Slow electron scattering from one-dimensional crystal models

SURl54CE SClENCE 9 (1968) 198-216 Q North-Holland

PubMing

Co., Amsterdam

SLOW ELECTRON SCATTERING FROM ONE-DIMENSIONAL CRYSTAL MODEIL’S* J. GERSTNER and 9. H. CUTLER** Department of Physics, T?tePennsylvania State Unirwsity, IJt;iyersityPurk, Pemwylvania 16802, US. A. Received 19 June 1967; revised manuscript received 18 July B967 calculations have been made for low energy elastic scattering of ekctrons from several one-dimensional models of a crystal. Two forms of the surfalce potential have been used: the square barrier and the cutoff classical image potential. Interaction of the incident electron with the ion-cores of the “crystal” is introduced by &functionpotentials at “lattice” sites. Two models of the lattice are considered: (i) a stxkintiite ;:lrray of identical S potentials and (ii) a small finitenumber of S-function of variable strength and spacing to represent limited penetration of low energy electrons. Refktion coe’ficients as a function of incident slectrun energy, i.e., 4-100 eV were calculated for all cases. The resulting CUPZSexhibit considemble structure. Comparison is made with experimental data (i.e., slow electron retkction, electron diffraction, etc.) and other one-dimensional model calculaticns.

1. hbrodnctiom

The reflection coeficient for electrons scattered elastically from normal incidence on single metal crystals of tungsten has been measured by Kahn et al. I), and P!!x -$j. These recent measurements, which employed LEED techniques. all show conkstent structure in the reflection verslrs energy curve. This structure 1s characteristic of the crystal face viewed and of the material employed. Sirnil;.r measureti6rits hOlvzbeen made by other iqvestigato-s on other materialss). A quantitative exp!anation elf these results would involve the formulatio,: of a three-dimensions8 crystal model with a surface, Attempts have been matie

by Darwin

and

klthers4) at treating

this pir&!em_, but, ip_ a----oe.nersrl --

neglecting surface potential scatteriag. This is a formidable proble:m involving both mathematical and conceptual difficulties which is as yet not solved. To isolate certain aspects of this prohlem and to mak.e them more: tractable, * Research supported by Cr. S. Air Forw OFFiceof Scientific Research, &ants numbers AFOSR-AF-2 13-63and AF -213-M. ** Address from Sept. 1SW to Jiune 19611:Laboratory for Electrophysics, The Technical Un’lverGty, Lyngby,, Denmark. 198

SLOW ELECTRON SC ~TTJERING

199

albeit qualitati~e, we have adopted a one-dimensional model of the surface and crystal. Using this model, we have calculated the elastic component in the reflection of slow electrons at normal incidence. Other one-dimensionalmodels have been proposed in the past. Several of these are ~tescribedin section 2. They have been unsuccessful, however, in explaining the structure found in the reflection experiments, yielding curves wMch are generally monotonically decreasing functions of energy. One exception to this is noted in section 2. The present model predicts such structure which is a consequence of interference between scattering planes in the crystal as modified by the surfacepotential. The model further suggests that the observed reflection is characteristic of limited penetration o f the crystal by low energy electrons, probably to a maximum depth of three or four atomic layers. In section 3 we describe the proposed model with a discussion of the methods employed in the calculation of the reflection coefficient. The results of these calculations are presented in section 4 and the conclusions are contained in section 5. 2. One-dimensional models The relevant features of the reflection curves reported in refs. 1-3 may be summarized as follows(l) a structure which is characteristic of the crystallographic face; (2) a small, but measurable, re:~te~;tionat zero incident electron energy, and (31}an apparent monotonic decrease at higher energies. Previous theories have attempted to qualitatively analyse the problem of elastic reflection of an electron :incident normally on a metal surface from a vacuum using one-dimensional models. To describe these models let the x-axis be normal to the crystal surface, which is located at x = 0 , with the vacuum in the region x > 0. Denote the potential experienced by the electron as Vext in the vacuum and Vi,t in the intzrior of the metal. The following forms for Vexthave been used" (a) Ve~t=0; a potential which is constant in x > 0 , and which changes abruptly at the surface, i.e., a square barrier. (b) Ve~t= --e2/4x; the classical image pote.n_da! for an electron in the vicinity of a perfect conductor. A modification of this to remove the singularity at the origin, viz. V,~t=-e2/4x, x~x¢ and V~,t=--Vo, 0~
zoo

