Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 361{364
c International Academic Publishers
Vol. 38, No. 3, September 15, 2002
Ionization of Atoms and the Thomas Fermi Model for the Electric Field in Crystal
Planar Channels
LIU Ying-Tai,2 ZHANG Qi-Ren,1 2 and GAO Chun-Yuan2
1 CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
2 Department of Physics, Peking University, Beijing 100871, Chinay
(Received April 26, 2002)
Abstract
The electric eld in the crystal planar channels is studied by the Thomas{Fermi method. The Thomas{
Fermi equation and the corresponding boundary conditions are derived for the crystal planar channels. The numerical
solution for the electric eld in the channels between (110) planes of the single crystal silicon and the critical angles of
channelling protons in them are shown. Reasonable agreements with the experimental data are obtained. The results
show that the Thomas{Fermi method for the crystal works well in this study, and a microscopic research of the channel
electric eld with the contribution of all atoms and the atomic ionization being taken into account is practical.
PACS numbers: 61.80.+m, 61.85.+p
Key words:
ionization of atoms, Thomas{Fermi method for the crystal, electric eld in channels
1 Introduction
The physics of particle channelling in crystals was developed rapidly in recent decades. Since the early exploration of the possibility of analyzing the high-energy
charged particles by means of bent crystal by Tsganov1]
in 1976, Biryukov2] and others have made fast progress
along this line. Recently experiments on SPS at CERN3]
have shown many applications of the particle channelling
in bent crystals. On the other hand, since Kumakhov4]
rst demonstrated the existence of intense spontaneous radiation from channelled particles in the x-ray and gammaray ranges, the study for channelling radiation has gained
plentiful and substantial results. More recently, Korol
with the Frankfurt group5] has investigated a new way
for generating gamma-ray from ultra-relativistic particles
channelling in a periodically bent crystal. Along this
line, a way for generating gamma-ray laser by particle
channelling in an acoustically excited crystal has been
proposed.5 6] All these developments show that the research of particle channelling is a robust and fruitful scientic activity.
In order to accurately describe the motion and radiation of channelled particles in crystals, we need a transparent and microscopic understanding of the detailed properties of the electric eld in crystal channels. In fact,
Lindhard7] published a comprehensive theoretical study
on this topic in 1965. His major emphasis was on the
axial channelling and introduced continuum potential describing electric eld in axial channels. The continuum
potential was obtained through a uniform \spreading out"
of the single atom Thomas{Fermi potentials in the lattice
over the section from ;d=2 to d=2 (d is the distance between atoms in the string). However, the integral has been
extended to the whole string for simplication. Dierent
analytical parametrizations of the atomic Thomas{Fermi
potential have been substituted. Among them, the channel potential obtained by the integration of the Moliere
potential8;10] has been widely used. The planar continuum potential was also obtained by the similar method.
These channel potentials successfully described many experimental phenomena. However, the polarization of the
atoms and the contributions from far away atoms were
ignored in them.
Based on the success of the above potentials, we try to
describe the electric eld in crystal planar channels purely
microscopically. According to the periodically translational symmetry, one needs only to calculate the electric
eld between two adjacent crystal planes. As the lattice
atoms uniformly \spread out" over crystal planes, which
has been assumed in previous studies, the continuum potential depends only on the distance of the point considered to one of the neighboring planes. Instead of \spreading out" the atomic Thomas{Fermi potential, we derive
a self-consistent Thomas{Fermi equation for the electron
density and the electric potential generated by them in
the crystal channels. The ionization and the polarization
of the lattice atoms are therefore considered. The contribution of the atoms on planes other than the neighboring
ones are considered by the periodic boundary conditions
for the eld. By solving the equation numerically under the boundary conditions on the neighboring crystal
The project supported by National Natural Science Foundation of China under Grant No. 10075004 and the Chinese High Performance
Computing Center (Beijing)
y Mailing address
362
LIU Ying-Tai, ZHANG Qi-Ren, and GAO Chun-Yuan
planes, we can obtain the potential. In this paper, we
calculate the potential between the (110) planes of single
crystal silicon and the corresponding critical angles of the
protons with dierent energies channelling in it. The results show that the method is eective and practical in
describing the characteristics of the electric elds between
crystal planes.
