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Electron diffraction at crystal surfaces
SURFACE
SCIENCE 11 (1968) 479-493 0 North-Holland
ELECTRON
DIRECTION
Publishing
AT CRYSTAL
Co., Amsterdam
SURFACES
I. GENERALIZATION OF DARWIN’S DYNAMICAL THEORY E. G. McRAE Bell Telephone Luboratories, Incorpornted, Murray Hill, New Jersey 07974, U.S.A. Received 16 January 1968; revised manuscript
received 2 April 1968
The dynamical diffraction theory put forward originally by C. G. Darwin in his treatment of X-ray reflection is developed in a more general form suitable for the treatment of electron diffraction at crystal surfaces. The theory applies to a model crystal consisting of two parts: a selvedge with twodimensional periodicity but otherwise arbitrary structure, and a semi-infinite substrate composed of identical and equally-spaced atom layers. In the treatment of this model, the wave field in a region of locally-constant scattering potential between neighboring layers is expressed as a superposition of all possible diffracted (propagating) waves and an arbitrary number of evanescent waves. The diffraction problem is formulated in terms of “transfer” matrices involving the scattering properties of individual atom layers parallel to the crystal surface. It is shown that the diffraction problem and the corresponding band-structure problem for the substrate may be reduced to a matrix eigenvalue problem involving the transfer matrix for a substrate layer. The diffraction amplitudes are expressed in terms of the eigenvectors of this transfer matrix.
In the last few years, the structure and chemical reactivity of crystal surfaces have been studied extensively by electron diffraction, including both low-energy electron diffraction (LEED)l) and the diffraction of high-energy electrons at nearly grazing incidence Z-4). To deal with the diffraction intensities observed in these experiments, it is necessary to use a dynamical theory that is valid in the case of strong attenuation of the electron wave field as a function of depth in the crystal. It has been shown recently in the case of LEED that multiple-~attering5-7) methods can be usefully applied to this type of problems). A complete multiple-scattering treatment of electron diffraction at crystal surfaces may be naturally divided into two parts: (a) calculation of the scattering by a single atom layer parallel to the crystal surface, the scattering properties of the individual atoms being given; (b) calculation of the diffraction by a semiinfinite sequence of atom layers, the scattering properties of the individual 479
480
E.G.
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layers being given. Some substantial progress has been made in Part (a) of the total multiple-scattering problem5,s), but Part (b) has received relatively little attention and is the subject of the present paper. The present treatment is a generalization of Darwin’sg) theory of diffraction, which was put forward over fifty years ago for the description of the reflection of X-rays by crystals. Darwin’s theory is a suitable starting point for the treatment of electron diffraction at crystal surfaces in that it deals explicitly with scattering by individual atom layers parallel to the surface of the crystal. One drawback of the theory as originally formulated is that it is only a two-beam theory - i.e., it is selfconsistent with respect to the primary beam and just one diffracted beam. The two-beam approximation is a useful one for X-ray diffraction, but in the diffraction of electrons at crystal surfaces the third Laue condition is not a sharp diffraction condition and so all possible diffracted beams have to be considered on an equal footing. Furthermore, the exact formulation of the multiple-scattering problem includes an infinite number of evanescent waves. At least a small number of such waves must be included for an accurate description of electron diffraction at crystal surfacess). The present treatment is a generalization of Darwin’s theory to include all possible diffracted beams (propagating waves) and an arbitrary number of evanescent waves. There is one other respect in which the present treatment is a generalization of Darwin’s theory. Darwin considered a model crystal composed of identical and uniformly-spaced atom layers. In the present paper we assume that this description applies to the bulk crystal, but we allow for the existence of a surface region - the selvedge - that is two-dimensionally periodic but otherwise of arbitrary structure and chemical composition. This extension is necessary in order to handle various topics of importance in surface science, such as the structures formed by the adsorption of gases on crystal surfaces. 2. Theory 2.1. MODEL
AND ASSUMPTIONS
We first introduce essentially the same set of assumptions as in the theories citeds-s), namely, (1) the primary wave field is a plane wave; (2) the crystal has perfect two-dimensional periodicity; (3) inelastic scattering processes may be represented as “absorption” of the elastic wave field; (4) the scattering potential has a single, constant value in a slablike reference region just above each atom layer; (5) scattering in the boundary region of the crystal, just above the topmost atom layer, may be neglected. To these assumptions, we add the specification that the crystal consist of two parts, namely a selvedge and a substrate, where the substrate structure is restricted as follows:
ELECTRON
(6) the substrate
consists
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AT CRYSTAL
of uniformly
spaced
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481
I
layers of identical
structure,
and the surface projections of the atom locations in any layer differ from those of the atom locations in the layer immediately below by a fixed translation q,,d/D, where q. is a net vector, d is the spacing of adjacent atom layers and D is the spacing of identical layers. The model crystal is shown schematically in fig. 1. LAYER NUMBER
SELVEDGE
SUBSTRATE
Fig. 1. Schematic section of the model crystal. Thick horizontal lines indicate atom layers in the selvedge. Filled circles indicate atoms in the substrate. The diagram is drawn for the case of qo in the plane of the paper and D = 2d.
2.2. GENERALIZED DARWIN THEORY Let us first consider the scattering of electrons by a single layer of atoms parallel to the crystal surface. We define a one-layer amplitude reflection coefficient P(K”K’) as follows (fig. 2). Let exp(iK’*R) denote the amplitude, at the point R, of a wave incident on an atom layer lying at depth z in the crystal. Then the amplitude of the wave diffracted by the layer, with propagation vector K”, is given by p (K”K’) exp {i [(K; - K;)
z
+
K” *RI},
where K; and K: are the projections of K’ and K”, respectively, on surface normal of the crystal. In the above definition, the surface of K’ is real but the normal projection K; may be either real or In the case of real KY we speak of a propagating wave, and in imaginary K; an evanescent wave. Similar remarks apply to the propagation vector K”. For conservation of electrons in scattering by a single atom
(1) the inward projection imaginary. the case of wave with layer,
it is
482
E.G.MCRAE
1 +
p(K’K’)
Fig. 2. Relationship between the amplitude of a plane wave incident on a single atom layer (thick horizontal line) and the amplitudes of the reflected and transmitted waves. The amplitude of the transmitted wave is the sum of the amplitudes of the incident and forward-scattered waves. In general there can be several reflected waves, but only one is shown. Phase factors are omitted in the expressions for the amplitudes; the complete expressions are given in the text.