3, C3ER!$TNERAND P. fz:cuTtsR

near tht: origin (the surface), going asy~~~toti~~~y *o %co~Wnt value EI.J&&the metal, wMe approaching the classkal il[fp8;8Lf p@tetititiiat large distanws loutsidet.he surface*This red t has bwa exterkkd by Juretschb@~ Etau&s an& Cut~er~)zand Sachs and Dexter?). The ti$t two Confirm 1 function

while tk ~ast~$&‘nat&$h6 &Spar* turn af the Bardeenresultfrmn the classicali~a~,~~~~t~~. (d) Q&r and @ibbons$‘J proposed a quantum ~~~di~~ image potential.. with t&e form VCxt = -(&4x)( 1 -q/x), x2x,, and Cgrl== - v& xeq, w&h inccrpcrrates t?teimpor~~t features of the potentials in (c) in a fuore tractahk analytk form. The &eld free reflection coeficients for electrons, iumployjngthese various ~ot~~~~a~s,were calculated by Cutler and 33avkP). Their results are iUustrated in figs. 44and 7 of ref. 10. The curses they obtain ak monotonic, ~th~ou,~structure, and indicate. that the form of the reflection coeflbicientas a funct~o~tof energy is relatively insensitive to the choice of externalpotential, ~rovid~~~it has the asymptotic behavior of the c1assicai image potential, and mx&s bounded at the origin. For tlnis reason, and because it reduces the the: *i~g~~~ ls&

tl;Bfm

of the

&&lam

result,

comput&ional work involved, we have employed only the cutofl image potential as a representative approximation to I?& ynz
The rnc~d~el, we propose is one-dimensional co~isting of external and interrrtallparts. The external part is essentially the .:ame as in those models dis<:uS;sedabove, with tthe same justi~~tion. The present model assumes, hovsesrer, that the electron now experiences at feast Some of the internal stzu&ure of the metal. ‘Thk is incorporated in the form of d&a functions repre?;e:nting isolated scatterers alnayed along the line of motion. The use of Ei-funr:tions ais scatterers allolws lebr both reflection from welLdefined sites, and phase shift at each s~~~tte~j.ng~ Tlheefkxt of each of these can be controlled

SLOW~LECTRONSCATTr~No

20!

by ,varying the strength, Vi, of the scatterers ~flthough they are not independently variable parameters. Eq. (8) below, indicates how the phase shift is produced by this potential. We represent this potential V(x) as follows: n

V (x) -.- ~,,, = - Vo - ]~ a,V~ (x + Sa,),

for

x ~
(1)

l=l

where

Sal = O,

i = 1,

i

S a t = ~ aj,

i .~ 1,

1=2

and

V(x)-- v,,,,

for x

o.

(2)

The argument of the &function in(l) locates these scatterers on the negative x-axiis with a spacing ai between them• The potentials used are illustrated in fig. 1.

v(x) t

(METAL)

I

I

I

(VACUUM)

x~o

__.

I

. . . . . .

VO

,'"

~X

If

t

....

-Vi :

Fig. 1.

•

.

Several potentials referred to in sections 2 and 3.

The elastic reflection coefficients have been calculated in the following cases: A. Square barrier with semi-inifinite array of equally spaced &function scatterers, i.e., ~ , t = 0 , in (2), and n - d o with all ai and ~ equal in (1). This constitutes essentiaiiy a Kronig-Penney model for the crystal, where the ion wells have been shrunk in width about the ion sites, but the total potential averaged over the space occupied by an ion (the atomic volume in three dimensions) has been kept constant. To calculatf~ the reflection coefficient, we solve the SchrSdinger equation dx 2 + (E - V (x))d' = 0

(3)

202

s. OERS'r~R A~'r~ P. s . ~

in unit~ in which

h=-ie 2= 2 m =

'

•

•

.>

.

