Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids
The integro-differential equations for cumulants of the distribution function that describes energy losses by fast ions during their propagation in solids have been obtained. The equations differ from those obtained by other authors by one new term. The term describes accurately the process of slowi...
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Ilyina, V.V. Makarets, M.V. 2010-11-05T14:28:59Z 2010-11-05T14:28:59Z 2010 Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids / V.V. Ilyina, M.V. Makarets // Укр. фіз. журн. — 2010. — Т. 55, № 2. — С. 236-243 . — Бібліогр.: 14 назв. — англ. 2071-0194 PACS 61.72.uf; 61.72.uj https://nasplib.isofts.kiev.ua/handle/123456789/13389 539.534.9 The integro-differential equations for cumulants of the distribution function that describes energy losses by fast ions during their propagation in solids have been obtained. The equations differ from those obtained by other authors by one new term. The term describes accurately the process of slowing down of an ion at the start of its path. The equations have been numerically solved for the first seven cumulants of the distribution function for both elastic and inelastic energy losses, and the results have been compared with the results for ion ranges. It has been found that: 1) for energies in the interval 1 keV-1 GeV, the average ranges with energy losses are approximately 30-90% of the ion ranges; 2) for low energies, the straggling of the distribution of energy losses are slightly larger than or equal to the straggling of the distribution of ion ranges, while, for high energies, the former can be 10 times as large as the latter; 3) for low energies, the skewnesses and excesses of the distributions of energy losses and ion ranges are approximately the same, while their changes for the former at higher energies are several orders smaller than those for the latter. This implies that the distribution of energy losses are wider and closer to the normal distribution than the distribution of ion ranges. We show that these properties of energy loss distributions are a result of the inclusion of the new terms in the equations which dominate at high energies. Отримано iнтегродиференцiальнi рiвняння для кумулянтiв функцiї розподiлу втрат енергiї швидких iонiв вздовж їх шляху у твердому тiлi, у яких враховано втрати енергiї починаючи iз точки старту. Рiвняння для перших семи кумулянтiв розподiлiв втрат енергiї у пружних, непружних та обох типах зiткнень розв’язанi чисельно за допомогою методу, розвинутого авторами ранiше, на iнтервалi енергiй iонiв 1 кеВ–1 ГеВ. Їх порiвняння з параметрами розподiлу пробiгiв iонiв показало, що: 1) середнiй шлях за розподiлом втрат енергiї становить 30–90% повного пробiгу iонiв; 2) при низьких енергiях страгглiнг за розподiлом втрат енергiї дещо бiльший або однаковий iз страгглiнгом за розподiлом пробiгiв iонiв, а при високих енергiях перший перевищує останнiй у десятки разiв; 3) скiсностi i ексцеси розподiлiв при низьких енергiях вiдповiдно близькi, у той час як при високих енергiях їх змiна для розподiлу втрат енергiї на кiлька порядкiв менша, нiж для розподiлу пробiгiв iонiв. Звiдси випливає, що розподiл втрат енергiї ширший i значно ближчий до нормального, нiж розподiл пробiгiв iонiв при всiх енергiях iмплантацiї. Показано, що цi властивостi розподiлу втрат енергiї зумовленi новими членами у рiвняннях, якi не враховувалися ранiше, але домiнують при високих енергiях. Получены интегродифференциальные уравнения для кумулянтов функции распределения потерь энергии быстрых ионов вдоль их пути в твердом теле, в которых учтены потери энергии начиная с точки старта. Уравнения для первых семи кумулянтов распределений потерь энергии в упругих, неупругих и обеих типах столкновений решены численно с помощью метода, развитого авторами ранее, на интервале энергий ионов 1 кэВ–1 ГэВ. Их сравнение с параметрами распределения пробегов ионов показало, что: 1) средний путь с потерей энергии составляет 30–90% полного пробега иона; 2) при низких энергиях страгглинг распределения потерь энергии немного больше или одинаков со страгглингом распределения пробегов ионов, а при высоких энергиях первый превышает последний в десятки раз; 3) скошенности и эксцессы распределений при низких энергиях соответственно близки, в то время как при высоких энергиях их изменение для распределения потерь энергии на несколько порядков меньше, чем для распределения пробегов ионов. Отсюда следует, что распределение потерь энергии более широкое и ближе к нормальному, чем распределение пробегов ионов для всех энергий имплантации. Показано, что эти свойства распределения потерь энергии обусловлены новыми членами в уравнениях, которые не учитывались ранее, но доминируют при высоких энергиях. en Відділення фізики і астрономії НАН України Тверде тіло Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids Розподіл втрат енергії швидких іонів вздовж їх шляху у твердому тілі Распределение потерь энергии быстрых ионов вдоль их пути в твердом теле Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids |
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Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids Ilyina, V.V. Makarets, M.V. Тверде тіло |
| title_short |
Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids |
| title_full |
Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids |
| title_fullStr |
Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids |
| title_full_unstemmed |
Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids |
| title_sort |
distributions of energy losses by fast ions along their propagation paths in solids |
| author |
Ilyina, V.V. Makarets, M.V. |
| author_facet |
Ilyina, V.V. Makarets, M.V. |
| topic |
Тверде тіло |
| topic_facet |
Тверде тіло |
| publishDate |
2010 |
| language |
English |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Розподіл втрат енергії швидких іонів вздовж їх шляху у твердому тілі Распределение потерь энергии быстрых ионов вдоль их пути в твердом теле |
| description |
The integro-differential equations for cumulants of the distribution function that describes energy losses by fast ions during their propagation in solids have been obtained. The equations differ from those obtained by other authors by one new term. The term describes accurately the process of slowing down of an ion at the start of its path. The equations have been numerically solved for the first seven cumulants of the distribution function for both elastic and inelastic energy losses, and the results have been compared with the results for ion ranges. It has been found that: 1) for energies in the interval 1 keV-1 GeV, the average ranges with energy losses are approximately 30-90% of the ion ranges; 2) for low energies, the straggling of the distribution of energy losses are slightly larger than or equal to the straggling of the distribution of ion ranges, while, for high energies, the former can be 10 times as large as the latter; 3) for low energies, the skewnesses and excesses of the distributions of energy losses and ion ranges are approximately the same, while their changes for the former at higher energies are several orders smaller than those for the latter. This implies that the distribution of energy losses are wider and closer to the normal distribution than the distribution of ion ranges. We show that these properties of energy loss distributions are a result of the inclusion of the new terms in the equations which dominate at high energies.