We derive the Thomas{Fermi equation and the corresponding boundary condition for electrons and their electric potential between crystal planes in Sec. 2, and solve
them numerically for channels between the (110) planes of
the single crystal silicon in Sec. 3. Section 4 is a discussion.
2 Formalism
The self-consistent Thomas{Fermi equation11] for an
electron system is
(r)]3 2 r2 (r) = e2me
(1)
2
3 h 3 "0
where (r) is the electric potential at point r generated
by electrons, m is the electron mass, "0 is the electric permittivity in vacuum, h is the Planck constant divided by
2. In a perfect crystal, there is a periodically translational symmetry along the direction perpendicular to a
given set of parallel crystal planes with the period of d ,
which is the distance between two adjacent planes. One
needs therefore only to calculate the electron distribution
and the electric potential between two neighboring crystal planes. One may naturally assume that the electrons
in the outmost shell(s) of the lattice atoms may move almost freely in the crystal, while electrons in the inner
shells of a lattice atom are bounded by the atomic nucleus to form a relatively compact atomic kernel. Notice
that in a not very light atom the radius of an inner shell
is usually smaller than that of an outer shell by orders of
magnitude. It means, one may assume that the atomic
kernel is a point, well located on the crystal plane. Now
we have to share the continuum approximation with our
predecessors. That is to \spread out" the electric charge of
lattice atom kernels on the crystal planes uniformly. This
makes the model translationally symmetric along any direction parallel the crystal plane under consideration, and
the problem becomes one-dimensional. Denote the distance between a point and the middle plane of the planar
channel by y. All stationary physical variables are functions of y only. Equation (1) in this case is simplied to
d 2 (y) = e2me(y)]3 2 :
(2)
d y2
32 h 3 "0
For an innite crystal, there is a symmetry of space inversion with respect to the middle plane of the channel. The
requirement of the smoothness for the electric potential
=
p
=
Vol. 38
provides the boundary condition on the middle planes,
d = 0 :
(3)
d y =0
On the crystal plane, according to the Gauss theorem, we
have the boundary condition
(4)
"0 ddy = 2 = 2 p
in which = Nd Z 0 e is the surface charge density, d is
the interplanar distance, N is the volume density of lattice
atoms, and Z 0 is the charge number of the lattice atom
kernel. One can determine the potential in planar channels by solving Eq. (2) under the boundary conditions (3)
and (4). To facilitate the solution of Eq. (2), we introduce
the dimensionless distance by
y = b
(5)
with a suitable constant b of length dimension. On the
other hand, we introduce the dimensionless potential u by
y
y
d =
p
p
= 2d
(6)
"0 u
with d being the lattice constant on the crystal plane under consideration. Equation (2) becomes
d 2 u( ) = 2b2 e med 3 2 u3 2 ( ) :
(7)
d2
32 h3 d "0
Setting
2b2 e med 3 2 = 1 (8)
32 h3 d "0
we have
s
3
3
2
0 b = 16 NdaZd
(9)
=
=
=
p
here a0 = h=mc is the Bohr radius, = e2 =4"0 hc is the
ne structure constant, and c is the speed of light in vacuum. By this transformation, equation (7) is simplied
to
d 2 u( ) = u3 2 ( ) :
(10)
d2
=
Equation (10) is the dimensionless Thomas{Fermi equation in the crystal planar channel. By the same transformation, the boundary conditions (3) and (4) become
u0 (0) = 0 and u0 ( )j =d p 2 = db :
(11)
In the following we shall solve Eq. (10) under the boundary conditions (11). For dierent crystals and dierent
crystal planes, we need only to change constants b, d, and
d.
