of about 10 100 '1 1 to MeV, the de '1 Broglie wavelengths of proton, neutron, and the light 0.5 —i, 0.5 1 nuclei are comparable to the nuclear radii, just as the ~ 1 electron wavelength is comparable to the atomic radii in the 100-eV range. Thus, there is a direct analogy P 11 between the oscillating cross sections and polarization curves found in nuclear scattering Lhere we quote only one typical example (Satchler, 1967) from the many papers treating these phenomenaj and the corresponding results in slow electron scattering. The only difference is that the nature of the forces which bring about these results is different (electromagnetic and nuclear forces, respectively). Therefore, in nuclear scattering there are relations between the differential cross section and the polarization very similar to those discussed before. I Rodberg (1960), Hiifner and De Shalit 05 I I (1965), I I 0 l20 I 30 I 40 150 and De Shalit (1966) showed how to express 8 these relations more quantitatively with certain approximations. FIG. 6. Quantitative example for the relation between the cross sections for spin-up and spin-down electrons and the polariza- Extending the results of De Shalit (1966), Buhring tion. (as=Bohr radius in hydrogen. ) (Data from Hoizwarth and Meister, 1964b.) ~ See the addendum. 10 REVIEWS OP MODERN PHYSICS JANUARY 1969

3.0

2.0—

FIG. 8. Contours of U(8, E) =const for Hg. U describes the rotation of the polarization out of its initial plane. Logarithmic energy scale. 0.5—

~ o~ l 30' 60' l20'I I l50'

—l.O» U» —0.8 o.o u —o.z —0.8» U» —0.6 or u-o4 — — 0.6 —U — 0.4 VÃEYXXA o.4 — u .- o.6 —OA» U» — c c p c c v7 0.2 o.6 - u-o8 -02- U —Oo o.s u —i.o

(1968) showed that valuable information on the elec- considerations is the signidcant quantitative dependence tron polarization in low-energy electron scattering of the polarization on the atomic model used to describe can be obtained when the scattering amplitude f(s) the screened Coulomb Geld. (with s= cos 8) is continued analytically into the Apart from the function 5 discussed so far, the complex s plane, and its properties in the neighborhood functions T and U defined in Eq. (10) are necessary of the real axis are studied. In particular, he studied for a complete description of the polarization eGects in the trajectories in the complex plane on which the electron scattering. These quantities, which describe zeros of f(s) move when the energy is varied. This led the rotation of the polarization vector during a scat- to some relations which allow one to interpolate the tering process, have seldom been discussed in theoretical extrema and the positions of the zeros of the polarization papers on electron polarization and never in experi- curve for a certain energy from the cross sections and mental papers. * The reason is that a measurement of polarizations near the intersections of these trajectories T and U would be at the margin of what can be done with the real axis. By these relations the combinations presently. Such a measurement would have to be a of energy and scattering angle for which total polariza- triple scattering experiment if the primary beam is tion occurs can be easily found. Buhring applied the unpolarized, whexeas measurements of S are made by results of his calculations to mercury, iodine, krypton, double scattering experiments (cf. Sec. III). By the argon, and neon with the unexpected consequence that 6rst scattering, a beam of well-defined polarization I'j even for the lightest of these atoms, total polarization should be attainable. Another interesting result of his * See the addendum. J. KEssLER E/ectron Spin Polarization by Low-Energy Scattering 11

would be produced, the second scattering would trans- I I I I I NEON fer Pi to P2 (the process to be studied), while the third a LLJ scattering would be necessary to analyze I'2. O From the tables of Holzwarth and Meister (1964b), LLI Q values of T(0) and U(0) could be determined for the K &o energies given there. Interpolation for the other energies O til leads to the contours of = const for given 0 N U(0, E) Hg W ~ in Fig. 8. They cover a smaller energy range than the I-— V) curves S(0, E) = const in Fig. 4 because fewer data Z 0- Q CL we&e available, particularly at the lowest energies con- I- K sidered. Comparison of Figs. 4 and 8 shows that the CQ EL 0.095 eV function U has its peaks near those areas in the E—8 I I I diagram where also has high values. I 1 I ~ very 15. 1 S ~ 15.8 9 16.0 16. 16.2 The contours T(E, 0) = const are not displayed ELECTRON ENERGY, eV here, since they can be determined from the corre- FIG. 9. Transmission of electrons by neon, showing the first sponding 6gures for S and U using the relation two resonances in the total cross section located about 0.5 eV S'+A+ U'= 1, which follows from Eqs. (5) and (10). below the first excited states of neon (from Kuyatt, Simpson, and Mielczarek, 1965). C. Resonance Scattering is not to be expected when only one state (for instance Again, the effect to be discussed has a direct analogy Ppip) contributes to the scattering cross section. On the to a phenomenon which is well known in nuclear contrary, in that case the polarization, which is a result scattering. Although spin polarization in resonance of the interference between the potential scattering scattering of nucleons was predicted as early as 1946 and the resonance scattering of the p waves from the (Schwinger, 1946) and investigated since then (Wolfen- split level, is zero. This general fact can be easily proved stein, 1949; Faissner, 1959; de Facio and Gammel, in the special case where all the partial waves with 1966), resonance scattering of electrons has been l&2 can be neglected. (The higher partial waves are detected only in recent years. Schulz (1963) reported a not very important, anyway, at the small electron strong resonance in the cross section for electron elastic energies where resonance scattering plays a role. ) scattering by He occurring at 19.3 eV, i.e., about Then we get from Eq. (1) 0.5 eV below the first excitation threshold. Similar and = — more-detailed results were found in many other cases f (i/2k) I 1 exp (2Qp) /for a list of pertinent references cf. the review of K. +L3—2 exp (2ibi+) —exp (2i0. ) ] cos 0I, Smith (1966)j confirming that the resonance results — from the temporary formation of a negative ion in a g= (i/2k) sin 0/exp (2i8i+) exp (2i0i ) j, (16) compound state analogous to the compound nuclei where we use notation b~+ and 8~ to show that the phase formed in nucleon scattering. shifts bi and b i i in Eq. (1) belong to the partial waves In the case of neon, Simpson and co-workers (Kuyatt, with j =l+ ', and j=l——', —(cf. Bethe and Salpeter, Simpson, and Mielczarek, 1965) were the first to show 1957). Substitution of Eq. (16) in Eqs. (5) and (9) can Qnd doublet that one a structure of the resonance yields, for the polarization of an initially unpolarized if the resolution is energy high enough (Fig. 9) . LMore beam, recent measurements have been made by Andrick and — Ehrhardt (1966).] This structure has been interpreted P= [2 sin 0 sin (8i+ Bi )/lpPI(0)] by Simpson and Fano (1963) as corresponding to a fine- &&Lsin 0p sin (0p —8i+ 8i ) —3 sin Bi+ sin 0i cos 0$, structure splitting of the compound ion with the config- urations (1s'2s'2p'3s')Pp~p i~,. Since there is not much (17) overlap between the cross sections of these two configu- where I(0) is defined in Eq. (4) . rations, which differ by the electron spin direction, it Ke see that the polarization disappears not only for seems plausible to expect an appreciable polarization of 6~+=8~, which is a trivial case because the level the electrons passing through one of these resonances, splitting disappears (Wolfenstein, 1949; Franzen and analogous to the polarization of neutrons and protons Gupta, 1965), but also if only one of the phase shifts is observed in resonance scattering from helium different from zero. (Schwinger, 1946; Wolfenstein, 1949; Faissner, 1959; A theoretical analysis of electron polarization by de Facio and Ganunel, 1966). This has been predicted resonance scattering in neon was made by Franzen and by Franzen and Gupta (1965), who calculated the Gupta (1965). The authors calculated the scattering polarization to be expected. amplitudes in the vicinity of the resonance. They It seems worth noting that maximum polarization determined phase shifts by making use of the results E~ms pp ODE~ P JAgg

i.o I j I tar risc to e electronn polarmat'a ion.. h. ects pf t is @nd. are onl y 1«sel y relatedr to g, problems too e discuussed here onnly briefl