Отримано iнтегродиференцiальнi рiвняння для кумулянтiв функцiї розподiлу втрат енергiї швидких iонiв вздовж їх шляху у твердому тiлi, у яких враховано втрати енергiї починаючи iз точки старту. Рiвняння для перших семи кумулянтiв розподiлiв втрат енергiї у пружних, непружних та обох типах зiткнень розв’язанi чисельно за допомогою методу, розвинутого авторами ранiше, на iнтервалi енергiй iонiв 1 кеВ–1 ГеВ. Їх порiвняння з параметрами розподiлу пробiгiв iонiв показало, що: 1) середнiй шлях за розподiлом втрат енергiї становить 30–90% повного пробiгу iонiв; 2) при низьких енергiях страгглiнг за розподiлом втрат енергiї дещо бiльший або однаковий iз страгглiнгом за розподiлом пробiгiв iонiв, а при високих енергiях перший перевищує останнiй у десятки разiв; 3) скiсностi i ексцеси розподiлiв при низьких енергiях вiдповiдно близькi, у той час як при високих енергiях їх змiна для розподiлу втрат енергiї на кiлька порядкiв менша, нiж для розподiлу пробiгiв iонiв. Звiдси випливає, що розподiл втрат енергiї ширший i значно ближчий до нормального, нiж розподiл пробiгiв iонiв при всiх енергiях iмплантацiї. Показано, що цi властивостi розподiлу втрат енергiї зумовленi новими членами у рiвняннях, якi не враховувалися ранiше, але домiнують при високих енергiях.
Получены интегродифференциальные уравнения для кумулянтов функции распределения потерь энергии быстрых ионов вдоль их пути в твердом теле, в которых учтены потери энергии начиная с точки старта. Уравнения для первых семи кумулянтов распределений потерь энергии в упругих, неупругих и обеих типах столкновений решены численно с помощью метода, развитого авторами ранее, на интервале энергий ионов 1 кэВ–1 ГэВ. Их сравнение с параметрами распределения пробегов ионов показало, что: 1) средний путь с потерей энергии составляет 30–90% полного пробега иона; 2) при низких энергиях страгглинг распределения потерь энергии немного больше или одинаков со страгглингом распределения пробегов ионов, а при высоких энергиях первый превышает последний в десятки раз; 3) скошенности и эксцессы распределений при низких энергиях соответственно близки, в то время как при высоких энергиях их изменение для распределения потерь энергии на несколько порядков меньше, чем для распределения пробегов ионов. Отсюда следует, что распределение потерь энергии более широкое и ближе к нормальному, чем распределение пробегов ионов для всех энергий имплантации. Показано, что эти свойства распределения потерь энергии обусловлены новыми членами в уравнениях, которые не учитывались ранее, но доминируют при высоких энергиях.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13389 |
| citation_txt |
Distributions of Energy Losses by Fast Ions along Their Propagation Paths in Solids / V.V. Ilyina, M.V. Makarets // Укр. фіз. журн. — 2010. — Т. 55, № 2. — С. 236-243 . — Бібліогр.: 14 назв. — англ. |
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| fulltext |
V.V. ILYINA, M.V. MAKARETS
DISTRIBUTIONS OF ENERGY LOSSES BY FAST IONS
ALONG THEIR PROPAGATION PATHS IN SOLIDS
V.V. ILYINA, M.V. MAKARETS
Taras Shevchenko National University of Kyiv
(64, Volodymyrs’ka Str.,01033 Kyiv, Ukraine; e-mail: mmv@ univ. kiev. ua )
UDC 539.534.9
c©2010
The integro-differential equations for cumulants of the distribu-
tion function that describes energy losses by fast ions during their
propagation in solids have been obtained. The equations differ
from those obtained by other authors by one new term. The term
describes accurately the process of slowing down of an ion at the
start of its path. The equations have been numerically solved
for the first seven cumulants of the distribution function for both
elastic and inelastic energy losses, and the results have been com-
pared with the results for ion ranges. It has been found that: 1)
for energies in the interval 1 keV-1 GeV, the average ranges with
energy losses are approximately 30-90% of the ion ranges; 2) for
low energies, the straggling of the distribution of energy losses are
slightly larger than or equal to the straggling of the distribution of
ion ranges, while, for high energies, the former can be 10 times as
large as the latter; 3) for low energies, the skewnesses and excesses
of the distributions of energy losses and ion ranges are approx-
imately the same, while their changes for the former at higher
energies are several orders smaller than those for the latter. This
implies that the distribution of energy losses are wider and closer
to the normal distribution than the distribution of ion ranges. We
show that these properties of energy loss distributions are a result
of the inclusion of the new terms in the equations which dominate
at high energies.
