INTRODUCTJON

Unified approach to photographic methods for obtaining the angles of Incidence In low-energy

1. INTRODUCTJON

From: DETERMINA

nON

OF SURFACE STRUCTURE BY LEEO EdIted by P.M. Marcus and F. Jona

(Plenum Publishing Corporation, 1986)

A MA'IrlrMATICAL FOUNDATION FOR AD HOC RELIABILI1Y FACTORS IN LEED*

A. C. Sobrero and W. H. Weinberlf

Division of Chemistry and Chemical Engineering California Institute of Technology

Pasadena.CA 91125

438 A. C. SOBRERO AND W. H. WEINBERG

2. TRADITIONAL i..EED INTENSITY ANALYSIS .• .

The usual method fo~ an,a).ysing.muIUple-scatteri:ng LEED datil ill a trial-and-error procedure.' This review of the method serves to introduce the definitions of the parameter manifold P. the intensity space Q. and the intensity operator I. The reconstructed Ir(110)-(lx2) surfaces is discussed to illustrate the notation.

The first step in the data analysis is to postulate a family of models.

M(P). for the structure. Here the point

is

is a member of the parameter space P which characterizes the family. In this way. each point in P represents a model of the surface according to the mapping

iT. =M(p). (l)

The models ih include the atomic positions. phase shifts. inner potentials.

and other factors which govern low-energy electron scatter.

Often. several different families Mi are proposed. each with its own parameter space Pi' Consider the two models for Ir(llO) illustrated in Fig- ure 1. The paired rows model M I has a three dimensional parameter space

Ph where CT. 151 • and Eo can vary (see the caption to Figure 1 for definitions).

The components of a point in the space PI are:

p =

^(CT. 15_1, Eo). Similarly.

the parameter space for the missing row model M 2 is four diInensional. with

p

= (151• 152 •

p.

Eo) when

p

^E P2 •

Clearly. not all the points in a parameter space Pi are acceptable.

There are two restrictions: the

p

must map to physically realizable struc- tures. and each structure should correspond to a unique point. The second condition insures that the mapping Mi is one-to-one. thus avoiding ambigui- ties in the parameterization of the crystal surface. For example. the spac- ing CT must always be less than or equal to a.

These constraints limit the choice of parameter values to a manifold Pi \: Pi. The bounds actually used with the paired rows model M I were:

CT E: [2.95

K.

3.55

K).

151 E [1.211t 1.81

K].

and Eo E [-10 eV. -20 eV). This manifold lies well within the above constraints.

The next step in the procedure is to calCUlate the intensity spectrum

q

for each model:

"

9

=I(Af\(P». (2)

The operator

1

denotes the compulational method used to obtain the spec- trum. This nonlinear operator constitutes a mathematical model for elec- tron diffraction.' The spectrum

q

is a point in the high-dimensional intensity space Q, "'iLh each component qj representing the intensity of one beam at a particular electron energy and angle of incidence. Thus. all the datu from a LEED experimenL defines one point.

q.,

in Q.

In the case of Ir(110). because I-V curves were computed for 18 beams at 2 eV intervals over an average range of 100 eV and for one angle of

AD· HOC RELIABILITY FACTORS IN LEED 439

incidence. the intensity space Q is 90D-dimensionaL Here. for example. the first 50 dimensions might repr\tsent the intensity of the (OI) beam at 30 eV.

32 eV ....• 128 eV.* The operator I consisted of the Reverse Scattering Pertur- bation with Layer Doubling.s

PAIRED

Row

^MODEL

MISSING

Row

MODEL

a

/3 20-/3

1st layer 2nd layer

1st layer 2nd layer 3rd layer

Figure 1. Two models for the reconstructed Ir(llO}-( 1x2} structure. Here.

a == 3.58

.a

is the bulk spacing between rows of atoms in the (OOl) direction.

while 6₁ is the distance from the first to the second layers of atoms. Tbe real part of tbe inner potential. Eo. is a non-structural parameter in the models. 01 her geometrical paramelers: for the paired rows model. cr is the spacing belween lhe adjacent close-packed top rows of atoms wtJich have moved toward e'lCh otber; in the missing row model. 02 is the distance from t: Sinl!e the experimental intensities were not recorded at exaclJy these values of the energy, it ?ras necessary lo interpolate the dala. This projection is a rnapping from the data space toQ.

440 A. C. SOBRERO AND W. H. WEINBERG the second to the third layer, whUe

fJ

is the spacing between the adjacent rows of atoms in. the second layer.

Henceforth, to simplify the notation, consider only on1' family of models M. and define the intensity operator J as the composition ToM. The operator J now provides a direct connection between the parameter manifold P and the intensity space Q:

9i

= J(p,) for jJ( € P. (3)

Since J describes a physical process. assume that this operator is a continu- ously differentiable function of jJ. Figure 2 illustrates the relationship between the various mappings.

The final stage in the structural analysis is to minimize the reliability factor.

Parameter Manifolds

Model Space

I

Intensity Space

Q

(4)

Figure 2. Schematic indicating the range and domain spaces for the opera- lors introduced in the text.

AD-HOC RELIABILITY FACTORS IN LEED 441

over the set

f4ct

of computed points and thus obtain the calculated spec- trum which best matches the observed one. The function D

(iJ

I, lJe) meas- ures a distance. in some sense. between members of the space Q. For con- venience. denote the best fitting spectrum by

iJ·

and the corresponding sur- face structure by

fJ·,

so that

iJ· =

I(fJ+). Also. define the vector of residuals as

(5) This vector contains the variation in the data which is not explained by the mathematical model I(fJ).