- Email: [email protected]
Surface plasmons at the surface of liquid mercury
Solid State Communications,
Vol. 14, PP. 1097—1100, 1974.
Pergamon Press.
Printed in Great Britain
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY Howard L. Lemberg* and Stuart A. Rice Department of Chemistry and the James Franck institute, University of Chicago, Chicago, Illinois 60637, U.S.A. and Paul H. Naylort and Aaron N. Bloch Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218, U.S.A. (Received 13 November 1973 by A.A. Maradudin)
We have evaluated the surface plasmon dispersion relation in the presence of the inhomogeneous surface zone which has been proposed for liquid mercury. Assuming that photon-plasmon coupling occurs through the mechanism of frustrated total reflection, we find the dispersion relation to be sensitive to the existence of such a zone, providing retardation effects are taken into account. The results, it is believed, define nonradiative surface plasmon absorption as a useful technique for the study of conducting surfaces.
IN RECENT years there has been much interest in surface plasma waves,1’2 the class of charge oscillations which are observed at the surface of a bounded conducting medium in addition to the bulk modes which exist in the interior. In order to understand the structure of a real metal surface, several experimental and theoretical attempts have been undertaken to investigate departures of the surface plasmon dispersion from the idealized behavior which is observed at a planar boundary of zero thickness between a homogeneous metal and a non-absorbing dielectric. Surface plasma waves in this idealized geometry are associated with electric and magnetic fields of the form F(x,y,z,t)
=
FoeI~_~t)e±I~12,
where
IW~2
\
— (~)fm,d)
(2) \ C and EmEd are the frequency-dependent dielectric constants of the metal and dielectric, respectively. The dispersion of these surface plasmons, determined by continuity of the tangential field components at the interface, is given implicitly by =
+ ~m
=
0.
,
(3)
Ed
Deviations from the dispersion relation (3) may result, on the one hand, from electronic interactions ofa sort similar to those that occur in the bulk conducting medium.3 A second class of departures arises from the presence of surface defects on a microscopically large scale, as in the case of metal surfaces which have been intentionally roughened to permit photon-plasmon coupling.4 Effects of this nature are also introduced by the existence of thin
(1)
_____________
*
/ Km,d
National Science Foundation Predoctoral Trainee,
t Present address: Department of Chemistry. University of Maryland, Baltimore Country, Baltimore, Md. 21228, U.S.A. 1097
1098
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY
dielectric layers, such as metal oxides, on conducting surfaces.5’6 In this note we consider surface phenomena of a third kind, which have previously been only briefly considered as an influence on surface plasma waves, We refer to the fundamental distinction between a sharp geometric boundary of zero thickness and the flat surface of a real metal, which may be characterized by electronic relaxation over several Angstroms. In the boundary region, the effects of interactions among the conduction electrons, as well as between electrons and metal ions, are qualitatively different from those in the bulk, because of the variation in dielectric screening near the surface. As a result, the surface region of a metal undergoes a well-known charge redistribution,7 and in certain cases may support localized surface states also.8 Two of us have suggested previously9 that in the case of liquid mercury, these effects may be so severe that the surface exhibits a liquid-state analog of surface reconstruction in a solid. This conclusion is supported by the available optical data,10”12 which have been obtained by the techniques of conventional reflection spectroscopy. In contrast, we address ourselves, in the present letter, to the effects that such surface structure would have on the dispersion of surface plasmonsin liquid mercury. By assuming a surface transition layer characterized by an Epstein profile for the electric conductivity, a(w)
=
W
0b
+ ~
w = _e2~2~,
(4)
Bloch and Rice9 have been able to reconcile the differences between the apparent optical constants of Hg as determined by normal-incidence reflectivity and ellipsometry. In equation (4) z 0b is the andcoordinate ; are the normal to the liquid metal surface: frequency-dependent bulk and surface contributions to the conductivity. An essential assumption of the model was that a~was described by a Drude frequency dependence o~= oo~/(l+ iwr) on the ‘bulk’ side of the conductivity maximum, and a Lorentzian form characteristic of localized states on the low density side, corresponding to a Mott transition as the Hgdensity falls from its bulk value to zero. The bulk 0b was assumed to follow a Drude form contribution throughout the transition zone.
