Calculation of energy and angular distribution of positronium ejected from surfaces

Nuclear

392

CAL~~~ON OF ENERGY AND ANGULAR FROM SURFACES Shigeru

and Methods

DIS~B~ION

in Physics

Research B33 (1988) 392-395 North-Holland, Amsterdam

OF POSI~ON~~

EJECmD

SHINDO

Department

Akira

instruments

of Physics, Tokyo Gakugei University, Koganei, Tokyo 184, Japnn

ISHII

The Hackett Laboratory,

Imperial College, Prince Consort Road, London SW 282,

We present a theory of positronium different processes, i.e. the direct process where the image-potential state positron positronium ejected from surfaces, which

UK

formation at solid surfaces. It is shown that positronium formation occurs by the two where the work function positron combines with a surface electron, and the indirect process combines with an electron. We calculate the energy and the angular distributions of the are compared with the experiments.

1. Inception

2. Theory of direct Ps fo~ation

It has recently been recognized that a low energy positron beam provides a powerful probe for studying solid surfaces [l]. We have recently developed a theory of the formation of the electron positron bound state, called positronium or Ps, from spontaneously ejected positrons from solid surfaces [z-4]_ Our theoretical results can be summarized as follows: (1) Ps is formed from positrons ejected from a metal surface, since the Ps state is unstable inside bulk metals. The kinetic energy of the ejected positron is the negative of the positron work function from the surface, which is called work function positron. (2) Ps formation can be theoretically described as a kind of resonant ion neutralization process at surfaces [5] because the energy level of the Ps is located inside the surface band. (3) The energy spectrum of the Ps so formed can be related to the chemisorption function A(e) which is proportional to the surface electron density of states in the first approximation. (4) The semiclassical (trajectory) theory of ion neutralization is not adequate for Ps formation; instead a quantum description for the motion of the positron and the Ps is needed. Higher order Born theory is important to explain the experimental Ps spectra. (5) The positrons trapped into the surface i~ge-potential state also combine with the electrons to form Ps, which is ejected spontaneously from surfaces. In the present paper, we calculate the energy and angular distribution of Ps ejected from surfaces and we compare it with experimental results [1,6,7]. We use the wide electron band approximation for the chemisorption function A(e), which may be valid for an aluminum surface.

The probability of the Ps formation from the work function positron, called direct Ps formation, per unit time is given by the golden rule formula for resonant charge exchange [4]. In the first Born approximation, it becomes

0168-583X/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

k f +e,-Q-e,)).

(1)

In this formula, e, represents an energy level of the Ps associated with the wave function 1a); E, is the kinetic energy of the Ps emitted from the surface associated with the wave function 1f}; I k > represents the initial state of the electron of energy level e,; and 1i) represents the initial state of the positron of energy (- @+) the negative of the positron work function. V in eq. (1) is the screened Coulomb interaction between the positron and the electron: Y= -e2

exp(-Dir--r+

])/]r--r+

],

(2)

where p is the screening constant, r_ and r+ denote the coordinates of the electron and the positron, respectively. N(e,)

=

[l + exp($_

+ e,),/k,T]-‘,

(3)

where T is the temperature of the solid, k, the Boltzmann constant, and c#_ is the electron work function from the surface. The matrix element in eq. (1) is assumed to be factorizable as

(f(aIYlk)li)= where

Y?k%Z

Vnk represents

the hopping

(4) term of the surface

393

S. Shindo, A. Ishii / Angular distributions of positronium from surfaces

electron with the Ps state. This can be related to the Ps chemisorption function A(e) as A(e)=~CI~~ak12S(e-ek).

(5)

&#

k

In our present calculation, the wide band electron model, where A(e) is independent of e, is used. We also assume a surface exponential decrease of the surface electron density. ui, in eq. (4) is the matrix element of the center of mass part, u(Z), of the hopping matrix Y in eq. (1) where Z denotes the center of mass normal coordinate of the relevant electron and positron. Since u(Z) is proportional to the electron density from the surface, we choose that u(Z) = exp( - CXZ), then

I~~i12~I~fIex~(-az)Ii~12 = [(Pi-P,)*+cq

L

8.

c

A =nftu

(6)