1. Introduction
The first system of equations for the distribution func-
tion of ions implanted into a solid was obtained over half
a century ago [1]. The methods to study these equations
have been developed in several stages [2,3]. In the recent
years, the authors of [4–6] have proposed the cumulant
approach and developed numerical methods that allow
cumulants up to the 6th order to be calculated and ana-
lyzed for the whole energy range, where the assumptions
of the classical implantation theory [1] hold.
The main advantage of this approach is that cumu-
lants of the distribution are smoother functions of the
ion energy than its moments – their amplitudes are less
by several orders. It has allowed one to find an exact
enough analytical approximation and construct a stable
numerical method. In general, the hardly soluble equa-
tions with the obvious physical meaning were reduced
to a more easily soluble equation but with a more fuzzy
physical meaning.
Now we use the cumulant approach for the investiga-
tion of energy losses of fast ions in a solid.
When we have applied this approach for the first time
to equations from [7] which describe the energy loss dis-
tributions of ions and knocked-out particles, we obtained
unexpected results. Namely, the norm and other cumu-
lants of the nuclear energy losses distribution were equal
to zero. On the contrary, the norm of the electronic en-
ergy losses distribution was equal to the ion energy E,
and its other cumulants were also non-zero. These re-
sults contradict the physical meaning, but the sum of
norms is in complete agreement with the energy conser-
vation law.
We felt obliged to investigate the reasons for having
the solution to the well-established equations that con-
tradicts the basic physics laws. This investigation [8] has
shown that the equations for the spatial energy losses
[2, 7] contain a hidden assumption that the Energy Loss
Distribution (ELD) function for an ion at the start of
its path has a zero value. This assumption is a con-
sequence of the even earlier assumption that the losses
inside an infinitely small volume δV are proportional
to δV .
The same assumption was made while obtaining the
equations for the ranges of implanted ions [1]. In this
case, however, the assumption is completely valid, be-
cause an ion cannot stop more than once inside δV .
Therefore, the number of random stops is δn = 1 if an ion
stopped inside δV and δn = 0 if an ion stopped outside
δV . Then, using the first-order approximation, it can be
assumed that an ion takes up a volume of ΔV ≈ 1/N0,
where N0 is the atom concentration of a solid. Then the
probability of the ion stopping in a volume δV ≤ ΔV is
defined as δP ≈ δnδV/ΔV ∼ δV and has a norm equal
to 1. Then we can introduce a density of implanted ions
as a limit Π (~r,E) = lim
δV→0
δP/δV . In the same way, one
can introduce the density of the ions distribution along
their paths by considering an infinitely small length δl.
236 ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2
DISTRIBUTION OF ENERGY LOSSES
Let us now consider the energy losses of an ion. While
an ion only stops once, it looses energy in many col-
lisions during its trajectory. Energy losses can occur
many times in the same volume δV if the characteristic
length of a collision-free path of an ion λ is small or if
ion’s trajectory is very complex. Therefore, an ion passes
through the same volume δV many times before leaving
it or stopping in it, and thus the energy lost in δV will
be equal to 0 if an ion did not cross δV or proportional
to a number of collisions δn if an ion crossed δV .
Now, the number of random collisions inside δV is
δn ≈ δl/λ, where δl ∼ (δV )1/3 is the ion path, and
we can assume in the first-order approximation that the
energy is transferred from an ion to a solid through a
cylinder with a volume ΔV ≈ πR2λ. The axis of the
cylinder connects the locations of two neighboring col-
lisions. For elastic collisions, the radius of the cylinder
Rn is equal to the half-distance between atoms of a solid
Rn ≈ a/2. For inelastic collisions, the radius Re depends
on the ion energy and the properties of the electronic
subsystem of a solid [9]. It can be defined generally as
Re ≡ R(E,N0, . . .).
If we denote the energy lost in one collision of the α-
type (where α = n, e for nuclear and electronic collisions)
as 〈Tα〉, then the probable loss of energy inside δV ≤
ΔV can be determined from δPα ≈ 〈Tα〉δnδV/ΔV .
This probability has a norm of κα0 (E), which is the en-
ergy lost in one collision of the α-type. It follows from
here that δPα ∼ (δV )4/3, and therefore we cannot deter-
mine the volume density for energy losses, at least within
the limits of this approach. However, we can introduce
the density of energy losses along an ion’s path δl if we
consider the probable loss of energy δPα ≈ 〈Tα〉δn ∼ δl.