necessary
that the inequality
7121
Id(K”, K’) +
L
K"
p(K”K’)IZ d 1
)
IAll
be satisfied, where K' is any real propagation vector and the summation extends over real K".The inequality sign allows for inelastic scattering. If we represent the probability of inelastic scattering by s(K')and put the left side of eq. (2) equal to 1-s(K’), we get - 2 Rep(K’K’)
= s(K')+
c
K”
where again the summation
extends
I-$ Ip(K"K')12, I
over real K".Eq. (3) is the counterpart
of the optical theorem for atomic scatteringlo). To examine the connection with the optical theorem, we take the pseudo-kinematical expression for the amplitude reflection coefficient, namelyri),
~(K"K')= il~,f(c~~) I in the case of one atom per unit mesh-layer (fis the atomic scattering factor, A the area per unit mesh) and then go to the limit of infinite atom separation. In this way we recover the usual form of the optical theorem provided s(K') is identified with the inelastic atomic scattering cross-section divided by the projection of the area per unit mesh in the direction of K'. In addition to the electron-conservation condition expressed by eq. (3),
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the one-layer
amplitude
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reflection
AT CRYSTAL
coefficients
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483
I
satisfy a symmetry
relation
deriving from assumption (4). This relation expresses the idea that the scattering of electrons from one side of an atom layer is equivalent to scattering from the other side. Let K’, 2’ denote two propagation vectors with the same magnitude and the same projection on the atom layer, but with oppositely-signed projections on the layer normal. Then the symmetry relation is p (K”K’) = p (Ix’).
(5)
We turn now to the treatment of diffraction by the whole crystal, treating this as the end result of a sequence of diffraction events in single atom layers. Let Kl denote the propagation vector of the primary wave in the region of locally-constant potential just above the topmost atom layer [see assumptions (1) and (4)]. Let K denote the modulus of Ki and let k, K,, denote the projections of K,f on the surface and inward surface normal, respectively. The components of Kl may be indicated briefly by writing Kz s(k, K,); a similar abbreviation is used elsewhere in this paper. It is shown in the Appendix that the total wave field (i.e., the wave function) at a point R in the region of locally-constant potential just above the ,uth atom layer may be expressed in the form II/(R) = c [a, (K:) ” where u is a vector b’s are independent K’
exp (iK: -R) f b, (KV) exp (iK; *R)]
,
(6)
of the reciprocal net of the crystal surface, the a’s and of R and the propagation vectors K* are defined by E (k + 27m, -&K,)
K, = I(K2 -
(7)
Ik + 27~uj~)~l (radical
= il(K2 - Ik + 27rr~1~)*1(radical
positive) negative).
(8)
The objective of the theory is to determine the total wave field in the region of locally-constant potential just above the topmost atom layer of the crystal. If this layer is numbered zero, the primary wave field is described by the statement a,(Kt>
= 0,
UfO.
(9)
The amplitude of the diffracted beam corresponding to reciprocal net vector u is given by b,(K,). The ratio of the intensity of this beam to that of the primary beam is I(K;K,+) According scattering
= ;
Jb, g;‘,l:. oao 0
to assumption (5), the beam intensities potential in the boundary region.
(10) are not affected
by the
E.G.
484
MCRAE
The wave amplitudes just above each atom layer in the crystal may be calculated by solving an equation representing self-consistency between the wave field incident on each atom layer (the effective field) and the wave field emitted by each layer. In order to obtain a physically meaningful solution, it is necessary to impose the condition that the total wave field is finite everywhere. We now derive the self-consistency equations and indicate a method of solution for the model crystal described under assumption (6). According to the proof given in the Appendix, we may express the effective field incident on the pth atom layer (p =O, 1,. . .) as V(R)
= 1 [a,(K:) ”
exp(iK:
*R) + b,+l (FL;) exp(i&
*R)],
(11)
where the a’s and b’s are constants having the same significance as in eq. (6). The summation is supposed to be carried over an arbitrary number, n, of reciprocal net points. Now each term in eq. (11) may be considered to contribute to the coefficient a,+,(K:) in the expression for @‘+l(R), and the individual contributions may be expressed in terms of the depth zP and the amplitude reflection coefficients pp for the pth layer, according to eq. (1). Similarly, each term in eq. (11) may be considered to contribute to the coefficient b,(K;) in the expression for $“-‘(R). In this way we obtain a set of difference equations that may be expressed in the following matrix form: a,+,
=Q:q,+Q:‘b,+,, (12)
b, = Qya, + Qyb,+l.
Here, a, is a column vector with n elements uy(K:), b, is a similar column vector with tz elements b,,(K,) and Q,!, Q’{, etc. are the following n-square matrices (matrices are denoted by a representative element in brackets): Q’, = [{6(&T,
C)
+ P,(CC))
ev{i(-
Qt’ = by WKJ
exp { - i K
Q1” = by (KK:)
exp {i (K, + K)
Q!,” = [{6 (K,
K)
Eq. (12) may be rearranged ap+l b P+l
or, with an obvious
+ J4
+ AK&J)
41 y
+ K)
G)I y
(13)
41,
ew G(K
-
~Jz,Il~
to read
1 [......, =
K
Qf!(Qy)-’ (Q:">-i.
Q: - Q:'(Q:")-'Q~' q(j;j1 @I
a,,
I[
b,
(14) 1
notation, c,+1=
Q,c,
.
(15)
We refer to Q, as the transfer matrix for the @h layer. In eq. (14), the matrices are symmetrically partitioned as indicated by the broken lines; thus, for
ELECTRON
example,
DIFFRACTION
the submatrices
AT CRYSTAL
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of Q are each n-square
for partitioned matrices are used elsewhere From eq. (15), we have
I
matrices.
485
Similar
notations
in this paper.
~,=~=$_~Qvco~b>Oo). This equation expresses above the topmost layer crystal. The generalized lower half of cO. The top example, if the primary taken to be exp(iKl - R),
(16)
the connection between the total of the crystal and the total wave Darwin problem is to solve eq. half a, of c,, describes the primary wave field just above the topmost this is expressed by a&)
wave field just field within the (16) for bO, the wave field. For atom layer is
= 6(K,+, K;).
