In this paper, we introduce an approach to increase integration rate of elements of a current source circuit. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/ or radiation defects should be optimized.

1. www.ajms.com 59
ISSN 2581-3463
RESEARCH ARTICLE
On Approach to Increase Integration Rate of Elements of a Current Source Circuit
E. L. Pankratov1,2
1
Department of Mathematics, Nizhny Novgorod State University, 23 Gagarin Avenue, Nizhny Novgorod,
603950, Russia, 2
Department of Mathematics, Nizhny Novgorod State Technical University, 24 Minin Street,
Nizhny Novgorod, 603950, Russia
Received: 25-06-2020; Revised: 10-07-2020; Accepted: 10-08-2020
ABSTRACT
In this paper, we introduce an approach to increase integration rate of elements of a current source circuit.
Framework the approach, we consider a heterostructure with special configuration. Several specific
areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/
or radiation defects should be optimized.
Key words: Analytical approach for modeling, current source circuit, optimization of manufacturing
INTRODUCTION
An actual and intensively solving problem of solid-state electronics is increasing of integration rate of
elements of integrated circuits (p-n-junctions, their systems et al.).[1-8]
Increasing of the integration rate
leads to necessity to decrease their dimensions. To decrease, the dimensions are using several approaches.
They are widely using laser and microwave types of annealing of infused dopants. These types of
annealing are also widely using for annealing of radiation defects, generated during ion implantation.[9-17]
Using the approaches gives a possibility to increase integration rate of elements of integrated circuits
through inhomogeneity of technological parameters due to generating inhomogenous distribution of
temperature. In this situation, one can obtain decreasing dimensions of elements of integrated circuits[18]
with account Arrhenius law.[1,3]
Another approach to manufacture elements of integrated circuits with
smaller dimensions is doping of heterostructure by diffusion or ion implantation.[1-3]
However, in this
case, optimization of dopant and/or radiation defects is required.[18]
In this paper, we consider a heterostructure. The heterostructure consists of a substrate and several
epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further, we
consider doping of these sections by diffusion or ion implantation. The doping gives a possibility to
manufacture field-effect transistors framework a current source circuit so as it is shown in Figure 1.
The manufacturing gives a possibility to increase density of elements of the current source circuit.[4]
After the considered doping, dopant and/or radiation defects should be annealed. Framework the paper,
we analyzed dynamics of redistribution of dopant and/or radiation defects during their annealing. We
introduce an approach to decrease dimensions of the element. However, it is necessary to complicate
technological process.
METHOD OF SOLUTION
In this section, we determine spatiotemporal distributions of concentrations of infused and implanted
dopants. To determine these distributions, we calculate appropriate solutions of the second Fick’s law.[1,3,18]
Address for correspondence:
E. L. Pankratov
E-mail: elp2004@mail.ru

2. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 60
C x y z t
t
, , ,
( )
∂
=











x
D
C x y z t
x y
D
C x y z t
y z
D
C x
C C C
, , , , , ,
( )





 +
( )





 +
,
, , ,
y z t
z
( )







(1)
Boundary and initial conditions for the equations are


C x y z t
x x
, , ,
( )
=
=0
0,


C x y z t
x x Lx
, , ,
( )
=
=
0,


C x y z t
y y
, , ,
( )
=
=0
0,


C x y z t
y x Ly
, , ,
( )
=
=
0,


C x y z t
z z
, , ,
( )
=
=0
0,


C x y z t
z x Lz
, , ,
( )
=
=
0, C (x,y,z,0)=f(x,y,z).(2)
The function C(x,y,z,t) describes the spatiotemporal distribution of concentration of dopant; T is the
temperature of annealing; DС
is the dopant diffusion coefficient. Value of dopant diffusion coefficient
could be changed with changing materials of heterostructure, with changing temperature of materials
(including annealing), with changing concentrations of dopant and radiation defects. We approximate
dependences of dopant diffusion coefficient on parameters by the following relation with account results
in Refs.[20-22]
D D x y z T
C x y z t
P x y z T
V x y z t
C L
= ( ) +
( )
( )





 +
(
, , ,
, , ,
, , ,
, , ,
1 1 1
ξ ς
γ
γ
)
)
+
( )
( )








V
V x y z t
V
* *
, , ,
ς2
2
2
(3)
Here, the function DL
(x,y,z,T) describes the spatial (in heterostructure) and temperature (due toArrhenius
law) dependences of diffusion coefficient of dopant. The function P(x,y,z,T) describes the limit of
solubility of dopant. Parameter γ ∈[1,3]
describes average quantity of charged defects interacted with
atom of dopant.[20]
The function V(x,y,z,t) describes the spatiotemporal distribution of concentration of
radiation vacancies. Parameter V*
describes the equilibrium distribution of concentration of vacancies.
The considered concentrational dependence of dopant diffusion coefficient has been described in details
Figure 1: The considered cascaded inverter[4]

3. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 61
in Kozlivsky.[20]
It should be noted that using diffusion type of doping did not generation radiation
defects. In this situation, ζ1
= ζ2
= 0. We determine spatiotemporal distributions of concentrations of
radiation defects by solving the following system of equations.[21,22]








I x y z t
t x
D x y z T
I x y z t
x y
D x y z
I I
, , ,
, , ,
, , ,
, ,
( )
= ( )
( )





 + ,
,
, , ,
T
I x y z t
y
( )
( )





 +






z
D x y z T
I x y z t
z
k x y z T I x y z t V
I I V
, , ,
, , ,
, , , , , ,
,
( )
( )





 − ( ) ( ) x
x y z t
, , ,
( )−
k x y z T I x y z t
I I
, , , , , , ,
( ) ( )
2
(4)








V x y z t
t x
D x y z T
V x y z t
x y
D x y z
V V
, , ,
, , ,
, , ,
, ,
( )
= ( )
( )





 + ,
,
, , ,
T
V x y z t
y
( )
( )





 +






z
D x y z T
V x y z t
z
k x y z T I x y z t V
V I V
, , ,
, , ,
, , , , , ,
,
( )
( )





 − ( ) ( ) x
x y z t
, , ,
( )+
k x y z T V x y z t
V V
, , , , , , ,
( ) ( )
2
.
Boundary and initial conditions for these equations are
∂ ρ
∂
x y z t
x x
, , ,
( )
=
=0
0,
∂ ρ
∂
x y z t
x x Lx
, , ,
( )
=
=
0,
∂ ρ
∂
x y z t
y y
, , ,
( )
=
=0
0,
∂ ρ
∂
x y z t
y y Ly
, , ,
( )
=
=
0,
∂ ρ
∂
x y z t
z z
, , ,
( )
=
=0
0,
∂ ρ
∂
x y z t
z z Lz
, , ,
( )
=
=
0,  (x,y,z,0)=fρ
(x,y,z).(5)
Here, ρ = I,V. The function I(x,y,z,t) describes the spatiotemporal distribution of concentration of
radiation interstitials; Dρ(x,y,z,T) is the diffusion coefficients of point radiation defects; terms V2
(x,y,z,t)
and I2
(x,y,z,t) correspond to generation divacancies and diinterstitials; kI,V
(x,y,z,T) is the parameter of
recombination of point radiation defects; kI,I
(x,y,z,T) and kV,V
(x,y,z,T) are the parameters of generation of
simplest complexes of point radiation defects.
Further, we determine distributions in space and time of concentrations of divacancies ΦV
(x,y,z,t) and
diinterstitials ΦI
(x,y,z,t) by solving the following system of equations.[21,22]








Φ Φ
Φ Φ
I
I
I
I
x y z t
t x
D x y z T
x y z t
x y
D x
, , ,
, , ,
, , ,
( )
= ( )
( )





 + ,
, , ,
, , ,
y z T
x y z t
y
I
( )
( )





 +


Φ




z
D x y z T
x y z t
z
k x y z T I x y z t
I
I
I I
Φ
Φ
, , ,
, , ,
, , , , , ,
,
( )
( )





 + ( ) 2
(
( )− ( ) ( )
k x y z T I x y z t
I , , , , , , (6)








Φ Φ
Φ Φ
V
V
V
V
x y z t
t x
D x y z T
x y z t
x y
D x
, , ,
, , ,
, , ,
( )
= ( )
( )





 + ,
, , ,
, , ,
y z T
x y z t
y
V
( )
( )