1, length is; measured in units of a0 03ohr

radius), and energy in rydbergs. In the region x>~0, the solution to (3) is (x) = e -l~x + r e 'k*

(4)

r,..presenfing incoming and reflected plane waves, with k=E ~ and the desired reflection coefficient R(E)= Irl a. To determine r, the complex amplitude of the refleet~ ~ave, this solution is matched at x = 0 with a solution obtained in the region x<~0 by a method due to Saxon and HutneraS) . They have calculated Bloch wave functions along ~m infinite line of pedodicaUy arrayed di-functions by matching the solutions at the 6-functions through the conditions of continuity of the functions, and discontinuity of the first derivative. This last condition can be obtained by examining the SehrSdinger equation (.3) with the potential (1) (the a~}] have been factored oult and set equal to

aV), dx ~ +

+ Vo + aV ]~ ~(x + Sa,

qJ = o .

i--I

(5)

lntegratipg this across some - x i = Sai we obtain -x~+~

i'

-xi+e

f

(dq()= d \dx/

- - X j - - C

oo

---'~J

)'t

{_E_Vo_aV~6(x+Sai

qtdx,

i= 1

--

(6)

£

0I"

--Xj+£

--Xj'sr~, OD

--xj-'~

--xJ -el=:1

The first term on the riight-hand side is zero in the limit e~0, and the s~.cond term is -aVqJ(-x~). q'he discontinuity in the first derivative is thus: ,p'> (x~) - qJ', (x~) - - a vq, ( x j ) .

(8)

The symbol>(<)iindicates the derivative from the right (left). B,.¢ this method the wave function in the region x ~
where g, (0) is ,a constant, k ' = (E + Vo)"~, x"= x -- [x,a],

-

a)'(

(~,)

J

[x/a]= larg~,~;t integer

e~x/a, ~ d r p it; d e t e ~ e d

by evaluating (9) at x = 0

aV

cos ~ta = cos k'a - 2-~ sh~ k'a.

(If))

Eq.~ (10) deter.mines(the-~lowed~ energies (band structure) by the condition t h a t r e a l v a l u e s 0 f ~ p~r ~ U i r~e:t h a t l c ° s ~ l i ~ ' L i ~• To determiner, e ~ a t e i ~ ( x ) f r o m : ( 4 ) and (9) at x=O, and use (8) to ~'vlate the deriVatiVes'of~,(x) fr0~ (4) and (9) atX=O, The result is

R ( E ) = lr*rl = ( 1 - Z)(I - Z*)/(I + Z ) ( I + Z*),

(l])

where k' sin p a aV Z= ..... -i . k sin ka 2k

This quantity was evaluated for various values of a, V, and Vo as a function of E. The results are presented infigs. 2-4. B. Square barrier with a finite number of internal 6-functic,ns. Again, with V~x,=O, x>~O, and n in (1) a small number, the Schr~dinger equation (3) III

I

'

q'

li!

i

II*

0.7'-

1

I1.

0.6-

i:!

0.5-

I"

,I

R 0.4"

0.3-

I

'1

:1

Jl

0.2-

!

'

0

J

%

I

2 3 E:NERGY (R~' D)

_J 4

5

]Fig. 2. Reflection coefficient R(E) for the case of a square barrier with a semi-infinite array of internal ~-furtctions. The effect of changing the st~mgth, V~ in eel. ( l i~,is illustrated. I[n each case Vo=0.662 ryd, a~=5.98 a0. In curve (a} V~=O.O01 ryd, (b} K~=O.05 ryd, (c) V~=0.075 ryd, (d) I~=0.1 :ryd.