2. Main Equations and Their Analysis
In this paper, we consider the distribution functions for
energy losses per unit length of an ion along its trajectory
in the solid. To be more precise, we study the cumulants
of these distributions.
Let Πα(E,R)dR be the most probable energy lost by
a fast ion with initial energy E in the α-type collision in
the vicinity of a point dR along ion’s path R, where α =
t, n, e correspond to the total, nuclear, and electronic
losses, respectively. We follow [1] to obtain the equations
for the density of the distribution:
∂Πα(E,R)
∂R
= −N0
∫
dσ(E, T ) {Πα(E,R) −
−Πα(E − T,R)} , (1)
where dσ(E, T ) is the differential cross-section of the
elastic and inelastic scattering process involving the ion
and a target atom, and T is the energy lost in this scat-
tering process.
This equation is identical in form to the equation for
the density distribution of ion ranges obtained in [1].
However, the meaning of the function Πα(E,R) is dif-
ferent from the one used in [7]. This leads to an initial
condition (at R = 0) for this function being different
from the one used in [2, 7].
Namely, the function Πe(E, 0)dR for electronic
losses is the most probable energy lost in inelas-
tic collisions during the path dR near the start
point of an ion. Then it follows that Πe(E, 0) =
Se(E), where Se(E) ≡
∫
Tedσe(E, Te) is the inelas-
tic energy loss per unit length of an ion with en-
ergy E or the electronic stopping, and dσe(E, Te)
is the differential cross-section of electronic scatter-
ing.
For nuclear losses, Πn(E, 0) = Sn(E), where Sn(E) ≡∫
Tndσn(E, Te) is the elastic energy loss per unit length
of an ion with energy E or the nuclear stopping, and
dσn(E, Tn) is the differential cross-section of nuclear
scattering. It is obvious that, for the total losses,
Πt(E, 0) = Se(E) + Sn(E).
Therefore, for the α-type losses, we can write down
Πα(E, 0) = Sα(E), where Sα(E) is the α-type stopping
for an ion with energy E. These initial conditions mean
that a fast ion will start to loose energy immediately
after the start.
We can also include the ion ranges distribution in
the common scheme of energy losses calculations, if
we taken its initial condition Π(E, 0) = 0 into ac-
count and then set the index α = i and formally put
Si(E) = 0. In this case, the most probable num-
ber of ions which are stopped at the start is zero, but
there is no reason to believe that this initial condition
holds for energy losses. Since this zero initial condi-
tion was incorrectly used in [2, 7] for the elastic energy
loss distribution of fast particles stopped into a unit vol-
ume, the corresponding equations have the above solu-
tions.
Let us now change to the Lindhard’s dimension-
less units by replacing R ⇒ ρ and E ⇒ ε following
[1, 2] in Eq. (1). We will also replace the function
Πα(ε, ρ) by its cumulants καk , where k = 0, 1, 2, . . .,
following [4]. Taking the norm of the energy distri-
bution function as the energy lost in an α-type col-
lision κα0 (ε), we can write down the following ex-
pression for the energy loss distribution density func-
ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2 237
V.V. ILYINA, M.V. MAKARETS
tion [4]:
Πα(ε, ρ)=
κα0
π Δρα
∞∫
0
exp
[
− 1
2!
s2+
Exα1
4!
s4−Ex
α
2
6!
s6+. . .
]
×
× cos
[
−rαs− Skα1
3!
s3 +
Skα2
5!
s5 − . . .
]
ds, (2)
where Δρα =
√
κα2 (ε)/κα0 (ε) is the straggling of the
mean path ρα(ε) = κα1 (ε)/κα0 (ε), rα ≡ rα(ρ, ε) is the
centered dimensionless mean ion path, and Skαj and
Exαj are the distribution’s skewness and excess, re-
spectively, which depend on the energy ε only. These
quantities can be expressed through the cumulants
by
rα =
ρ− ρα(ε)
Δρα(ε)
, Skαj =
κα2j+1(ε)
κα0 (ε) [Δρα(ε)]2j+1
,
Exαj =
κα2j+2(ε)
κα0 (ε) [Δρα(ε)]2j+2
, j = 1, 2, . . . (3)
At a fixed ion energy ε, rα(ρ) is the dimensionless
distribution’s variable, and Skαj and Exαj are dimen-
sionless distribution’s parameters. The cumulant of
the jth order has the units of (length)j multiplied by
the units of κα0 . It follows from Eq. (2) that the cu-
mulant κα0 defines the amplitude of the distribution,
while its form depends on the parameters χαj (ε) =
καj (ε)/κα0 (ε), where j = 1, 2, . . . Equations (2) and
(3) remain true for the ion ranges distribution, since
κi0(ε) ≡ 1.
Following [4] and using Eq. (1), we obtain the equa-
tions for the cumulants using the initial condition
Πα(E, 0) = Sα(ε). For the norm of the distribution
function κα0 and the mean path distribution κα1 , these
equations become
L̂ {κα0 } = Sα(ε), (4)
L̂ {κα1 } = κα0 (ε), (5)
where the left-hand operator is defined as
L̂ {καk} ≡ N̂ {καk (ε− γτ)}+ Se(ε)
∂καk (ε)
∂ε
, (6)
N̂ {. . .} ≡ − 1
γ
ε∫
0
σ(ε, τ) ∂∂τ {. . .} dτ, (7)
where N̂ is the nuclear scattering operator, σ(ε, τ) is
defined in [4] as the so-called summarized cross-section
of ion’s nuclear scattering on the target atom, γ =
4M1M2/ (M1 +M2)
2, and M1,2 are the masses of an ion
and the target atom, respectively.
