Influence of photon generation on the low-energy electron band-reflection
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
1999
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| Cite this: | Influence of photon generation on the low-energy electron band-reflection / I.E. Tikhonenkov // Вопросы атомной науки и техники. — 1999. — № 3. — С. 110-112. — Бібліогр.: 12 назв. — англ. |
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Tikhonenkov, I.E. 2015-05-14T20:21:28Z 2015-05-14T20:21:28Z 1999 Influence of photon generation on the low-energy electron band-reflection / I.E. Tikhonenkov // Вопросы атомной науки и техники. — 1999. — № 3. — С. 110-112. — Бібліогр.: 12 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81358 en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Influence of photon generation on the low-energy electron band-reflection Влияние излучения фотонов на зонное отражение электронов низких энергий Article published earlier |
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Influence of photon generation on the low-energy electron band-reflection |
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Influence of photon generation on the low-energy electron band-reflection Tikhonenkov, I.E. |
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Influence of photon generation on the low-energy electron band-reflection |
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Influence of photon generation on the low-energy electron band-reflection |
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Influence of photon generation on the low-energy electron band-reflection |
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Influence of photon generation on the low-energy electron band-reflection |
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influence of photon generation on the low-energy electron band-reflection |
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Tikhonenkov, I.E. |
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Tikhonenkov, I.E. |
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1999 |
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English |
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Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Влияние излучения фотонов на зонное отражение электронов низких энергий |
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1562-6016 |
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Influence of photon generation on the low-energy electron band-reflection / I.E. Tikhonenkov // Вопросы атомной науки и техники. — 1999. — № 3. — С. 110-112. — Бібліогр.: 12 назв. — англ. |
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AT tikhonenkovie influenceofphotongenerationonthelowenergyelectronbandreflection AT tikhonenkovie vliânieizlučeniâfotonovnazonnoeotraženieélektronovnizkihénergii |
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INFLUENCE OF PHOTON GENERATION ON THE LOW-ENERGY
ELECTRON BAND-REFLECTION
I.E.Tikhonenkov
NSC KIPT, Kharkov, Ukraine
1. INTRODUCTION
An interest to the small angle reflection of
relativistic particles from the crystal surface was
initiated due to Kumakhov’s papers [1, 2] where a
possibility to carry the charged particle beams by
reflection from the solid surface was shown. For
reflection of positive particles with intermediate and
high energies under glancing incidence the classical
description is applicable and if the conditions of planar
channeling are satisfied the mirror reflection will be
observed at incidence angle less than the critical angle
of planar channeling [3-5]. During reflection a positive
particle is deflected by the crystal surface and
electromagnetic radiation is generated. First theoretical
consideration of this phenomenon was performed by
Rozhkov [3, 4]. Under axial semichanneling particles
are expected to be scattered by laying on a surface of
atomic rows. When works [1, 2] were published the
reflection of low-energy electrons due to scattering
from surface atomic strings was studied by computer
simulation [6] based on the pair collision model. It was
shown that a significance part of particles (up to 15%)
may be reflected owing to a correlative interaction.
Later the mechanism of small-angular reflection of
electrons and positrons, the main feature of which is the
rainbow scattering on atomic strings, was proposed in
[7] and the processes which cause electromagnetic
radiation have been investigated [8-9]. However, from
viewpoint of classical theory the reflection of low-
energy electrons under planar semichanneling is
impossible and the quantum consideration is required.
Authors of [10] proposed a specific quantum
mechanism of small-angular reflection of electrons
from crystas surface i.e band reflection. The full
theoretical and numerical investigation if this effect was
made in [11]. It was established, that at some energies
and crystals one can observe a band reflection for
angles up to half of Lindhard angle. In addition, it has
been noted that a band reflection is a result of
coherence interaction of electron with a system parallel
to surface atomic planes. That objection may follow to
increase a radiation transition probability. In our work
[12] this problem was studied and a new quantum
coherent mechanism of electromagnetic radiation under
planar semichanneling of electrons has been
manifested. It has been shown that the transitions with
caption of particles in surface channeling have a higher
probability and it is expected to change the number of
reflected particles. In present paper the influence of
radiation processes on band reflection appearance is
considered and showed that these effects have a
maximum magnitude at quite different conditions.