204

0,6-

0.5" R

o,y 0.2"

t" I

~ 2 3 EI~ERGY (RYD)

'

4

5

]Fig. 3a.

!

O.

'

'i ....

|

"

7-

0.6

0.5

0.4

R 0.3-

O2

~_v

.1-I

0

.

0

.

.

.

I

H- .........

i

2

I ............

3 ENERGY (RYD)

F~g, 3b.

I

........... I!

4

5

21B [

; i

0 jl~ ~

':r.

I

4

I

i

_

•

•

,

•

!

-'=--

;

•

.]

•

I 1

06

1 ]

m

, •

•'/

:.L "

:,"

:r

: :. . . .

.

'

-

:

i

o s •

OA

i

m

!

0.3-

.2

-~

0.1,

0 0

...I I

I t

I

i i

I i

2 3 ENERGY (RYD)

4

5

*

Fig. 3¢. Fig. 3. Reflection coetTgient R(E) for the case of a square, barrier wdth a semi-infufite array of internal 6-functions. The effect of changing the spacing, a~ in eq. (!), is illustrated. In each case I,'o=0.662 ryd and V1=0.075 "ryd. In curve (a) a~=2.99 4:~, {b} a~=8.65 ao, (c) a~= 17.94 ao. Compare with fig. 2c. 0.7

l

i

0.6-

0.5-

0.4 R 0.5

0,2

°tl

J

O,

I

0

I

I 2

I

3 ENERGY (RYD)

4

5

Fig. 4. Reflection coefficient R(E) for the case of a square barrier with a se~ai-infinite axray of internal ~functions. The effect of different well depths, Vo in eq. (1), ,is illustrated. a,=5.98 ao, ~ - 0 . 0 7 5 ryd, and Vo=0.331 ryd. Compare with fig. 2c.

206 ~" ....

". . . . . . . . .

I

.............

I

.... ,

,

.

'

:

t

'. '~ , , ~ ''~ :, :1 .....

0,6-

0.4R

il

2"

C,I

i

O! 0

I

I

T

2

3

ENERGY (RYD)

Fig. 5a.

I

I

I

I

0.6-

0.5-

0,'~

0.3 N~

0

.

0

.

.

.

I

.

.

.

.

2 ~) ENERGY (RY?,

Fig. 5b.

4

5

0.6-

g 0.50.4'

!::i r~.

m

0.3 . . . .

t

0.20-

i,,~ I

II fl

I

2 3 ENERGY (RYD)

4

5

Fig. ~.

I

I

I

I

O.

0.6

/!

0.4 R

/

O.

0.1 ,,~_1

%/7. . . . . . . . . . . . . . . o

I I

I

4

2

5

ENERGY (RYD)

Fig. 5d. Fig. 5. Reflection coefficient, R(E), for the case of a square barrier with various numbers of 6-functions. In each case a,=5.98 a0, and V0=0.662 ryd. (a) 2 6-functions, V,=0.075 [yd for each; (b) 2 ~-functions, V1=0.075 ryd, V2=0.055 ryd; (c) 3 ~-functions, V,=0.075 ryd for each; (d) 4 S.functions, V,=0.075 ryd for each. Curve (b) differs from (a) in that the strength V~ is reduced.

"

2

•

".

.

"

,

must be sol~ed for x~
¢(x) = t, e - i k ' x + ~, @ik'x .

-a2<

O(x) = ta e - ~

- (a2 + a3) < x < - a , ,

+ r2 ei~'~,

x<~O,

(12)

<. - ( s a . ) .

q, (x) - t, e -'''~,

In (12) and (4), k=E t, k'f(E+ Vo)~, and r, tx, ..., rl,•., are compiex transmission and reflection amplitudes• These functions must be joined a t x - 0 , -al,-(at +a2),...,-Sa, by applying the, conditions of continuity and (8). Upon separating the real and imaginaxy parts of the resulting equations one Obtains a set of 4~ linear equations inRe(r), Ira(r), RV(tl),Im(tt),':.. These equations were solved numerically by a matrix inversion routine employing a Gauss-Jordan elimination method with maximum element selection. The reflection coeffici~mtwas obtained from RCE) = (Reef)) 2 + (Im(r))2 •

The results of these +:alculations are presented in figs. ~:and 6.