The equations for the cumulants of the k ≥ 2 order
are:
L̂ {καk} = N̂
{
κα0 (ε− γτ)Fk
(
χα1 , . . . , χ
α
k−1
)}
+
+Sα(ε)fk
(
χα1 , . . . , χ
α
k−1
)
, (8)
where the functions Fk(χα1 , . . . , χ
α
k−1) and
fk(χα1 , . . . , κ
α
k−1) do not depend on the type of a
collision explicitly. In particular, for the cumulants of
the second-sixth orders which define the straggling, two
skewnesses, and two excesses, the functions Fk do not
depend on the norm κα0 explicitly:
F2 = [Δα
1 (ε, τ)]2 , (9)
F3 = Δα
1 (ε, τ)
{
[Δα
1 (ε, τ)]2 + 3Δα
2 (ε, τ)
}
, (10)
F4 = Δα
1 (ε, τ)
{
[Δα
1 (ε, τ)]3 + 6Δα
2 (ε, τ)Δα
1 (ε, τ) +
+4Δα
3 (ε, τ)
}
+ 3 [Δα
2 (ε, τ)]2 , (11)
F5 = Δα
1 (ε, τ)
{
[Δα
1 (ε, τ)]4 + 10Δα
2 (ε, τ) [Δα
1 (ε, τ)]2 +
+10Δα
3 (ε, τ)Δα
1 (ε, τ) + 15 [Δα
2 (ε, τ)]2 +
+5Δα
4 (ε, τ)
}
+ 10Δα
2 (ε, τ)Δα
3 (ε, τ), (12)
F6 = Δα
1 (ε, τ)
{
[Δα
1 (ε, τ)]5 + 15Δα
2 (ε, τ) [Δα
1 (ε, τ)]3 +
+20Δα
3 (ε, τ) [Δα
1 (ε, τ)]2 + 15Δα
4 (ε, τ)×
×Δα
1 (ε, τ) + 45 [Δα
2 (ε, τ)]2 Δα
1 (ε, τ) +
+60Δα
2 (ε, τ)Δα
3 (ε, τ) + 6Δα
5 (ε, τ)
}
+
+15 [Δα
2 (ε, τ)]3 + 10 [Δα
3 (ε, τ)]2 +
+15Δα
2 (ε, τ)Δα
4 (ε, τ), (13)
where Δα
i (ε, τ) = χαi (ε− γτ)− χαi (ε), and Δα
i (ε, 0) = 0
for all i. The functions fk are obtained from Fk by the
substitution χαj (ε− γτ) = 0 for j = 1, 2, . . . , k − 1.
Note that the equations for the cumulants contain the
terms which are proportional to Sα(ε) on the right-hand
side and take the energy losses of an ion at the start of
its path into account.
Equations (4), (5), and (8) are inhomogeneous second-
order Volterra-type integral equations with integrable
singularity core σ(ε, τ → 0)→∞ [4] which describe the
first derivative of the cumulants. They have non-trivial
solutions only if the right-hand sides are non-zeros [10].
The equations should be solved one by one, since Eq. (4)
has the explicitly given right-hand side.
The ion ranges distribution is an example where the
contribution of Sα (ε) is equal to zero. In this case, Eq.
238 ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2
DISTRIBUTION OF ENERGY LOSSES
(4) for the norm becomes homogeneous with respect to
its derivative,
ε∫
0
σ(ε, τ)
{
∂κi0(ε− γτ)
∂ε
}
dτ + Se(ε)
∂κi0(ε)
∂ε
= 0, (14)
and has the trivial solution ∂κi
0(ε)
∂ε = 0. Using the norm
conditioning it follows that κi0(ε) ≡ 1, and Eq. (5) re-
duces to the equations for ranges obtained in [1]. For
the cumulants of order k ≥ 2, the right-hand sides of
Eqs. (8) do not contain the second term, and the inte-
grals in Eqs. (9)–(11) coincide with the results obtained
in [4].
Therefore, we have shown that while the equations
for the ion ranges distribution functions and the ELD
functions are the same, the equations for their cumu-
lants, moments, and central moments differ. The reason
for this is that while it is very unlikely that an ion will
stop right at the beginning of its path, it is almost cer-
tain that it will loose some energy at the beginning of
its path. The consequence of this difference is the differ-
ent initial conditions for the distribution functions which
define the right-hand sides of the equations for the cu-
mulants of these distributions.
3. Results and Discussion
The first-order series expansions of the integral func-
tions, Eqs. (4) and (5), for the energy loss γτ allows
us to obtain the analytical approximations for the norm
of the distribution function and for the mean path:
κα0 (ε) ≈
ε∫
0
Sα(ε′)
Sn(ε′) + Se(ε′)
dε′, (15)
ρα(ε) ≈ 1
κα0 (ε)
ε∫
0
κα0 (ε′)
Sn(ε′) + Se(ε′)
dε′. (16)
Approximation (15) satisfies the energy conservation
law, because κt0(ε) = ε, and the integrands in this equa-
tion can be interpreted as the probability of the energy
losses of the α-type per unit path of an ion with the en-
ergy ε′. To our knowledge, the equations have not been
presented elsewhere. Expression (16) is a generalization
of a well-known result for the ion mean range [1].