O.g- ' g of electron s from atpoms with a preferential population o certain s in ' a«s Lwh ma e Produced b. the Stern-g, p r nent or p 'c» pum p a»&ed light (I s» a transfe s to 0— ground of t is rans ferr has been d be»yrnecussed b and F go (1965 ; cxpcrim nts have ™adeby Far and S'iegmann (1966) Since n epretical d t «e availabl o consult th e fe~ experiment 1 -O.g ' estlga, tjpns emstin gso far (Dehe elt, (9/8.'Franken ' ', nds, and Hobo (9g8 Collins, art, ' gold stein & g edcrspn, a d R bin, 196y L,Lichten and Schult ~ for ~ " ' 59) t e i/ values p the spin „ha"ge crosss sections ' t I &.0 0''' -8 -6 -2 y Naturall y c exch an e e ect does rpm 2 6 ~ ~ an cx llc''t SPin dependence pf t e scatterr»g potent' ~ ut from th e «ct th at the in n the atom. ~ ~zat' FxG go ectron Poa zation P averagedve over three ~erent ~ solid angjeses as a funct on of electron ener ~5 for elastic ectron are Particles pi requixin 0, v' of; 2y by 9 neon in the i inity of th ' g resonance a Pp«priate sy trizatlon of ctions. The V H ') ~~& ~here j. ra &965). results pf g ey (1,962 who treated the ~ta . - '. ence of the pin direction f incident omic ' (F. d'ff electrons electron-h d g n scatterining, shpw a't of Slmnnpson ' & and of the absolute erent;al c th ~ e spin deependence f the fore«es due tpo the cross sections ' r the scatt ring of ~5 9-CV electrpns b ' pau 1'i prin c~ple cann h av 'quiteu. remarl able egeccts. neon easured by Ramsauerr and Kollath (1932 . p ~ ' LTh.is is alsso known frpm M 11 rom the semiem pirlcal'. scatt erlng Phtudes th us er scatterin' g (Frauen feld er and St )96~ ~ Recentl h obtained th e authp rs calculatede the polar&sationsp for egen, y~ these results been ~xtended b 19967) to scatte ring from scattering lntp Qniite solid annges1 ccnteered arpund 0 dwire h. He+, but m ch ~or~ is left to be donee ssince np dat 0= 9O vlj, ich are given ln Fig y0 ~ The electr pns are avaga ble for pth elements a d or electr~~ron energies ssumed to b ttered 'int p the solid an gle subtended ~ . . ' higherh' than th e lowest exexcitation ene by a pyramid of q"are cross section whose a;a»s is perPendicular t the ln' "'a dircctipn &,=90') and ' . ~ EXpERIM RESULTS whose faces mme th . d'cated an g1 s with;ts . e avera. e polarization pyra~d»tensive e P nnent». studiess . of th e P risation the the effects c»ected ~th& potential sccattering to & ~ alues for g e three solid angles stron g1y screened Coulomb ge}d h ave been™adein th e . . ~ indicated ~t,'s»«restin gtpnote t}at p ch anges ver Present dec de. It is th e main pu rppse p th» secti'on tp '. y ' ' ra id with th e elect rpn energ but depend only survrvey thes c ex eriments ~ th ejr thepreo etlcal aspec p' y y . p ' slj'ghtly on the so»d angle Th.ls behavivior is cpn trar avlng been iscussed Secs. ~ A and ~ Electr pn ' y II II to that f the plaarizatlpn curves n ln F' 4 larizatipn ln»scatterin frrpm singl e atoms m olecu les p g p ' . us in th e ]atte r case the energy respluo utipn can be ]00 and sphdsol' target has bee»tudied will e treate times worse than 'n the respn nce scatterinring without succe s»vely. . ' a@ecting th e measurede polarizat' curv c~ while th pp ar»ation in respnnance sca,tte g has beeen studied anngular resplutu lpnm st be about ten timess better th an ln only one experiment sosp far (Re;cl nd Deichsel in the respna»ce ca,se.' 196~ (cf. Sec. fff A ' ) though oth er world i prep ~ ' The curv es .pf Fig. 10are the only thepre results ' n (Franzen) on thc polarization 'n resonance»«icalt 'ng of slo eectrons ~h h r available at er &. scattering b7 &toms ~ 'nvestigat;on s are de b The o s whichc "ave bee scussedd ln~ Sec II . ' d mth sc t erlng fromm sing]e atoms..Th Crefpre an Re&ateg F&ects ~ . perlment with . ectrons wh ' is to be compare Slow electronsrons can be polarized by ex g h these th les is m ade preferabl. y ~ith t~~g~te s pf terin is not the gas a density. +esides, th +eo«tical po arization of the results on ~e pola ion effe t hold f«clastjc J. Kxssxzit EIeetrors Spil Polarssotsers by Lore Er-iergy Scatterirrg 13 scattering only; thus, the inelastically scattered electrons must be eliminated in the experiments to make possible a direct comparison. Electron polarization in inelastic scattering has been studied neither theoreti- caQy nor experimentally, although this would be an interesting subject. I. Merclry