0.7

i

I

I

I

0.6

0.5

R0'4

°-+T !/;

°2t o ,i 0

I

A

Uj I

I.... ~

I

2

,,

II

,5 ENERGY (RYD)

Fig. 6a.

5

"

.~': ~' i~¸i~'¸~%¸¸¸¸~¸''~',!::: :i: :: :'~

~~ ~'ii~i~i

,

i ...... : / : : : ..... = .i:: ¸:..,:

|

vo

:~::

I

I

,

~

I

• '

L

(1

~=.

Jt

0.I"

o:

I

0

i

l

.

I

2

:3 ENERGY (RYD)

4

5

Fig. 6b. 09,

!

I

|

I

I

0.6-

0.5-

0.4-

R 0.3

0.2

0.1

0 - 0

I

I

2

3 ENERGY(RYD)

4

5

Fig. 6c. Fig. 6. Reflection coefficient, R(E), for the case of a square barrier with four internal 6-fianctions, I n each case.. V0=0.662 ryd. In curves (a) and Co) the effect of altering the strengths, V~, are illustrated. In curve (c) the spacings are changed. (a) V1= Va =0.075 ryd, Vg:=V4=O.OSryd, at=5.98ao; (b) V1=0.075 ryd, V9=0.C65 ryd, Va=0.055 ryd, V4=0.045 ryd, a~=3.98 ao; (c) V~=0.075 ryd, al=aa=5.98 ao. In curves (a) and (b) the effec~ of ~IteHng the strengths, V~, are. illustrated, In curve (c) the spacings are changed.

0.05 T . . . . . . .

~

I

!

....

-"

i:

--"

|

t

0,04']-

o.o2+',

...... /

~

_

' ',,.-o

o

I

-~I

0

,,

~

,

t"

7 .....

2

'I

:5

4

ENERGY(RYD)

,5,

Fig. 7a.

0.3

"~

I

I '

I

'"

I '

0.2-

R

0

i

2 3 ENF__ZRGY(RYD)

4

5

Fig. 7b.

0.4

t'

"--'-1-

.

.

.

.

I

I

0"31 f:,2 O.2"-~

!

0

I

2

3 ENERGY (RYD) Fig. 7c.

4

5

SLOW~ o N

0.51

~,

s e a ~

,. . . . . . .

~.

211 .

.

.

,

0.4 0.3 R

0.2

0

i

2 3 ENERGY (RYD) Fig. 7(1.

4

5

Fig. 7. Reflection coefficient, R(E), for the case of a cutoff image potential with various numbers of ~-functions. In ~ c h ease m=5.98 a0, I,'o-0.662 ryd, V~=0.075 tyd in eq. (1). The dashed curves represent the reflection coefficient calculated for the cutoff image potential with a constant interior potential of 0.662 ryd. (a) one J-function (note change of scale); (b) 2 &functions; (e) 3 &functions; (d) 4 &functions.

C. Cutoff image potential, i.e., Ve~t= - ½X, X>~Xc and the internal structure represented by (1), with n~< 4, in x ~
(13)

The desired quantity is then 10) R(E) = (B*B/A*A).

(14)

Knowing the value of the wave function, ~ (x), at some x < Sa. we can determine the values of d and B by the following argument. The Wronskian of two functions is W (q,, ~) = ( ~ , ' - ~,'~,), (! 5) where the prime denotes differentiation with respect to x. With the appropriate choices of ¢p, (13) and (15) yield W (~, e ~k''~) = 2 i k ' B ,

(16a)

W(~, e-ik'x) = - 2 i k ' A .