We have solved Eqs. (4), (5), and (8) numerically,
by using the method developed in [4, 5]. Within this
method, the left-hand sides of the equations are trans-
formed into a system of linear equations for the expan-
sion coefficients καi (ε). The expansion is carried out
for the energy intervals, whose lengths grow geomet-
rically until they cover the range of energies, where
the assumptions of binary elastic ion-atom collisions
hold.
For the lower energy limit, we assume the energy, for
which the distance between an ion and an atom during
the head-on collision is 1/4 of the half-distance of atoms
in the target. For the upper energy limit, we assume
the energy, for which the distance between an ion and
an atom during the head-on collision is 1/4 of the sum
of their radii and the radius of the nuclear force action.
For inelastic and elastic collisions, we use, respectively,
the approximation of [11] and the potential proposed by
Ziegler–Biersack–Littmark [12].
In Fig. 1, we show the dependences κt,n,e0 (ε) and
κt,n,e0 (ε)/ε calculated for the implantation of phospho-
rus ions into a silicon target. The lower and upper en-
ergy limits are chosen to represent energies, where the
assumptions of [1] hold. They are equal to 2.8 keV and
281 MeV, respectively. We have chosen this ion-target
combination because of its practical applications, but
also because γ ≈ 1 for this combination, and the expan-
sion in γ in integrands has the worst asymptotic behav-
ior.
One can see from Fig. 1,a that, for large energies,
nearly all losses are caused by the electronic scattering,
however the nuclear losses slowly increase as well. We
can see from Fig. 1,b that the energy lost in elastic,
inelastic, and all collisions satisfies the energy conserva-
tion law, though it has not been used in obtaining the
equations. We believe that the accuracy, to which the
law is satisfied (more than 0.2%), can be interpreted as
the indirect evaluation of the numerical method accu-
racy.
To approximate the equations for cumulants of the
order k ≥ 2 analytically, we have to take into account
the expansion of the integrands Fk in Eq. (8) in γτ up
to the second-order terms. Therefore, the approximation
of the integral begins with the straggling of the nuclear
scattering W (ε) =
∫
τ2dσ(ε, τ) = 2
∫
τσ(ε, τ)dτ . The
right-hand sides have the term proportional to Sα(ε),
and we can rewrite the equations as
{Sn(ε) + Se(ε)}
∂καk (ε)
∂ε
≈ Sα(ε)fk
(
χα1 , . . . , χ
α
k−1
)
+
+
1
2
W (ε)κα0 (ε)F (2)
k
(
χα1 , . . . , χ
α
k−1
)
, (17)
where F (2)
k is the second derivative of Fk with respect
to τ at τ = 0, which is a polynomial of the second
order in the first derivatives of the cumulants of the
(k − 1)th order inclusive. In case of the distribution
ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2 239
V.V. ILYINA, M.V. MAKARETS
Fig. 1. Dependence of the norm of the energy losses function – а and the relative norm – b on the implantation energy of phosphorus ions
into a silicon target. Letters n, e, and t next to the curves denote the distributions of nuclear, electronic, and total losses, respectively
of ranges, when κα0 (ε) ≡ 1 and Si(ε) ≡ 0, the right-
hand sides of Eq. (17) can be simplified, and the first
derivative of the cumulant of the kth order can be ex-
pressed through the moments of nuclear scattering ex-
plicitly [4]. In the case of energy losses, the second
term on the right-hand side of Eq. (17) is proportional
to the cumulants of the (k − 1)th order, and this can
change the dependence of cumulants on energy signifi-
cantly.
Figure 2 presents the graphs of dependences of pa-
rameters from Eqs. (2) and (3) on the ion energy for
the distributions α = n, e, t and i for the combina-
tion phosphorus-silicon. The arrow and the number
in Fig. 2,с–f near the curve for the parameter of the
ions distribution show the direction of its monotonic
change and its value for an ion energy of 1 GeV, re-
spectively.
One can see from Fig. 2,a that the mean of the
total ELD is smaller than the mean of the ions dis-
tribution corresponding to the picture of the energy
losses of an ion just before it stops. One can also see
that the mean of the elastic losses distribution and the
ranges distribution have similar values for high ener-
gies suggesting that this mechanism dominates at the
end of the ion path, when its energy is low. Strag-
glings of losses of all types, Fig. 2,b, exceed the strag-
gling of the ranges distribution, by suggesting that the
energy losses are distributed more evenly during the
ion path. Therefore, the ELDs will have maxima to-
ward the beginning of its path, and the distributions
will be wider than the ranges distribution. In addi-
tion, the ratio of the straggling to the mean Δρα/ρα
has a maximum for the total losses distribution and
has a minimum for the ranges distribution. Therefore,
these distributions will be narrower and wider, respec-
tively.