Extensive measurements of electron polarization in low-energy elastic scattering from mercury have been made in two laboratories during the past few years: In the group (Deichsel, 1961; Deichsel and Reichert, 1965; Steidl, Reichert, and Deichsel, 1965; Deichsel, Reichert, and Steidl, 1966) of Kollath, who published the first ideas on measurements of this kind (1949), at the University of Mainz, and in the author's group CONDENSER (Jost and Kessler, 1965; 1966; Eitel, Jost, and Kessler, 1967; 1968) at the University of Karlsruhe. In both Fro. 11. Schematic diagram of double scattering experiment for measuring the polarization (from Jost and Kessler, 1966). cases double scattering experiments were made, the polarization due to the 6rst scatterer having been known value of S together with the measured ratio analyzed by the second scattering process. As an can be determined example for the basic experimental arrangement, Fig. Ni/N„P It is not necessary to discuss here the elimination of 11 gives a schematic diagram of the apparatus used by systematic errors like instrumental asymmetries or the the Karlsruhe group. of due 6nite thickness of A beam of electrons crosses an atomic beam which change S to the gold foil; this is now standard technique and can found scatters some of the elections. The scattering angle 8 be in original and reviews (Frauenfelder and mould be varied continuously between 0' and ~150' many papers Steffen, 1965). The statistical error of the polarization by rotating the electron gun about the axis of the atomic by* beam. The scattered-electron beam is expected to be is given polarized, the polarization P being perpendicular to 6P= $(S ' P')/(N&+N —) ]'" (19) the scattering plane Pcf. Eq. (9)j as indicated in Fig. 11. The measurement of the polarization was performed which follows from Eq. (18) together with the law of by a "Mott detector" which, because of the many error propagation. In most cases 1/S'»P' is fulfilled and we obtain investigations on polarization in P decay (van Klinken, 1966; Frauenfelder and Steffen, 1965; Schopper, 1966), hP [S(Ni+N„) "j (20) has become the most accurate device for this purpose. The best results with this method are obtained for The apparatus of the Mainz group divers from that electron energies above 100 keV. Therefore, the of Fig. 11 mainly in the second scattering process: polarized electrons which passed through a 61ter lens The polarization is analyzed by scattering the electrons (Simpson and Marton, 1961; Kessler and I.indner, at low energies from a mercury atomic beam by ~90' 1964) for removing inelastically scattered. electrons (in the newer papers). The asynunetry function for were accelerated to 120 keV before they entered the mercury at 90' and low energies, which is needed in Mott detector. There they were scattered for a second order to calculate the polarization according to Eq. time. In the geometrical arrangement of Fig. 11, the (18), is determined by the authors themselves: If ratio of Sg and E&, the number of electrons counted in scattering angle, electron energy, and target are the the left and right counter, respectively, is, according to same in both scattering processes, we see from Eqs. Eq. (7), given by (18) and (9) that Ni/N, = (1+PS)/(1 —PS) . Ni/N, = (1+S')/(1—S'), (21) and can be determined measurement ~ a of the Here S is the asymmetry function of the target in the ~ S by left— Mott analyzer and is well known at Z= 79 (gold) and right asymmetry. electron energies E above 100 keV, particularly at The main advantage of this arrangement is that the electrons do have be 121 keV where van Klinken (1966) made very accurate not to accelerated to high energies 100 after the first whereas measurements. Also, at this energy and 8=120, there () keV) scattering, in the arrangement of 11 either the is only a slight dependence of S on E and 8 (cf. Fig. 3), Fig. scattering chamber so that Z= E= 120 and 8= 120' were chosen 79, keV, as e The plus sign in Jost and Kessler (1966) in the numerator parameters for the Mott detector in Fig. 11.From the of this formula is a misprint. 14 REvIEws oF MoDERN PHYsIcs JANUARY ~969

40 /o work. This can be seen from Fig. 23, which shows 200 8 more recent experimental data by Eitel, Jost, and Kessler (1967) together with theoretical results (Holzwarth and Meister, 1964a; 1964b; Schonfelder, 30 1966; Bunyan, 1963; Bunyan and Schonfelder, 1965); P(E, 90') is shown for mercury in the energy range 50—2500 eV covered by the measurements of the 20 Karlsruhe group. Since we are concerned with the polarization I' arising from the scattering of an unpolarized— beam throughout this section, we have P= S. Therefore, 200 I" should give the data of Fig. IO 22, which is about right for E)2300 eV. The agreement between the new experimental data and the theoretical predictions is remarkably good, as Fig. 23 shows. This is a general feature of the results for mercury and will be discussed later in this section. 0.5 I O KeY I 5 2.0 The angular dependence of the polarization has also I' IG. 12. The first experimental proof of electron polarization been measured extensively for mercury. In these in low-energy scattering. Energy dependence of the scattering experiments, the energy has been varied in small steps S') for double 90' asymmetry 2006(8= scattering by mercury between 3.5 eV (Deichsel, Reichert, and Steidl, (from Deichsel, 1961). 1966) and 1700 eV (Jost and Kessler, 1966). The following figures show a few typical examples of these measure- or the Mott detector must be at high potential (in the ments. Practically all the experiments have been Inade experiments of the Karlsruhe group, it was the Mott in the angular 30' 250', 30' detector). range to for below the polarization is so small that measurements On the other hand, it appears worthwhile to face this appeared not very appealing, whereas measurements complication because of the advantages of the conven- far beyond 250' were not possible with the layout of the tional Mott detector. In this case the asymmetry func- present scattering chambers. tion S has been determined very accurately by various Figure 14(a) shows for 1700 the authors, so that Eq. (18) yields very accurate values P(8) eV, highest energy for which a complete for I', and the eKciency of the detector~ is much larger. polarization curve from 30' to 250' has been measured. At this The ratio of the numbers of incident and scattered energy, only a few theoretical values are available; electrons was about 10 in our case (S was 0.26) and they agree quite well with the experimental results. even 6&&10 in van Klinken's experiment (1966). The 900-eV curve shown in Fig. has even more The large eKciency of the Mott detector makes it 14(b) pronounced minima and maxima. At this the possible to analyze even the polarization of low-intensity energy highest polarization measured so far was observed: beams quite accurately. Therefore, the polarization I'= —85% at 8=101'. It is quite obvious that in curves could be measured with high angular resolution Fig. 24 and some of the other examples (~1' in most cases) which is equivalent to low intensity given here, the experimental peak values of the Karlsruhe of the polarized beam because of the small angular group are slightly lower than the theoretical ones. However, divergence of primary and scattered beam. High this is not a failure of the theory. Because of the 6nite angular resolution is crucial, especially for measurements near the higher values of the polarization, which

changes very rapidly with 8 as shown in Fig. 4. ll ll ll 8 =90' The first experimental proof that low-energy electrons 0.5. I ji can be polarized by scattering was given by Deichsel I t — (1961) and is reproduced in Fig. 12. The left right pt$ )

asynnnetry was measured in a double scattering experi- l r ment with tI=90' and Z=80 in both scattering pro- r cesses, yielding P according to Eq. (21). Although l I quantitatively the results for 8= S' do not agree for all p' energies with the theoretical and experimental data IlP t found later on (cf. Bunyan, 1963, for a thorough dis- -0.4 I . I 0 )Q2 cussion of these very first measurements), the peak at 5 x)03 2.5 keV is compatible with the results of subsequent FIr.. 13. Energy dependence of the polarization P for 8=90' Z=80. Solid circles, experimental values with statistical errors; * As can be seen from Eq. (20). an adequate measure for the dashed line, theory. Pseudologarithmic energy scale dered by efficiency of a Mott detector is the ratio S'(A'&+X,)jX, where lV log (1+8/100 eV) (8 in electron volts) (from Eitel, jost, and js the ngrnber of incident electrons. Kessler, 1967). J. KEssLER Electron Spirt Polarizationby Lore En-ergy Scatterirtg 15 angular resolution of the measurement, the experimental 0.9 data are average values over an angular range of ~1'. P 500 eV I resolution is almost I This angular everywhere good ) I enough to make possible a direct comparison of the ' li experimental results with the theoretical polarization curves which have been calculated for an ideal angular resolution; the only exceptions are peaks of the very III