(16b)

Letting W* denote the Wronskian of the complex conjugates of the argument functions, we can solve eq. (16) to obtain A = - W (d/, e-'k'x)/2ik ' , B = W ( ¢ , e~k'x)/2ik ' ,

A* = W* (~b, e-~k'::)/2ik';

B* = - W*(~, e~k x)/2ik ' .

(17)

s. O

212

AND P, H.

~

Combining eqs. (14) and (17) we have

w'<+,:'=) w(+,:'=! I

R(E) = w* (q,, e

w(q,,

(18)

Using (16), and writing (13) as t//-t//R+il]/i, where qta a n d ~l arc the real and imagm ry parts of the wave function,, respectively, we obtain .

i

a

!(k'#~. - 4~;)2 + (k'~z + ~ ) z

(19)

R (E) = ! ( k ~ + ~k;)2 + (k'~, - ~a) i ' z" ~a and ~q, along with their derivatives are evaluated at a point x < - S a n . The functions ~a, ~ki, ~ , ~} were evaluated numerically by selecting a

point in the region x >i x~ at which the WKB condition, dk'/dx

2k,~ t ~ I , is satisfied 15). This can be transformed by using

dV/dxi 2k'3 t ~< c,

. . . .

k' (x) = (E -- V (x)) + into

where c is a small constant of the order of 10 -a. It was found that the solutions obtair ed were relativeily insensitive to this number in the.• range 10 - 2 10 -4 so the larger value of c was used to reduce computation: time. When the WKB criterion is satisfied at some x = x o the solution to (3) which represents a wave traveling to the left can be written 15) .4

,4

d~}

+ (x) - (k, (x))+ exp {- i f g' (e)

.

(20)

~o

choose A =(k'(xo)) +, thus fixing the amplitude and phase, and obtain ¢:R(xo)= 1, ~ ( X o ) = 0 . With h - a small increment, values of thes,~ functions at x = X o - h are found by evaluating (20). With the values of ~ka and ~i thus obtained at two points the solution to (3) was continued by a three-point integration program 14). The first two values of the derivatives ~ and ~were obtained by differentiating (20). The solution for these derivative functions was continued in a manner s i m i l a r to t h a t n,ged f a r ~b.. a n d dr., h v t h a same program. These functions were then generated toward the origin unlil the first f-function was encountered, at which point qtR and ~b~ were continued, and condition (8) w'as used to evaluate the discontinuity in the functions qJ~ and 1' In this manner the solutions were continued into the region beyond ~,~. - S a , , , where (19) was invoked in order to evaluate the reflection coefficient. The results of these calculations are presented in ~ •,gs. 7 and 8. We

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

T

IN;. . . . .

"1" li~

~J

..-v

213

SLOW ELECTRON SCATTERINO

0.6

......

• ,,,,

,

=

,'

,,,

,

,

i. . . . . . .

,

,,

t'

,

='

,

,

,

= ,,

,

,

i

,

,

"-r-

"

0.5.

R 0"4~ 0.3,

=1

0.20.1 o

k

0

!

0.1

0.2

|'~

0.3 0.4

0.5 0.6 0.7 0.8 ENERGY(RYD)

0.9

1.0

I.I

-

I.?.

Fig. 8a.

0.6

i'

,,

!

!

I

,

I

,

I

i

,

I

I

,

a

I

I

I.I

1.2

0.5~ 0.4,. ~l"

0~2O.!

0

0.1

0.2

0.3

0.4

0.5 0.6 0.7 0.8 ENERGY(RYD)

0.9

1.0

Fig. 8b. Fig. 8. Reflection coefficient, R(E), for the ease of a cutoff image potential with four internal t~-funetions. In each ease a~=5.98 a0, Vo=0.662 ryd. The effect of different strengths, V~ in eq. (1), is illustrated. In curve (a) V~--=-0.075 ryd for each, (b) Vx=0.005 ryd, V9=0.05 ryd, 7a=0.075 ryd, ~q=0.1 ryd. The scale of energy is expanded.