Vol. 14, No. 11
Without reiterating the justification of this simplified phenomenological model of the diffuse surface region, we remark that recent calculations’3 of the self-consistent electron density at a metal surface show an oscillatory behavior which, we conjecture, in a local theory, would produce a peak in the surface conductivity similar to the one associated with the Epstein profile. More sophisticated treatments of the liquid metal surface, including the evaluation of dynamical properties in the presence of both retardation and nonuniform charge densities, are clearly required, however. In the present investigation we have performed calculations based on the skewed conductivity profile suggested by reference 9. The nonradiative surface plasmon (NRSP) dispersion has been determined by applying Maxwell’s equations for incident p-polarized radiation to the inhomogeneous zone, using the frustrated total reflection (FIR) configuration’4 to couple light waves to the surface excitation. In evaluating the optical response RTM, we have used the method of Herpin matrices,15 which approximates the full nonuniform region by cutting it up into a large number of thin films, each assumed to be homogeneous. I
I
I
I
E~25OeV, d~28OO I
0.8 06
- - -
--
dr~fuse sur~oce ellipsornefac sharp su,tace D’~de sharp surthce
I
~4 ~
~
°2L 00L
28°
32°8
36°
40°
44°
1. Reflectivity calculations threeEpstein models of aFIG. liquid Hg surface: diffuse, with for skewed profile; sharp surface with ellipsometric optical constants; sharp surface with Dnide optical constants. Arrows the abscissae corresponding to the for angles oflocate minimum reflectivity. All computations frustrated total reflection geometry with prism refractive index = 2.53, spacer index = 1.39, spacer thickness = d. O~is the critical angle for total reflection. The profile parameters which we have used for this and the next two figures are as follows: ao= ~ TS=O.9 Tb, 2ir~A 1=IA andbulk, 2ir~, Tb 15 the Drude relaxation time in the and 2A. the static bulk conductivity, Gob, is9.35X 10’5 sec”.
Vol. 14, No. ii
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY
Typical results are illustrated in Fig. 1, which shows that the surface plasmon at a diffuse interface should behave quite differently from the NRSP at a perfectly sharp boundary with effective optical constants as determined by ellipsometry or reflectivity measurements. The ellipsometric curve of Fig. 1 is based upon the measurements of Faber and Smith,’° which extended to a maximum energy of about 3 eV. To evaluate ‘ellipsometric’ sharp surface reflectivities at at higher energies, it was first necessary to compute a set of apparent optical constants characterizing the Hg surface. The constants calculated in this manner represent the effective values of Re (Em) and Im (Em) that would be measured in an ellipsometric experiment for E> 3 eV and 78°incidence if the skewed profile were completely accurate.
The NRSP dispersion relation may be determined over a range of energies in the optical region by measuring the angle of minimum reflectivity, conesponding to maximum coupling to the excitation, from several sets of calculations. The effect of the surface zone, most noticeable at high energies, has been evaluated from 3.5 eV to 5.5 eV (where interband effects start to become important),’6 and is shown in Fig. 3. We note, in particular, the shift of the dispersion relation towards the radiative regime (left of the light line). This finding is consistent with the higher effective electron density in the transition region, and also agrees with a simpler metal film-metal subsrate model in which the film plasma frequency is somewhat greater than that of the substrate.17
_______________________________________ I I I I
28°
32°
I
—~-
-
~::____
~°~:urf0Ce
-
I
8.0
E’400eV, d~IO5O1
OC
1099
36°
400
1
440
50-
3.0 I
3.5
4.0
4.5
5.0
I0”5k, [cm-Il FIG. 2. Reflectivity calculations for three models of a liquid Hg surface: diffuse, with skewed Epstein profile; sharp surface which is ellipsometrically equivalent to the profile; sharp surface with Drude optical constants. FIR parameters as in Fig. 1. The calculated optical response of Hg to NRSP excitation at 400 eV is shown in Fig. 2 for the full profile, a zero thickness surface free-electron model, and a zero thickness surface with optical parameters ‘equivalent’ to the profile as discussed above. Surface plasmon absorption should be measurably different for the three cases. The difference between the lower two curves demonstrates convincingly that the shift observed in Fig. 1 is due structure the transition region, andtois the notsurface an artifact of theof slight deviation in the Bloch and Rice fit to the ellipsometric data. It also underscores the enhanced sensitivity of NRSP absorption to surface properties in relation to the sensitivity of classical optical methods.