 +


Φ

4. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 62




z
D x y z T
x y z t
z
k x y z T V x y z t
V
V
V V
Φ
Φ
, , ,
, , ,
, , , , , ,
,
( )
( )





 + ( ) 2
(
( )− ( ) ( )
k x y z T V x y z t
V , , , , , , .
Boundary and initial conditions for these equations are [19,23]
∂
∂
ρ
Φ x y z t
x x
, , ,
( )
=
=0
0,
∂
∂
ρ
Φ x y z t
x x Lx
, , ,
( )
=
=
0,
∂
∂
ρ
Φ x y z t
y y
, , ,
( )
=
=0
0,
∂
∂
ρ
Φ x y z t
y y Ly
, , ,
( )
=
=
0,
∂
∂
ρ
Φ x y z t
z z
, , ,
( )
=
=0
0,
∂
∂
ρ
Φ x y z t
z z Lz
, , ,
( )
=
=
0,
I
(x,y,z,0)=fΦ
I
(x,y,z), V
(x,y,z,0)=fΦ
V
(x,y,z).(7)
Here, DΦρ
(x,y,z,T) is the diffusion coefficients of the above complexes of radiation defects; kI
(x,y,z,T) and
kV
(x,y,z,T) are the parameters of decay of these complexes.
We calculate distributions of concentrations of point radiation defects in space and time by recently
elaborated approach.[18]
The approach based on transformation of approximations of diffusion coefficients
in the following form: Dρ
(x,y,z,T)=D0ρ
[1+ερ
gρ
(x,y,z,T)], where D0ρ
is the average values of diffusion
coefficients, 0≤ερ
1, |gρ
(x, y,z,T)|≤1, ρ =I,V. We also used analogous transformation of approximations
of parameters of recombination of point defects and parameters of generation of their complexes:
kI,V
(x,y,z,T)=k0I,V
[1+εI,V
gI,V
(x,y,z,T)], kI,I
(x,y,z,T)=k0I,I
[1+εI,I
gI,I
(x,y,z,T)] and kV,V
(x,y,z,T)=k0V,V
[1+εV,V
gV,V
(x,y,z,T)], where k0ρ1,ρ2
is the their average values, 0≤εI,V
1, 0≤εI,I
1, 0≤εV,V
1, | gI,V
(x,y,z,T)|≤1,
|gI,I
(x,y,z,T)|≤1, |gV,V
(x,y,z,T)|≤1. Let us introduce the following dimensionless variables:

I x y z t I x y z t I
, , , , , , *
( ) = ( ) , 
V x y z t
, , ,
( ) = = ( )
V x y z t V
, , , *
,  = L k D D
I V I V
2
0 0 0
, ,
Ω  
= L k D D
I V
2
0 0 0
, , ϑ = D D t L
I V
0 0
2
, χ = x/Lx
, η = y /Ly
, φ = z/Lz
. The introduction leads to
transformation of Equation (4) and conditions (5) to the following form
∂ χ η ϕ
∂
∂
∂ χ
ε χ η ϕ
∂ χ η ϕ
 
I D
D D
g T
I
I
I V
I I
, , ,
, , ,
, , ,
ϑ
ϑ
ϑ
( )
= + ( )

 

( )
0
0 0
1
∂
∂ χ
∂
∂ η
ε χ η ϕ






+ + ( )

 
 ×
{ 1 I I
g T
, , ,
∂ χ η ϕ ϑ
∂ η
∂
∂ ϕ
ε χ η ϕ

I D
D D
D
D D
g T
I
I V
I
I V
I I
, , ,
, , ,
( )


+ + ( )

 

0
0 0
0
0 0
1
∂
∂ χ η ϕ ϑ
∂ ϕ
χ η ϕ ϑ


I
I
, , ,
, , ,
( )






− ( ) ×
ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
1 1
+ ( )

 
 ( )− + ( )

I V I V I I I I I
g T V g T
, , , ,
, , , , , , , , ,
 Ω 
 
 ( )

I 2
χ η ϕ ϑ
, , , (8)
∂ χ η ϕ
∂
∂
∂ χ
ε χ η ϕ
∂ χ η ϕ
 
V D
D D
g T
V
V
I V
V V
, , ,
, , ,
, , ,
ϑ
ϑ
ϑ
( )
= + ( )

 

( )
0
0 0
1
∂
∂ χ
∂
∂ η
ε χ η ϕ






+ + ( )

 
 ×
{ 1 V V
g T
, , ,
∂ χ η ϕ ϑ
∂ η
∂
∂ ϕ
ε χ η ϕ

V D
D D
D
D D
g T
V
I V
V
I V
V V
, , ,
, , ,
( )


+ + ( )

 

0
0 0
0
0 0
1
∂
∂ χ η ϕ ϑ
ϕ
χ η ϕ ϑ


V
I
, , ,
, , ,
( )
∂






− ( ) ×
ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
1 1
+ ( )

 
 ( )− + ( )

I V I V V V V V V
g T V g T
, , , ,
, , , , , , , , ,
 Ω 
 
 ( )

V 2
χ η ϕ ϑ
, , ,

5. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 63
∂ ρ χ η ϕ
∂ χ χ
 , , ,ϑ
( )
=
=0
0 ,
∂ ρ χ η ϕ
∂ χ χ
 , , ,ϑ
( )
=
=1
0,
∂ ρ χ η ϕ
∂ η η
 , , ,ϑ
( )
=
=0
0 ,
∂ ρ χ η ϕ
∂ η η
 , , ,ϑ
( )
=
=1
0,
∂ ρ χ η ϕ
∂ ϕ ϕ
 , , ,ϑ
( )
=
=0
0,
∂ ρ χ η ϕ
∂ ϕ ϕ
 , , ,ϑ
( )
=
=1
0, 
ρ χ η ϕ
χ η ϕ
ρ
ρ
, , ,
, , ,
*
ϑ
ϑ
( ) =
( )
f
(9)
We determine solutions of Equation (8) with conditions (9) framework recently introduced approach,[18]
that is, as the power series
 
ρ χ η ϕ ε ω ρ χ η ϕ
ρ ρ
, , , , , ,
ϑ ϑ
( ) = ( )
=
∞
=
∞
=
∞
∑
∑
∑ i j k
ijk
k
j
i
Ω
0
0
0
.(10)
Substitution of the series (10) into Equation (8) and conditions (9) gives us possibility to obtain equations
for initial order approximations of concentration of point defects 
I000 χ η ϕ
, , ,ϑ
( ) and 
V000 χ η ϕ
, , ,ϑ
( ) and
corrections for them 
Iijk χ η ϕ
, , ,ϑ
( ) and 
Vijk χ η ϕ
, , ,ϑ
( ), i ≥1, j ≥1, k ≥1. The equations are presented in
the Appendix. Solutions of the equations could be obtained by standard Fourier approach.[24,25]
The
solutions are presented in the Appendix.[34]
Now, we calculate distributions of concentrations of simplest complexes of point radiation defects in
space and time. To determine the distributions, we transform approximations of diffusion coefficients in
the following form: DΦρ
(x,y,z,T)=D0Φρ
[1+εΦρ
gΦρ
(x,y,z,T)], where D0Φρ
is the average values of diffusion
coefficients. In this situation, the Equation (6) could be written as
∂
∂
∂
∂
ε
∂
∂
Φ Φ
Φ Φ Φ
I
I I I
I
x y z t
t
D
x
g x y z T
x y z t
x
, , ,
, , ,
, , ,
( )
= + ( )

 

( )

0 1






+ ( ) ( )+
k x y z T I x y z t
I I
, , , , , , ,
2
D
y
g x y z T
x y z t
y
D
z
I I I
I
I
0 0
1 1
Φ Φ Φ Φ
Φ
∂
∂
ε
∂
∂
∂
∂
+ ( )

 

( )






+ +
, , ,
, , ,
ε
ε
∂
∂
Φ Φ
Φ
I I
I
g x y z T
x y z t
z
, , ,
, , ,
( )

 

( )






−
k x y z T I x y z t
I , , , , , ,
( ) ( )
∂
∂
∂
∂
ε
∂
Φ Φ
Φ Φ Φ
V
V V V
V
x y z t
t
D
x
g x y z T
x y z t
x
, , ,
, , ,
, , ,
( )
= + ( )

 

( )


