J. Plasma Fusion Res. SERIES, Vol.7 (2006) 183-186

Single Electron Capture by Back Scattered Protons from
High-Melting Temperature Metals
KATO Daiji, SUNO Hiroya, KATO Takako, KIMURA Mineo1 and TANABE Tetsuo2†
National Institute for Fusion Science, Toki 509-5292, Japan
1 Kyushu University, Fukuoka 812-8581, Japan
2 Nagoya University, Nagoya 464-8603, Japan（Received: 5 October 2004 / Accepted: 21 December 2005）

Abstract
Formation of excited hydrogen atoms above a tungsten surface was studied by means of the semi-classicaltheory. Occupation probabilities of the excited levels were calculated for proton velocities of 0.1 – 1.0 a.u.
Keywords:
ion-surface interaction, electron capture, non-adiabatic transition
1．Introduction
Excited state abundance in neutrals reflected at refractory metals has been an issue of study concerning
the plasma-wall interaction in nuclear fusion devices.
Dα emission has been observed from the reflected neutrals of a deuteron beam at a molybdenum surface [1].
However, the mechanism of the excited state formation
has been little understood. Excited levels of the atomic
hydrogen are above the Fermi level of the molybdenum
by a few eV. Surface electrons hardly transfer to suchthe shallow levels via tunneling or classical-over-thebarrier, unless some surface impurities decrease the effective work function substantially. The eﬀect of thesurface impurities has not been studied in the presentwork; we have investigated other (non-trivial) mechanism first.
In the present work, the non-resonant single-electrontransfer was studied by means of the semi-classical theory (electronic transition was treated quantum mechanically, while proton motion was represented by classical trajectories). Early studies on the electron transfer
have been undertaken with the aid of simplified interaction matrix elements and/or the adiabatic approximation, e.g. Brako and Newns [2] and Burgdörfer et al.
[3]. The present study is based upon direct numericalsolutions of the time-dependent Schrödinger equation
including accurate interaction potentials. We calculated
occupation probabilities of the excited levels due to thenon-resonant single-electron capture by a proton from atungsten surface.
Atomic units ( = me = 1) are used throughout unless otherwise stated.

2．Theoretical method
A hydrogen atom (a proton and an electron) above
a metal surface is illustrated in Fig. 1. The electron
moves under influences of three potentials: theCoulomb attractive potential of a proton, the attractive
potential of the surface dipole layer and the exchangecorrelation eﬀect (VeI ), and repulsive potential of a pileof electron density at the surface induced by the proton (V pI ). The semi-empirical formula is used for VeI oftungsten proposed by Jennings et al. [4],⎧
λ(x−x0 )⎪/2 (x − x0 ) , x < x01−e⎪1⎪⎨
VeI (x) = ⎪⎪2⎪⎩ −2V0 1 + Ae−B(x−x0 ) −1 , otherwise,(1)where A = 4V0 /λ − 1 and B = 4V0 /A are constants de-Fig. 1 Coordinate system of a hydrogen atom above ametal surface.

C 2006 by The Japan Society of Plasma

Corresponding author’s e-mail: kato.daiji@nifs.ac.jp† Present address: Kyushu University, Fukuoka 812-8581, Japan

Science and Nuclear Fusion Research183

Single Electron Capture by Back Scattered Protons from High-Melting Temperature Metals

solved numerically using the discrete-variable representation [8] of the Jacobi-polynomial. Figure 3 shows thecalculated eigen-energies (real part of k 2 ) and broadening of the atomic levels (imaginary part of −k 2 ) asa function of D for the tungsten surface. The eigenenergies separate into energy levels of the atomic hydrogen and discretized energy levels of the conduction
electrons at large D values. On the other hand, near
the surface, the atomic levels are broadened as well as
shifted by the surface potentials.
The motion of the proton is represented by classical trajectories, as the de Broglie wavelength is much
shorter than the atomic scale for proton energies considered here (>100 eV). Once excited hydrogen atoms
dive into the metal, electrons at the excited levels are
stripped immediately due to the strong screening eﬀectof conduction electrons. Thus, only outgoing part of

Fig. 2 Effective potential of electron for tungsten surface
along x-axis (θ = 0 geometry). Bold solid line issum of the three potentials. D = 10 a.u.

termined by matching the potential and its derivative at
x = x0 . x0 ≈ −0.93 is the position of the image planeand |x0 | was determined using an empirical formula by
Ossicini et al. [5]. V0 ≈ 0.565 a.u. is given by the sumof the Fermi energy and the work function of tungsten.
λ ≈ 1 is obtained by fitting Eq. (1) to an planar averaged potential of the linearized augmented-plane-wave
calculation [4]. V pI is obtained from the static linear
density response function of the conduction electrons
whose Kohn-Sham wave functions are calculated using
the VeI [6]. The potentials are shown in Fig. 2 for tungsten.
Electronic wave functions are written in the spherical coordinates whose origin is located at the proton. Geometry of the electronic system is restricted toθ = 0; the angular freedom of motion is omitted in thepresent model. Because of the lowest eﬀective potential barrier, the electron transfer takes place most probably along x-axis (θ = 0 geometry). Accordingly, theone-dimensional Schrödinger equation for each protonsurface distance D is written as,
11 d2I
I+V−(r;D)+V(r;D)Φ(r; D)−ep
2 dr2 r=  (D) Φ (r; D).