(17)
We now describe the solution of eq. (16) for the model crystal specified by assumption (6) (fig. 1). We suppose that the selvedge consists of v layers, so the r]th layer is the topmost layer of the substrate. Also we put z,, =0 and p,,=p. In the. substrate, the vth atom layer differs from the r]th only by a translation qoz,/D, so we have (v 2 q) .
py (K”K’) = p (K”K’) exp [i (K’ - K”)*q,z,/D], If we put this into eq. (13) and then introduce
the abbreviations
= exp [i (K, + 2rcu .qo/D) d] ,
0:
13; = exp [i(K, e, (K:)
= (~:>‘-“a,
f”(KJ
= (O;)+%”
(19)
- 2rcv~q0/O) d], (K:),
(20)
(R;))
we get after rearrangement
I[ P+l
ep+l
[
f
where e, and tively, and
RI _
RII(RIV)-lRIII
p(/-p)-l
=
f,are column
vectors
R’ = [{6(C
+ P(CC)} o.+w,]
R”’ = [p (K;K:)] R’” = [{S (K;,
1
with 12elements
C)
R” = [p (K:K;)
I[ 1 eg
pIV)-
- w)-‘~“’
Eq. (21) may be written,
(18)
fp
’
(21) respec-
4,
)
+ P (K;K;)}
=
49 ’
(22)
with an obvious g,,,
2
e,(K:),f,(K,)
) K;)
(v
notation,
4%.
co;] .
as (23)
486
E.G. MCRAE
We refer to R as the reduced transfer matrix. Now we have g, = c,, according to eq. (20), so that by combining g, = RX,
eqs. (16) and (23) we get
(Y= 93
g,=R"-" fi Qvco, (P2YY?>O). v=q-1 This is the special form of the more general self-consistency equation [eq. (16)] that applies to the model specified by assumption (6). Before indicating a method of solution of eq. (24), we establish a special property of the reduced transfer matrix R, namely that its eigenvalues may be grouped in pairs such that the product of the members of each pair is of modulus unity. This property arises from the three-dimensional periodicity of the model substrate and the existence of a mirror symmetry plane parallel to the crystal surface [see assumption (6)]. The proof is as follows. In view of the three-dimensional periodicity of the substrate, the total wave field at any point R in the substrate can be expressed as a superposition of waves of the Bloch form @(K, R) = u(R) exp(irc*R), (25) where u(R) is a periodic function and the wavevector K has the same surface projection k as the primary propagation vector Kt . Let K, denote the inward surface normal projection of K, let R’ =(I-‘, z’) denote a point in a region of locally constant potential, let t denote a 2n-element vector whose elements are the coefficients of the expansion of @(rc, R’) in terms of the functions exp(iK2 * R’), and put p = D/d. Then the relation @(K, r’, 2’ + D) = @(fc, Y’, 2’) exp (i@), which follows from eq. (25), leads to the matrix eigenvalue RPt = exp (iK,D)
t.
(26) equation (27)
Eq. (27) is a statement of the band-structure problem for the substrate. The existence of a mirror plane parallel to the surface implies that for every Bloch state with wave vector K =(k, K,) there exists another one of the same -K,), i.e., for every eigenvalue exp(iic,D) energy with wave vector K’ =(k, of Rp there is another eigenvalue exp( - irc,D). Now the eigenvalues of R are pth roots of those of RP. It follows that for every eigenvalue i of R there is another eigenvalue I’ such that 12;1’1=l. This is the proof of the statement at the beginning of the paragraph. However, all that is necessary for the following development is that half the eigenvalues lie inside and half lie outside the unit circle in the complex plane (the case in which one or more pairs of eigenvalues lie on the unit circle is here considered as a limiting case which is actually realized only in the case of purely elastic scattering).
ELECTRON
We now proceed
DIFFRACTION
with the method
AT CRYSTAL
of solution
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487
of eq. (24). It is shown in
books on matrix algebraia) that any square matrix can be reduced by a similarity transformation to a normal form in which the eigenvalues appear on the principal diagonal and all elements below the principal diagonal are zero. Let T denote the matrix that transforms R to its normal form, and let T be so chosen as to place the eigenvalues of R that lie outside the unit circle on the right-hand end of the principle diagonal: T-'RT=h
,
(28)
(29)
I&2+I)(“+l)l~
l&“+2)(“+2)1, . ..v ILZnl ’ 1 *
(30)
We may now write eq. (24) as T-'g, =A'-"T-k,,
(pa q).
(31)
In the matrix A@-“, the diagonal elements are A:;“, A:;“, . . ., ,I;,& and all elements below the principal diagonal are zero. Keeping in mind the stated property that half the eigenvalues of R are of modulus greater than unity, we find that the condition that the total wave field be finite everywhere can only be satisfied if the coefficient of A&‘!! in eq. (31) vanishes. Similarly, it is necessary that the coefficient of IfZJ,‘!_l,,Czn- 1J vanishes, and so on up to y-” The final result may be expressed conveniently in terms of the (n+l)(n+l)* n-square submatrices of T- '; using the notation indicated by (32) we have b, = - i,-,li,,a,.
(33)
Let us put X=
fi v=?l-l
Q,
(34)
and partition X in the same way as the other square matrices. eqs. (16) and (33) we get b, = -
(X22+ i2;1?21X,2)-1 (X,,+ f;;T,,XJ
This is the formal solution treated in this paper.
of the generalized
Darwin
problem
On combining
a,.
(35)
for the model
488
E. G. MCRAE
3. Discussion The main result of this paper is that the problem of calculating diffraction intensities according to the “layer-by-layer” scheme first introduced by Darwin is essentially equivalent to a matrix eigenvalue problem. The present generalization of Darwin’s theory involves the replacement of a 2 x 2 matrix eigenvalue problem by one of arbitrarily larger dimension. The presence of a crystal selvedge of arbitrary structure can be handled by multiplying together a sequence of matrices, one for each layer in the selvedge. Thus there is no special difficulty in treating surface layers that differ in structure, spacing and chemical composition from atom layers in the bulk crystal. The input to a numerical calculation of diffraction intensities along the lines indicated in this paper comprises a set of layer spacings and a set of one-layer amplitude reflection coefficients. In the case of high-energy electron diffraction, it is possible that a pseudo-kinematical calculation of one-layer amplitude reflection coefficients (i.e., a calculation neglecting inter-atomic multiple scattering) might be useful, but this possibility needs further study. In the case of low-energy electron diffraction, a pseudo-kinematical treatment is likely to give inaccurate or even absurd results. The reason for this is that low-energy electrons are sufficiently strongly scattered atoms by that inter-atomic multiple scattering makes a very important contribution to the effective wave field incident on each atoms). To handle the diffraction of low-energy electrons by a layer of atoms, it is necessary to use a fully selfconsistent method such as that proposed recently by Kambec). In discussing the effects of the assumptions listed in section 2.1, we note first that the limitation of the treatment to substrates consisting of identical and uniformly-spaced atom layers is not unduly restrictive; fee, bee, hcp, NaCl and graphite structures, for example, are included in this description. Other crystal structures, such as that of ZnO, could be treated by a simple extension of the scheme given here. The model requirement that the scattering potential have a single, constant value between each neighboring pair of atom layers - and just above the topmost layer - is an unrealistic one for some systems, such as a crystal covered with a layer of foreign atoms. It would probably not be very difficult to modify the theory to allow for different scattering potentials between different pairs of atom layers, and this would be a desirable refinement for quantitative applications. For the treatment of diffraction by atomically clean crystals of ideal structure, the present treatment does not appear to offer any special advantage over the standard diffraction theory as first formulated by Bethei4). If a selvedge is present, however, the “layer-by-layer” approach is definitely the
ELECTRON
more suitable
DIFFRACTION
AT CRYSTAL
one in that it makes possible
SURFACES.