= b
p
3 Numerical Results
We have solved the potential in the (110) planar channels of the single crystal silicon numerically. The solution
is done under the assumption that electrons in the 1s and
2s subshells are bounded with the silicon nucleus to form
an atomic kernel, and these kernels \spread out" over the
No. 4
Ionization of Atoms and the Thomas{Fermi Model for the Electric Field in Crystal Planar Channels
363
a positively charged particle in a planar channel,7]
s
e(Eymax ) (13)
(110) planes to form uniformly charged thickless crystal
planes. Based on this consideration, the parameters of
(110) planes for perfect silicon crystal are calculated to
be Z 0 = 10, d = 1:92 A b = 0:2366911 A d = 5:43 A
= 4:055919 d=2"0 = 470:6877 V. The numerical solution for the potential and the electric eld strength are
shown in Figs. 1 and 2 respectively. The ordinate of Fig. 1
represents the dierence
= (y) ; (0) :
(12)
Now let us compare our results with experimental data of
the critical angles. The critical angle is dened as the
smallest angle, so that a particle incident with an angle
larger than could not be channelling. This is an observable quantity. Theoretically, it may be calculated for
c
p
in
here e and Ein are the charge and the energy of the incident particle, ymax is the maximum distance from the
middle plane. One may set ymax = dp =2 ; a, a is a screening distance in some sense. In Moliere potential,12] it is
chosen to be a = aTF = 0:194 A. In our case, it should
be the radius of the atomic kernel. It is the radius of the
2s shell of the silicon atom. The radius of the ns shell
for an atom may be estimated to be n2 a0 =Z 00 , in which
a0 is the Bohr radius, and Z 00 is the charge number of the
nucleus-electron system inside the ns shell. Setting n = 2
and Z 00 = 12, we have a = a0 =3.
c
c
Fig. 1 The interplanar potential for Si channels (110).
Fig. 2 The interplanar electric eld for Si channels (110).
Table 1 Comparison of the calculated critical angle with the experimental critical angle for proton
of dierent energies.
E (MeV)
in
2.8
4.8
5.0
6.8
8.8
2c (experimental data)
0:29
2c (Moliere)
0:2469733
0:1886320
0:1848209
0:1584826
0:1393317
0:20
0:19
0:18
Moliere potential12 13] in a crystal planar channel is
2c (Thomas{Fermi)
0:2864789
0:2188019
0:2143812
0:1838303
0:1615959
(y) = 2ZeNdp
with
3
exp; y aTF
=1 X
i
i
i
i
= (0:1 0:55 0:35) = (6:0 1:2 0:3) :
The data in the third column of Table 1 were calculated by this formula.
(14)
364
LIU Ying-Tai, ZHANG Qi-Ren, and GAO Chun-Yuan
Vol. 38
The critical angles for protons calculated by our method and those by use of Moliere potential are compared with
corresponding experimental values in Table 1 (experimental data are taken from Ref. 14]). Our theoretical result
seems better than those from Moliere potential in this example.
4 Discussion
From the above results, we see that the eects of polarization and ionization of the lattice atoms as well as the
contributions of atoms other than those on the neighboring crystal planes may be important for understanding the
details of particle channelling in crystals. It may be the reason which makes our theoretical results better than those
obtained from Lindhard standard potential and Moliere potential. One thing may be worthy to emphasis. It is
that in our theory there is no adjustable parameter. We have indeed introduced some approximations. They are
Thomas{Fermi approximation, continuum approximation which spreads out the electric charge of atom kernels on the
crystal planes uniformly, the zero temperature approximation, and the nite procedure approximation in numerical
calculations. Therefore, to improve our result, we need to eliminate these approximations as more as possible. Thomas{
Fermi formulation for the crystal lattice itself may eliminate the continuum approximation. We would try it in the
following work. Hartree{Fock theory for the crystal lattice is also attractive. We hope it may be realized in the near
future. The nite temperature treatment may be realized at dierent levels of complexities. It has been realized at the
phenomenological level.12] We hope a temperature-dependent Thomas{Fermi or Hartree{Fock formulation may also
be realized later.
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