0 1





+ ( ) ( )+
k x y z T I x y z t
I I
, , , , , , ,
2
D
y
g x y z T
x y z t
y
D
z
V V V
V
V
0 0
1 1
Φ Φ Φ Φ
Φ
∂
∂
ε
∂
∂
∂
∂
+ ( )

 

( )






+ +
, , ,
, , ,
ε
ε
∂
∂
Φ Φ
Φ
V V
V
g x y z T
x y z t
z
, , ,
, , ,
( )

 

( )






−
k x y z T I x y z t
I , , , , , ,
( ) ( ).
Farther, we determine solutions of above equations as the following power series
Φ Φ
Φ
ρ ρ ρ
ε
x y z t x y z t
i
i
i
, , , , , ,
( ) = ( )
=
∞
∑
0
.(11)
Now, we used the series (11) into Equation (6) and appropriate boundary and initial conditions. The using
gives the possibility to obtain equations for initial order approximations of concentrations of complexes

6. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 64
of defects Φρ0
(x,y,z,t), corrections for them Φρi
(x,y,z,t) (for them i ≥1) and boundary and initial conditions
for them. We remove equations and conditions to the Appendix. Solutions of the equations have been
calculated by standard approaches[24,25]
and presented in the Appendix.
Now, we calculate distribution of concentration of dopant in space and time using the approach, which
was used for analysis of radiation defects. To use the approach, we consider the following transformation
of approximation of dopant diffusion coefficient: DL
(x,y,z,T)=D0L
[1+ εL
gL
(x,y,z,T)], where D0L
is the
average value of dopant diffusion coefficient, 0≤εL
1, |gL
(x,y,z,T)|≤1. Farther, we consider solution of
Equation (1) as the following series:
C x y z t C x y z t
L
i j
ij
j
i
, , , , , ,
( ) = ( )
=
∞
=
∞
∑
∑ε ξ
1
0
.
Using the relation into Equation (1) and conditions (2) leads to obtaining equations for the functions
Cij
(x,y,z,t) (i ≥1, j ≥1), boundary and initial conditions for them. The equations are presented in the
Appendix.Solutionsoftheequationshavebeencalculatedbystandardapproaches(see,forexample,[24,25]
).
The solutions are presented in the Appendix.
We analyzed distributions of concentrations of dopant and radiation defects in space and time analytically
using the second-order approximations on all parameters, which have been used in appropriate series.
Usually, the second-order approximations are enough good approximations to make qualitative analysis
and to obtain quantitative results. All analytical results have been checked by numerical simulation.
DISCUSSION
In this section, we analyzed spatiotemporal distributions of concentrations of dopants. Figure 2 shows
typical spatial distributions of concentrations of dopants in neighborhood of interfaces of heterostructures.
We calculate these distributions of concentrations of dopants under the following condition: Value of
dopant diffusion coefficient in doped area is larger, than value of dopant diffusion coefficient in nearest
areas. In this situation, one can find increasing of compactness of field-effect transistors with increasing
of homogeneity of distribution of concentration of dopant at 1 time. Changing relation between values
of dopant diffusion coefficients leads to opposite result [Figure 3].
Figure 2a: Dependences of concentration of dopant, infused in heterostructure from Figure 1, on coordinate in direction,
which is perpendicular to interface between epitaxial layer substrate. Difference between values of dopant diffusion
coefficient in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in
the epitaxial layer is larger, than value of dopant diffusion coefficient in the substrate

7. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 65
It should be noted that framework the considered approach one shall optimize annealing of dopant and/
or radiation defects. To do the optimization, we used recently introduced criterion.[26-33]
The optimization
based on approximation real distribution by step-wise function  (x,y,z) [Figure 4]. Farther, the required
values of optimal annealing time have been calculated by minimization the following mean-squared
error.
Figure 2b: Dependences of concentration of dopant, implanted in heterostructure from Figure 1, on coordinate in direction,
which is perpendicular to interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient
in layers of heterostructure increases with increasing of number of curves. Value of dopant diffusion coefficient in the epitaxial
layer is larger, than value of dopant diffusion coefficient in the substrate. Curve 1 corresponds to homogenous sample and
annealing time Θ = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0
. Curve 2 corresponds to homogenous sample and annealing time Θ = 0.0057
(Lx
2
+Ly
2
+Lz
2
)/D0
. Curves 3 and 4 correspond to heterostructure from Figure 1; annealing times Θ = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0
and Θ = 0.0057 (Lx
2
+Ly
2
+Lz
2
)/D0
, respectively
Figure 3a: Distributions of concentration of dopant, infused in average section of epitaxial layer of heterostructure from
Figure 1 in direction parallel to interface between epitaxial layer and substrate of heterostructure. Difference between values
of dopant diffusion coefficients increases with increasing of number of curves. Value of dopant diffusion coefficient in this
section is smaller, than value of dopant diffusion coefficient in nearest sections

8. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 66
U
L L L
C x y z x y z d z d y d x
x y z
L
L
L z
y
x
= ( )− ( )

 

∫
∫
∫
1
0
0
0
, , , , ,
Θ  (12)
We show optimal values of annealing time as functions of parameters on Figure 5. It is known that
standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal case
after finishing the annealing, dopant achieves interface between layers of heterostructure. If the dopant
Figure 3b: Calculated distributions of implanted dopant in epitaxial layers of heterostructure. Solid lines are spatial
distributions of implanted dopant in system of two epitaxial layers. Dashed lines are spatial distributions of implanted dopant
in one epitaxial layer. Annealing time increases with increasing of number of curves
Figure 4a: Distributions of concentration of infused dopant in depth of heterostructure from Figure 1 for different values of
annealing time (curves 2–4) and idealized step-wise approximation (curve 1). Increasing of number of curve corresponds to
increasing of annealing time

9. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 67
has no enough time to achieve the interface, it is practicably to anneal the dopant additionally. Figure 5b
shows the described dependences of optimal values of additional annealing time for the same parameters
as for Figure 5a. Necessity to anneal radiation defects leads to smaller values of optimal annealing of
implanted dopant in comparison with optimal annealing time of infused dopant.
Figure 4b: Distributions of concentration of implanted dopant in depth of heterostructure from Figure 1 for different values
of annealing time (curves 2–4) and idealized step-wise approximation (curve 1). Increasing of number of curve corresponds
to increasing of annealing time
Figure 5a: Dimensionless optimal annealing time of infused dopant as a function of several parameters. Curve 1 describes
the dependence of the annealing time on the relation a/L and  = γ = 0 for equal to each other values of dopant diffusion
coefficient in all parts of heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter ε
for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of the annealing time on value of parameter  for a/L=1/2 and
ε = γ = 0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0

10. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 68
CONCLUSIONS
In this paper, we introduce an approach to increase integration rate of element of a current source
circuit. The approach gives us possibility to decrease area of the elements with smaller increasing of the
element’s thickness.
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16. Shishiyanu ST, Shishiyanu TS, Railyan SK. Shallow pn-junctions in Si prepared by pulse photon annealing.
Figure 5b: Dimensionless optimal annealing time of implanted dopant as a function of several parameters. Curve 1 describes
the dependence of the annealing time on the relation a/L and ξ = γ = 0 for equal to each other values of dopant diffusion
coefficient in all parts of heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter ε
for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of the annealing time on value of parameter ξ for a/L=1/2 and
ε = γ = 0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L=1/2 and ε = ξ = 0

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Semiconductors 2002;36:611-7.
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Sci Semicond Proc 2015;34:260-8.

12. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 70
APPENDIX
Equations for the functions 
Iijk χ η ϕ
, , ,ϑ
( ) and 
Vijk χ η ϕ
, , ,ϑ
( ), i ≥0, j ≥0, k ≥0 and conditions for them
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
I D
D
I I
I
V
000 0
0
2
000
2
2
000
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
(
( )
∂
+
∂ ( )
∂






η
χ η ϕ
ϕ
2
2
000
2

I , , ,ϑ
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
V D
D
V V
V
I
000 0
0
2
000
2
2
000
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
(
( )
∂
+
∂ ( )
∂






η
χ η ϕ
ϕ
2
2
000
2

V , , ,ϑ
;
∂
∂
∂
∂
∂
∂
  
I D
D
I I
i I
V
i i
00 0
0
2
00
2
2
00
χ χ η ϕ
χ
χ η ϕ
η
, , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
( )
2
2
2
00
2
0
0
+
( )





 + ×
∂
∂

I D
D
i I
V
χ η ϕ
ϕ
, , ,ϑ
∂
∂
∂
∂
∂
∂
∂
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
g T
I
g T
I
I
i
I
i
, , ,
, , ,
, , ,
( )
( )





 + ( )
−
 
100 −
− ( )





 +





100 χ η ϕ ϑ
η
, , ,
∂
∂
∂
∂
∂
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
g T
I
I
i
, , ,
, , ,
( )
( )











−

100
, i ≥1,
∂
∂
∂
∂
∂
∂
  
V D
D
V V
i V
I
i i
00 0
0
2
00
2
2
00
χ χ η ϕ
χ
χ η ϕ
η
, , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
( )
2
2
2
00
2
+
( )





 + ( ) ×



∂
∂
∂
∂

V
g T
i
V
χ η ϕ
ϕ χ
χ η ϕ
, , ,
, , ,
ϑ
∂
∂
∂
∂
∂
 
V D
D
D
D
g T
V
i V
I
V
I
V
i
− −
( )

 + ( )
100 0
0
0
0
100
χ η ϕ ϑ
χ η
χ η ϕ
χ
, , ,
, , ,
,
, , ,
, , ,
η ϕ ϑ
η ϕ
χ η ϕ
( )





 + ( ) ×



∂
∂
∂
g T
V
∂ ( )
∂



−

V D
D
i V
I
100 0
0
χ η ϕ ϑ
ϕ
, , ,
, i ≥1,
∂
∂
∂
∂
∂
  
I D
D
I I
I
V
010 0
0
2
010
2
2
010
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
010
2

I , , ,ϑ
1 000 000
+ ( )

 
 ( ) ( )
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I V I V
g T I V
, , , , , , , , , , ,
 
∂
∂
∂
∂
∂
  
V D
D
V V
V
I
010 0
0
2
010
2
2
010
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
010
2

V , , ,ϑ
1 000 000
+ ( )

 
 ( ) ( )
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I V I V
g T I V
, , , , , , , , , , ,
  ;

13. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 71
∂
∂
∂
∂
∂
  
I D
D
I I
I
V
020 0
0
2
020
2
2
020
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
020
2

I , , ,ϑ
1 010 000 000
+ ( )

 
 ( ) ( )+
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ χ
I V I V
g T I V I
, , , , , , , , , , ,
   ,
, , , , , ,
η ϕ ϑ χ η ϕ ϑ
( ) ( )

 


V010
∂
∂
∂
∂
∂
  
V D
D
V V
I
V
020 0
0
2
020
2
2
020
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
020
2

V , , ,ϑ
1 010 000 000
+ ( )

 
 ( ) ( )+
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ χ
I V I V
g T I V I
, , , , , , , , , , ,
   ,
, , , , , ,
η ϕ ϑ χ η ϕ ϑ
( ) ( )

 


V010 ;
∂
∂
∂
∂
∂
  
I D
D
I I
I
V
001 0
0
2
001
2
2
001
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
001
2

I , , ,ϑ
1 000
2
+ ( )

 
 ( )
ε χ η ϕ χ η ϕ ϑ
I I I I
g T I
, , , , , , , ,

∂
∂
∂
∂
∂
  
V D
D
V V
V
I
001 0
0
2
001
2
2
001
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
001
2

V , , ,ϑ
1 000
2
+ ( )

 
 ( )
ε χ η ϕ χ η ϕ ϑ
I I I I
g T V
, , , , , , , ,
 ;
∂
∂
∂
∂
∂
  
I D
D
I I
I
V
110 0
0
2
110
2
2
110
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 + ×
∂
∂
∂
η
χ η ϕ
ϕ
2
2
110
2
0
0

I D
D
I
V
, , ,ϑ
∂
∂
∂
∂
∂
∂
∂
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
g T
I
g T
I
I I
, , ,
, , ,
, , ,
( )
( )





 + ( )
 
010 010 χ
χ η ϕ ϑ
η ϕ
χ η ϕ
, , ,
, , ,
( )





 + ( ) ×






 ∂
∂
∂
g T
I
∂
∂

  
I
I V I
010
100 000 0
χ η ϕ ϑ
ϕ
χ η ϕ ϑ χ η ϕ ϑ
, , ,
, , , , , ,
( )







− ( ) ( )+ 0
00 100
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
( ) ( )

 
 ×

V
1+ ( )

 

ε χ η ϕ
I I I I
g T
, , , , ,
( ) ( ) ( ) ( )
2 2 2
110 110 110 110
0V
2 2 2
0I
V , , , V , , , V , , , V , , ,
D
D
χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ
ϑ χ η ϕ
 
∂ ∂ ∂ ∂
= + + +
 
∂ ∂ ∂ ∂
 
 
   
D
D
g T
V
g T
V
I
V V
0
0
010
∂
∂
∂
∂
∂
∂
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
, , ,
, , ,
, , ,
( )
( )





 + ( )
 ∂
∂
∂

V010 χ η ϕ ϑ
η
, , ,
( )





 +






14. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 72
∂
∂
∂
∂
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
ε χ η ϕ
g T
V
g
V V V V V
, , ,
, , ,
, ,
, ,
( )
( )











− +

010
1 ,
,T
( )

 
 ×
   
V I V I
100 000 000 100
χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ
, , , , , , , , , , , ,
( ) ( )+ ( ) ( )


 
 ;
∂
∂
∂
∂
∂
  
I D
D
I I
I
V
002 0
0
2
002
2
2
002
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
002
2

I , , ,ϑ
1 001 000
+ ( )

 
 ( ) ( )
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I I I I
g T I I
, , , , , , , , , , ,
 
∂
∂
∂
∂
∂
  
V D
D
V V
V
I
002 0
0
2
002
2
2
002
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 −
∂
∂
∂
η
χ η ϕ
ϕ
2
2
002
2

V , , ,ϑ
1 001 000
+ ( )

 
 ( ) ( )
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
V V V V
g E V V
, , , , , , , , , , ,
  ;
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
I D
D
I I
I
V
101 0
0
2
101
2
2
101
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
(
( )
∂
+
∂ ( )
∂





 +
η
χ η ϕ
ϕ
2
2
101
2

I , , ,ϑ
D
D
g T
I
g T
I
V
I I
0
0
001
∂
∂
( )
∂ ( )
∂





 +
∂
∂
( )
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
, , ,
, , ,
, , ,
 ∂
∂ ( )
∂





 +






I001 χ η ϕ ϑ
η
, , ,
∂
∂
∂
∂
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
ε χ η ϕ
g T
I
g T
I I I
, , ,
, , ,
, , ,
( )
( )











− + ( )

001
1


 
 ( ) ( )
 
I V
100 000
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
∂
∂
∂
∂
∂
  
V D
D
V V
V
I
101 0
0
2
101
2
2
101
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 +
∂
∂
∂
η
χ η ϕ
ϕ
2
2
101
2

V , , ,ϑ
D
D
g T
V
g T
V
I
V V
0
0
001
∂
∂
( )
∂ ( )
∂





 +
∂
∂
( )
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
, , ,
, , ,
, , ,
 ∂
∂ ( )
∂





 +






V001 χ η ϕ ϑ
η
, , ,
∂
∂
∂
∂
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
ε χ η ϕ
g T
V
g T
V V V
, , ,
, , ,
, , ,
( )
( )











− + ( )

001
1


 
 ( ) ( )
 
I V
000 100
χ η ϕ ϑ χ η ϕ ϑ
, , , , , , ;
∂
∂
∂
∂
∂
  
I D
D
I I
I
V
011 0
0
2
011
2
2
011
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 − ( ) ×
∂
∂
∂
η
χ η ϕ
ϕ
χ η ϕ
2
2
011
2 010


I
I
, , ,
, , ,
ϑ
ϑ

15. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 73
1 1
000
+ ( )

 
 ( )− + ( )

ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
I I I I I V I V
g T I g T
, , , ,
, , , , , , , , ,


 
 ( ) ( )
 
I V
001 000
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
∂
∂
∂
∂
∂
  
V D
D
V V
V
I
011 0
0
2
011
2
2
011
χ η ϕ χ η ϕ
χ
χ η ϕ
, , , , , , , , ,
ϑ
ϑ
ϑ ϑ
( )
=
( )
+
(
( )
+
( )





