Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals
We analyzed the emission of the conduction electrons in metals caused by any nuclear decay. The refraction of the electron wave at the crystal surface, as well as its attenuation due to scattering by phonons, are taken into account. It is shown that the energy distribution of ejected shake-off elect...
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| Cite this: | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals / A.Ya. Dzyublik, V.Yu. Spivak // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 426-430. — Бібліогр.: 21 назв. — англ. |
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| citation_txt | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals / A.Ya. Dzyublik, V.Yu. Spivak // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 426-430. — Бібліогр.: 21 назв. — англ. |
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| description | We analyzed the emission of the conduction electrons in metals caused by any nuclear decay. The refraction of the electron wave at the crystal surface, as well as its attenuation due to scattering by phonons, are taken into account. It is shown that the energy distribution of ejected shake-off electrons contains a peak at the energy of the order of 1 eV, whose intensity falls down with growing temperature. The dependence of the yield of conduction electrons on the thickness of a radioactive source is studied as well.
Проаналiзовано емiсiю iз металу електронiв провiдностi, спричинену ядерним розпадом. Враховано заломлення електронної хвилi на поверхнi кристала та затухання, викликане розсiянням на фононах. Показано, що енергетичний розподiл випромiнених електронiв струсу має пiк при енергiї порядку 1 еВ, iнтенсивнiсть якого спадає зi зростанням температури. Вивчено також залежнiсть виходу електронiв провiдностi вiд товщини зразка.
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A.YA. DZYUBLIK, V.YU. SPIVAK
TEMPERATURE DEPENDENCE OF THE SHAKE-OFF
EFFECT FOR CONDUCTIVITY ELECTRONS IN METALS
A.YA. DZYUBLIK, V.YU. SPIVAK
Institute for Nuclear Research, Nat. Acad. of Sci. of Ukraine
(47, Prosp. Nauky, Kyiv 03028, Ukraine; e-mail: dzyublik@ ukr. net )
PACS 25.20.Dc
c©2010
We analyzed the emission of the conduction electrons in metals
caused by any nuclear decay. The refraction of the electron wave
at the crystal surface, as well as its attenuation due to scattering
by phonons, are taken into account. It is shown that the en-
ergy distribution of ejected shake-off electrons contains a peak at
the energy of the order of 1 eV, whose intensity falls down with
growing temperature. The dependence of the yield of conduction
electrons on the thickness of a radioactive source is studied as well.
1. Introduction
For a long time, the problem of the emission of low-
energy electrons from targets, following any nuclear
transmutation, attracts a great attention [1–18]. The
electrons around the nucleus apprehend its charge al-
teration as a sudden perturbation of the Coulomb field,
which gives rise to their emission from the target. The
energy of emitted shake-off electrons equals a few eV.
Previously, all theorists have been concentrating on the
shake-off effect for electrons initially bound on deep K
and L levels of an isolated atom. The emission proba-
bility of these electrons is too small as compared with
that from experimental data. In particular, the estima-
tions in [3] show that the probability for the emission of
a K electron after the beta-decay only is 3/4Z2, where Z
stands for the atomic number. At the same time, exper-
iments [12] indicate that the average yield of low-energy
electrons after the β decay of a single nucleus 154Eu in a
thin source is nβ = 0.5. Such a discrepancy can be easily
understood, by applying standard formulas of the sud-
den perturbation theory [19] which predict a quick de-
crease of the electron emission probability with increase
in their binding energy. Therefore, it can be stated that
experimentalists mainly observe the shake-off electrons
initially bound on the upper levels. In metals, such levels
belong to the conduction band.
In our previous paper [18], the shake-off effect was first
analyzed for valent electrons in metal crystals within the
simplest model. The electrons were treated as noninter-
acting particles moving in a rectangular potential well
U(r) =
{
−U0, inside the crystal,
0, outside it (1)
with wave vectors q. The depth of the potential well
equals
U0 = εF +A, (2)
where εF = ~2q2F/2m is the Fermi energy, and A is the
work function.