4. Results

Since this model was proposed to explain qualitatively the observed elastic reflection of electrons from tungsten, we eva~uate our results for values of the parameters which might be characteristic of this material. Specifically, the crystal structure is Ice with a = 5.98 ao (3.16 A). Viewed normal to the (100) face, the interplanar spacing is ½a= 2.99 ao, and identical planes are spaced a distance a = 5.98 ao apart. The parameter Vo was taken to be approximately equal to the Fermi energy plus the average work

214

J. ( j l ~ l ~ r ~

AI~I:) P, ,tl, C U ' I ~ R

function. A value of I+o=0.662 ryd (10 eV) was chosen as a convenient one. T , scatterer strength, r'~., was taken from the results presented in fig. 2. L = 0 . 0 7 5 ryd was selected as a figure which produces reasonable results An ~ttempt to justit~ this on physical grounds, viz. by equating the quantity a+t~ to the integrated poten,.iai in a Fermi-Thomas model, was not,,uccessful. From fig. 2 we have the result tl~at scattering from a square step potential and a semi-infinite array of scatterers produces a series of peaks (complete reflectit~n) located at energies which are characteristic of the spacing, a~. The values a~= 5.98 ao and Vo=0.662 ryd, were used in this example. It is observed that width of the peaks is a fimction of V+, the strength of the scatterers. The curve rises to unity at E = 0 , which is characteri.,+tic of the step function. It i~ observed in fig. 3, that: an effect of changing the spacing, a~, is to alter the locations of the peaks, l+a fig. 4, which is obtained by using a~= 5.98 ao (as in fig. 1), the effect of changing the inner potential Vo is seen to move the pe.aks, also. In this latter case, the shift of the peak, AEm, is numerically equal to the change in I+~,i.e., AEm = - A I%. These last two effecls can be discussed in terms of interfierence in the reflected beam between two scatterers which produce a phase shift and reflection. The condition for constructive interference is that the wavelength of the incident wave satisf~

J~tl

n),, = 2a i + 2Acp 2~'

(21)

x~here pt is the order nun,.ber of the maxima, a+ the spacing between the scatterers, and Acp the phase'shift on transmission. Since k=2rc/2---(E+ Vo) ~ for the case of plane waves, this reduces to Em+ Vo =

•

(22)

If Aq~ is independent of Vo, then d E m / d V o = - l , in agreemenl with the results shown in fig. 4. Eq. (22) further il!ustrates that the order number of reflection maxima depends on the choice of a~. Specifically, one could interpret fig. 1 as integral order refk~tion maxima for a~= 5.98 ,,, (spacing between identical planes), or, alternatively; as half-integer order maxima for a,=2.99 a 0 (',pacing between adjace.nt planes). ~n either case, the distance between maxima ~s the same. In this sense the distance between interfering planes would b.~ determined by the separation of the maxima. This is ~.ot, however, an unambiguous conclusion of this model, as will be seen below. In fig. 5 we see the effect of a step function and a small number of scatterers, i.e., a limited penetration of a low energy electron into a crystal. The