I

I

I

7

-2

0

I

I

I

I

I 2

I

I 0

I

I

b

c

IU/i12

&

I

P,,imrad)

where Pi is the initial momentum of the center of mass, is the final momentum, P, = Pi = (+2M++)“2, P, 2ME cos2 e) 1/2 M is the mass of Ps, and 13 the emission angle of Ps measured from normal to the surface. By normalizing eq. (1) by the initial flux of the center of mass of the relevant positron and electron, u, = ( -2+,/M), we obtain the normal energy distribution of the direct Ps as dn dE,

I 2

I -2

-I

P,, I mrad )

(7)

x~~~~N(~,+E,+E,,+O~)~E,~,

where E” = 2sh*/(MS,), So is the unit area for the surface electron, E,f= = Min.(E’, E, tan2v), and v is the detector’s acceptance angle. We note that the kinetic energy of the ejected Ps is limited by E < -Gps, where

I

I

I

I

I

I

I

I

I

c

~p,=~-+~++ee,.

2

P,,Cmrad

I

7

-2

II

1

Fig. 2. Contour map of the ejected momentum distribution [3]: (a) calculated using the Born approximation; (b) using normalized Born approximation; (c) experimental contour map by Lynn et al. [7]. Parameters are chosen as a = 0.5, and A = 0.2.

I / ‘I/I III,,

2

4

6

8

I ’ 10

,

I 12

Fig. 1. Normal energy distribution the formed positronium [4]. Dashed line is the contribution from the direct process of eq. (7) and dotted line is that from the indirect process of eq. (13). Experimental Ps energy spectra using Al(111) surface by Mills at al. [l] are marked by vertical lines. The parameters are Eb = -0.045 (a.u.), @p, = 0.095 (a.u.), A = 0.2 (a.u.). a = 0.5 (a.u.) and r in eq. (13) is 4.0 (a.u.).

In previous papers [2,3], we have shown that this first Born theory is insufficient to explain the low energy region of the experimental Ps spectra but that higher order Born effects are important. The higher order term can be represented by the normalization of the Born approximation as [2,3] d ,,N.B. dE,

dnBom dE,

where dnBom/dEl

1

(8)

[I + .aom,/4]* ’

is the Born energy distribution VI. ION-SURFACE

INTERACTION

394

S. Shindo, A. Ishii /Angular

distributions of posiironium from surfuces

calculated in eq. (7) and n Bom is its integral where the initial momentum of the center of mass is (2ME,)“‘. In fig. 1, we show our theoretical calculation of eq. (8) by the dashed line and compare it with the experiment of Mills et al. [l]. We see that the agreement is qualitatively good except in the low-energy region, This discrepancy will be discussed in section 3. The two-dimensional momentum distribution of the ejected Ps can be expe~ment~y measured by using the two-dimensional angular correlation of the annihilation radiation (ZD-ACAR) [6,7]. This quantity can be calculated from our theory as follows: d2n dP, dP, =(A/?rti~)(2?r)-‘(P~,-(P:

+P:))]uj,]‘.

(9)

where A2 P&J2 M = Gps. In fig. 2, we show our calculation and compare it with the Born calculation and the experiments by Lynn et al. [7]. It is asserted that our normalized Born calculation agrees much better than the Born calculation in the low-momentum region. This indicates that the higher Born effect cannot be ignored in this region.

unit time is given by the golden rule formula in the first Born approximation,

k f xS(Eb+ek-+e,)),

(10)

where I ii,) represents the positron image potential binding state and Eb is the energy level. We assumed here that the trapping of the positron into the image potential state and emission of the positron in the form of Ps are statistically independent. This assumption is valid when the transition time of the positron into the image potential state r( = m,Ga’) is much shorter than the lifetime of the positron in that state. For the surface electron, the lifetime of the surface image-potential state is calculated by Echenique et al. [13] and it is much shorter than r. Hence indirect Ps formation probability can be written as

ni, = 7 x Pi,.

(11)

The energy spectra of Ps formed by the indirect process dnim/dE is

dnim

A =Q--

ahu

dE,

xfEi;‘““N(e,+E, +E,,-Eb) 0

dE,,.