 − ( ) ×
∂
∂
∂
η
χ η ϕ
ϕ
χ η ϕ
2
2
011
2 010


V
V
, , ,
, , ,
ϑ
ϑ
1 1
000
+ ( )

 
 ( )− + ( )

ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
V V V V I V I V
g T V g t
, , , ,
, , , , , , , , ,


 
 ( ) ( )
 
I V
000 001
χ η ϕ ϑ χ η ϕ ϑ
, , , , , , ;
∂
∂

ρ χ η ϕ
χ
ijk
x
, , ,ϑ
( )
=
=0
0,
∂
∂

ρ χ η ϕ
χ
ijk
x
, , ,ϑ
( )
=
=1
0 ,
∂
∂

ρ χ η ϕ
η η
ijk , , ,ϑ
( )
=
=0
0,
∂
∂

ρ χ η ϕ
η η
ijk , , ,ϑ
( )
=
=1
0,
∂
∂

ρ χ η ϕ
ϕ ϕ
ijk , , ,ϑ
( )
=
=0
0 ,
∂
∂

ρ χ η ϕ
ϕ ϕ
ijk , , ,ϑ
( )
=
=1
0 (i ≥0, j ≥0, k ≥0);

ρ χ η ϕ χ η ϕ ρ
ρ
000 0
, , , , , *
( ) = ( )
f , 
ρ χ η ϕ
ijk , , ,0 0
( ) = (i ≥1, j ≥1, k ≥1).
Solutions of the above equations could be written as

ρ χ η ϕ χ η ϕ
ρ ρ
000
1
1 2
, , ,ϑ ϑ
( ) = + ( ) ( ) ( ) ( )
=
∞
∑
L L
F c c c e
n n
n
,
where F nu nv n w f u v w d wd vd u
n n
ρ ρ
ρ
π π π
= ( ) ( ) ( ) ( )
∫
∫
∫
1
0
1
0
1
0
1
*
cos cos cos , , , cn
(χ) = cos (π n χ),
( ) ( )
2 2
0 0
exp
ϑ ϑ
nI V I
e n D D

= − , e n D D
nV I V
ϑ ϑ
( ) = −
( )
exp π2 2
0 0 ;

I
D
D
nc c c e e s u c
i
I
V
n nI nI n n
00
0
0
2
χ η ϕ π χ η ϕ τ
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) v
v
I u v w
u
i
n
( )
∂ ( )
∂
×
−
=
∞
∫
∫
∫
∫
∑

100
0
1
0
1
0
1
0
1
, , ,τ
ϑ
c w g u v w T d wd vd u d
D
D
nc c c e e
n I
I
V
n nI nI
( ) ( ) − ( ) ( ) ( ) ( ) −
, , , τ π χ η ϕ ϑ τ
2 0
0
(
( ) ( ) ( ) ×
∫
∫
∫
∑
=
∞
c u s v
n n
n 0
1
0
1
0
1
ϑ
c w g u v w T
I u v w
v
d wd vd u d
D
D
nc
n I
i I
V
n
( ) ( )
∂ ( )
∂
−
−
∫ , , ,
, , ,

100
0
1
0
0
2
τ
τ π χ
χ η ϕ ϑ τ
ϑ
( ) ( ) ( ) ( ) −
( ) ×
∫
∑
=
∞
c c e e
nI nI
n 0
1
c u c v s w g u v w T
I u v w
w
d wd vd u d
n n n I
i
( ) ( ) ( ) ( )
∂ ( )
∂
−
∫ , , ,
, , ,

100
0
1
0
1


∫
∫
∫
0
1
, i ≥1,

V
D
D
nc c c e e s u c
i
V
I
n nV nI n n
00
0
0
2
χ η ϕ π χ η ϕ τ
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) v
v g u v w T
V
n
( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
, , ,
0
1
0
1
0
1
0
1
ϑ

16. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 74
c w
V u
u
d wd vd u d
D
D
nc c c e e
n
i V
I
n nV nI
( )
∂ ( )
∂
− ( ) ( ) ( ) ( )
−

100 0
0
,τ
τ χ η ϕ ϑ −
−
( ) ( ) ( ) ×
∫
∫
∫
∑
=
∞
τ
ϑ
c u s v
n n
n 0
1
0
1
0
1
2 2
100
0
1
0
0
π
τ
τ π χ
c w g u v w T
V u
v
d wd vd u d
D
D
nc
n V
i V
I
n
( ) ( )
∂ ( )
∂
− (
−
∫ , , ,
,

)
) ( ) ( ) ( ) ×
=
∞
∑ c c enV
n
η ϕ ϑ
1
e c u c v s w g u v w T
V u
w
d wd vd u d
nI n n n V
i
−
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
τ
τ
τ
, , ,
,

100
0
1
∫
∫
∫
∫
∫ 0
1
0
1
0
ϑ
, i ≥1,
where sn
(χ) = sin (π n χ);

ρ χ η ϕ χ η ϕ τ
ρ ρ
010 2
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
c c c e e c u c v c w
n n n n n n n n (
( ) ×
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
n
1 000 000
+ ( )

 
 ( ) ( )
ε τ τ
I V I V
g u v w T I u v w V u v w d wd vd u d
, , , , , , , , , , ,
  τ
τ ;

ρ χ η ϕ χ η ϕ τ
ρ ρ
020
0
0
2
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( )
D
D
c c c e e c u c
I
V
n n n n n n n v
v c w
n I V
n
( ) ( ) +

 ×
∫
∫
∫
∫
∑
=
∞
1
0
1
0
1
0
1
0
1
ε ,
ϑ
g u v w T I u v w V u v w I u v w
I V
, , , , , , , , , , , , ,
( )
 ( ) ( )+ ( )
  
010 000 000
   

V u v w d wd vd u d
010 , , , 
( )

 
 ;

ρ χ η ϕ χ η ϕ τ
ρ ρ
001 2
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
c c c e e c u c v c w
n n n n n n n n (
( ) ×
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
n
1 000
2
+ ( )

 
 ( )
ε ρ τ τ
ρ ρ ρ ρ
, , , , , , , ,
g u v w T u v w d wd vd u d
 ;

ρ χ η ϕ χ η ϕ τ
ρ ρ
002 2
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
c c c e e c u c v c w
n n n n n n n n (
( ) ×
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
n
1 001 000
+ ( )

 
 ( ) ( )
ε ρ τ ρ τ
ρ ρ ρ ρ
, , , , , , , , , , ,
g u v w T u v w u v w d wd vd u d
  τ
τ ;

I
D
D
nc c c e e s u
I
V
n n n nI nI n
110
0
0
2
χ η ϕ π χ η ϕ τ
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v c u
n n
n
( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
g u v w T
I u v w
u
d wd vd u d
D
D
nc c
I
i I
V
n n
, , ,
, , ,
( )
∂ ( )
∂
− ( ) ( )
−

100 0
0
2
τ
τ π χ η c
c e
n nI
n
ϕ ϑ
( ) ( ) ×
=
∞
∑
1

17. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 75
e c u s v c u g u v w T
I u v w
v
d wd vd u
nI n n n I
i
−
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
τ
τ
, , ,
, , ,

100
d
d
D
D
I
V
τ π
ϑ
0
1
0
1
0
1
0
0
0
2
∫
∫
∫
∫ − ×
n e e c u c v s u g u v w T
I u v w
nI nI n n n I
i
ϑ τ
τ
( ) −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
, , ,
, , ,

100
w
w
d wd vd u d
n
τ
ϑ
0
1
0
1
0
1
0
1
∫
∫
∫
∫
∑
=
∞
×
c c c c e c c e c u c v c v
n n n n nI n n nI n n n
χ η ϕ χ ϑ η ϕ τ
( ) ( ) ( )− ( ) ( ) ( ) ( ) −
( ) ( ) ( )
2 (
( ) + ×


∫
∫
∫
∫
∑
=
∞
1
0
1
0
1
0
1
0
1
ε
ϑ
I V
n
,
g u v w T I u v w V u v w I u v w
I V
, , , , , , , , , , , , ,
( )
 ( ) ( )+ ( )
  
100 000 000
   

V u v w d wd vd u d
100 , , , 
( )

 