FIG. 3. Calculated dispersion relations for surface plasmons at a liquid Hg-dielectric interface, for FTR geometry. Dashed curve is dispersion relation for full inhomogeneous profile; dot-dashed line, for an ellipsometrically equivalent sharp surface. Absorption at a Drude sharp surface is heavily damped at these energies, so the resulting dispersion is not shown. Solid curve represents the light line w = ck~/n 3in the dielectric spacer layer.
In conclusion, it is to be noted that many of the investigations of surface plasmons at imperfect surfaces have concentrated on the high energy (E ~ hw8~),high momentum (hk ~ 8hc~~/c) region, and RPA’9 for in this range the hydrodynamic’ treatments of surface plasmons yield to solution after retardation is neglected or high-frequency expansions are made for the response functions of the electron gas. As yet, however, there has been no satisfactory treatment of the nature of surface plasma waves in the
1100
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY
low energy region, at a planar metal surface with a nonuniform transition zone. It might be objected that the surface plasmon at these energies should be insensitive to the detials of surface structure because the mode is primarily photon-like, its dispersion curve asymptotic to the light line of the contact medium as w 0. But for the case of liquid mercury, we find that there is an intermediate energy range, between E 4 hw9 and E h~.,5 within which the NRSP -~
~
,
p . . dispersion appears sensitive to nonuniformities in the electronic surface structure. Additional results for liquid Hg will appear in a later paper. -
.
-
Vol. 14, No. 11
Acknowledgements — This research has been supported by a grant (S.A.R.) from the National Science Foundation, H.L.L. and S.A.R. have also benefitted ~ ~~i~~ion Foundation. RH.N. and A.N.B. acknowledge the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. We acknowledge helpful conversations with Mr. Daniel Guidotti in the execution of this work.
REFERENCES
1.
KLIEWER K.L. and FUCHS R.,Phys. Rev. 153. 498 (1967).
2.
ECONOMOU EN.,Phys. Rev. 182, 539 (1969).
3. 4.
FUCHS R. and KLIEWERK.L.,Phys. Rev. 83, 2270(1971). ENDRIZ J.G. and SPICER WE.,Phys. Rev. 84,4144(1971).
5.
STANDORD J.L.,J. Opt. Soc. Am. 60,49 (1970).
6.
RAETHER H., in The Structure and Chemistri’ ofSolid Surfaces, p. 10—1 (edited by SOMORJAI G.A.) John Wiley, New York, (1969). BARDEEN J., Phys. Rev. 49, 653 (1936). FORSTMANN F. and PENDRY J.B.,Z. Phvs. 235, 75(1970).
7. 8. 9. 10.
BLOCH A.N. and RICE S.A., Phys. Rev. 185,933(1969). FABER T.E. and SMITH NV.,.!. Opt. Soc. Am. 58, 102 (1968).
11.
HODGSON J.N.,Phil. Mag. 4, 189(1959).
12.
CROZIER E.D. and MURPHY E.,Can. .1. Phys. 50, 1914(1972).
13.
