Variational calculation of the intensity-energy spectra of low - energy electron diffraction by the nickel (100) surface

Volume 56A, number 5

PHYSICS LETTERS

19 April 1976

VARIATIONAL CALCULATION OF THE INTENSITY-ENERGY SPECTRA OF LOW -ENERGY ELECTRON DIFFRACTION BY THE NICKEL (100) SURFACE Y. HAMAUZU Research Institute for Catalysis, Hokkaido University, Sapporo, Japan Received 17 November 1975 A new perturbation-variation method for calculating intensities of low-energy electron diffraction (LELD) is proposed and applied to the calculation of the LEED intensities from the nickel (100) surface. The results are remarkably better than those by the t-matrix perturbation method.

The intensity-energy spectra of low-energy electron diffraction (LEED) at a crystal surface have been extensively calculated by various perturbation methods, namely, the t-matrix perturbation method [1, 2], the r-matrix perturbation method [3 81 and the renormalized forward scattering (RFS) perturbation method [91.The f-matrix perturbation method is the simplest and fastest scheme for calculating LEED spectra, and is suitable for calculating LEED spectra from complicated surfaces such as those consisting of several subplanes [10] in a plane. However the method is less accurate in comparison with the T-matrix and RFS perturbation methods. Therefore it is highly desirable to develope a new efficient method which is more accurate than and as simple as the f-matrix perturbation method. It is the purpose of the present paper to develope a new method (“perturbation-variation method”) on the basis of the Lippmann Schwinger variational method [11], where it is attempted to miniirii~ethe errors caused by the cut-off at the Nth order term in the Born expansion series for the outgoing scattered wave i,1i(~)(r).We use a trial function of the form, N =

where

~

c~~(r),

(1)

x~(r)is the nth iterated solution to the integral

equation obeyed by li(~)(r).Using the trial functions, eq. (1) withN= 1 and N 2, we obtain the diffracted amplitude T(Kg K~)designated by a reciprocal vector g

T(Kg K~)=

T~, 1

(2)

rg

and

r~’~(1rW) 2r~~~(1 rt2~)+r~2~(lr~3~) T(K K~)=g g g g g g ~ g (1 r(1))r(2)(l r~3~ r(1)(l —r(2))2 g (3

respectively, where r~ = ~ l)/T~n)and ~ 7~2), are the contribution to the amplitude T(Kg K+) from the single, double, scatterings, respective~y. Equations (2) and (3) will be referred to as the second and fourth order perturbation-variation methods. Incidentally the third order perturbation-variation method is obtained from eq. (3) by putting r~3~0. To test the convergence of the f-matrix perturbation method and the perturbation-variation method, LEED spectra of the (00), (10) and (11) beams from the nickel (100) surface are calculated for a normally incident beam and the isotropic inelastic scattering model [4] by using 1) the matrix inversion method [10, 12], 2) the f-matrix perturbation method and 3) the perturbation-variation method. For the present model the method 1) is exact and methods 2) and 3) are topotential method l).The s-wave phase shiftapproximations ~ and the inner V 0 + iV0 are treated as parameters, which are adjusted so that the best fit of calculated LEED spectra to the experimental data by Andersson and Kasemo [l3]is obtained. The result is: = 0.5 ir, V~= 12 eV and V~= 5 eV. It is found from figs. 1 3 that the perturbation-variation calcula...

...

—

tions are far superior to the f-matrix perturbation calculations when compared with the exact calculations. 417

Volume 56A, number S

19 April 1976

1~

5%

PHYSICS LETTERS

:

~x

%

(00)Ni(I00)

I

~

X

XX

~

10-

8~=O5~r 9=~,=0 V~=5eV

~

:

~ ~It

X.

Xl1

-~

X

.JX

I ~:

(00) Ni (100)

-

&~0 3~05ir

~XX

V~ 5eV

\ X

Q5.

X

,.~X\

l0• 05~

.

X

20

I

I

40

60

I

80

00eV

Fig. 1. LEED spectra of the (00) beam are shown for a normally incident beam (0 — ~ 0). Full curve, the matrix inversion method (MIM); broken curve, the second order perturbation method (20PM); crosses, the second order perturbation-variation method (2OPVM). Peak height at 18 eV and 40 eV indicated by arrows are 1.67 and 2.70, respectively.

5%

iV 40

60

80

100eV

Fig. 3. Full curve, MIM; broken curve, 4OPM; crosses, 4OPVM. Peak height at 37 eV indicated by a arrow is 2.79.

The author would like to express his sincere thanks to Prof. T. Toya and Prof. T. Nakamura for their

r.

I

I

20

encouragement throughout the course of the present

(00) Ni (lao)

fellowship from the Japan Society for the Promotion I

=

=

~ eV

work. of Science. This work was done during the tenure of the

Ill

References

IX

05

-

[1] J.A. Strozier and R.O. Jones, Phys. Rev. B3 (1971) 3228. [21 R.H. Tait, S.Y. Tong and T.N. Rhodin, Phys. Rev. Lett. 28 (1972) 553. I

20

40

60

80

100 eV

Fig. 2. Full curve, MIM; broken curve, 3OPM; crossi..s, 3OPVM. Peak height at 45 eV indicated by a arrow is 1.67.

Also for the (10) and (11) beams, which are not reported here, similar results are obtained. Thus the perturbationvariation method proves to be a very efficient tool for calculating LEED spectra. The details of the present work will be published elsewhere.

418

[3] E.G. McRae and D.E. Winkel, Surface Sci. 14 (1969) 407. 141 C.W. Tucker, Ji. and C.B. Duke, Surface Sci. 24 (1971) 31. [5] S.Y. Tong and TN. Rhodin, Phys. Rev. Lett. 26 (1971) 711. [6] M.R. Martin and G.A. Somorjai, Phys. Rev. B7 (1973)

3607. [7] V. Hoffstein and G. Albinet, Surface Sci. 38 (1973) 506. [8] P.J. Jennings, Surface Sd. 41(1974) 67. [9] J.B. Pendry, J. Phys. C4 (1971) 3095. [10] J.L. Beeby, J. Phys. Cl (1968) 82. [11] B.A. Lippmann and J. Schwinger, Phys. Rev. 79 (1950) 469. [12] K. Kambe, Z. Naturforsch. 22a (1967) 322. [13] S. Andersson and B. Kasemo, Surface Sci. 25 (1971) 273.