,4 ', l.O / I I l700 eV I I 0 44 I I I il 60o I 90o lll20 8

4I 0.5 I ~4

a 4 I 0 I ~ I (a) 60' ~ 90' " l20 e 4 C Pro. 15, Angular dependence of the polarization I' for 300 eV, Z= 80. Experimental and theoretical values (from Jost and Kessler, 1966). -0.5 are practically identical. The theoretical curves plotted in the papers of the Mainz and Karlsruhe groups, respectively, look different only because in the former - l.O work the theoretical data have been convoluted with the I.O angular resolution of &3'. This gives rise to an appreciable modification of the curves near the II peaks. 900 eV I Although the agreement between experimental and tl theoretical polarizations is generally very good, some I slight deviations exist. An example is given in Fig. 15 0.5 I' f which shows P(8) for 300 ev. The experimental curve l I is shifted by about 2' towards larger angles. Some of the I following figures show a similar behavior at other P" 60o SO 120 8 energies. The angular shift can be seen even better in 4 a . I (b) 0 'T Fig. 16, where the strong energy dependence of the - ~44 I polarization peak in the neighborhood of 300 eV is

I 44( I I shown. At the time when these measurements were I I I published (jost and Kessler, 1966), no theoretical -0.5 tt

I I.O II 260eV 280eV 300eV 320eV 340eV P II

It} A -l.0 II I I il II 0.5 Ia FIG. 14. (a, b) Angular dependence of the polarization I for I I 1700 and 900 eV, Z=80. Experimental and theoretical values I 'I I I ~t4 (from Jost and Kessler, 1966). I I I I I) I I I I 0 8 IIOo 8 IIOo e iso el uo' Oo ' o ISO' 13 I30 ~ IBO' ' I 130' narrow I I4t l 1 maxima and minima with widths comparable I I to &j. . This fully explains the somewhat lower experi- II' I I mental peak values which occasionally occur. The theoretical values used in this section are either -0.5 those of the Munich group (Holzwarth and Meister, 1964a; 1964b) or of the Melbourne group (Bunyan, 1963; Bunyan and Schonfelder, 1965), depending on FrG. 16. Energy and angular dependence of the polarization for Z=80 in the neighborhood of 300 120'. what was availaMe at the energies considered. For those P eV, Experimental values from Jost and Kessler (1966); dashed line, unpublished energies where data of both groups are available, they theoretical values by Coulthard Lsee Footnote (*),p. 16j. 16 Rzvrzws os MODERN PHYSICS JANUARY 1969

cross sections. It is worth noting that Fig. 17 shows very 0.3- clearly how the interference minima and maxima disappear with increasing energy because the electron wavelength becomes less and less comparable to the 0.2— atomic radius. The few examples shown here, together with the more comprehensive results of the original $4Q Owl papers, I- ABSCISSA give evidence of the good correspondence between FOR theoretical and experimental results if we regard the 0 keV 0 !. angular shift as a minor difference. On the other hand, O discrepancies between theory and experiment are to be I.2 kev 0 anticipated if the electron energy is low enough so that the conditions of experiment are no longer compatible I.54V~ O with the assumptions of theory. As we pointed out in O Sec. II.B, the present theories treat the scattering by a I.8kev't static potential without taking into account that the electron cloud of the atom is distorted during the 2.0 4V scattering process if the incident electrons are slow enough. Furthermore, exchange and correlation eGects a 0 —.5 keV due to electron spin have been neglected. No doubt, these eGects have an appreciable inQuence at lower 3.5 kev energies; the question is at what energy does this inhuence begin. 4.0 kev 0 Discrepancies between theory and experiment have 0 50 IOO l50 SCATTERING ANGLE e (DEGREES) been claimed for energies below 500 eV by Schonfelder (1966), whose considerations are based on some of the Fxo. 17. DiGerential cross sections for electron scattering by results of one of the experimental mercury between 1000 and 4000 eV. Solid line, experiment; groups (Steidl, dashed line, theory. At the points marked by a circle, the ordinate Reichert, and Deichsel, 1965). Schonfelder proposed to of each experimental curve has been normalized to the corres- explain these discrepancies by the above-mentioned ponding theoretical curve. (as —Bohr radius in hydrogen. ) (From Kessler and Lindner, 1965.) theoretical approximations. However, this disagreement could not be conlrmed in extensive measurements of results for 260, 280, 320, and 340 eV were available. our own (Eitel, Jost, and Kessler, 1967) where the The theoretical data of Fig. 16 have been calculated polarization below 500 eV has been studied. The subsequently by Coulthard. * measurements were made down to 100 eV, which was An angular shift has been found in the cross-section the lowest energy for which numerical evaluations of measurements as well and is demonstrated in Pig. 17. the present theories were available. Figure 18 gives a Again some of the experimental curves Lfull line, plotted comparison of the experimental and theoretical results continuously by an X-F recorder (Kessler and Lindner, for this energy, showing that the correspondence is as 1965)j are shifted by about 2' to the right. t More good as at the higher energies discussed before. We experimental cross-section curves can be found in some found this to be true for all the energies studied between of the papers on polarization measurements (Deichsel, 100 and 500 eV (Eitel, Jost, and Kessler, 1967). 1961; Deichsel and Reichert, 1965; Steidl, Reichert, These results are considered to be particularly and Deichsel, 1965; Deichsel, Reichert, and Steidl, reliable for two reasons. First, the angular resolution 1966;Eitel, Jost, and Kessler, 1968) and also in Kessler with is and Lindner (1965), where comparison theory f00 eV made. The relations between the cross section and P polarization curves which have been discussed in Sec. Il.a (polarization maxima near cross section minima, I I I etc. have been confirmed in all the experiments. I Fzo. 18. Angular de- ) I I pendence of the polari- Because of these close relations, it is quite natural that o zation I' for 100 I eV, angular shifts between the theoretical and experimental I I Z= 80. Experimental curves, if any, occur in both the polarizations and the and theoretical values -P 0 4'~ (from Eitel, Jost, and J I 9'O' ' 'l20' Kessler, 1967). 30' 760' P I 150' ~ The author is indebted to Dr. Coulthard for kindly making 8 available his unpublished results. While the ordinate of each experimental cross-section curve t Il & has been normalized to the corresponding theoretical curve, no 'I such adaption has been made with the polarizations; they have been measured absolutely. J. Kzssr.Ea E/estroII Sfis'II PolarssuIIoII by Loui-ZIIergy SssIIersIIg 1f w as high enough to make possible a direct comparison 0,4 3.5+ l eV with the theoretical results. Therefore, with the pronounced structure of the curves, any discrepancies p&e& should have been even more distinct than in a presenta- 0.3— tion where the theoretical curves are convoluted with a lower angular resolution. Second, it was ascertained 0.2- that at each energy the density of the mercury atomic beam was low enough to avoid plural scattering for II II ~ which the results would no longer be comparable with O.t— the theoretical values calculated for single scattering. This turned out to be particularly important at the I s I I « 1 « l i I I e lower energies. Figure 19 gives an example, showing the 304 604 904 I204 1504 rapid change of the polarization curve for 300 eV with in- increasing temperature of the mercury oven, i.e. -O. i— creasing density of the atomic beam. Under the conditions of the experiment (Eitel, Jost, and Kessler 1968), the results of Fig. 15 could be obtained only for -0.2— oven temperatures up to 160'C. At higher temperatures the shape of the curve changes drastically because of a -0.3— strong reduction of the polarization peaks by plural scattering. Since the maxima of a polarization curve are close to Fxc. 20. Angular dependence of the polarization P for 3.5 eV, Z=80 (from Deichsel, Reichert, and Steidl, 1966).