LANG N.D.arid KOHN W.,Phys. Rev. B1,4555 (1970).
14.
OTTO A., Z. Phys. 216, 398 (1968); BARKER A.S., Jr., Phys. Rev. Lett. 28, 892 (1972).
15. 16.
HERPIN A.,Compt. Rend. 225, 182(1947). WILSON E.G. and RICE S.A.,Phys. Rev. 145,55(1966).
17.
GUIDOTTI D., RICE S.A. and LEMBERG H.L.,Phys. Rev. Lett., to be published.
18.
BENNETT A.J.,Phys. Rev. BI, 203 (1970).
19.
FEIBELMAN P.J., Phys. Rev. Lett. 30, 975 (1973). ~lbiOUCHII.1I4 o~uowemie ~1i1C[]~~CI1I1rio~epXHOCTb - n,1a3M0H np~i HU.iI1’41111 3OHbI Heo~i11opO1iHOi~noBepxHOCTlI, KoTopoe ObI,1O iipei.ioseIIO J.IB il~i1~KOIlpA~yTu. llpeirlo.la.laB, ~TO cB~33b 4X~TOHB C n.~a3MoHoM UpOM3XO.111T 3U (.q~- IIapyueHHOFO flO.1HOFO BI1~T~CHHOCOoTpaaeHMH, Nibi OTK~bLi1I ~1yBCTB1ITe.ThHOCTh oTHoae1~I5B mcnepcim }~C~U1CCTBOB~HI111T1-1TO aKOU :3oHbI. lIpJi}IIIMaB B CI-leT Hepa~n1auI1oHHyiOrIOBCpXHOCTh ~34)4)eF~Tbl 3amnL~bIBaHi1B. Mbl B~~BM pe3yJIhTaTb orlpeleiBloT rl.1a3MOH dOCOpOUII~O K~KHO1C3HV}O TCXHIIKY ]~1U Hc-c.eLI0BaHIIB flpOBOLIlllliIIX noBepxHocTeIl.
Vol. 14, PP. 1097—1100, 1974.
Pergamon Press.
Printed in Great Britain
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY Howard L. Lemberg* and Stuart A. Rice Department of Chemistry and the James Franck institute, University of Chicago, Chicago, Illinois 60637, U.S.A. and Paul H. Naylort and Aaron N. Bloch Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218, U.S.A. (Received 13 November 1973 by A.A. Maradudin)
We have evaluated the surface plasmon dispersion relation in the presence of the inhomogeneous surface zone which has been proposed for liquid mercury. Assuming that photon-plasmon coupling occurs through the mechanism of frustrated total reflection, we find the dispersion relation to be sensitive to the existence of such a zone, providing retardation effects are taken into account. The results, it is believed, define nonradiative surface plasmon absorption as a useful technique for the study of conducting surfaces.
IN RECENT years there has been much interest in surface plasma waves,1’2 the class of charge oscillations which are observed at the surface of a bounded conducting medium in addition to the bulk modes which exist in the interior. In order to understand the structure of a real metal surface, several experimental and theoretical attempts have been undertaken to investigate departures of the surface plasmon dispersion from the idealized behavior which is observed at a planar boundary of zero thickness between a homogeneous metal and a non-absorbing dielectric. Surface plasma waves in this idealized geometry are associated with electric and magnetic fields of the form F(x,y,z,t)
=
FoeI~_~t)e±I~12,
where
IW~2
\
— (~)fm,d)
(2) \ C and EmEd are the frequency-dependent dielectric constants of the metal and dielectric, respectively. The dispersion of these surface plasmons, determined by continuity of the tangential field components at the interface, is given implicitly by =
+ ~m
=
0.