a treatment
489
I
of scattering
in the
substrate and selvedge from the same view point - i.e., in terms of one-layer scattering properties. Since the Darwin type of diffraction theory is essentially a matrix eigenvalue problem, it should be possible to use standard approximation methods for such problems to obtain a simplified description of diffraction intensities. The use of perturbation theory in this connection will be described in Part II.
Acknowledgment The writer is pleased to thank G. H. Wannier and J. J. Lander for drawing his attention to the Darwin method for diffraction problems, and V. Heine, P. J. Jennings and H. Scher for many helpful discussions.
Appendix It is required to show that the total field and effective field are given by eqs. (6) and (ll), respectively. TotalJield: The total field at any point R in a crystal may be expressed in general as 15) $ (R) = exp (5,’
exp(iK
*R) cs R,
IR - R’I) ---- T(R,, R’) tiefl (R,, R’) dR’ , (A. 1) 47-cIR - R’I
where R, denotes the position vector of the sth atom in the crystal, T(R,, R’) denotes the transition operator for the sth atom and I/,~(R,, R’) denotes the effective field incident on the sth atom. The other notation in eq. (A.l) is as in the text. Let q denote a vector of the crystal surface net. Then we have
T(Rs+ 4, R’ + q) = T(R,,R’), J~~~(R, + 4, R’ + q) = tieE(R,,R’) exp(ik*q).
64.2) (A-3)
Upon interchanging the order of summation and integration in eq. (A.l), using eqs. (A.2) and (A.3) and transforming the integrand by means of the two-dimensional analog of Ewald’s theta-function transformation5)
c
exp [iK(lq + rlz + z2)* - ik*q] (14 + t-1’ + z’)+
4
with R=(r,
z), we get
i27c = 7
exp [i(K” Iz/ - k*r)] c ”
K
,
(A-4)
490
E.G.
+(R)=exp(iK,i.R)--iALGmC’
x ”
exp [Xi
X
MCRAE
”
RS
*(R - R’)] T(R,R’) Gefi (R,R’) dR’ +
[f
r’
+
zI>z exp [iK;
*(R - R’)] T(R,R’) t,ber(R,R’) dR’
s
1 ,
(A3
where c’ denotes summation over one unit mesh and R’-(Y’, z’). For a point R lying in a region of locally-constant potential, the integrals in eq. (A.5) are independent of R and eq. (A.5) reduces to an expression of the form given by eq. (6)9. EffectiveJield: Let $&,(R) denote the wave field emitted by the ,uth atom layer of a crystal. The effective field t,P(R) incident on the layer is defined as the total field minus the emitted fieldl5): V’(R) = II/(R) - K,,
(R).
G4.6)
At a point R in the region of locally-constant potential layer, the wave field emitted by the layer is of the form C,, (R) = c P, (Rb) exp (K ”
.R) 2
where the p’s are constants. This may be proved, eq. (A.5) to a single layer. Now we have obviously b,+, CR;) = WC) and the combination
just above the ,&h
- P/JR,)>
e.g., by application
(4.7) of
(A.81
of this result with eqs. (A.6) and (6) leads to eq. (11). References
1) J. J. Lander, Progr. Solid State Chem. 2 (1965) 26. 2) 3) 4) 5) 6) 7) 8)
9. 10) 11) 12)
G. W. Simmons, D. F. Mitchell and K. R. Lawless, Surface Sci. 8 (1967) 130. E. Menzel and 0. Schott, Surface Sci. 8 (1967) 217. B. M. Siegel and J. F. Menedue, Surface Sci. 8 (1967) 206. E. G. McRae, J. Chem. Phys. 45 (1966) 3467. K. Kambe, 2. Naturforsch. 22a (1967) 322. M. Macrakis, unpublished. Other methods of treatment include the mixed representation and band-model methods. See K. Hirabayashi and Y. Takeishi, Surface Sci. 4 (1966) 150; P. M. Marcus and D. W. Jepson, Phys. Rev. Letters 20 (1968) 925; D. S. Boudreaux and V. Heine, Surface Sci. 8 (1967) 426; F. Hofman and H. P. Smith, Phys. Rev. Letters 19 (1967) 1472. C. G. Darwin, Phil. Mag. 27 (1914) 675. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961) p. 499. P. W. Palmberg and W. Peria, Surface Sci. 6 (1967) 57. F. E. Hohn, Elementary Matrix Algebra (Macmillan, New York, 1958).
ELECTRON
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AT CRYSTAL
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491
13) The ordering expressed by eq. (22) automatically applies to the diagonal elements of R in the limit of weak elastic scattering, since the expression 1 +p(K’K’) is to be interpreted as a one-layer amplitude transmission coefficient and so is necessarily of modulus less than unity. 14) H. A. Bethe, Ann. Physik [4] 87 (1928) 55. 15) M. Lax, Rev. Mod. Phys. 23 (1951) 287; Phys. Rev. 85 (1952) 621. 16) The result for the effective field was stated in ref. 5 but the derivation given there is strictly correct only for s-wave scatterers. The error arises from the fact that the partial wave expansion coefficients of the “effective” atomic scattering factor are not independent of the space coordinates, as was implicitly assumed in the derivation. The author thanks Jar-MO Chen for pointing this out.
Note added in proof. Since this paper was submitted for publication, the writer learned that K. Kambe had independently derived the principal properties of the reduced transfer matrix (called the “scattering matrix” by Kambe). Kambe’s theory, which is more general than the present one in some important respects, will be published in Z. Naturforsch. under the title “Theory of Low-Energy Electron Diffraction. II. Cellular Method for Complex Monolayers and Multilayers”. The writer thanks K. Kambe for a helpful correspondence.