09[.: o 'p'. 300eV - the minima of the corresponding cross-section curve, it ~BOoe W can be easily understood that plural scattering has this I I effect. As shown in Fig. 19(b), plural scattering begins to afFect the diGerential cross section mainly by Glling I / up the minima (for further discussion see Eitel, Jost, prt ) I 0- ~ w~t4 I ~ I r I r r 5 r and Kessler, 1968). This means that electrons are 60' I 90o ~f20 8 IM. tenno scattered into the angular range of a polarization peak 9---& 9 i which, under the conditions of single scattering, would

I I I I reach adjoining angles. The polarization connected -0$ with these adjoining angles is much smaller than the or even has so the 0.9 i1 peak polarization opposite sign, that II 300ev II II maximum polarization is reduced considerably by p II Fro. 19. (a) Effect of plural I ' 200C I I I plural scattering. scattering on polarization curve I I I for 300 Z=80 and different I Summarizing the results obtained between 100 and eV, I I oven temperatures. Dashed I h I 1700 eV for mercury, we can say that scattering by a line, theoretical values for II I static Hartree potential is a good description of the single scattering. (b) Effect of I't I plural scattering on differential scattering process in this energy range. No appreciable cross section for 300 eV, Z =80 60' ' 90O "uOoeit inQuence of atomic polarizability, exchange and and various oven temperatures I,r t t.t t (S=normalization The it/ correlation e6ects can be found. Theoretical investiga- point). I II I 160'C I values for correspond to I tions are being made (Walker, private communication) I single scattering (from Eitel, III II to find out whether the slight angular shift between Jost, and Kessler, 1968). -0.6 s theory and experiment can be explained these eGects. (a) by 2g: N There is a great interest in theoretical studies which 00eV take these eGects into account. Experimental results l.0 on polarizations (Deichsel, Reichert, and Steidl, 1966) and cross sections (Deichsel, Reichert, and Steidl, 1966; 05 Eitel, Jost, and Kessler, 1968) at energies far below 100 eV, where this improvement of the theory is expected to be important, are already available. The ylBO 'C O.l rl60 'C measuremeots of the Mainz group have been made down to 3.5 eV, the lowest energy for which a polarization

~ curve has been measured. result is shown 20. ~ The in 0 I r ~ I Fig. 3OO 90' SP e 60. The polarization, which reaches values up to 0.3, changes (b) less rapidly with the scattering angle because at these 18 REVIEWS OZ MODERN PHYSICS ' JANUARY 1969

5x lo& which such measurements have been made is argon*; the results of Mehr (1967) for I'(0) at 40 eV are reproduced in Fig. 22. These were obtained with the Mainz apparatus; the Qrst mercury atomic beam was replaced by the argon beam. The theoretical data inserted into this figure are those of Coulthard. t' For other free atoms, experimental polarization data are not available~; the theoretical situation is not much better, as pointed out in Sec. II.B. Not only are the polarization data missing, but in many cases there are no data even for cross sections at low energies. The situation at higher energies (large range around 100 keV) is IO more favorable. Calculations of polarization have been made for several elements scattered over the periodic table, and it is easier to make interpolations because of the smoother dependence of I' on the atomic number. I I I I 50' 60 90 I20' l50' Cross sections, at least theoretical ones, can be found 8 for all elements across the periodic table. Although Fxo. 21. Survey of angular and energy range covered by these cross sections are based on the Born approximation measurements of electron polarization in elastic low-energy (Motz, Olsen, and Koch, 1964) in many cases, f they scattering from mercury. Dashed lines, measurements of the are, within certain limits (Fink and Kessler, 1966), Mainz group (Deichsel, 1961;Deichsel and Reichert, 1965; Steidl, Reichert, and Deichsel, 1965; Deichsel, Reichert, and Steidl, accurate enough for most purposes. This more favorable 1966); dash-and-dot lines, measurements of the Karlsruhe group situation at higher energies is due to the fact that (Jost and Kessler, 1965; 1966; Eitel, Jost, and Kessler, 1967; electron diffraction in this has been a 1968); solid line, overlapping measurements of both groups. For energy range better orientation areas E&0.5 (checkered) and I'&0.5 (hatched) valuable tool in the study of matter for a long time. are indicated.

2— low energies only a few partial waves of lowest order con- 0. tribute to the scattering process. Theoretical results for comparison do not exist as yet. Since the present review can display only a few examples from the many measurements which have been made for mercury, Fig. 21 gives a survey of the O.I— energies and angles where electron polarization in elastic low-energy scattering from mercury has been measured. The measurements cover the whole region thoroughly enough to provide consistent information I' on the validity of the theory between 100 and, say, 0 5 II/ 2000 eV. At the energies below 100 eV, calculations and I 4) experiments were made at different energies so that a II comparison is not feasible. II The calculations of Holzwarth and Meister II .. (1964a, II II II 1964b) have been made for a large energy range ex- II il II II tending up to several hundred thousand electron volts. -0 I— If II Since these higher energies are of no particular interest II II in the present paper, we merely mention that at these II energies and at large scattering angles the agreement of the theoretical results with new experimental cross t ~ ~ & . ~ I I I I l I I I I section data (Kessler and Weichert, 1968) is not as -02-- 20 40 IOO 150 good as at low energies. For further discussion cf. Kessler and Weichert (1968) and Buhring (1968). FIG. 22. P (9) for 40 eV, Z = 18. Experimental values from Mehr (1967), theoretical values by Coulthard for ideal angular resolution. Z. Other Elements * See the addendum. We have discussed spin polarization in elastic electron f The author is indebted to Dr. M. A. Coulthard for kindly scattering from only mercury, by far the most in- making available his unpublished results. vestigated atom to date. Information on the other f Recently, R. A. Bonham and H. L. Cox (1967) calculated elastic electron scattering amplitudes for atoms using the partial- elements is insufficient. The only other free atom for wave method between 10 and 100 keV from S=1 to Z=54. J. KzssLzR Electron Strt'n Potart'sateen by Low Z'n-ergy Scattering 19

For interpretation of such experiments, knowledge of at least the cross sections was necessary. Now the oo 0 0 technique of low-energy electron diGraction is becoming 0 0 0 0 more and more important, and data of cross sections 0 0 o 0 0 0 oo and polarizations in low-energy electron scattering are o 0 oo oo Pl 0 0 0 CD o 0 not only of fundamental interest (appealing enough by )C 0 o 0 oo oooo 0 itself) but also of great value for the study of matter. ,-95 Sections and C will examples. A effort I I I B give strong to I5.90 ISO I620 eV overcome the lack of data in this field is needed.