When a nucleus decays at some distance zs from the
surface of the crystal slab, it suddenly perturbs the
Coulomb field at this point and gives rise to a spher-
ical outgoing electron wave with the origin at the nu-
cleus. Such spherical wave may be decomposed into the
plane waves eiKr which are refracted at the surface, bear-
ing the waves eikr in vacuum. If the shake-off electron
has the energy E = ~2k2/2m, then the obvious equality
holds:
E = ε− U0. (3)
Here, ε = ~2K2/2m represents the kinetic energy of the
electron inside the crystal.
Due to the inelastic scattering of an electron wave
by vibrating ions of the crystal, the wave vector K at-
tributes the imaginary part. As a result, the intensity
of the electron beam which passed the distance x in the
medium exponentially decreases:
I(x) = I(0)e−µx. (4)
Here, the attenuation coefficient depending mainly on
the scattering by phonons is
µ = σin/v0, (5)
where σin is the inelastic scattering cross section of elec-
trons by phonons referred to one atom, and v0 is the
volume of the elementary cell (we assume that it con-
tains one atom).
In this article, we will study the role of the attenuation
of a shake-off electron wave in the crystal. For this aim,
426 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
TEMPERATURE DEPENDENCE OF THE SHAKE-OFF EFFECT
we calculate the inelastic cross section σin for the scatter-
ing of electrons with absorption or emission of phonons
which depends significantly on the temperature. Ear-
lier, σin was calculated in the long-wave approximation
for low-energy conduction electrons, regarding the crys-
tal as a continuous medium (see, e.g., [20]). But, in our
case, the electrons inside the crystal have kinetic energy
ε of the order of 10 eV and higher, which forces us to
give a more refined derivation. Having found formulas
for σin, we are able then to analyze the effects related to
the attenuation of shake-off electrons in the medium.
It is worth to note that a similar picture arises when
fast ions flying through microstrip metal detectors pro-
vide the emission of a great number of low-energy elec-
trons [21]. Therefore, the investigation of various aspects
of low-energy electron emission from metal films become
today the most actual.
2. Attenuation of an Electron Wave
In this section, we will calculate the inelastic scattering
cross section σin of electrons by phonons in a perfect
crystal which enters the attenuation coefficient (5). We
recall that we consider the crystal with one atom per
unit cell, whose position is defined by the vector
Rl = l + ul, (6)
where l is the lattice vector, and ul is a displacement of
the ion from its equilibrium position.
The perturbation operator is given by
V̂ (r) =
∑
l
[vc(|r−Rl|)− vc(|r− l|)], (7)
where vc(r) is the Coulomb interaction energy of the
electron with an ion,
vc(r) = −Ze
2
r
e−r/r0 , (8)
depending on the screening length r0.
The initial state of the system (crystal lattice + elec-
tron) is described by the function
|a〉 = |{νκj} 〉 eiKr, (9)
where νκj denotes the number of phonons specified by
a quasiwave vector κ, branch number j, and frequency
ωj(κ). The final wave function will be
|b〉 = |{ν′κj} 〉 eiK
′r. (10)
The cross section for the transition from |a〉 to |b〉 is
given by
σa→b =
2π
~v
|Vba|2δ(Eb − Ea), (11)
where v = ~K/M is the velocity of incident electrons,
and the matrix element in the Born approximation is
determined by the expression
Vba =
4πZe2r20
1 +Q2r20
∑
l
〈
{ν′κj}|eiQ(l+ul)|{νκj}
〉
(12)
with the scattering vector
Q = K−K′. (13)
In the single-phonon approximation, the inelastic scat-
tering cross section of electrons by a crystal is given by
σ
(N)
in =
2π
~v
∫
dK
′
(2π)3
∑
κj
(
4πZe2r20
1 +Q2r20
)2
×
×e−2W (Q) ~
2NMωj(κ)
|Q ej(κ)|2 ×
×
[ ∣∣∣∣∣∑
l
ei(Q+κ)l
∣∣∣∣∣
2(
ν̄j(κ)
2
)
δ (ε′ − ε+ ~ωj(κ)) +
+
∣∣∣∣∣∑
l
ei(Q−κ)l
∣∣∣∣∣
2(
ν̄j(κ) + 1
2
)
δ (ε′ − ε− ~ωj(κ))
]
, (14)
where ε = ~2K2/2m and ε′ = ~2K ′2/2m are the initial
and final values of the electron kinetic energy inside the
crystal formed by N atoms, exp(−2W (Q)) is the Debye–
Waller factor, and ν̄j(κ) is the mean number of phonons.