,
(3)
Ed
Deviations from the dispersion relation (3) may result, on the one hand, from electronic interactions ofa sort similar to those that occur in the bulk conducting medium.3 A second class of departures arises from the presence of surface defects on a microscopically large scale, as in the case of metal surfaces which have been intentionally roughened to permit photon-plasmon coupling.4 Effects of this nature are also introduced by the existence of thin
(1)
_____________
*
/ Km,d
National Science Foundation Predoctoral Trainee,
t Present address: Department of Chemistry. University of Maryland, Baltimore Country, Baltimore, Md. 21228, U.S.A. 1097
1098
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY
dielectric layers, such as metal oxides, on conducting surfaces.5’6 In this note we consider surface phenomena of a third kind, which have previously been only briefly considered as an influence on surface plasma waves, We refer to the fundamental distinction between a sharp geometric boundary of zero thickness and the flat surface of a real metal, which may be characterized by electronic relaxation over several Angstroms. In the boundary region, the effects of interactions among the conduction electrons, as well as between electrons and metal ions, are qualitatively different from those in the bulk, because of the variation in dielectric screening near the surface. As a result, the surface region of a metal undergoes a well-known charge redistribution,7 and in certain cases may support localized surface states also.8 Two of us have suggested previously9 that in the case of liquid mercury, these effects may be so severe that the surface exhibits a liquid-state analog of surface reconstruction in a solid. This conclusion is supported by the available optical data,10”12 which have been obtained by the techniques of conventional reflection spectroscopy. In contrast, we address ourselves, in the present letter, to the effects that such surface structure would have on the dispersion of surface plasmonsin liquid mercury. By assuming a surface transition layer characterized by an Epstein profile for the electric conductivity, a(w)
=
W
0b
+ ~
w = _e2~2~,
(4)
Bloch and Rice9 have been able to reconcile the differences between the apparent optical constants of Hg as determined by normal-incidence reflectivity and ellipsometry. In equation (4) z 0b is the andcoordinate ; are the normal to the liquid metal surface: frequency-dependent bulk and surface contributions to the conductivity. An essential assumption of the model was that a~was described by a Drude frequency dependence o~= oo~/(l+ iwr) on the ‘bulk’ side of the conductivity maximum, and a Lorentzian form characteristic of localized states on the low density side, corresponding to a Mott transition as the Hgdensity falls from its bulk value to zero. The bulk 0b was assumed to follow a Drude form contribution throughout the transition zone.
Vol. 14, No. 11
Without reiterating the justification of this simplified phenomenological model of the diffuse surface region, we remark that recent calculations’3 of the self-consistent electron density at a metal surface show an oscillatory behavior which, we conjecture, in a local theory, would produce a peak in the surface conductivity similar to the one associated with the Epstein profile. More sophisticated treatments of the liquid metal surface, including the evaluation of dynamical properties in the presence of both retardation and nonuniform charge densities, are clearly required, however. In the present investigation we have performed calculations based on the skewed conductivity profile suggested by reference 9. The nonradiative surface plasmon (NRSP) dispersion has been determined by applying Maxwell’s equations for incident p-polarized radiation to the inhomogeneous zone, using the frustrated total reflection (FIR) configuration’4 to couple light waves to the surface excitation. In evaluating the optical response RTM, we have used the method of Herpin matrices,15 which approximates the full nonuniform region by cutting it up into a large number of thin films, each assumed to be homogeneous. I
I
I
I
E~25OeV, d~28OO I
0.8 06
- - -
--
dr~fuse sur~oce ellipsornefac sharp su,tace D’~de sharp surthce
I
~4 ~
~
°2L 00L
28°
32°8
36°
40°
44°
1. Reflectivity calculations threeEpstein models of aFIG. liquid Hg surface: diffuse, with for skewed profile; sharp surface with ellipsometric optical constants; sharp surface with Dnide optical constants. Arrows the abscissae corresponding to the for angles oflocate minimum reflectivity. All computations frustrated total reflection geometry with prism refractive index = 2.53, spacer index = 1.39, spacer thickness = d. O~is the critical angle for total reflection. The profile parameters which we have used for this and the next two figures are as follows: ao= ~ TS=O.9 Tb, 2ir~A 1=IA andbulk, 2ir~, Tb 15 the Drude relaxation time in the and 2A. the static bulk conductivity, Gob, is9.35X 10’5 sec”.