SCIENCE 11 (1968) 479-493 0 North-Holland
ELECTRON
DIRECTION
Publishing
AT CRYSTAL
Co., Amsterdam
SURFACES
I. GENERALIZATION OF DARWIN’S DYNAMICAL THEORY E. G. McRAE Bell Telephone Luboratories, Incorpornted, Murray Hill, New Jersey 07974, U.S.A. Received 16 January 1968; revised manuscript
received 2 April 1968
The dynamical diffraction theory put forward originally by C. G. Darwin in his treatment of X-ray reflection is developed in a more general form suitable for the treatment of electron diffraction at crystal surfaces. The theory applies to a model crystal consisting of two parts: a selvedge with twodimensional periodicity but otherwise arbitrary structure, and a semi-infinite substrate composed of identical and equally-spaced atom layers. In the treatment of this model, the wave field in a region of locally-constant scattering potential between neighboring layers is expressed as a superposition of all possible diffracted (propagating) waves and an arbitrary number of evanescent waves. The diffraction problem is formulated in terms of “transfer” matrices involving the scattering properties of individual atom layers parallel to the crystal surface. It is shown that the diffraction problem and the corresponding band-structure problem for the substrate may be reduced to a matrix eigenvalue problem involving the transfer matrix for a substrate layer. The diffraction amplitudes are expressed in terms of the eigenvectors of this transfer matrix.
In the last few years, the structure and chemical reactivity of crystal surfaces have been studied extensively by electron diffraction, including both low-energy electron diffraction (LEED)l) and the diffraction of high-energy electrons at nearly grazing incidence Z-4). To deal with the diffraction intensities observed in these experiments, it is necessary to use a dynamical theory that is valid in the case of strong attenuation of the electron wave field as a function of depth in the crystal. It has been shown recently in the case of LEED that multiple-~attering5-7) methods can be usefully applied to this type of problems). A complete multiple-scattering treatment of electron diffraction at crystal surfaces may be naturally divided into two parts: (a) calculation of the scattering by a single atom layer parallel to the crystal surface, the scattering properties of the individual atoms being given; (b) calculation of the diffraction by a semiinfinite sequence of atom layers, the scattering properties of the individual 479
480
E.G.
MCRAE
layers being given. Some substantial progress has been made in Part (a) of the total multiple-scattering problem5,s), but Part (b) has received relatively little attention and is the subject of the present paper. The present treatment is a generalization of Darwin’sg) theory of diffraction, which was put forward over fifty years ago for the description of the reflection of X-rays by crystals. Darwin’s theory is a suitable starting point for the treatment of electron diffraction at crystal surfaces in that it deals explicitly with scattering by individual atom layers parallel to the surface of the crystal. One drawback of the theory as originally formulated is that it is only a two-beam theory - i.e., it is selfconsistent with respect to the primary beam and just one diffracted beam. The two-beam approximation is a useful one for X-ray diffraction, but in the diffraction of electrons at crystal surfaces the third Laue condition is not a sharp diffraction condition and so all possible diffracted beams have to be considered on an equal footing. Furthermore, the exact formulation of the multiple-scattering problem includes an infinite number of evanescent waves. At least a small number of such waves must be included for an accurate description of electron diffraction at crystal surfacess). The present treatment is a generalization of Darwin’s theory to include all possible diffracted beams (propagating waves) and an arbitrary number of evanescent waves. There is one other respect in which the present treatment is a generalization of Darwin’s theory. Darwin considered a model crystal composed of identical and uniformly-spaced atom layers. In the present paper we assume that this description applies to the bulk crystal, but we allow for the existence of a surface region - the selvedge - that is two-dimensionally periodic but otherwise of arbitrary structure and chemical composition. This extension is necessary in order to handle various topics of importance in surface science, such as the structures formed by the adsorption of gases on crystal surfaces. 2. Theory 2.1. MODEL
AND ASSUMPTIONS
We first introduce essentially the same set of assumptions as in the theories citeds-s), namely, (1) the primary wave field is a plane wave; (2) the crystal has perfect two-dimensional periodicity; (3) inelastic scattering processes may be represented as “absorption” of the elastic wave field; (4) the scattering potential has a single, constant value in a slablike reference region just above each atom layer; (5) scattering in the boundary region of the crystal, just above the topmost atom layer, may be neglected. To these assumptions, we add the specification that the crystal consist of two parts, namely a selvedge and a substrate, where the substrate structure is restricted as follows:
ELECTRON
(6) the substrate
consists
DIFFRACTION
AT CRYSTAL
of uniformly
spaced
SURFACES.
481
I
layers of identical
structure,
and the surface projections of the atom locations in any layer differ from those of the atom locations in the layer immediately below by a fixed translation q,,d/D, where q. is a net vector, d is the spacing of adjacent atom layers and D is the spacing of identical layers. The model crystal is shown schematically in fig. 1. LAYER NUMBER
SELVEDGE
SUBSTRATE
Fig. 1. Schematic section of the model crystal. Thick horizontal lines indicate atom layers in the selvedge. Filled circles indicate atoms in the substrate. The diagram is drawn for the case of qo in the plane of the paper and D = 2d.
2.2. GENERALIZED DARWIN THEORY Let us first consider the scattering of electrons by a single layer of atoms parallel to the crystal surface. We define a one-layer amplitude reflection coefficient P(K”K’) as follows (fig. 2). Let exp(iK’*R) denote the amplitude, at the point R, of a wave incident on an atom layer lying at depth z in the crystal. Then the amplitude of the wave diffracted by the layer, with propagation vector K”, is given by p (K”K’) exp {i [(K; - K;)
z
+
K” *RI},
where K; and K: are the projections of K’ and K”, respectively, on surface normal of the crystal. In the above definition, the surface of K’ is real but the normal projection K; may be either real or In the case of real KY we speak of a propagating wave, and in imaginary K; an evanescent wave. Similar remarks apply to the propagation vector K”. For conservation of electrons in scattering by a single atom
(1) the inward projection imaginary. the case of wave with layer,
it is
482
E.G.MCRAE
1 +
p(K’K’)
Fig. 2. Relationship between the amplitude of a plane wave incident on a single atom layer (thick horizontal line) and the amplitudes of the reflected and transmitted waves. The amplitude of the transmitted wave is the sum of the amplitudes of the incident and forward-scattered waves. In general there can be several reflected waves, but only one is shown. Phase factors are omitted in the expressions for the amplitudes; the complete expressions are given in the text.