-0.3 3. Resonance Scattering from 1Veon (b)

The theoretical predictions by Franzen and Gupta

(1965) on spin polarization by elastic resonance scat- -O.l tering from neon (cf. Sec. II.C) have been confirmed by Reichert and Deichsel (1967). A schematic diagram I I of their apparatus is given in Fig. 23. An electron beam IGLOO l6.IO cV

—1 of 2)(10 ' A with an energy half-width 68=40 meV 0. and an angular divergence of &2 emerges from a 127' cylindrical monochromator and is scattered by a neon atomic beam. The electrons scattered by 8=90' are --0.3 accelerated to 300 eV and scattered by a mercury atomic beam in order to analyze their polarization by the method described in Sec. III.A. 1. It is the same Fxo. 24. (a) Experimental results for the resonances of the polarization analyzer used by the authors in earlier cross section for 0=90' in neon. (b) Energy dependence of the work. polarization in resonance scattering from neon for 8=90 (from Reicherf and Deichsel, 1967) Figure 24 gives the experimental results. The reso- . nance of the cross section for 0=90 is shown in Fig. Molecules 24(a). It was recorded with the detector "north, " the 3. Scattering by mercury beam having been switched off. Figure 24(b) Experiments on spin polarization in electron scat- shows the energy dependence of the spin polarization. tering from molecules have been started recently The measurements have been checked by rotating (Hilgner and Kessler, 1967), prompting interesting primary beam and monochromator by 180'. The theoretical studies (Yates, 1968a; 1968b) * in this points thus obtained are labeled by K. direction. Comparison of the results with the experimental According to Eqs. (5) and where no special theoretical predictions of Franzen and Gupta (Fig. 10) (9), assumptions on the nature of the scattering center have shows good agreement. Since the resonance peaks are been made, the polarization E obtained by the scattering only 95 meV apart, the curves calculated by Franzen of an unpolarized electron beam by a molecule is and Gupta for an ideal energy resolution can be checked only with an energy half-width AE((95 meV. Polariza- iF* — ( GF*G)l(l ~ I'+ I G I'), (22) tion measurements with increased energy resolution where and are the amplitudes of the wave scattered are highly desirable but very dificult because of in- F G the molecule. can be expressed in terms of tensity problems; an experiment with this goal is being by They and the amplitudes of the wave scattered the done by Franzen and Gupta (private communication) . f; g;, by jth atom, as F=gf, exp (isr;), MONOCHROMATOR MONITOR DETECTOR WEST G= Qgt exp (isr;), (23)

ACCELERATION DETECTOR NORTH ro 3II ~v shown textbooks on electron diffraction I as in the (see, I e.g., Pjnsker, 1953); s is the momentum transfer in units of ti, s= (4tr/1I, ) sin —', II; r, is the radius vector from the origin of coordinates to the center of the jth atom. ELECTRONVELVE T DETECTOR EAST Equations (23) hold under the assumption that the molecule is composed of free atoms which are inde- FIG. 23. Schematic diagram of apparatus for measuring pendent of each other. Substitution of Eqs. (23) into polarization in resonance scattering from neon (from Reichert and Deichsel, 1967). *See the addendum. 20 REVIEWS OE MODERN PHYSIOS JANUARY 1969