Fig. 3 (a) Eigen-energies for tungsten surface as a function of D. Solid curves are the eigen-energies,
dashed curves atomic hydrogen levels shiftedby the classical image potential, a dotted curve
the top of the potential ridge (see Fig. 2), and adashed-dotted line the Fermi level of tungsten(about –0.19 a.u.).

(2)

A set of discrete eigen-energies  are obtained by solving Eq. (2) in a finite radial coordinate range, r ∈[0, R], under boundary conditions of Φ (0; D) = 0 andΦ−1 dΦ/dr|r=R = a, where a is a constant. To represent
dissipation of the atomic state near the surface, the outgoing wave boundary condition is imposed to the wave
√function, (d/dr − ik) Φ|r=R = 0, where k = 2 measured from −V0 . As a result, the Hamiltonian matrixof Eq. (2) becomes quadratic of k. It is transformed tothe linear non-Hermitian matrix equation of k [7], andFig. 3 (b) Level broadening of atomic hydrogen abovetungsten surface as a function of D. Dottedlines are ﬁtting to exponential functions of D.
184

Single Electron Capture by Back Scattered Protons from High-Melting Temperature Metals

the trajectory is relevant to the excited hydrogen atoms
emitted from the surface. For simplicity, we use thestraight line trajectory normal to the surface. The velocity change of the proton due to the image force andelectronic transition is omitted, since it may be not significant for the proton energies considered here.
At the starting time t0 , the proton is placed just above
the surface with a small distance of D0 , and then recedesfrom the surface with a constant velocity of v < 0. The
initial electronic state of the hydrogen-tungsten system
is assumed to be in the ground state: All electronic levels below the Fermi level of the tungsten are equally occupied, while higher levels are empty. The initial wave
function is expanded in terms of the eigen-state wave
function at D0 ,
Ψ (x, t0 ) = e−ivx+iv /2t02

vanishingly small as expected in the adiabatic approximation. As the velocity increases, the probabilities ofthe excited levels grow, while that of the ground leveldeclines. The enhancement of the non-resonant electrontransfer to the higher excited levels may be explained
as follows. Near the surface, top of the potential ridge
lying between the proton and the surface dives below
the Fermi level (see Fig. 3(a)). As the proton recedesfrom the surface, the top rises out of the Fermi level.
Faster the proton recedes from the surface, more theelectronic wave function is kept sitting astride of thepotential ridge and elevated to the higher atomic level.
The wave function astride of the potential ridge represents a diabatic state which is promoted to the higher
excited levels along the top of the potential ridge.
A hump is seen in the n = 2 occupation probability
at low velocities. It may be attributed to the oscillatorytransition between quasi-resonant states of the n = 2
level and a discrete conduction level of the metal slab.
With the larger slab width (the higher number density of
the conduction levels), the hump becomes less significant. For the semi-infinite metal, the hump should disappear after the transition amplitude is integrated overthe continuous conduction band in which the atomic
levels are embedded (bandwidth eﬀect).
As a concluding remark, it is noted that the present
model is based upon the dispersion relation of the quasifree electron-gas. However, transition metals like tungsten have substantially diﬀerent density of states fromthat of the free electron-gas. An improved model taking account of a correct density of states is under development. The Auger deexcitation might depopulate
the excited levels, while it is left for future studies. To
compare the present occupation probability of the n = 3
level (Fig. 4) with the Dα emission measurement, i.e.
Ref. [1], the velocity distribution of reflected hydrogenatoms must be taken into account. The comparison will
be made in future studies.

Φk (r; D0 )eiσk Θ(F − k (D0 )),k(3)isakinematicalfactortomeetthewhere e
Galilean invariance of the time-dependent Schrödingerequation. In the present study, we included about 300
eigen-states in the expansion. The random phase approximation is applied to the initial state. The wave
function is propagated numerically in the time domainusing the split-operator spectral method [9]. At a sufficiently large D (after a long time t), the occupation
probabilities of the atomic levels are obtained by projecting Ψ (t) on to Φk (D(t)), and those of the conduc2
tion levels by projecting on to e−ivx+iv /2t Φk (D (t)).
−ivx+iv2 /2t0

3．Results and discussion
Present calculations were performed for a tungsten
slab whose width is about 100 a.u. (about 5 nm). Figure 4 shows the occupation probabilities of the atomiclevels as a function of the proton velocity. At low velocities, the occupation probability of the ground levelis almost unity, whereas those of the excited levels are

Acknowledgements
The authors thank to Prof. R. Janev and Prof. R.
More for fruitful discussions. D. K. thanks to Dr. X.
-M. Tong for providing us with his own version of theKohn-Sham equation solver code.

[1][2][3]

Fig. 4 Level occupation of atomic hydrogen emitted fromtungsten surface.

185

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