SLOW ~ L ~ C ~ . ~ s c A ' r r m a n ~

2U5

maj,~r peaks are related to those obtained on the first model, although shifted somewhat. We note, however, that the inclusion of more than two scatterers inlroduces approximately ,equally spaced intermediate peaks of lesser magnitt~de. In view of the above discussion we could assign an order number to, the major maxima, but the intermediate peaks might also be assigned order numbers, making the interpretation of observed reflection data ambiguous. This is particularly accentuated in fig. 6 where the strength of each scatterer is different. This in general alters the relative siizes of the adjacent peaks. In fig. 6c the different spacing bet~teen equal strength scatterers destroys the identity of major and subsidiary peaks. The inclusion of a more realistic external potential, the cutoff image potential, produces a much lower value of the reflection coefficient at E = 0 , with substantially the same behavior of the peaks (cf. fig. 7). In fig. 8 we present, an example of the overlapping of these separate peaks caused by a variation in the strength, V~. In ref. 1 the reflection curve for W(100) shows maxima at 4 eV(0.30 ryd), 8 eV(0.59 ryd) and 17 eV(1.3 ryd) at which points the coefficients of reflection are approximately 0.28, 0.42, and 0.17 respectively. This apparently include~, one major peak and two minor peaks. None of the models presented here car reproduce all the characteristics of this curve, but that shown in fig. 7c, i.e., ~ cutoff image potential with three internal scat~terers, would predict the correct structure up to about 2 ryd if the following parameters were used: Vo=0.56 r.~/d, V1=0.01 ryd, ~ = Vs=0.075 ryd, and a~=:5.98 ao. 5. Conclusion It is noteworthy that this one-dimensional crystal model incorporating as few as two scattering centers, i.e., "crystal phmes", is capable of producing considerable structure in the reflection curve. This results from interference between internal scatterers and between the scatterers and parts of the external potential. The locations of the reflection maxima are primarily determined by the spacing of the internal scatterers and by tlhe internal potential. The energy difference between adjacent peaks is a function of the spacing, ai, while intermediate peaks are introduced when three or more scatterers are used. Although only semi-quantitative agreement was obtained with this model, it is suggested that such a model which employs well located scattering ceners (as the delta-functions provide) with adjustable individual reflection and transmission coefficients, and phase shifts can be employed in the qualitative interpretation of some of the rapidly accumulatir~g data from low energy elastic scattering of electror~s from metallic surfaces.

216

J. GEP.Sl'I~It AND P. H. c t r r ~ l t

Acknowledgment We wish to thank the staff of the Computation Center at The Pennsylvania State University for their support and cooperation. References t) I. H. Kahn, I. P. Hobson and R. A. Armstrong, Phys. Rev. 129(1963) 1513. 2) R. J. Zollweg, in: Solid Surfaces, Ed. H. C. Gatos (North-Holland Publ. Co., Amsterdam, 1~+4). 3) See, for example: L. H. Germer, C. D. Hattman, A. U. Mac, Rae ~nd E. J. Scheibner, Lo,+,-Ehergy Electron Diffraction, Bell Telephone System Tech. Publ. Monog~'aph 4364 (1963), and references iisted there; and R. Niederr~ayer and J. E~olzl, Phys. Status Solidi 11 (! 965) 651. 4) C. G. Darwin, Phil. Mag. 27 ,{1914) 675; J. J. Lander and J. Morrison, J. Appl. l."hys. 36 (1965) 1706; K. Hirabayashi and Y. Takeishi, Surface Sci. 4 (1966) 150; E. G. McRae, J, Chem. Phys. 45 (1966) 3258; and K. Kambe, Z. Naturforsch. 22a (1967) 322. 5) J. Bardeen, Phys. Re~,",44 (1936) 653. 6) H. J. Juretschke, 'Phys. Rev. 92 (1953) 1140. 7) T. L. Lou,-ks and P. H. Cutler, J. Phys. Chem. Solids 25 (1964) 105. 8) R . G . Sachs and I). L. Dexter J. Appl. Phys. 21 (1950) 1307. 9) P. H. Cut~' and ~I.J. Gibbons, Phys. Rev. 111 (1958) 39,4. 10) P. H. Cutle~: and ~I.C. Davis, St xface Sci. 1 (1964) 194. 1!) L.A. MacColi, 1~:11System Tex;h. J. 30 (1951) $88. ] 2) T. Hayasi, :~..Okada, H. Watanabe, Sci. Rept. Tokyo Univ. 1 (1964) 87. 13) D. S. Saxon and R. A. Hutnet, Philips Res. Rept. 4 (1949) 81. 14) Handbook of Mathematical Fu,~ctions, Eds. M. Abramowitz and I. A. Stegun (NBS, AMS55, 1964) p. 897, sex:. 25.5.21. 15) L. I. Schiff, QuaPrtu,,a Mechanics (McGraw-Hill Book Co., New York, 19.'i5) P. 186.