V
D
D
nc c c e e s u
V
I
n n n nV nV n
110
0
0
2
χ η ϕ π χ η ϕ τ
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v c u
n n
n
( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
g u v w T
V u v w
u
d wd vd u d
D
D
nc c
V
i V
I
n n
, , ,
, , ,
( )
∂ ( )
∂
− ( ) ( )
−

100 0
0
2
τ
τ π χ η c
c e
n nV
n
ϕ ϑ
( ) ( ) ×
=
∞
∑
1
e c u s v c u g u v w T
V u v w
v
d wd vd u
nV n n n V
i
−
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
τ
τ
, , ,
, , ,

100
d
d
D
D
V
I
τ π
ϑ
0
1
0
1
0
1
0
0
0
2
∫
∫
∫
∫ − ×
ne e c u c v s u g u v w T
V u v w
nV nV n n n V
i
ϑ τ
τ
( ) −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
, , ,
, , ,

100
w
w
d wd vd u d
n
τ
ϑ
0
1
0
1
0
1
0
1
∫
∫
∫
∫
∑
=
∞
×
c c c c e c c e c u c v
n n n n nI n n nV n n
χ η ϕ χ ϑ η ϕ τ ε
( ) ( ) ( )− ( ) ( ) ( ) ( ) −
( ) ( ) ( ) +
2 1 I
I V I V
n
g u v w T
, , , , ,
( )

 
 ×
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
c w I u v w V u v w I u v w V u v w
n ( ) ( ) ( )+ ( )
   
100 000 000 100
, , , , , , , , , , ,
   ,
, 
( )

 
 d wd vd u d ;

I
D
D
nc c c e e s u
I
V
n n n nI nI n
101
0
0
2
χ η ϕ π χ η ϕ τ
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v g u v w T
n I
n
( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
, , ,
0
1
0
1
0
1
0
1
ϑ
c w
I u v w
u
d wd vd u d
D
D
nc c c e
n
I
V
n n n nI
( )
∂ ( )
∂
− ( ) ( ) ( )

001 0
0
2
, , ,τ
τ π χ η ϕ ϑ
ϑ
( ) ×
=
∞
∑
n 1
s v c w g u v w T
I u v w
v
d wd vd u d
D
n n I
I
( ) ( ) ( )
∂ ( )
∂
−
∫
∫ , , ,
, , ,

001
0
1
0
1
0
2
τ
τ π
D
D
ne c c c
V
nI n n n
n
0 1
ϑ χ η ϕ
( ) ( ) ( ) ( ) ×
=
∞
∑

18. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 76
e c u c v s w g u v w T
I u v w
w
d wd vd u d
nI n n n I
−
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
τ
τ
τ
, , ,
, , ,

001
0
0
1
0
1
0
1
0 1
2
∫
∫
∫
∫ ∑
− ( ) ( ) ( ) ×
=
∞
ϑ
χ η ϕ
c c c
n n n
n
e e c u c v c w g u v w T I u
nI nI n n n I V I V
ϑ τ ε
( ) −
( ) ( ) ( ) ( ) + ( )

 

1 100
, , , , , ,
 v
v w V u v w d wd vd u d
, , , , ,
τ τ τ
ϑ
( ) ( )
∫
∫
∫
∫ 
000
0
1
0
1
0
1
0

V
D
D
nc c c e e s u
V
I
n n n nV nV n
101
0
0
2
χ η ϕ π χ η ϕ τ
, , ,ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( )
ϑ c
c v g u v w T
n V
n
( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
, , ,
0
1
0
1
0
1
0
1
ϑ
c w
V u v w
u
d wd vd u d
D
D
nc c c e
n
V
I
n n n nI
( )
∂ ( )
∂
− ( ) ( ) ( )

001 0
0
2
, , ,τ
τ π χ η ϕ ϑ
ϑ τ
ϑ
( ) −
( ) ( ) ×
∫
∫
∑
=
∞
e c u
nV n
n 0
1
0
1
s v c w g u v w T
I u v w
v
d wd vd u d
D
n n I
I
( ) ( ) ( )
∂ ( )
∂
−
∫
∫ , , ,
, , ,

001
0
1
0
1
0
2
τ
τ π
D
D
ne c c c
V
nI n n n
n
0 1
ϑ χ η ϕ
( ) ( ) ( ) ( ) ×
=
∞
∑
e c u c v s w g u v w T
V u v w
w
d wd vd u d
nV n n n V
−
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
τ
τ
τ
, , ,
, , ,

001
0
0
1
0
1
0
1
0 1
2
∫
∫
∫
∫ ∑
− ( ) ( ) ( ) ×
=
∞
ϑ
χ η ϕ
c c c
n n n
n
e e c u c v c w g u v w T I u
nV nV n n n I V I V
ϑ τ ε
( ) −
( ) ( ) ( ) ( ) + ( )

 

1 100
, , , , , ,
 v
v w V u v w d wd vd u d
, , , , ,
τ τ τ
ϑ
( ) ( )
∫
∫
∫
∫ 
000
0
1
0
1
0
1
0
;

I c c c e e c u c v c w
n n n nI nI n n n
011 2
χ η ϕ χ η ϕ τ
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( ) (
( ) ( )
{ ×
∫
∫
∫
∫
∑
=
∞

I u v w
n
000
0
1
0
1
0
1
0
1
, , ,τ
ϑ
1 1
010
+ ( )

 
 ( )+ + ( )

ε τ ε
I I I I I V I V
g u v w T I u v w g u v w T
, , , ,
, , , , , , , , ,


 
 ( ) ( )}
 
I u v w V u v w d wd vd u d
001 000
, , , , , ,
τ τ τ

V c c c e e c u c v c w
n n n nV nV n n n
011 2
χ η ϕ χ η ϕ τ
, , ,ϑ ϑ
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( ) (
( ) ( )
{ ×
∫
∫
∫
∫
∑
=
∞

I u v w
n
000
0
1
0
1
0
1
0
1
, , ,τ
ϑ
1 1
010
+ ( )

 
 ( )+ + ( )

ε τ ε
I I I I I V I V
g u v w T I u v w g u v w T
, , , ,
, , , , , , , , ,


 
 ( ) ( )}
 
I u v w V u v w d wd vd u d
001 000
, , , , , ,
τ τ τ .
Equations for functions Φρi
(x,y,z,t), i ≥0 to describe concentrations of simplest complexes of radiation
defects.







Φ Φ Φ
Φ
I
I
I I
x y z t
t
D
x y z t
x
x y z t
y
0
0
2
0
2
2
0
2
2
, , , , , , , , ,
( )
=
( )
+
( )
+
Φ
ΦI x y z t
z
0
2
, , ,
( )





 +

k x y z T I x y z t k x y z T I x y z t
I I I
, , , , , , , , , , , , ,
( ) ( )− ( ) ( )
2

19. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 77







Φ Φ Φ
Φ
V
V
V V
x y z t
t
D
x y z t
x
x y z t
y
0
0
2
0
2
2
0
2
2
, , , , , , , , ,
( )
=
( )
+
( )
+
Φ
ΦV x y z t
z
0
2
, , ,
( )





 +

k x y z T V x y z t k x y z T V x y z t
V V V
, , , , , , , , , , , , ,
( ) ( )− ( ) ( )
2
;







Φ Φ Φ
Φ
I i
I
I i I i
x y z t
t
D
x y z t
x
x y z t
y
, , , , , , , , ,
( )
=
( )
+
( )
+
0
2
2
2
2
2
Φ
ΦI i x y z t
z
, , ,
( )








+
 2
D
x
g x y z T
x y z t
x y
g x y z T
I I
I i
I
0
1
Φ Φ Φ
Φ






, , ,
, , ,
, , ,
( )
( )





 + ( )
− 


ΦI i x y z t
y
− ( )





 +





1 , , ,




z
g x y z T
x y z t
z
I
I i
Φ
Φ
, , ,
, , ,
( )
( )











−1
, i≥1,
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
∂
Φ Φ Φ
Φ
V i
V
V i V i
x y z t
t
D
x y z t
x
x y z t
y
, , , , , , , , ,
0
2
2
2
2
2
Φ
ΦV i x y z t
z
, , ,
( )
∂








+
2
D
x
g x y z T
x y z t
x y
g x y z T
V V
V i
V
0
1
Φ Φ Φ
Φ






, , ,
, , ,
, , ,
( )
( )





 + ( )
− 


ΦV i x y z t
y
− ( )