025 difference being that the polarization values for the P dA bismuth atom are much higher. In Bi(CSHS)((, the 20- 0. polarization of the electrons scattered from Bi is "diluted" by those scattered from the low-Z part of 0.15 the molecule, which have a very small polarization. The latter fact was veri6ed by the measurements made 0.10- FM. 25. Experimental results for differential cross with molecules consisting of light atoms, like HSO, section and polarization of which showed that the polarizations are not greater 0.05- 900-eV electrons scattered by Ili(C4Hsl, (from Hilgner, than 0.01. A more thorough evaluation of the measure- " and to be o„-.ti-t- e Kessler, Steeb, ments with Bi(CSHS) s (which could not be completed published) . before conclusion of this article) shows that the in- o -0.05- o Quence of the bismuth atom increases at higher electron (I o energies, whereas the light atoms play an important -0.10- role at lower energies. For full details see Hilgner, Kessler, and Steeb (1969). -0.15 As an example of what one can learn. from electron polarization measurements with molecules, Fig. 26 gives a comparison of the polarization curves for ethyl Eq. (22) gives the polarization of an electron beam iodide and iodine at the same electron energy. Minima scattered from a molecule with a certain spatial orienta- and maxima are at the same angles in the two cases but tion. The experiments to be discussed here were made are more pronounced for I2. Since the electrons scattered with beams of unoriented molecules. The mean value from the light atoms, C and H, are virtually unpolarized, of the polarization for arbitrary orientation of the we can consider the scattered intensity in the case of molecules (P) is obtained by a straightforward integra- CSHSI as being composed of two parts. One part comes tion of Eq. (22) analogous to the evaluation of the mean from the iodine atom and is marked by its polarization, scattered intensity in electron diBraction from molecules given by the values of Fig. 26(b) . (The iodine molecule (Pinsker, 1953). The result is (cf., e.g., Hilgner and gives rise to the same polarization as a single atom if Kessler, 1969) certain conditions are fulllled. ) It can therefore be distinguished from the unpolarized part coming from P= P,s~;s/ Q o;s, (24) g C and H. The resulting polarization is determined by where the ratio of the polarized and unpolarized fractions of the scattered intensity. an old P'= I:~(fgs* fs*g() j/(—f~'+g gs*) So, question (Hughes f and McMillen, 1933) in slow electron scattering can be — o,;A (ffq*+g;gs*) (sin sr;s/sr, s) . If the molecule consists of atoms of the same kind, all P,R are identical, P;A=PA, and Eq. (24) reduces to 0.20 0t2C P P P=P~ Q ~;s/ Q ~3A=P~, (25) 0.10- 0.10- i.e., the molecule gives the same polarization as a single atom. The apparatus used for the polarization experiments $3$ 150' with molecular targets was basically the same as that 120ft( l' H') used for the measurements with mercury targets (Fig. i, 11). So far, measurements have been made (Hilgner, -0.10- -0.10- Kessler, and Steeb, 1969) for Is, CIHSI, HSO, CC14, (I Bi(CSHS) s, and CSHs. ( 3 As a typical example for the results obtained with a -020- -020- 3 I ( 3 ( I molecule, Fig. 25 gives experimental curves of polariza- ( diGerential tion and cross section for electron scattering (I 1( 94':-0.41 from Bi(CSHS) s at 900 eV. One can clearly see polariza- -0.3C -0.3C )Min (b) tion peaks near the minima of the cross section. The (a) polarization curve has exactly the same structure as FIG. 26. (a, b) Angular dependence of cross section and that of the bismuth atom calculated Walker, * the polarization at 400 eV for C~Hf;I and Ig. Solid lines, experimental by cross sections (open circles, normalization points); dash-and-dot, lines, theoretical cross sections; solid circles, experimental polariza- *The author is indebted to Dr. D. %.Walker for kindly making tions; dashed lines, theoretical polarisations (from Hilgner and available his unpublished results. Kessler, 1969). J. KzssLER Electron Spin I'olarizatiort by Lozo Ea-orgy Scattering 21 easily answered in the example of CzHzi; namely, polarization of the scattered electrons give additional How much of the scattered intensity comes from one information on the molecules which complements the part of the molecule and how much comes from another knowledge obtained by cross section measurements partP (cf. Sec. IV). For evaluation of future measurements A quantitative theoretical analysis of these measure- with molecules, it is necessary to know the polarization ments has been made by Yates (1968a; 1968b), whose curves for the atoms. Calculations across the periodic results are also shown in Fig. 26. The good agreement table are desirable to overcome the lack of data men- of theoretical and experimental data shows that it was tioned in Sec. III.A.2. a good approximation to make the following assumptions in the calculations: The molecule consists of free C. Scattering by Solid Targets atoms which are independent of each other (the in- fluence of binding is negligible), and scattering of an From what has already been said about scattering electron does not occur more than once within the from atoms and molecules, one will readily anticipate molecule. Unpublished theoretical results of Walker* ~ that slow electrons scattered from solid surfaces are and Buhring* at other energies also agree satisfactorily polarized, too. For instance, in low-energy electron diffraction with the experimental results, though yielding higher (LEED) experiments, one should obtain polarization peaks for iodine. However, according to electrons of appreciable polarization and intensity Suhring, there is a significant dependence of these peaks whenever a diGraction spot appears under an angle for on the scattering potential used for the calculations. which P(0) for the single atom has a maximum. Apart from the 6rst According to the calculations, the mixed terms j&k very attempt by Davisson and Germer in Eq. (24) are relatively unimportant for the molecules (1929), polarization in LEED with single crystals has been and the angular range considered here. This can be not investigated. But polarization studies were made with foils of and quite diferent for molecules consisting of medium or tungsten, platinum, gold (Loth, and a solid heavy atoms, where not all the mixed products contain 1967) mercury target (Eckstein, 1967). small factors due to light atoms. Theoretical estimates These experiments have been made with the apparatus of the Mainz show (Bonham, private communication) that these group (cf. Sec. IILA.1) after re- the atomic in molecules can give rise to an enhancement of the placing mercury beam the erst scattering chamber the solid polarization over that obtained from the single atoms. by target under investigation. The metal foils Many other interesting points have not yet been couM be heated in order to clean their studied experimentally because investigations on surfaces, whereas the mercury was condensed con- polarization in electron-molecule scattering began tinuously on a liquid-air-cooled supporting substrate. The evaporation rate of the was only recently. Another point which will be studied in mercury chosen low the near future is the following: Since the polarization enough to make the intensity of the electrons scattered into the is very sensitive to plural and multiple scattering by mercury vapor angular range studied small (Eitel, Jost, and Kessler, 1968), effects of multiple compared to the intensity scattered by the solid target. elastic intramolecular scattering which were treated The angular resolution of the polarization measurement is given the authors as 60=8'. 65' theoretically by many authors (Bonham, 1965) and are by Angles between and 155' were covered. expected to be quite important at low energies in With the mercury target, the electron molecules containing several heavy atoms should be energies were between 300 and 900 eV, while 900-eV electrons were used for the other easily detectable by polarization measurements. targets. As in all the discussed in Actually, multiple intramolecular scattering is probably experiments this review, only elastically scattered electrons responsible for the small deviations found between were observed. theoretical and experimental polarization peaks. This Polarization of the scattered electrons was found with all these solid targets. As an is discussed by Hilgner and Kessler (1969). example, the points in the Massey and Burhop (1952) write in their well-known Fig. 27 give measured polarization values at 900 eV for the solid book, E/ectronic and Ionic ImPact Phenomena that mercury target. The measurements with this "much interesting information about the outer fields of target have the advantage that they can be com- with molecules could be derived from a systematic study of pared the results obtained for free mercury atoms. the elastic scattering of slow electrons in conjunction For this purpose the dashed line in Fig. 27 represents with approximate theories, but a much wider range of the theoretical polarization curve Lwhich was conlrmed experimentally (Deichsel and and molecular types would have to be investigated. " This Reichert, 1965; Jost for the statement is underlined by the fact, not known when Kessler, 1966)j free atom convoluted with the of 68=8' that book was written, that measurements of the angular resolution of the experiment under discussion )for the unconvoluted curve cf. Fig. 14(b)j. The two results of Fig. 27 have the same qualitative ~The author is indebted to Dr. W. Buhring and Dr. D. W. Walker for kindly making available their unpublished results. angular dependence. Most of the polarization values $ See the addendum. for the solid target are smaller, however, than those for 22 REviEws oR MODERN PHYsIcs ~ JANUARY 1969

IV. APPLICATIONS AND OUTLOOK

SOLID MERCURY Electron scattering experiments have been an in- 5- 0. valuable tool for studying the structure of matter. One

I I of the fundamental quantities measured in these I I experiments was the diGerential cross section for 3— I 0. I an unpolarized beam I / I / I d~/dQ= I'+ I I f gI', (26) O.I— which gives information on the structure of the sca,t-

0 I I I I f I I tering center. 90O I20o T 8 I50 I I If one observes the polarization of the scattered -OI— electrons„one gets additional information on the ~ / I o scattering center which is independent of that obtained o/ I by cross section measurements: The polarization of an I I -03- I un- I electron beam resulting from scattering of an I I polarized beam is given by

