Surface Science 103 (1981) L104-L108 © North-Holland Publishing Company

SURFACE SCIENCE LETTERS

DISPERSION RELATIONS OF SURFACE PHONONS FROM ATOMIC SCATTERING

Ricardo AVILA * EEAP Departrnent, Case Western Reserve University, Cleveland, Ohio 44106, USA

and

Miguel LAGOS lnstituto de Fisica, Universidad Catolica de Chile, Cas. 114-D, Santiago, Chile

Received 13 August 1980; accepted for publication 16 September 1980

We show the existence of a spatial forbidden zone for one-phonon atomic scattering by sur- faces. In case of scattering by surface modes, this region is bounded by sharp observable max- ima of the scattering intensity, whose angular locations depend only on the dispersion relation of the modes involved. A method for obtaining the dispersion relation of surface modes needing no measurement of scattering energies then follows. The method is applied to the He-NaF sys- tem.

Atomic scattering off a crystal surface is certainly expected to contain a wealth of information about surface characteristics and on the a tom-surface interaction in general. Of special interest is the possibility of measuring the dispersion relation of surface excitations, a purpose for which the atomic scattering probe seems to be specially suited. We present here a method which makes use of sizeable inelastic intensity structures than can only be ascribed to the surface phonons contribution [1,2] to obtain their dispersion relation.

The method depends upon the conservation of energy and conservation of pseudo surface momentum which dgtermine'a boundary for the angular spread of the inelastic events. In the surface phonon contribution, this boundary is observable as the density of allowed inelastic processes is singular in this direction. This singu- larity arises from the vanishing of the Jacobian transforming from surface phonon coordinate to detector angular coordinates, a corresponding singularity of the den- sity of phonon states is not necessary [3].

We present next the details of this new method, showing how this singularity arises and showing that the bulk modes do not yield a corresponding singularity. In addition, the method is applied to the scattering of 4He from the (001) surface of

* Research supported in part by the NSF under Grant DMR79-01658.

L104

R. Avila, M. Lagos / Dispersion relations L105

NaF [2]. In agreement with these authors [1,2], we identify the observed intensity streaks as single surface phonon inelastic scattering, but our method of analysis does not depend upon assuming zero for the phonon wave vector component in the plane of incidence. Our results show, in fact, that this assumption is incorrect and we obtain a velocity of long wavelength Rayleigh waves in agreement with theory based on bulk elastic constants.

Conservation of surface pseudo-momentum and energy for single phonon pro- cesses are expressed in the equations

k = (X i + ~O + G, kz), (1)

(h2/2m) (k 2 - k~) = ~hco(q), (2a)

where k i = (Ki, kz) and k are the incident and outgoing wave-vectors of the atom of mass m, G is a reciprocal lattice vector, 6o(q) is the frequency of the phonon of wave-vector q = (Q, qz) and ~ = +1 for annihilation and creation respectively. Under the normal range of experimental conditions, k i ~,/3q with/3 = oJ(q)/qoi, where oi = hki/m, eq. 2(a) can then be approximated as

k ~ k i + ~/3q. (2b)

Eqs. (1) and (2b) have solutions only when certain inequalities are satisfied. To show this, eliminate Qx and Qy from the equations, express Kz and Ky in terms of k and the polar angles 0, ¢, express Kix + Gx and Kiy + Gy in terms of k i and the polar angles 0 a, Ca for the elastic G.diffraction beam, and choose the x-axis to lie in the incident plane. We then obtain the equation

(1 -/32 sin20) k = ki{1 -/32 sin 0 sin Oa cos ¢'

+/3[sin20 + sin206 _/~2 sin20 sin2Oa sin2¢, _ 2 sin 0 sin 0 a cos ~'

+ (1 -/32 sin20) (qz/ki)2] 1/2 } , O)

where ¢' = ¢ - eG. For real k and hence allowed scattering the discriminant A in (3) must be non-negative. For scattering by surface phonons with qz = 0, the dis- criminant condition Ao i> 0 establishes a region of allowed scattering direction (O, ¢) dependent upon fl, the equality determines the boundary ¢~(0,0 a,/3)

, 1 + [1 +/32~2 sin20 sin20G _ sin20 _ sin20G)] i/2 (4) cos ~F = /~2 sin 0 sin 0 G '

and scattering is allowed for I¢'1 ~ ¢~. The locus (0, ¢F) will be called the Frontier of the Allowed Zone (FAZ) for scattering by qz = 0 phonons. For ¢~ to be real we require # ) M a x (1/sin 0, 1/sin OG}. Otherwise Ao is never negative, the FAZ does not exist, and inelastic scattering is allowed all around the elastic peak.

As the probability of inelastic scattering is a slowly varying functicCn of q [4], we expect the FAZ to be experimentally observable as a sudden cutoff in the inelastic intensity. By solving Ao = 0 for /3F(0, ¢~, Oa), we can determine/3F from an ob-

L106 R. Avila, M. Lagos / Dispersion relations

served cutoff, and then qx, qy a r e determined along the FAZ from the conservation equations.