Figure 2,с–f demonstrates the common property for
all parameters of the order k ≥ 3 for the ELDs, which
is their relatively small change of amplitudes. Namely,
the changes in amplitudes for the ELDs are several
times smaller than those for the ranges distribution.
Then it follows that these distributions are closer to
the normal distribution, as all cumulants of the or-
der k ≥ 3 are equal to zero [13]. We can also see
that the parameters for the total loss distribution for
low energies are similar to parameters for the elas-
tic loss distribution, while, for high energies, they be-
come similar to the parameters for the inelastic ener-
gies distribution. This corresponds to the fact that
different loss mechanisms dominate at different ener-
gies.
We can now conclude that, for the ELDs for different
types of collisions, we can use the normal distribution as
the zero-order approximation. An indirect criterion for
its applicability will be the initial condition Πβ(E, 0) =
Sβ(E) for β = n, e, t which can be written explicitly as
κβ0 (ε)√
2π Δρβ(ε)
exp
{
−1
2
[
ρβ(ε)
Δρβ(ε)
]2}
≈ Sβ(E). (18)
This condition links three first cumulants. We have also
calculated the ratio of the left-hand side of Eq. (18) to its
right-hand side for the phosphorus/silicon combination.
The value of this ratio shows that the normal distribu-
tion at |Sk|, |Ex| ≤ 1 gives values of the ELDs at the
coordinate origin that are approximately 1/3 of the true
values. If |Sk| and |Ex| are large, then the difference
increases further, and it becomes necessary to use one of
240 ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2
DISTRIBUTION OF ENERGY LOSSES
Fig. 2. Dependence on the implantation energy of phosphorus ions into a silicon target for: а, b – mean path distribution and its
straggling; c, d – the first and second skewnesses; e, f) – the first and second excesses. The letter i near the curves denotes the ion
distribution
the Pearson distributions [2] or the Johnson distribu-
tion [14].
We have also carried out the numerical analysis of the
contributions of both members on the right-hand side of
Eq. (8). The analysis has shown that the stopping mem-
ber with Sα(ε) dominates, while the nuclear scattering
only contributes significantly at low energies. This jus-
tifies the use of the approximation
χαk (ε) ≈ 1
κα0 (ε)
ε∫
0
Sα(ε′)fk
(
χα1 (ε′), . . . , χαk−1(ε
′)
)
Sn(ε′) + Se(ε′)
dε′,
(19)
for k ≥ 2. The analysis has shown that, for all ion-
target combinations, the error of approximations (15),
(16), and (17) is the largest for k = 2, when M1 ≈ M2.
For all types of losses, the error of the norm can be
negative or positive and generally does not exceed 5%.
For the mean and the straggling, the results of using this
approximation gives values which are smaller than true
ones by no more than 15% and 25%, respectively. The
skewnesses and excesses change the sign, and the errors
can be large.
In Fig. 3, we show the comparison of the numerical
results obtained by solving Eqs. (4), (5), and (8) and the
results obtained using approximations for the implanta-
tion of boron ions in a germanium target.
One can see from Fig. 3 that the approximate values
of the parameters of the order k ≥ 3 are very close to
the calculated values for high energies. For low energies,
the approximate values have the correct order of mag-
nitude and the general dependence structure. The max-
imum errors for the mean and the straggling occur for
the same energies. Therefore, we can use the approxima-
tions in Eqs. (15), (16), and (17) or (19) for estimating
the cumulants for the whole energy range used in the
implantation.
The results above show that, for the ELDs of vari-
ous types, we can use the Gaussian distribution for the
zero-order approximation. This approximation will be
ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2 241
V.V. ILYINA, M.V. MAKARETS
Fig. 3. Parameters for the ELDs for the implantation of boron ions into the germanium target. Solid lines denote the numerical
solutions, and the dotted lines denote the approximations
true for the whole energy range used in the implanta-
tion. If the parameters of higher order are large enough
to distort the Gaussian distribution [2], we can use the
Pearson-4 distribution.
4. Conclusions
In the present work, it is shown that it is necessary to
take into account in equations for the cumulants of the
energy loss distribution (ELD) function that the energy
loss is a continuous process, while the stop of an ion is an
event. Therefore, the energy losses happen continuously
along the whole path of an ion with high probability,
while the probability of the ion stopping at the beginning
of its path is very low.
Accounting for these differences leads to different ini-
tial conditions that must be used in the equations for
the range distributions and ELDs. As a consequence,
the equations for the cumulants of the loss distribution
functions have additional terms on the right-hand side
which are proportional to the stopping per unit length.
This term gives the main contribution into the values of
the cumulants and the corresponding parameters. On
the other hand, the equations for the cumulants of the
range distribution function do not contain this term and
account only for nuclear scattering, which makes a con-
tribution to the values of the cumulants of the ELD func-
tion to be insignificant.
1. J. Lindhard, M. Scharf, and H.E. Schiott, Kgl. Danske
Videnskab. Selskab, Mat.-Fys. Medd. 33, No. 14, 1
(1963).
2. A.F. Burenkov, F.F. Komarov, M.A. Kumakhov, and
M.M. Temkin, Tables of Parameters of the Spatial Dis-
tribution of an Implanted Admixture (Bel. State Univ.,
Minsk, 1980) (in Russian).