Vol. 14, No. ii
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY
Typical results are illustrated in Fig. 1, which shows that the surface plasmon at a diffuse interface should behave quite differently from the NRSP at a perfectly sharp boundary with effective optical constants as determined by ellipsometry or reflectivity measurements. The ellipsometric curve of Fig. 1 is based upon the measurements of Faber and Smith,’° which extended to a maximum energy of about 3 eV. To evaluate ‘ellipsometric’ sharp surface reflectivities at at higher energies, it was first necessary to compute a set of apparent optical constants characterizing the Hg surface. The constants calculated in this manner represent the effective values of Re (Em) and Im (Em) that would be measured in an ellipsometric experiment for E> 3 eV and 78°incidence if the skewed profile were completely accurate.
The NRSP dispersion relation may be determined over a range of energies in the optical region by measuring the angle of minimum reflectivity, conesponding to maximum coupling to the excitation, from several sets of calculations. The effect of the surface zone, most noticeable at high energies, has been evaluated from 3.5 eV to 5.5 eV (where interband effects start to become important),’6 and is shown in Fig. 3. We note, in particular, the shift of the dispersion relation towards the radiative regime (left of the light line). This finding is consistent with the higher effective electron density in the transition region, and also agrees with a simpler metal film-metal subsrate model in which the film plasma frequency is somewhat greater than that of the substrate.17
_______________________________________ I I I I
28°
32°
I
—~-
-
~::____
~°~:urf0Ce
-
I
8.0
E’400eV, d~IO5O1
OC
1099
36°
400
1
440
50-
3.0 I
3.5
4.0
4.5
5.0
I0”5k, [cm-Il FIG. 2. Reflectivity calculations for three models of a liquid Hg surface: diffuse, with skewed Epstein profile; sharp surface which is ellipsometrically equivalent to the profile; sharp surface with Drude optical constants. FIR parameters as in Fig. 1. The calculated optical response of Hg to NRSP excitation at 400 eV is shown in Fig. 2 for the full profile, a zero thickness surface free-electron model, and a zero thickness surface with optical parameters ‘equivalent’ to the profile as discussed above. Surface plasmon absorption should be measurably different for the three cases. The difference between the lower two curves demonstrates convincingly that the shift observed in Fig. 1 is due structure the transition region, andtois the notsurface an artifact of theof slight deviation in the Bloch and Rice fit to the ellipsometric data. It also underscores the enhanced sensitivity of NRSP absorption to surface properties in relation to the sensitivity of classical optical methods.
FIG. 3. Calculated dispersion relations for surface plasmons at a liquid Hg-dielectric interface, for FTR geometry. Dashed curve is dispersion relation for full inhomogeneous profile; dot-dashed line, for an ellipsometrically equivalent sharp surface. Absorption at a Drude sharp surface is heavily damped at these energies, so the resulting dispersion is not shown. Solid curve represents the light line w = ck~/n 3in the dielectric spacer layer.
In conclusion, it is to be noted that many of the investigations of surface plasmons at imperfect surfaces have concentrated on the high energy (E ~ hw8~),high momentum (hk ~ 8hc~~/c) region, and RPA’9 for in this range the hydrodynamic’ treatments of surface plasmons yield to solution after retardation is neglected or high-frequency expansions are made for the response functions of the electron gas. As yet, however, there has been no satisfactory treatment of the nature of surface plasma waves in the
1100
SURFACE PLASMONS AT THE SURFACE OF LIQUID MERCURY
low energy region, at a planar metal surface with a nonuniform transition zone. It might be objected that the surface plasmon at these energies should be insensitive to the detials of surface structure because the mode is primarily photon-like, its dispersion curve asymptotic to the light line of the contact medium as w 0. But for the case of liquid mercury, we find that there is an intermediate energy range, between E 4 hw9 and E h~.,5 within which the NRSP -~
~
,
p . . dispersion appears sensitive to nonuniformities in the electronic surface structure. Additional results for liquid Hg will appear in a later paper. -
.