To show that the inelastic intensity from acoustic surface phonons is singular on the FAZ, we project the density of modes from surface wave-vector space to the space of the detector (0, ¢)). The projection equations as obtained from the conser- vation equations are:

tan q~ = k i sin 0 G sin q5 G + ~qy (5) k i sin 0 G cos q5 G + ~qa- '

sin 0 = ki~sin OG-c-os-fPG + ~qx . (6) k cos q5

The density of surface projections k(q) in the detector space is

Dks = d s [ l J */-1 + IJ -[-1] , (7)

where d s = LxLy/(27r) 2. J is the Jacobian J = 3(0, ~))/a(qx, qy) which from (5) and (6) is

k 2s in0 cos0 1 - q~-qx +/3 cos

( ° '>1} + q - - + i 3 sinO sin0 (8) aq),

where the -+ correspond to that of eq. (3). Under the condition that the Frontier of the Allowed Zone exists, J~ = 0 and

Dk s -+ oo on the FAZ provided the derivatives of/3 with respect to the wavevector can be neglected. This strong focussing of the scattered beam in this outgoing direc- tion should be experimentally observable as a sharp maximum of the inelastic inten- sity along the FAZ.

In fig. 1 we assume a linear dispersion relation to show a plot of the Frontier of the Allowed Zone for typical experimental parameters. This boundary should be observable as an intensity streak projecting away from the diffraction peak. The nature of the focussing at the FAZ is shown as an inset in this figure in which Dks from eqs. (7) and (8) is plotted for fixed 0.

Shown also in the inset in fig. 1 is the density of projections Dkt ~ from bulk modes. The significance of this plot is that the bulk modes do not produce streaks and that observed streaks can then be attributed to surfaces modes only. It is also observed in this figure that, since ~ for the bulk modes is larger than that for the surface modes, the region of allowed scattering from the bulk modes lies within that for the surface modes, no bulk contribution being superimposed on the FAZ of the surface contribution.

In a separate work [5] we will consider a realistic (nonlinear) dispersion relation showing that the qualitative behavior of D ~ is unchanged and establishing the con-

R. Avila, M. Lagos /Dispersion relations L107

q y [ A - ' ] \

x \ ° ~, ,

o a / ' e o ~ e + aN

S° °", ( ) q × [ i - ' ] /

• t . 0

41°~ ~ 7 4~'-I ~ " , , - - FAZ

2 i t " 48" ~ ~, ~ i t i i

Qj [t0~3 ~.~ ] rad .~OJ R <I00>

<~0> % /

N o F ( O 0 t ) + ; / G" z~ (00)

~ . ~ a~ / o (~-)Ann • (~'f) Creahon + (07) Ann

ql~a o (07) Creation

.¥ (b) /

/ /

I 5

q[~-'] I ~,~

~.0

Fig. 1. The Allowed Zone (shaded area) for inelastic scattering by long-wave surface modes near diffraction peak G. The inset shows a plot of Dks and DkB (arbitrary units) versus <~' for fixed 0 along the dashed line. The dispersion relation is assumed linear with ~s urf= 2.0, flBulk = 2.2. The elastic peak is at 0 G = 45 ° .

Fig. 2. Calculated surface phonon dispersion surfaces for NaF (001) using experimental data from ref. [31. Wave vectors q are shown in (a), and (b) shows the corresponding frequencies. Dotted lines in (a) stand for the First Brillouin Zone boundary and the (110> direction and in (b) for the Rayleigh linear dispersion relation in the (100) and ( 110> directions (the difference is too small to be shown for NaF). The key to the inelastic streaks in Williams' data is indicated in (b).

nection between the present work and Benedek's [3] in plane study. We apply these results to the experimental data on the scattering of a 4He beam

by a (001) surface of NaF [2] in which streaks are observed extending in particular directions outward from the Bragg angle. The positions of these streaks are shown

in fig. 15 of ref. [2]. Identifying these streaks as singularities of the surface phonon contribution, we calculate the dispersion curve of surface modes in a NaF(001) sur-

face with the results shown in fig. 2. We obtain 6o = Vsq with vs = (3.4-+ 0 .2 )× 10 s cm/s, in good agreement with the long wavelength region of Rayleigh modes [6] (dashed line in fig. 2b), calculated from bulk elastic constants [7] and from

L108 R. Avila, M. Lagos /Dispersion relations

shell model calculations of Chen et al. [8]. A small upward deviation of our result can be expected since we have made no a t tempts to incorporate the nonmonochro- matici ty of the incident beam and finite detector size in our calculations. Similar analysis of 4He/LiF (001) [1] data yields v s = (4.6 + 0 . 6 ) × l0 s cm/s, in similar agreement with theory.

The authors wish to thank the Universidad Catolica de Chile for the hospitali ty enjoyed there during the main part of this work. We are also indebted to Dr. B.R. Williams for providing us with his experimental data and to Professor Eric Thomp- son for discussions on the subject and restyling of the manuscript.

References

[1] B.R. Williams, J. Chem. Phys. 55 (1971) 3220. [2] B.F. Mason and B.R. Williams, J. Chem. Phys. 61 (1974) 2765. [3] G. Benedek, Phys. Rev. Letters 35 (1975) 234. [4] M. Lagos and L. Birstein, Surface Sci. 51 (1975) 469. [5] Contr. to the Intern. Conf. on Ordering in Two Dimensions, Lake Geneva, Wisconsin, May

28-30, 1980. [6] D.N. Gazis, R. Herman and R.F. WaUis, Phys. Rev. 119 (1960) 533. [7] American Institute of Physics Handbook (McGraw-Hill, New York, 1963) pp. 2-52. [8] T.S. Chen, F.W. de Wette and G.P. AUdredge, Phys. Rev. B15 (1977) 1167.