It is determined by the Bose–Einstein distribution
ν̄j(κ) =
[
exp
(
~ωj(κ)
kBT
)
− 1
]−1
. (15)
Since the minimum electron kinetic energy ε ∼ 10 eV,
while the maximum phonon energy ~ω ∼ 0.1 eV, one can
neglect ~ω in the δ functions. Following Ziman [20], we
consider only the so-called normal scattering of electrons
without any diffraction. In addition, we approximate the
phonon spectrum by the Debye model and believe that
the sound velocity s is the same for all three branches of
the acoustic vibrations. So that, we have
ωj(κ) = ω(κ) = sκ, (16)
where the wave vector κ varies from 0 to the bound value
κD, given by [20]
κD =
(
6π2/v0
)1/3
. (17)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 427
A.YA. DZYUBLIK, V.YU. SPIVAK
The corresponding maximum frequency is
ωD = kBθD/~, (18)
where θD is the Debye temperature. These quantities
determine the sound velocity:
s = ωD/κD. (19)
In what follows, the average number of phonons ν̄j(κ)
depending only on |κ| will be designated by ν̄(κ).
Moreover, the Debye model yields
2W (Q) =
3~2Q2T 2
MkBθ3D
θD/2∫
0
[
1
ez − 1
+
1
2
]
zdz. (20)
In (14), we first integrated over κ and after that over
the spherical coordinates K ′, ϑ, φ of the vector K′, using
the equality
Q2 = 2K2t, t = 1− cosϑ. (21)
Here, ϑ represents the angle between the wave vectors
K′ and K.
Then the inelastic scattering cross section of electrons
referred to one atom of the crystal,
σin = σ
(N)
in /N, (22)
becomes
σin = 4π
(
mZe2r20
~2
)2
×
×
(
~K
Ms
) tmax∫
0
dt
√
2te−2W (K
√
2t)
(1 + 2K2r20t)2
[
ν̄(K
√
2t) +
1
2
]
, (23)
where tmax = κ2
D/2K
2.
3. Energy Spectrum
Let a crystal film be formed by Np crystal planes spaced
by distance d. They are numerated by the number n =
0, 1, 2, ..., Np − 1, where the number n = 0 is associated
with the plane on the face surface. The thickness of such
a film equals D = Npd.
The energy and angular distribution of shake-off elec-
trons emitted from the crystal after the decay of a nu-
cleus embedded in the n-th plane is described by a func-
tion wn(E, θ), so that the average number of electrons
emitted in the energy interval ΔE at a solid angle ΔΩ
after the decay of one nucleus in the n-th plane is
ΔN (n)
e =
∫
ΔE
dE
∫
ΔΩ
dΩwn(E, θ). (24)
This distribution is defined by the expression [18]
wn(E, θ) = w0(E, θ) exp {−µ(E)nd/ cos θ0} , (25)
where w0(E, θ) is the distribution of electrons ejected
after the decay of a nucleus lying on the surface, θ0 or
θ is, respectively, the angle between the electron wave
vector K or k and the z axis which is perpendicular to
the surface of the crystal film. They are connected by
the relation [18]
cos θ0 =
(
1− E
E + U0
sin2 θ
)1/2
. (26)
The distribution w0(E, θ) written in terms of the di-
mensionless parameters
K̃ = Kr0, q̃ = qr0, (27)
has the form [18]
w0(E, θ) = T (E)
(r0
a
)5 1
E0
√
2E
E0
8
π3
×
×
q̃max∫
0
ñ(q̃)q̃2dq̃
[K̃2 − q̃2][1 + 2(K̃2 + q̃2) + (K̃2 − q̃2)2]
, (28)
where a = ~2/me2 and E0 = e2/a are the Bohr radius
and the atomic unit of energy, respectively,
T (E) =
4(1 + U0/E cos2 θ)1/2
[1 + (1 + U0/E cos2 θ)1/2]2
(29)
is the transmission coefficient of an electron wave
through the surface, and
ñ(q̃) =
[
exp
{
αq̃2 − εF
kBT
}
+ 1
]−1
(30)
with
α = ~2/2mr20 (31)
represents the Fermi distribution for conduction elec-
trons depending on the dimensionless parameter q̃.