 +





1 , , ,




z
g x y z T
x y z t
z
V
V i
Φ
Φ
, , ,
, , ,
( )
( )











−1
, i≥1;
Boundary and initial conditions for the functions take the form
∂
∂
Φρi
x
x y z t
x
, , ,
( )
=
=0
0,
∂
∂
Φρi
x L
x y z t
x
x
, , ,
( )
=
=
0,
∂
∂
Φρi
y
x y z t
y
, , ,
( )
=
=0
0,
∂
∂
Φρi
y L
x y z t
y
y
, , ,
( )
=
=
0,
∂
∂
Φρi
z
x y z t
z
, , ,
( )
=
=0
0,
∂
∂
Φρi
z L
x y z t
z
z
, , ,
( )
=
=
0, i≥0; Φρ
0
(x,y,z,0)=fΦρ
(x,y,z),
Φρ
i
(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
Φ Φ Φ
  
0
1
1 2
x y z t
L L L L L L
F c x c y c z e t
x y z x y z
n n n n n
n
, , ,
( ) = + ( ) ( ) ( ) ( )
=
∞
∑
∑ ∑
+ ( ) ( ) ( ) ×
=
∞
2
1
L
n c x c y c z
n n n
n
e t e c u c v c w k u v w T I u v w
n n n n n I I
Φ Φ
ρ ρ
τ τ
( ) −
( ) ( ) ( ) ( ) ( ) ( )−

 , , , , , , ,
2
0
L
L
L
L
t z
y
x
∫
∫
∫
∫ 0
0
0

20. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 78
k u v w T I u v w d wd vd u d
I , , , , , ,
( ) ( )

  ,
where F c u c v c w f u v w d wd vd u
n n n n
L
L
L z
y
x
Φ Φ
 
= ( ) ( ) ( ) ( )
∫
∫
∫ , ,
0
0
0
, e t n D t L L L
n x y z
Φ Φ
ρ ρ
π
( ) = − + +
( )




− − −
exp 2 2
0
2 2 2
,
cn
(x) = cos (π n x/Lx
);
Φ Φ Φ
ρ
π
τ
ρ ρ
i
x y z
n n n n n n
x y z t
L L L
nc x c y c z e t e s u
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) (
2
2 )
) ( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
c v g u v w T
n
L
L
L
t
n
z
y
x
Φρ
, , ,
0
0
0
0
1
c w
u v w
u
d wd vd u d
L L L
nc x c y c z e
n
I i
x y z
n n n
( )
( )
− ( ) ( ) ( )
−
∂ τ
∂
τ
π
ρ
Φ 1
2
2
, , ,
Φ
Φ Φ
ρ ρ
τ
n n
t
n
t e
( ) −
( ) ×
∫
∑
=
∞
0
1
e c u s v c w g u v w T
u v w
v
d wd vd
n n n n
I i
Φ Φ
Φ
ρ ρ
ρ
τ
∂ τ
∂
−
( ) ( ) ( ) ( ) ( )
( )
−
, , ,
, , ,
1
u
u d
L L L
n
L
L
L
t
x y z n
z
y
x
τ
π
0
0
0
0
2
1
2
∫
∫
∫
∫ ∑
− ×
=
∞
e t e c u c v s w
u v w
w
g u v w T
n n n n n
I i
Φ Φ Φ
Φ
ρ ρ
ρ
ρ
τ
∂ τ
∂
( ) −
( ) ( ) ( ) ( )
( )
−1 , , ,
, , ,
(
( ) ×
∫
∫
∫
∫ d wd vd u d
L
L
L
t z
y
x
τ
0
0
0
0
c x c y c z
n n n
( ) ( ) ( ), i ≥1,
where sn
(x) = sin (π n x/Lx
).
Equations for the functions Cij
(x,y,z,t) (i ≥0, j ≥0), boundary and initial conditions could be written as






C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
00
0
2
00
2 0
2
00
2
, , , , , , , , ,
( )
=
( )
+
( )
+ D
D
C x y z t
z
L
0
2
00
2


, , ,
( )
;







C x y z t
t
D
C x y z t
x
C x y z t
y
C
i
L
i i
0
0
2
0
2
2
0
2
2
, , , , , , , , ,
( )
=
( )
+
( )
+ i
i x y z t
z
0
2
, , ,
( )





 +

D
x
g x y z T
C x y z t
x
D
y
g x y z T
L L
i
L L
0
10
0






, , ,
, , ,
, , ,
( )
( )





 + ( )
− 


C x y z t
y
i− ( )





 +
10 , , ,
D
z
g x y z T
C x y z t
z
L L
i
0
10




, , ,
, , ,
( )
( )






−
, i ≥1;






C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
01
0
2
01
2 0
2
01
2
, , , , , , , , ,
( )
=
( )
+
( )
+ D
D
C x y z t
z
L
0
2
01
2


, , ,
( )
+
D
x
C x y z t
P x y z T
C x y z t
x
D
y
L L
0
00 00
0
∂
∂
∂
∂
∂
∂
γ
γ
, , ,
, , ,
, , ,
( )
( )
( )





 +
C
C x y z t
P x y z T
C x y z t
y
00 00
γ
γ
, , ,
, , ,
, , ,
( )
( )
( )





 +
∂
∂

21. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 79
D
z
C x y z t
P x y z T
C x y z t
z
L
0
00 00
∂
∂
( )
( )
∂ ( )
∂








, , ,
, , ,
, , ,
;






C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
02
0
2
02
2 0
2
02
2
, , , , , , , , ,
( )
=
( )
+
( )
+ D
D
C x y z t
z
L
0
2
02
2


, , ,
( )
+
D
x
C x y z t
C x y z t
P x y z T
C x y z t
L
0 01
00
1
00
∂
∂
∂
∂
, , ,
, , ,
, , ,
, , ,
( )
( )
( )
( )
−
γ
γ
x
x y
C x y z t
C x y z t
P x y z T





 + ( )
( )
( )
×







−
∂
∂
01
00
1
, , ,
, , ,
, , ,
γ
γ


∂
∂
∂
∂
C x y z t
y z
C x y z t
C x y z t
P x y z T
00
01
00
1
, , ,
, , ,
, , ,
, , ,
( )

 + ( )
( )
−
γ
γ
(
( )
( )











+
∂
∂
C x y z t
z
00 , , ,
∂
∂
∂
∂
C x y z t
y z
C x y z t
C x y z t
P x y z T
00
01
00
1
, , ,
, , ,
, , ,
, , ,
( )

 + ( )
( )
−
γ
γ
(
( )
( )











+
( )
∂
∂
∂
∂
C x y z t
z
D
x
C x y z t
P x y z T
L
00
0
00
, , , , , ,
, , ,
γ
γ
(
( )
×








∂
∂
∂
∂
∂
C x y z t
x y
C x y z t
P x y z T
C x y z t
01 00 01
, , , , , ,
, , ,
, , ,
( )

 +
( )
( )
(
γ
γ
)
)





 +
( )
( )
( )





∂
∂
∂
∂
∂
y z
C x y z t
P x y z T
C x y z t
z
00 01
γ
γ
, , ,
, , ,
, , ,







;






C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
11
0
2
11
2 0
2
11
2
, , , , , , , , ,
( )
=
( )
+
( )
+ D
D
C x y z t
z
L
0
2
11
2


, , ,
( )
+
∂
∂
∂
∂
x
C x y z t
C x y z t
P x y z T
C x y z t
x
10
00
1
00
, , ,
, , ,
, , ,
, , ,
( )
( )
( )
( )


−
γ
γ




 + ( )
( )
( )
×








−
∂
∂ y
C x y z t
C x y z t
P x y z T
10
00
1
, , ,
, , ,
, , ,
γ
γ
∂
∂
∂
∂
C x y z t
y z
C x y z t
C x y z t
P x y z T
00
10
00
1
, , ,
, , ,
, , ,
, , ,
( )

 + ( )
( )
−
γ
γ
(
( )
( )











+
∂
∂
C x y z t
z
D L
00
0
, , ,
D
x
C x y z t
P x y z T
C x y z t
x y
C
L
0
00 10 00
∂
∂
∂
∂
∂
∂
γ
γ
, , ,
, , ,
, , ,
( )
( )
( )





 +
γ
γ
γ
x y z t
P x y z T
C x y z t
y
, , ,
, , ,
, , ,
( )
( )
( )