(12)

in eq. (12) is calculated using the wave function of the image potential state positron [4] as

3. Indirect Ps fo~tion

ufi

The positrons ejected from a surface with the kinetic energy of the work function lose energy due to the inelastic scattering from electrons and phonons at the surface. Some of those positrons, which lose more than their kinetic energy, will be trapped into the positron bound state by the induced image-potential at the surface. The Ps formation from such surface imagepotential positrons, which we call the indirect process, is interesting in that it yields information about the energy level of the image-potential positron bound state. The energy level of the surface positron imagepotential state has been calculated [X-10] where it has been shown that the energy level is close to Eb = l/32 (a.u.) which is a value obtained using the classical static electric image potential (V= e2/4z+). On the other hand, the energy level of the surface electron imagepotential state has been measured using the inverse photoemission method [ll] and it is close to Eb = l/32 (a.u.). Because of the similarity between the positron and the electron surface image-potential, the energy level may approximately be described by Eb = l/32 (a.u.) for the positron case. Then, this image-potential positron will be ejected spontaneously due to recombination with the surface electron, since the electron work function from the aluminum surface is 0.156 a.u. The probability for the indirect Ps formation per

l”fi12=

I(flexP(-~z)Iiim)12 (13)

where aa is the Bohr radius. Using the same normalization with eq. (8), the higher order Born effect can be taken into account for the indirect process. It should be noted that the kinetic energy of the indirect Ps is limited by E<

-cp_-e,+Eb.

114

The calculated Ps spectrum due to the indirect process is shown in fig. 1 by the dotted line. It is asserted that the experimental sudden increase at the low energy region, which is also seen in the recent experiment by Howell et al. [13], can be explained by the contribution from the indirect image potential process. In fact, the sudden increase occurs for energies given by wndition (16). In fig. 1, the binding energy of the positron, Eb, is chosen as -0.045 (a.u.) to fit with the experiment.

4. Concluding renmrks We have developed the theory of positronium (Ps) formation at surfaces. We have shown that Ps formation

S. Shindo, A. Ishii / Angular distributions of positronium from surfaces

occurs via two processes, i.e. the direct Ps formation where the work function positron combines with an electron, and the indirect image potential state process where the positron trapped into the image potential state combines with an electron. From the experimental Ps spectra, we may determine the value of the positron work function from the solid and the binding energy of the surface image potential state. The binding energy of the image-potential state of the aluminum (111) surface, Eb, is determined as -0.045 (a.u.) from our model. This value is close to the binding energy given by the electrostatic image-potential V= e2/4z+. In our present calculation of the indirect Ps formation, we assumed that the parallel momentum of the positron trapped into the image potential state is zero. According to the experiment by Howell et al. [12], the contribution from the image potential positron to the Ps energy spectra is more important when the detector’s acceptance angle is larger. This indicates that the parallel momentum of the image state positron cannot be ignored. However, in the experiments of the momentum distribution of Ps using 2D-ACAR, there does not appear evidence for the indirect surface image-potential Ps formation. We consider that the momentum distribution of indirect Ps is widely distributed and that it was experimentally subtracted as the surface chemisorbed part of Ps momentum.

VI. ION-SURFACE

INTERACTION

395

This study is partly supported by the Special Project Research of Ion Beam Interaction with Solids from the Ministry of Education, Science, and Culture of Japan, and by Japan Society of Promotion of Science.

References

VI A.P. Mills, Jr., L. Pfeiffer and P.M. Platzman, Phys. Rev. Lett. 51 (1983) 1085. PI S. Shindo and A. Ishii, Phys. Rev. B35 (1987) 8360. [31 S. Shindo and A. I&ii, Phys. Rev. B36 (1987) 709. [41 S. Shindo and A. Ishii, Phys. Scripta to be published. [51 S. Shindo and R. Kawai, Surf. Sci. 165 (1986) 477. 161 R.H. Howell, P. Meyer, I. Rosenberg and M.J. Fluss, Phys. Rev. Lett. 54 (1985) 1698. 171 K.G. Lynn, A.P. Mills, Jr., R.N. West, S. Berko, K. Canter and L.O. Roelling, Phys. Rev. Lett. 54 (1985) 1702. PI N. Barberan and P.M. Echenique, Phys. Rev. B19 (1979) 5431. 191 G. Barton, J. Phys. Cl4 (1982) 4727. [lOI D.B. Lanbrick and G. Siopsis, Surf. Sci. 120 (1982) L491. [Ill V. Dose, W. Altmann, A. Goldmann, U. Kolac, and J. Rogozik Phys. Rev. Lett. 52 (1984) 1919; D. Staub and F.J. Himpsel, ibid 52 (1984) 1922. 1121 R.H. Howell, I.J. Rosenberg, M.J. Fluss, R.E. Goldberg and R.B. Laughlin, Phys. Rev. B35 (1987) 5303. 1131 P.M. Echenique, F. Flores and F. Sols, Phys. Rev. Lett. 55 (1985) 2348.