necessary
that the inequality
7121
Id(K”, K’) +
L
K"
p(K”K’)IZ d 1
)
IAll
be satisfied, where K' is any real propagation vector and the summation extends over real K".The inequality sign allows for inelastic scattering. If we represent the probability of inelastic scattering by s(K')and put the left side of eq. (2) equal to 1-s(K’), we get - 2 Rep(K’K’)
= s(K')+
c
K”
where again the summation
extends
I-$ Ip(K"K')12, I
over real K".Eq. (3) is the counterpart
of the optical theorem for atomic scatteringlo). To examine the connection with the optical theorem, we take the pseudo-kinematical expression for the amplitude reflection coefficient, namelyri),
~(K"K')= il~,f(c~~) I in the case of one atom per unit mesh-layer (fis the atomic scattering factor, A the area per unit mesh) and then go to the limit of infinite atom separation. In this way we recover the usual form of the optical theorem provided s(K') is identified with the inelastic atomic scattering cross-section divided by the projection of the area per unit mesh in the direction of K'. In addition to the electron-conservation condition expressed by eq. (3),
ELECTRON
the one-layer
amplitude
DIFFRACTION
reflection
AT CRYSTAL
coefficients
SURFACES.
483
I
satisfy a symmetry
relation
deriving from assumption (4). This relation expresses the idea that the scattering of electrons from one side of an atom layer is equivalent to scattering from the other side. Let K’, 2’ denote two propagation vectors with the same magnitude and the same projection on the atom layer, but with oppositely-signed projections on the layer normal. Then the symmetry relation is p (K”K’) = p (Ix’).
(5)
We turn now to the treatment of diffraction by the whole crystal, treating this as the end result of a sequence of diffraction events in single atom layers. Let Kl denote the propagation vector of the primary wave in the region of locally-constant potential just above the topmost atom layer [see assumptions (1) and (4)]. Let K denote the modulus of Ki and let k, K,, denote the projections of K,f on the surface and inward surface normal, respectively. The components of Kl may be indicated briefly by writing Kz s(k, K,); a similar abbreviation is used elsewhere in this paper. It is shown in the Appendix that the total wave field (i.e., the wave function) at a point R in the region of locally-constant potential just above the ,uth atom layer may be expressed in the form II/(R) = c [a, (K:) ” where u is a vector b’s are independent K’
exp (iK: -R) f b, (KV) exp (iK; *R)]
,
(6)
of the reciprocal net of the crystal surface, the a’s and of R and the propagation vectors K* are defined by E (k + 27m, -&K,)
K, = I(K2 -
(7)
Ik + 27~uj~)~l (radical
= il(K2 - Ik + 27rr~1~)*1(radical
positive) negative).
(8)
The objective of the theory is to determine the total wave field in the region of locally-constant potential just above the topmost atom layer of the crystal. If this layer is numbered zero, the primary wave field is described by the statement a,(Kt>
= 0,
UfO.
(9)
The amplitude of the diffracted beam corresponding to reciprocal net vector u is given by b,(K,). The ratio of the intensity of this beam to that of the primary beam is I(K;K,+) According scattering
= ;
Jb, g;‘,l:. oao 0
to assumption (5), the beam intensities potential in the boundary region.
(10) are not affected
by the
E.G.
484
MCRAE
The wave amplitudes just above each atom layer in the crystal may be calculated by solving an equation representing self-consistency between the wave field incident on each atom layer (the effective field) and the wave field emitted by each layer. In order to obtain a physically meaningful solution, it is necessary to impose the condition that the total wave field is finite everywhere. We now derive the self-consistency equations and indicate a method of solution for the model crystal described under assumption (6). According to the proof given in the Appendix, we may express the effective field incident on the pth atom layer (p =O, 1,. . .) as V(R)
= 1 [a,(K:) ”
exp(iK:
*R) + b,+l (FL;) exp(i&
*R)],
(11)
where the a’s and b’s are constants having the same significance as in eq. (6). The summation is supposed to be carried over an arbitrary number, n, of reciprocal net points. Now each term in eq. (11) may be considered to contribute to the coefficient a,+,(K:) in the expression for @‘+l(R), and the individual contributions may be expressed in terms of the depth zP and the amplitude reflection coefficients pp for the pth layer, according to eq. (1). Similarly, each term in eq. (11) may be considered to contribute to the coefficient b,(K;) in the expression for $“-‘(R). In this way we obtain a set of difference equations that may be expressed in the following matrix form: a,+,
=Q:q,+Q:‘b,+,, (12)
b, = Qya, + Qyb,+l.
Here, a, is a column vector with n elements uy(K:), b, is a similar column vector with tz elements b,,(K,) and Q,!, Q’{, etc. are the following n-square matrices (matrices are denoted by a representative element in brackets): Q’, = [{6(&T,
C)
+ P,(CC))
ev{i(-
Qt’ = by WKJ
exp { - i K
Q1” = by (KK:)
exp {i (K, + K)
Q!,” = [{6 (K,
K)
Eq. (12) may be rearranged ap+l b P+l
or, with an obvious
+ J4
+ AK&J)
41 y
+ K)
G)I y
(13)
41,
ew G(K
-
~Jz,Il~
to read
1 [......, =
K
Qf!(Qy)-’ (Q:">-i.
Q: - Q:'(Q:")-'Q~' q(j;j1 @I
a,,
I[
b,
(14) 1
notation, c,+1=
Q,c,
.
(15)
We refer to Q, as the transfer matrix for the @h layer. In eq. (14), the matrices are symmetrically partitioned as indicated by the broken lines; thus, for
ELECTRON
example,
DIFFRACTION
the submatrices
AT CRYSTAL
SURFACES.
of Q are each n-square
for partitioned matrices are used elsewhere From eq. (15), we have
I
matrices.
485
Similar
notations
in this paper.
~,=~=$_~Qvco~b>Oo). This equation expresses above the topmost layer crystal. The generalized lower half of cO. The top example, if the primary taken to be exp(iKl - R),
(16)
the connection between the total of the crystal and the total wave Darwin problem is to solve eq. half a, of c,, describes the primary wave field just above the topmost this is expressed by a&)
wave field just field within the (16) for bO, the wave field. For atom layer is
= 6(K,+, K;).