The integration in (28) is performed over all bound
states |q〉 of the conduction electrons in the potential
well with the depth U0. Therefore, the upper limit of
integration q̃max in (28) should be taken a bit less than√
2mU0r0/~.
428 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
TEMPERATURE DEPENDENCE OF THE SHAKE-OFF EFFECT
0
2
4
6
N
e
(D
)
N
e
(D
)
0 20 40 60
D, ÅD, Å
Fig. 1. Average yield of shake-off electrons Ne from a copper crys-
tal after the single decay of a radioactive nucleus as a function of
the crystal thickness D. Our calculations are drawn by the solid
line, while the data of [16] are presented by dots. The dashed line
indicates the experimental background
4. Yield of Electrons
By averaging (25) over all crystal planes, we get a distri-
bution of shake-off electrons emitted from the slab after
the single nuclear decay at any point of the crystal:
w(E, θ) = w0(E, θ)
1
Np
1− e−µ(E)D/ cos θ0
1− e−µ(E)d/ cos θ0
. (32)
For a thick crystal, when µD � 1, this expression is
reduced to
w(E, θ) = w0(E, θ)
1
Np
1
1− e−µ(E)d/ cos θ0
. (33)
The energy distribution of all electrons emitted from
the metal is determined by the integral over the angles:
W (E) = 2π
π/2∫
0
w(E, θ) sin θdθ. (34)
The average number of low-energy electrons ejected
from the crystal following the decay of one nucleus lo-
cated at an arbitrary point of the crystal is given by the
integral
Ne =
∞∫
0
W (E)dE +B, (35)
where B denotes any experimental background.
2
4
W
(E
)
×
10
4
,
eV
−
1
W
(E
)
×
10
4
,
eV
−
1
0 10 20 30
E, eVE, eV
T=600K
T=300K
T=0K
Fig. 2. Energy distribution of shake-off electrons emitted from a
copper film at various temperatures
0
20
40
60
λ
,
Å
λ
,
Å
0 10 20 30
E, eVE, eV
T=0K
T=300K
T=600K
Fig. 3. Energy dependence of the free path length for electrons in
a copper crystal at various temperatures
The number Ne of low-energy electrons emitted from
a copper film as a function of the film thickness D has
been recently measured in [16]. Such electrons escape
mainly from the conduction band, since they are most
weakly bound as compared with inner electrons of ions.
We calculated the function Ne(D), by using the follow-
ing parameters for the copper film: v0 = 1.2 × 10−23
cm3, εF = 7.0 eV, A = 4.4 eV, θD = 315 K, U0 = 11.4
eV, and r0 = 0.55 Ȧ. From Eqs. (32)–(35), one sees
that the number of emitted shake-off electrons per one
nuclear decay falls down with increase in Np due to the
attenuation of the electron wave inside the crystal. The
attenuation coefficient has been calculated with the aid
of Eqs. (5) and (23). Our results are compared with the
experimental data in [16] in Fig. 1.