-
Vol. 14, No. 11
Acknowledgements — This research has been supported by a grant (S.A.R.) from the National Science Foundation, H.L.L. and S.A.R. have also benefitted ~ ~~i~~ion Foundation. RH.N. and A.N.B. acknowledge the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. We acknowledge helpful conversations with Mr. Daniel Guidotti in the execution of this work.
REFERENCES
1.
KLIEWER K.L. and FUCHS R.,Phys. Rev. 153. 498 (1967).
2.
ECONOMOU EN.,Phys. Rev. 182, 539 (1969).
3. 4.
FUCHS R. and KLIEWERK.L.,Phys. Rev. 83, 2270(1971). ENDRIZ J.G. and SPICER WE.,Phys. Rev. 84,4144(1971).
5.
STANDORD J.L.,J. Opt. Soc. Am. 60,49 (1970).
6.
RAETHER H., in The Structure and Chemistri’ ofSolid Surfaces, p. 10—1 (edited by SOMORJAI G.A.) John Wiley, New York, (1969). BARDEEN J., Phys. Rev. 49, 653 (1936). FORSTMANN F. and PENDRY J.B.,Z. Phvs. 235, 75(1970).
7. 8. 9. 10.
BLOCH A.N. and RICE S.A., Phys. Rev. 185,933(1969). FABER T.E. and SMITH NV.,.!. Opt. Soc. Am. 58, 102 (1968).
11.
HODGSON J.N.,Phil. Mag. 4, 189(1959).
12.
CROZIER E.D. and MURPHY E.,Can. .1. Phys. 50, 1914(1972).
13.
LANG N.D.arid KOHN W.,Phys. Rev. B1,4555 (1970).
14.
OTTO A., Z. Phys. 216, 398 (1968); BARKER A.S., Jr., Phys. Rev. Lett. 28, 892 (1972).
15. 16.
HERPIN A.,Compt. Rend. 225, 182(1947). WILSON E.G. and RICE S.A.,Phys. Rev. 145,55(1966).
17.
GUIDOTTI D., RICE S.A. and LEMBERG H.L.,Phys. Rev. Lett., to be published.
18.
BENNETT A.J.,Phys. Rev. BI, 203 (1970).
19.
FEIBELMAN P.J., Phys. Rev. Lett. 30, 975 (1973). ~lbiOUCHII.1I4 o~uowemie ~1i1C[]~~CI1I1rio~epXHOCTb - n,1a3M0H np~i HU.iI1’41111 3OHbI Heo~i11opO1iHOi~noBepxHOCTlI, KoTopoe ObI,1O iipei.ioseIIO J.IB il~i1~KOIlpA~yTu. llpeirlo.la.laB, ~TO cB~33b 4X~TOHB C n.~a3MoHoM UpOM3XO.111T 3U (.q~- IIapyueHHOFO flO.1HOFO BI1~T~CHHOCOoTpaaeHMH, Nibi OTK~bLi1I ~1yBCTB1ITe.ThHOCTh oTHoae1~I5B mcnepcim }~C~U1CCTBOB~HI111T1-1TO aKOU :3oHbI. lIpJi}IIIMaB B CI-leT Hepa~n1auI1oHHyiOrIOBCpXHOCTh ~34)4)eF~Tbl 3amnL~bIBaHi1B. Mbl B~~BM pe3yJIhTaTb orlpeleiBloT rl.1a3MOH dOCOpOUII~O K~KHO1C3HV}O TCXHIIKY ]~1U Hc-c.eLI0BaHIIB flpOBOLIlllliIIX noBepxHocTeIl.