(17)
We now describe the solution of eq. (16) for the model crystal specified by assumption (6) (fig. 1). We suppose that the selvedge consists of v layers, so the r]th layer is the topmost layer of the substrate. Also we put z,, =0 and p,,=p. In the. substrate, the vth atom layer differs from the r]th only by a translation qoz,/D, so we have (v 2 q) .
py (K”K’) = p (K”K’) exp [i (K’ - K”)*q,z,/D], If we put this into eq. (13) and then introduce
the abbreviations
= exp [i (K, + 2rcu .qo/D) d] ,
0:
13; = exp [i(K, e, (K:)
= (~:>‘-“a,
f”(KJ
= (O;)+%”
(19)
- 2rcv~q0/O) d], (K:),
(20)
(R;))
we get after rearrangement
I[ P+l
ep+l
[
f
where e, and tively, and
RI _
RII(RIV)-lRIII
p(/-p)-l
=
f,are column
vectors
R’ = [{6(C
+ P(CC)} o.+w,]
R”’ = [p (K;K:)] R’” = [{S (K;,
1
with 12elements
C)
R” = [p (K:K;)
I[ 1 eg
pIV)-
- w)-‘~“’
Eq. (21) may be written,
(18)
fp
’
(21) respec-
4,
)
+ P (K;K;)}
=
49 ’
(22)
with an obvious g,,,
2
e,(K:),f,(K,)
) K;)
(v
notation,
4%.
co;] .
as (23)
486
E.G. MCRAE
We refer to R as the reduced transfer matrix. Now we have g, = c,, according to eq. (20), so that by combining g, = RX,
eqs. (16) and (23) we get
(Y= 93
g,=R"-" fi Qvco, (P2YY?>O). v=q-1 This is the special form of the more general self-consistency equation [eq. (16)] that applies to the model specified by assumption (6). Before indicating a method of solution of eq. (24), we establish a special property of the reduced transfer matrix R, namely that its eigenvalues may be grouped in pairs such that the product of the members of each pair is of modulus unity. This property arises from the three-dimensional periodicity of the model substrate and the existence of a mirror symmetry plane parallel to the crystal surface [see assumption (6)]. The proof is as follows. In view of the three-dimensional periodicity of the substrate, the total wave field at any point R in the substrate can be expressed as a superposition of waves of the Bloch form @(K, R) = u(R) exp(irc*R), (25) where u(R) is a periodic function and the wavevector K has the same surface projection k as the primary propagation vector Kt . Let K, denote the inward surface normal projection of K, let R’ =(I-‘, z’) denote a point in a region of locally constant potential, let t denote a 2n-element vector whose elements are the coefficients of the expansion of @(rc, R’) in terms of the functions exp(iK2 * R’), and put p = D/d. Then the relation @(K, r’, 2’ + D) = @(fc, Y’, 2’) exp (i@), which follows from eq. (25), leads to the matrix eigenvalue RPt = exp (iK,D)
t.
(26) equation (27)
Eq. (27) is a statement of the band-structure problem for the substrate. The existence of a mirror plane parallel to the surface implies that for every Bloch state with wave vector K =(k, K,) there exists another one of the same -K,), i.e., for every eigenvalue exp(iic,D) energy with wave vector K’ =(k, of Rp there is another eigenvalue exp( - irc,D). Now the eigenvalues of R are pth roots of those of RP. It follows that for every eigenvalue i of R there is another eigenvalue I’ such that 12;1’1=l. This is the proof of the statement at the beginning of the paragraph. However, all that is necessary for the following development is that half the eigenvalues lie inside and half lie outside the unit circle in the complex plane (the case in which one or more pairs of eigenvalues lie on the unit circle is here considered as a limiting case which is actually realized only in the case of purely elastic scattering).
ELECTRON
We now proceed
DIFFRACTION
with the method
AT CRYSTAL
of solution
SURFACES.
I
487
of eq. (24). It is shown in
books on matrix algebraia) that any square matrix can be reduced by a similarity transformation to a normal form in which the eigenvalues appear on the principal diagonal and all elements below the principal diagonal are zero. Let T denote the matrix that transforms R to its normal form, and let T be so chosen as to place the eigenvalues of R that lie outside the unit circle on the right-hand end of the principle diagonal: T-'RT=h
,
(28)
(29)
I&2+I)(“+l)l~
l&“+2)(“+2)1, . ..v ILZnl ’ 1 *
(30)
We may now write eq. (24) as T-'g, =A'-"T-k,,
(pa q).
(31)
In the matrix A@-“, the diagonal elements are A:;“, A:;“, . . ., ,I;,& and all elements below the principal diagonal are zero. Keeping in mind the stated property that half the eigenvalues of R are of modulus greater than unity, we find that the condition that the total wave field be finite everywhere can only be satisfied if the coefficient of A&‘!! in eq. (31) vanishes. Similarly, it is necessary that the coefficient of IfZJ,‘!_l,,Czn- 1J vanishes, and so on up to y-” The final result may be expressed conveniently in terms of the (n+l)(n+l)* n-square submatrices of T- '; using the notation indicated by (32) we have b, = - i,-,li,,a,.
(33)
Let us put X=
fi v=?l-l
Q,
(34)
and partition X in the same way as the other square matrices. eqs. (16) and (33) we get b, = -
(X22+ i2;1?21X,2)-1 (X,,+ f;;T,,XJ
This is the formal solution treated in this paper.
of the generalized
Darwin
problem
On combining
a,.
(35)
for the model
488
E. G. MCRAE
3. Discussion The main result of this paper is that the problem of calculating diffraction intensities according to the “layer-by-layer” scheme first introduced by Darwin is essentially equivalent to a matrix eigenvalue problem. The present generalization of Darwin’s theory involves the replacement of a 2 x 2 matrix eigenvalue problem by one of arbitrarily larger dimension. The presence of a crystal selvedge of arbitrary structure can be handled by multiplying together a sequence of matrices, one for each layer in the selvedge. Thus there is no special difficulty in treating surface layers that differ in structure, spacing and chemical composition from atom layers in the bulk crystal. The input to a numerical calculation of diffraction intensities along the lines indicated in this paper comprises a set of layer spacings and a set of one-layer amplitude reflection coefficients. In the case of high-energy electron diffraction, it is possible that a pseudo-kinematical calculation of one-layer amplitude reflection coefficients (i.e., a calculation neglecting inter-atomic multiple scattering) might be useful, but this possibility needs further study. In the case of low-energy electron diffraction, a pseudo-kinematical treatment is likely to give inaccurate or even absurd results. The reason for this is that low-energy electrons are sufficiently strongly scattered atoms by that inter-atomic multiple scattering makes a very important contribution to the effective wave field incident on each atoms). To handle the diffraction of low-energy electrons by a layer of atoms, it is necessary to use a fully selfconsistent method such as that proposed recently by Kambec). In discussing the effects of the assumptions listed in section 2.1, we note first that the limitation of the treatment to substrates consisting of identical and uniformly-spaced atom layers is not unduly restrictive; fee, bee, hcp, NaCl and graphite structures, for example, are included in this description. Other crystal structures, such as that of ZnO, could be treated by a simple extension of the scheme given here. The model requirement that the scattering potential have a single, constant value between each neighboring pair of atom layers - and just above the topmost layer - is an unrealistic one for some systems, such as a crystal covered with a layer of foreign atoms. It would probably not be very difficult to modify the theory to allow for different scattering potentials between different pairs of atom layers, and this would be a desirable refinement for quantitative applications. For the treatment of diffraction by atomically clean crystals of ideal structure, the present treatment does not appear to offer any special advantage over the standard diffraction theory as first formulated by Bethei4). If a selvedge is present, however, the “layer-by-layer” approach is definitely the
ELECTRON
more suitable
DIFFRACTION
AT CRYSTAL
one in that it makes possible
SURFACES.