In addition, the energy distribution W (E) for elec-
trons emitted from a copper film is presented in Fig. 2
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 429
A.YA. DZYUBLIK, V.YU. SPIVAK
2
4
6
N
e
(T
)
×
10
3
N
e
(T
)
×
10
3
0 200 400 600
T, KT, K
Fig. 4. Temperature dependence of the integral electron yield from
a copper film
at the temperatures T = 0, 300, and 600 K. For defi-
niteness, we took the number of crystal planes Np = 40.
We see that, as T grows, the shape of the curve W (E)
changes due to the strong dependence of µ on the energy
E.
The energy dependence of the free path length λ =
µ−1 for low-energy electrons in copper is displaced in
Fig. 3 at the same temperatures. It is seen that λ de-
creases with increase in the temperature due to growing
the average number of phonons and, respectively, the
amplitude of vibrations.
The total yield of electrons Ne from the same copper
film, as is shown in Fig. 4, decreases with increase in the
temperature due to raising the attenuation µ of electron
waves.
We thank Profs. V.M. Pugatch and V.I. Sugakov for
the helpful discussion of the results.
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Received 19.06.09
ТЕМПЕРАТУРНА ЗАЛЕЖНIСТЬ ЕФЕКТУ СТРУСУ
ЕЛЕКТРОНIВ ПРОВIДНОСТI В МЕТАЛАХ
О.Я. Дзюблик, В.Ю. Спiвак
Р е з ю м е
Проаналiзовано емiсiю iз металу електронiв провiдностi, спри-
чинену ядерним розпадом. Враховано заломлення електронної
хвилi на поверхнi кристала та затухання, викликане розсiян-
ням на фононах. Показано, що енергетичний розподiл випро-
мiнених електронiв струсу має пiк при енергiї порядку 1 еВ, iн-
тенсивнiсть якого спадає зi зростанням температури. Вивчено
також залежнiсть виходу електронiв провiдностi вiд товщини
зразка.
430 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
|
| id | nasplib_isofts_kiev_ua-123456789-13433 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T16:42:23Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Dzyublik, A.Ya. Spivak, V.Yu. 2010-11-08T17:21:40Z 2010-11-08T17:21:40Z 2010 Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals / A.Ya. Dzyublik, V.Yu. Spivak // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 426-430. — Бібліогр.: 21 назв. — англ. 2071-0194 PACS 25.20.Dc https://nasplib.isofts.kiev.ua/handle/123456789/13433 We analyzed the emission of the conduction electrons in metals caused by any nuclear decay. The refraction of the electron wave at the crystal surface, as well as its attenuation due to scattering by phonons, are taken into account. It is shown that the energy distribution of ejected shake-off electrons contains a peak at the energy of the order of 1 eV, whose intensity falls down with growing temperature. The dependence of the yield of conduction electrons on the thickness of a radioactive source is studied as well. Проаналiзовано емiсiю iз металу електронiв провiдностi, спричинену ядерним розпадом. Враховано заломлення електронної хвилi на поверхнi кристала та затухання, викликане розсiянням на фононах. Показано, що енергетичний розподiл випромiнених електронiв струсу має пiк при енергiї порядку 1 еВ, iнтенсивнiсть якого спадає зi зростанням температури. Вивчено також залежнiсть виходу електронiв провiдностi вiд товщини зразка. We thank Profs. V.M. Pugatch and V.I. Sugakov for the helpful discussion of the results. en Відділення фізики і астрономії НАН України Тверде тіло Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals Температурна залежність ефекту струсу електронів провідності в металах Article published earlier |
| spellingShingle | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals Dzyublik, A.Ya. Spivak, V.Yu. Тверде тіло |
| title | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals |
| title_alt | Температурна залежність ефекту струсу електронів провідності в металах |
| title_full | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals |
| title_fullStr | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals |
| title_full_unstemmed | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals |
| title_short | Temperature Dependence of the Shake-off Effect for Conductivity Electrons in Metals |
| title_sort | temperature dependence of the shake-off effect for conductivity electrons in metals |
| topic | Тверде тіло |
| topic_facet | Тверде тіло |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/13433 |
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