3. D.G. Ashworth, M.D.J. Bowyer, and R.J. Oven, J. Phys.
D: Appl. Phys. A 24, 1376 (1991).
4. M.V. Makarets and S.N. Storchaka, Ukr. Fiz. Zh. 46, 486
(2001).
5. V.V. Ilyina and M.V. Makarets, Ukr. Fiz. Zh. 49, 815
(2004).
6. M.V. Makarets, V.V. Ilyina, and V.V. Moskalenko, Vac-
uum. 78, 381 (2005).
7. J. Lindhard, V. Nielsen, M. Scharff, and P.V. Thomsen,
Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 33, No.
10, 1 (1963).
8. V.V. Ilyina and M.V. Makarets, Abstr. of the 2nd Con-
ference FMMN’2008 (SCPT, Kharkiv, 2008), 2, p. 534.
9. E.G. Gamaly and L.T. Chaddernon, Proc. Roy. Soc.
Lond. 449, 381 (1995).
10. A.D. Polyanin and A.V. Manzhirov, Handbook of Integral
Equations (CRC Press, Boca Raton, 1998).
11. www.srim.org
12. J.F. Ziegler, J.P. Biersack, and U. Littmark, The Stop-
ping and Ranges of Ions in Solids (Pergamon Press, New
York, 1985), V.1.
13. W. Feller, An Introduction to Probability Theory and Its
Applications (Wiley, New York, 1970).
14. M.D.J. Bowyer, D.G. Ashworth, and R. Oven, J. Phys.
D: Appl. Phys. 29, 1274 (1996).
Received 07.07.09
242 ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2
DISTRIBUTION OF ENERGY LOSSES
РОЗПОДIЛ ВТРАТ ЕНЕРГIЇ ШВИДКИХ IОНIВ
ВЗДОВЖ ЇХ ШЛЯХУ У ТВЕРДОМУ ТIЛI
В.В. Iльїна, М.В. Макарець
Р е з ю м е
Отримано iнтегродиференцiальнi рiвняння для кумулянтiв
функцiї розподiлу втрат енергiї швидких iонiв вздовж їх шля-
ху у твердому тiлi, у яких враховано втрати енергiї починаючи
iз точки старту. Рiвняння для перших семи кумулянтiв розпо-
дiлiв втрат енергiї у пружних, непружних та обох типах зi-
ткнень розв’язанi чисельно за допомогою методу, розвинутого
авторами ранiше, на iнтервалi енергiй iонiв 1 кеВ–1 ГеВ. Їх по-
рiвняння з параметрами розподiлу пробiгiв iонiв показало, що:
1) середнiй шлях за розподiлом втрат енергiї становить 30–90%
повного пробiгу iонiв; 2) при низьких енергiях страгглiнг за
розподiлом втрат енергiї дещо бiльший або однаковий iз страг-
глiнгом за розподiлом пробiгiв iонiв, а при високих енергiях
перший перевищує останнiй у десятки разiв; 3) скiсностi i екс-
цеси розподiлiв при низьких енергiях вiдповiдно близькi, у той
час як при високих енергiях їх змiна для розподiлу втрат енер-
гiї на кiлька порядкiв менша, нiж для розподiлу пробiгiв iонiв.
Звiдси випливає, що розподiл втрат енергiї ширший i значно
ближчий до нормального, нiж розподiл пробiгiв iонiв при всiх
енергiях iмплантацiї. Показано, що цi властивостi розподiлу
втрат енергiї зумовленi новими членами у рiвняннях, якi не
враховувалися ранiше, але домiнують при високих енергiях.
РАСПРЕДЕЛЕНИЕ ПОТЕРЬ ЭНЕРГИИ БЫСТРЫХ
ИОНОВ ВДОЛЬ ИХ ПУТИ В ТВЕРДОМ ТЕЛЕ
В.В. Ильина, Н.В. Макарец
Р е з ю м е
Получены интегродифференциальные уравнения для куму-
лянтов функции распределения потерь энергии быстрых ио-
нов вдоль их пути в твердом теле, в которых учтены потери
энергии начиная с точки старта. Уравнения для первых семи
кумулянтов распределений потерь энергии в упругих, неупру-
гих и обеих типах столкновений решены численно с помощью
метода, развитого авторами ранее, на интервале энергий ионов
1 кэВ–1 ГэВ. Их сравнение с параметрами распределения про-
бегов ионов показало, что: 1) средний путь с потерей энергии
составляет 30–90% полного пробега иона; 2) при низких энерги-
ях страгглинг распределения потерь энергии немного больше
или одинаков со страгглингом распределения пробегов ионов,
а при высоких энергиях первый превышает последний в деся-
тки раз; 3) скошенности и эксцессы распределений при низких
энергиях соответственно близки, в то время как при высоких
энергиях их изменение для распределения потерь энергии на
несколько порядков меньше, чем для распределения пробегов
ионов. Отсюда следует, что распределение потерь энергии бо-
лее широкое и ближе к нормальному, чем распределение про-
бегов ионов для всех энергий имплантации. Показано, что эти
свойства распределения потерь энергии обусловлены новыми
членами в уравнениях, которые не учитывались ранее, но до-
минируют при высоких энергиях.
ISSN 2071-0194. Укр. фiз. журн. 2010. Т. 55, №2 243
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