a treatment
489
I
of scattering
in the
substrate and selvedge from the same view point - i.e., in terms of one-layer scattering properties. Since the Darwin type of diffraction theory is essentially a matrix eigenvalue problem, it should be possible to use standard approximation methods for such problems to obtain a simplified description of diffraction intensities. The use of perturbation theory in this connection will be described in Part II.
Acknowledgment The writer is pleased to thank G. H. Wannier and J. J. Lander for drawing his attention to the Darwin method for diffraction problems, and V. Heine, P. J. Jennings and H. Scher for many helpful discussions.
Appendix It is required to show that the total field and effective field are given by eqs. (6) and (ll), respectively. TotalJield: The total field at any point R in a crystal may be expressed in general as 15) $ (R) = exp (5,’
exp(iK
*R) cs R,
IR - R’I) ---- T(R,, R’) tiefl (R,, R’) dR’ , (A. 1) 47-cIR - R’I
where R, denotes the position vector of the sth atom in the crystal, T(R,, R’) denotes the transition operator for the sth atom and I/,~(R,, R’) denotes the effective field incident on the sth atom. The other notation in eq. (A.l) is as in the text. Let q denote a vector of the crystal surface net. Then we have
T(Rs+ 4, R’ + q) = T(R,,R’), J~~~(R, + 4, R’ + q) = tieE(R,,R’) exp(ik*q).
64.2) (A-3)
Upon interchanging the order of summation and integration in eq. (A.l), using eqs. (A.2) and (A.3) and transforming the integrand by means of the two-dimensional analog of Ewald’s theta-function transformation5)
c
exp [iK(lq + rlz + z2)* - ik*q] (14 + t-1’ + z’)+
4
with R=(r,
z), we get
i27c = 7
exp [i(K” Iz/ - k*r)] c ”
K
,
(A-4)
490
E.G.
+(R)=exp(iK,i.R)--iALGmC’
x ”
exp [Xi
X
MCRAE
”
RS
*(R - R’)] T(R,R’) Gefi (R,R’) dR’ +
[f
r’
+
zI>z exp [iK;
*(R - R’)] T(R,R’) t,ber(R,R’) dR’
s
1 ,
(A3
where c’ denotes summation over one unit mesh and R’-(Y’, z’). For a point R lying in a region of locally-constant potential, the integrals in eq. (A.5) are independent of R and eq. (A.5) reduces to an expression of the form given by eq. (6)9. EffectiveJield: Let $&,(R) denote the wave field emitted by the ,uth atom layer of a crystal. The effective field t,P(R) incident on the layer is defined as the total field minus the emitted fieldl5): V’(R) = II/(R) - K,,
(R).
G4.6)
At a point R in the region of locally-constant potential layer, the wave field emitted by the layer is of the form C,, (R) = c P, (Rb) exp (K ”
.R) 2
where the p’s are constants. This may be proved, eq. (A.5) to a single layer. Now we have obviously b,+, CR;) = WC) and the combination
just above the ,&h
- P/JR,)>
e.g., by application
(4.7) of
(A.81
of this result with eqs. (A.6) and (6) leads to eq. (11). References
1) J. J. Lander, Progr. Solid State Chem. 2 (1965) 26. 2) 3) 4) 5) 6) 7) 8)
9. 10) 11) 12)
G. W. Simmons, D. F. Mitchell and K. R. Lawless, Surface Sci. 8 (1967) 130. E. Menzel and 0. Schott, Surface Sci. 8 (1967) 217. B. M. Siegel and J. F. Menedue, Surface Sci. 8 (1967) 206. E. G. McRae, J. Chem. Phys. 45 (1966) 3467. K. Kambe, 2. Naturforsch. 22a (1967) 322. M. Macrakis, unpublished. Other methods of treatment include the mixed representation and band-model methods. See K. Hirabayashi and Y. Takeishi, Surface Sci. 4 (1966) 150; P. M. Marcus and D. W. Jepson, Phys. Rev. Letters 20 (1968) 925; D. S. Boudreaux and V. Heine, Surface Sci. 8 (1967) 426; F. Hofman and H. P. Smith, Phys. Rev. Letters 19 (1967) 1472. C. G. Darwin, Phil. Mag. 27 (1914) 675. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961) p. 499. P. W. Palmberg and W. Peria, Surface Sci. 6 (1967) 57. F. E. Hohn, Elementary Matrix Algebra (Macmillan, New York, 1958).
ELECTRON
DIFFRACTION
AT CRYSTAL
SURFACES.
I
491
13) The ordering expressed by eq. (22) automatically applies to the diagonal elements of R in the limit of weak elastic scattering, since the expression 1 +p(K’K’) is to be interpreted as a one-layer amplitude transmission coefficient and so is necessarily of modulus less than unity. 14) H. A. Bethe, Ann. Physik [4] 87 (1928) 55. 15) M. Lax, Rev. Mod. Phys. 23 (1951) 287; Phys. Rev. 85 (1952) 621. 16) The result for the effective field was stated in ref. 5 but the derivation given there is strictly correct only for s-wave scatterers. The error arises from the fact that the partial wave expansion coefficients of the “effective” atomic scattering factor are not independent of the space coordinates, as was implicitly assumed in the derivation. The author thanks Jar-MO Chen for pointing this out.
Note added in proof. Since this paper was submitted for publication, the writer learned that K. Kambe had independently derived the principal properties of the reduced transfer matrix (called the “scattering matrix” by Kambe). Kambe’s theory, which is more general than the present one in some important respects, will be published in Z. Naturforsch. under the title “Theory of Low-Energy Electron Diffraction. II. Cellular Method for Complex Monolayers and Multilayers”. The writer thanks K. Kambe for a helpful correspondence.