2. REFLECTION COEFFICIENT
If spine effects ares neglected, the wave function
of electron under planar semichanneling is a solution of
well- known Schrodinger equation with the relativistic
electron mass:
( )
θ
ψψ
pp
M
p
E
xxUEx
M
==
=−+′′
⊥
⊥
⊥
⊥
,
2
,0)()()(
2
2
2
. (1)
Here γmcEM == 2 is the relativistic mass of
electron; E and m are its energy and rest mass; θ is the
glancing angle; function U(x) is equal to 0, if x < 0
(vacuum) and has the period dp, if x > 0 (crystal), where
dp is the distance between atomic planes being parallel
to the surface. Under condition x > 0 a solution of
Eq. (1) satisfies the Bloch theorem:
( ) ( ) ( )
( ) ( ) ( )xiqddx
xiqddx
pp
pp
ψψ
ψψ
′=+′
=+
exp
,exp
(2)
In a forbidden band of spectrum of Eq. (1) the
quasi-wave number q is complex and the amplitude of
ψ(x) decreases as the distance x from surface increases.
In this case a total reflection of particles from surface is
realized [10]. The positions of forbidden bands are
determined by condition D > 1, where the discriminant
D is defined by the formula
( ) ( )( )
1)0(,0)0(,0)0(,1)0(
2
1
2211
21 ,
=′==′=
′= +
ψψψψ
ψψ pp ddD
.(3)
The probability of radiation transition is dependent
on a matrix element of the flax operator. It has been
shown in [12] that the main part of this element arises
from the «crystal» wave function ψcr or, in other words,
the function ψ(x) under x > 0. The value dxcr
2
ψ is
proportional to a probability that an electron in the
interval [x, x + dx] will be found. Let us consider N0
particles in the same quantum state. In this situation
dxcr
2
ψ is proportional to a number of particles dN(x),
which have been detected in the interval [x, x + dx]:
∫
∞
=
0
22
0 )()()( dxxdxxNxdN ψψ .
If the transverse energy lies in a forbidden band
a total reflection occurs and all N0 will be reflected by
crystal. In this case dN(x) defines the number of
particles which have penetrated from the surface up to a
distance x and then undergo reflection in [x, x + dx].
Therefore, а value dN(x)/N0 is equal to the probability
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. №3.
Серия: Ядерно-физические исследования. (34), с. 110-112.
110
of such events. Denoting dN(x)/N0 as Wref dx and taking
into account (2), one can obtain that the density Wref(x)
is defined from the following formulae
( )
( )
∫=
−−
−−
=
=
pd
pref
p
pcr
e
ref
pe
ref
dxx
d
dq
dNq
N
dN
x
xW
0
22
2
2
)(1
,
2exp1
2exp1
,
)(
)(
ψψ
ψ
ψ
(4)
where Ncr is the common number of atomic planes
being parallel to the surface. If electron has been
reflected at the distance x, it moved in the crystal during
⊥= υxt 2 , where 12 −=⊥ γ
γ
θυ c is the classical
velocity along x direction. If, the function Wref(x) is
known then it is easily to derive the analogous
distribution of particles with respect to time intervals t:
ref
pedN
t
tWref 2
2
2
)(~
2)(
ψ
υυψ ⊥⊥= .
It follows from this equation that the value =)(~ tNd
dttWN ref )(~
0= is the number of electrons which will be
in a crystal during the time t and cross the crystal-
vacuum interfaceat the moment between t and t + dt.
Some of this electrons will make radiation transition.
Let )(~ tNd rad denote the number of such transitions
( )( ) )(~exp1)(~ tNdtWtNd radrad −−= ,
where Wrad is full probability of photon radiation.
Common number of radiation transitions radN~ is equal
to
( )( )
∫
−
−=
=∫ −−=∫=
∞
⊥
∞∞
0
0
00
)(
2
exp1
)(~exp1)(~~
dxxW
xW
N
tNdtWtNdN
ref
rad
radradrad
υ
.(5)
Using (4) one can obtain from (5) the part of
electrons which have made the transition
ref
rad
e
r
N
N
n
rad 2
2
1
ψ
ψ
−= . (6)
and the part of elastic scattered electrons:
rad
rad
e
r
rad N
N
n
2
2
ψ
ψ
= . (7)
In (6)-(7) used were the values Nr and
rad
2
ψ which
are expressed from the formulae
∫ −=
+−−
+−−
=
⊥
⊥
⊥
pd
rad
prad
p
rad
cr
rad
r
dxx
W
x
d
d
W
q
dN
W
q
N
p
0
22
2exp)(1
,
2exp1
2exp1
υ
ψψ
υ
υ
(8)
Let in the initial f-state the electron wave vector
is fk
. The g-state of electron after transition and the
coefficient of reflection ( )κfgR are determined by fk
and by the wave vector κ of photon radiated.