&=i(fg* —I'+ I') -05- f*g)/(I f I g . (27) One can easily see that a measurement of P gives information on the complex functions f and g and, thus, Fxo. 27. Experimental results for the polarization of 900-eV on the scattering center, which basically cannot be electrons scattered by a solid mercury target. Dashed line, theoretical curve for free mercury atoms convoluted with the obtained by a measurement of the cross section Eq. experimental angular resolution i@=8 (from Eckstein, 1967). (26). Since the complex scattering amplitudes f and g are equivalent to four independent real functions, measurements of the two quantities da/dQ and I' still do not this difference the free atom, being particularly pro- give all the information attainable about the and g. of the curve in f nounced near the peaks Fig. 27. The However, this information could be obtained by meas- same follows from a comparison of the corresponding urements of T and U defined by Eq. (10), i.e., by results for the other energies studied. measuring the rotation of the polarization in a scat- This behavior of the polarization curve for the solid tering process. Because of the relation S'+T'+ U'= 1, target can be easily understood if we recall the diGer- it is impossible to make four independent measurements atomic beams of low and ences in scattering from high for determining the and g. This is a consequence of in 19. Plural scattering in the f density presented Fig. the fact that the phase qi of the wave function P= beams of high density resulted in deviations from the s'"'LI b which a I PiI2+ I I exp (i'm)P iIs5 describes an theoretical curve which were signihcant near the electron beam of arbitrary spin direction (f&Is and polarization peaks and of the same kind as those of mean orthogonal spin functions) cannot be Fig. 27. Since plural and multiple scattering un- three and determined; only the parameters I a I, I b I, doubtedly an important role in the experiments play q~ are accessible to measurement. Measurements of the with the solid targets, no agreement with the results rotation of P are at the margin of what can be done for free atoms can be expected. presently, as we saw in Sec. II.B. The results for tungsten, platinum, and gold look On the other hand, measurements of th.. polarization very similar. Figure 28 gives an example. The measure- produced by scattering can be made with great accu- ment was made at a temperature of 1200'C of the platinum target. This was important since the results turned out to depend very sensitively on the purity of PLATINUM the target surfaces. Also in th. measurements with the p(e) indicated 01- solid mercury target, impurities are by the I I author to be an additional reason for the differences FIG. 28. Experimental results shown in Fig. 27. 70 90 l20O l50 8 for the polarization of 900-eV Polarization studies in LEED are planned in several electrons scattered by a heated platinum target (from Loth, laboratories. Single-crystal targets would be of particu- 1967). lar interest for this purpose. It would also be very -o~- helpful for these experiments to have as a guide theo- Ij retica, l polarization data for a large number of elements. J. Kzssx.ER Electron Spin Polarization by J.on&-Energy Scattering 23 racy. They not only provide information which is out this article, we will not go into the question of independent of that obtained by cross-section measure- which problems can be attacked with polarized elec- ments but also make possible a very accurate check of trons in high-energy physics. Instead, we mention a the models used to describe the scattering center; for few interesting applications at low energies. the theoretical polarizations depend very sensitively on Pepinsky (1966) discussed the possibility of studying these models. (Cf. Meister and Weiss, 1968.) This can eITects of ordered arrangements of magnetic atoms at be made plausible by recalling the fact shown in Fig. 6; crystal surfaces by low-energy electron diffraction P is determined by the difference of two cross sections (LEED) experiments with polarized electrons. Since which are not very much different from one another scattering due to electron spins (magnetic dipole so that small changes of the cross sections due to differ- scattering and exchange scattering) is small compared ent theoretical models can result in great changes of I'. to Coulomb scattering, it is generally masked by the The advantages mentioned suggest that it is re- latter. However, in "spin-only" diffraction maxima warding to make many more studies of the spin polar- (which can occur when the chemical unit cell and the ization in electron scattering, thus obtaining informa- magnetic structure cell differ from one another), tion which cannot be obtained by the more conventional Coulomb scattering is repressed by interference; investigations of cross sections. reflections arising solely from the magnetic dipole Not only do measurements of the polarization of the distribution should be detectable even when the scattered electrons yield interesting knowledge of the incident beam is unpolarized. But by analogy with scattering center, but also the polarized electrons can neutron-scattering (Bacon, 1966), more information be used as probes for studying matter. Measurements will be attainable through use of polarized electrons. of the scattering asymmetry or of the change of the For example, magnetic form factors describing the dis- polarization vector in a scattering process would give tribution of unpaired electron spin density in the atom new insight into many problems of atomic, molecular, or directions of magnetic moments in the sample could and nuclear physics. This is one of the reasons for the be obtained. Since exchange scattering makes an great many attempts made in various laboratories to essential contribution to the "spin-only" maxima of the construct sources of polarized electrons* (Miiller, LEED pattern, these experiments will also be an Obermair, and Siegmann, 1967; Hofmann, Regenfus, interesting way of examining the exchange interaction. Scharpf, and Kennedy, 1967; Hughes, Lubell, Posner, There is an even more direct way of studying the and Raith, 1967; Donally, Raith, and Becker, 1968; electron exchange interaction: Polarized electrons are Krisciokaitis and Robinson, 1965; McCusker, Hat- scattered by atoms and the change of the polarization field, Kessler and Walters, 1968). So far, the working is observed. It was not until a few years ago that a sources yield either small polarization and satisfactory general method for exploring electron exchange colli- current or high polarization and rather small current. sions was developed (Collins, Goldstein, Bederson, and One of the best results is that of Hughes, Lubell, Posner, Rubin, 1967; Lichten and Schultz, 1959) based on the and Raith (1967), who obtained 85% polarization with use of polarized atomic beams. The use of polarized 3&(107 electrons per pulse by photoionization of a electron beams would be a different approach to this polarized lithium atomic beam. As long as entirely new problem, feasible also for elements which are inappro- approaches resulting in much higher intensities are not priate for producing polarized atomic beams. On the found, polarization by low-energy scattering compares other hand, in those cases where polarized atomic favorably with the methods existing. It is a simple beams can be produced, the use of polarized electron method giving a reasonable direct current of polarized beams would make possible experiments with electrons electrons: In 1965, Steidl, Reichert, and Deichsel which can be distinguished from each other; one has obtained currents of 10 9 A with 17% polarization. only to choose the direction of polarization in electron More favorable results are attainable by producing the beam and atomic beam opposite to one another. primary electron beam with high-intensity guns such It should be one of the objectives of a review article as those used in electron accelerators. In contrast to to give perspective to the work which would be inter- foils, there are no limits for the power of the primary esting or even exciting to pursue. Pertinent remarks are beam hitting the gaseous target. The use of gaseous scattered throughout the present survey, but it may be targets is possible (and necessary) because of the large useful to compile in a condensed list the problems the scattering cross section of the slow electrons. A source attack of which seems rewarding: of polarized electrons along this line is being prepared at Stanford University and intended for use with an (1) Calculation of the scattering amplitudes f and g accelerator. across the periodic table using a static potential and/or Since we were concerned with slow electrons through- better approximations including polarizability of the atoms, exchange, and other effects which play a role at ~ low energies. (Cf. Secs. II. III.A. III. and III. . Literature on this subject up to $965 is quoted by Farago B, 2, B, C) (1965). (2) Polarization measurements at lower energies in 24 REvrmws ott MODERN PBxsrcs ' JANUARV 1969

conjunction with approximate theories to ind out the here with whom I held discussions, e.g., Dr. W. Biihring, inhuence of polarizability of the atom, exchange, etc.; Professor R. Pepinsky, my colleagues at Mainz, my measurements for various Z (cf. Secs. III.A.1 and co-workers, and particularly Professor R. A. Bonham III.A.2) . from Indiana University; the latter called my attention (3) Experimental check of all the results predicted to many of the points discussed in Sec. III.B.I am also by theory by studying not only the polarization of a grateful for the help of Professor P. L. Donoho, Rice scattered beam or the scattering asymmetry of a University, in the struggle with linguistic problems. polarized beam but also the change of the polarization The work was supported in part by the Deutsch: vector in scattering (cf. Secs. II.B and IV). Forschungsgemeinschaft and the Bundesministerium (4) Connection between chemical binding in a fur Wissenschaf tliche Forschung. molecular target and polarization of scattered electron beam; range of validity of independent-atom model for REFERENCES molecules; inQuence of molecular structure on polar- A. A. Abrahamson, Phys. Rev. 123, 538 (1961);D. Andrick and ization; enhancement of polarization by molecules over H. Ehrhardt, Z. Physik 192, 99 (1966). V. A. Apalin, I. Y. Kutikov, I, I. Lukashevich, L. A. Mikaelyan, that by single atoms; indication of internal multiple G. V. Smirnov, and P. Y. Spivak, Nucl. Phys. 31, 657 (1962). scattering in molecules by decrease of polarization F. L. Arnot, Proc. Roy. Soc. (London) 130, 655 (1931). peaks (cf. Sec. III. G. E. Bacon, Advan. Structure Res. DiGraction Methods 2, 1 B). (1966). (5) Direct measurements of cross sections for H. A. Bethe and E. E. Salpeter, Pandbuch der Physi&, S. 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