Therefore the effective coefficient of reflection in the f-
state may be calculated as
( ) ( )∫+= κκκ
fgf
rad
rad
refffrad RWd
W
n
nRR 3
0 , (9)
where R0f is the coefficient of reflection in the initial
state, ( )κfW is the probability of transition from state
(f) if the photon with the wave vector κ is emitted.
Integrating in (9) is performed over all the possible
forms of f-state radiation transitions [12].
3. DISCUSSION
The band reflection has a significance value if
there is the wide forbidden band in the E⊥ > 0 region the
low bound of which lies near the E = 0 level. An
example of such a situation is the band «+1» in Fig. 1
Here the curve D(ε) (ε = (E⊥/Up)1/2sign(E⊥),
Up = 2pZnpe2r0 is the typical value of a barrier of
continuous plane potential, Z denotes the atom number
of crystal atoms, np is the surface atom density of
reflecting plane, r0 is a screening length for potential of
an isolated atom in the Thomas-Fermi model for the
case of incidence of 0.819 MeV electrons on (111)
surface of silicon (the value D was defined in (3)). It
has been shown in [12] that only for neighbouring states
radiation transitions will occur as an initial state belong
to the band «+1». As the band «+1» has a large width
for all states after transition with the sufficient accuracy
the condition ( )κfgR ≈1 will be satisfied. Using (6)-(7)
and that for the band «+1» R0f = 1 we have from (9) that
10 =+≈ radrefffrad nnRR .
111
Fig. 1
It follows from this relation that under band
reflection conditions the appreciable decreasing in the
reflection coefficient owing to radiation processes is
unexpected.
Let the transverse energy of initial state belongs
to the band similar to that of «+2» in Fig. 1. In this case
the band reflection actually is absent because of the
very small forbidden bandwidth. For such states the
probability of radiation transition is large and electrons
goes to states with a small negative value of E⊥. Such
transitions correspond to radiation caption of particles
into surface channeling motion. Because the state with
E⊥ < 0 has a coefficient of reflection equal to zero, it is
reasonable to substitute in (9) the relation ( )κfgR ≈0
and after that action we have
refrefffrad nnRR =≈ 0 .
Using (4), (7), (8) and taking into account that |q|
Ncr>>1, it is possible to find for value of nref the
following condition
( )
( )( ) 1
||2exp1
||2exp1
<
+−−
−−
≤
⊥ prad
p
ref dWq
dq
n
υ . (10)
We can see that the reflection coefficient is less
than 1, although the transverse energy of initial state
lies in a forbidden band. Expression (10) demonstrates
that at certain energies and an angle of incidence the
number of reflected particles may be changed if the
radiation processes ares very appreciable. However, our
consideration shows, that the band reflection and
electromagnetic radiation under planar semichanneling
will appear in a different situation and the independent
experimental examination for these interesting quantum
effects is possible.
REFERENCES
1. Kumakhov М. А. // Pisma Zn. Techn. Fiz. 1979. V.
5. N 11. P. 689-692.
2. Kumakhov М. А. // Pisma Zn. Techn. Fiz. 1979. V.
5. N 24. P. 1530-1533.
3. Rozhkov V. V. // VANT. Ser.: FRP and RM. 1981. N
1(12). P. 83-85.
4. Rozhkov V. V. // Zn. Techn. Fiz. 1981. V. 51. P.
1740-1741.
5. Kumakhov M. A., Komarov F. F. // Rad. Eff. 1985.
V. 90. P. 269-281.
6. Taranin A. M., Vorobiev S. A. // Izv. Vuzov. Fizika.
1979. V. 22. N3. P. 85-90.
7. Gann V. V., Vit’ko V. I., Duld’ya S. V., Nasonov
N.N., Rozhkov V. V. // VANT. Ser.: FRP and RM.
1983. N 5(28). С. 72-80.
8. Vit’ko V. I., Voytsen’ya A. V., Duld’ya S. V.,
Nasonov N. N., Rozhkov V. V. // Pisma Zn. Techn.
Fiz. 1982, V. 8. N 15. P. 921-923.
9. Dyul’dya S.V., Nasonov N.N., Rozhkov V.V., Vit’ko
V. I., Vojtsenja A. V. // Radiat. Eff. 1983. V. 69. N
3-4. P. 293-297.
10. Khlabutin V. G., Pivovarov Yu. L., Vorobiev S. A.
// Poverhnost’ FHM. 1983. N 10. P. 46-48.
11. Duld’ya S. V., Rozhkov V. V., Tikhonenkov I. E. //
Poverhnost FHM. 1998. N 3. С. 66-72.
12. Duld’ya S. V., Rozhkov V. V., Tikhonenkov I. E. //
Poverhnost RNSI. 1999. N 10. N. C. 74-86.
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