 +





∂
∂
10
∂
∂
∂
∂
∂
z
C x y z t
P x y z T
C x y z t
z
D L
00 10
0
γ
γ
, , ,
, , ,
, , ,
( )
( )
( )











+
∂
∂
∂
∂
x
g x y z T
C x y z t
x
L , , ,
, , ,
( )
( )





 +





01







y
g x y z T
C x y z t
y z
g x y z T
C x y
L L
, , ,
, , ,
, , ,
, ,
( )
( )





 + ( )
01 01 z
z t
z
,
( )












;


C x y z t
x
ij
x
, , ,
( )
=
=0
0,


C x y z t
x
ij
x Lx
, , ,
( )
=
=
0,


C x y z t
y
ij
y
, , ,
( )
=
=0
0,


C x y z t
y
ij
y Ly
, , ,
( )
=
=
0,

22. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 80


C x y z t
z
ij
z
, , ,
( )
=
=0
0,


C x y z t
z
ij
z Lz
, , ,
( )
=
=
0 , i ≥0, j ≥0;
C00
(x,y,z,0)=fC
(x,y,z), Cij
(x,y,z,0)=0, i ≥1, j ≥1.
Functions Cij
(x,y,z,t) (i ≥0, j ≥0) could be approximated by the following series during solutions of the
above equations
C x y z t
F
L L L L L L
F c x c y c z e t
C
x y z x y z
nC n n n nC
n
00
0
1
2
, , ,
( ) = + ( ) ( ) ( ) ( )
=
∞
∑
∑ .
Here ( ) 2 2
0 2 2 2
1 1 1
nC C
x y z
e t exp n D t
L L L

 
 
= − + +
 
 
 
 
 
, F c u c v f u v w c w d wd v d u
nC n n C n
L
L
L z
y
x
= ( ) ( ) ( ) ( )
∫
∫
∫ , ,
0
0
0
;
C x y z t
L L L
n F c x c y c z e t e s u
i
x y z
nC n n n nC nC n
0 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
c v g u v w T
n L
L
L
L
t
n
z
y
x
, , ,
0
0
0
0
1
c w
C u v w
u
d wd vd u d
L L L
n F c x c y c z
n
i
x y z
nC n n n
( )
∂ ( )
∂
− ( ) ( ) (
−10
2
2
, , ,τ
τ
π
)
) ( ) −
( ) ×
∫
∑
=
∞
e t e
nC nC
t
n
τ
0
1
c u s v c v g u v w T
C u v w
v
d wd vd u d
n n n L
i
L
L z
y
( ) ( ) ( ) ( )
∂ ( )
∂
−
∫ , , ,
, , ,
10
0
0
τ
τ
∫
∫
∫ ∑
− ( ) ×
=
∞
0
2
1
2
L
x y z
nC nC
n
x
L L L
n F e t
π
c x c y c z e c u c v s v g u v w T
C u v
n n n nC n n n L
i
( ) ( ) ( ) −
( ) ( ) ( ) ( ) ( )
∂ −
 , , ,
, ,
10 w
w
w
d wd vd u d
L
L
L
t z
y
x
,

( )
∂
∫
∫
∫
∫ 0
0
0
0
, i ≥1;
C x y z t
L L L
n F c x c y c z e t e s u
x y z
nC n n n nC nC n
01 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( )×
∫
∫
∫
∫
∑
=
∞
c v c w
n n
L
L
L
t
n
z
y
x
0
0
0
0
1
C u v w
P u v w T
C u v w
u
d wd vd u d
L L L
n
x y z
00 00
2
2
γ
γ
τ τ
τ
π
, , ,
, , ,
, , ,
( )
( )
∂ ( )
∂
− F
F c x c y c z e t
nC n n n nC
n
( ) ( ) ( ) ( ) ×
=
∞
∑
1
e c u s v c w
C u v w
P u v w T
C u v w
nC n n n
−
( ) ( ) ( ) ( )
( )
( )
∂ (
τ
τ τ
γ
γ
00 00
, , ,
, , ,
, , , )
)
∂
− ( )×
∫
∫
∫
∫ ∑
=
∞
v
d wd vd u d
L L L
n e t
L
L
L
t
x y z
nC
n
z
y
x
τ
π
0
0
0
0
2
1
2
F c x c y c z e c u c v s w
C u v w
P u v
nC n n n nC n n n
( ) ( ) ( ) −
( ) ( ) ( ) ( )
( )
τ
τ
γ
γ
00 , , ,
, ,
, ,
, , ,
w T
C u v w
w
d wd vd u d
L
L
L
t z
y
x
( )
∂ ( )
∂
∫
∫
∫
∫
00
0
0
0
0
τ
τ ;
C x y z t
L L L
n F c x c y c z e t e s u
x y z
nC n n n nC nC n
02 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
c v c w
n n
L
L
L
t
n
z
y
x
0
0
0
0
1

23. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 81
C u v w
C u v w
P u v w T
C u v w
u
d wd vd
01
00
1
00
, , ,
, , ,
, , ,
, , ,
τ
τ τ
γ
γ
( )
( )
( )
∂ ( )
∂
−
u
u d
L L L
F c x c y
x y z
nC n n
n
τ
π
− ( ) ( )×
=
∞
∑
2
2
1
nc z e t e c u s v C u v w
C u v w
P
n nC nC n n
( ) ( ) −
( ) ( ) ( ) ( )
( )
−
τ τ
τ
γ
γ
01
00
1
, , ,
, , ,
u
u v w T
C u v w
v
L
L
L
t z
y
x
, , ,
, , ,
( )
∂ ( )
∂
×
∫
∫
∫
∫
00
0
0
0
0
τ
c w d wd vd u d
L L L
n F c x c y c z e t e c u
n
x y z
nC n n n nC nC n
( ) − ( ) ( ) ( ) ( ) −
( )
τ
π
τ
2
2 (
( ) ( ) ×
∫
∫
∫
∑
=
∞
c v
n
L
L
t
n
y
x
0
0
0
1
s w C u v w
C u v w
P u v w T
C u v w
w
n ( ) ( )
( )
( )
∂ ( )
∂
−
01
00
1
00
, , ,
, , ,
, , ,
, , ,
τ
τ τ
γ
γ
d
d wd vd u d
L L L
n c x
L
x y z
n
n
z
τ
π
0
2
1
2
∫ ∑
− ( ) ×
=
∞
F c y c z e t e s u c v c w C u v w
C u
nC n n nC nC n n n
( ) ( ) ( ) −
( ) ( ) ( ) ( ) ( )
∂
 
01
00
, , ,
,
, , ,
v w
u
L
L
L
t z
y
x

( )
∂
×
∫
∫
∫
∫ 0
0
0
0
C u v w
P u v w T
d wd vd u d
L L L
n F c x c y
x y z
nC n n
00
1
2
2
γ
γ
τ
τ
π
−
( )
( )
− ( ) ( )
, , ,
, , ,
c
c z e t e c u
n nC nC n
L
t
n
x
( ) ( ) −
( ) ( ) ×
∫
∫
∑
=
∞
τ
0
0
1
s v c w C u v w
C u v w
P u v w T
C u v w
n n
( ) ( ) ( )
( )
( )
∂
−
01
00
1
00
, , ,
, , ,
, , ,
, , ,
τ
τ
γ
γ
τ
τ
τ
π
( )
∂
− ×
∫
∫ ∑
=
∞
v
d wd vd u d
L L L
n
L
L
x y z n
z
y
0
0
2
1
2
F c x c y c z e t e c u c v s w C u v w
nC n n n nC nC n n n
( ) ( ) ( ) ( ) −
( ) ( ) ( ) ( ) ( )
τ τ
01 , , ,
C
C u v w
P u v w T
L
L
L
t z
y
x
00
1
0
0
0
0
γ
γ
τ
−
( )
( )
×
∫
∫
∫
∫
, , ,
, , ,
∂
∂
C u v w
w
d wd vd u d
L L L
F c x c y c z e t e
x y z
nC n n n nC
00
2
2
, , ,τ
τ
π
( )
− ( ) ( ) ( ) ( ) n
nC n
L
t
n
s u
x
−
( ) ( ) ×
∫
∫
∑
=
∞
τ
0
0
1
n c v c w
C u v w
P u v w T
C u v w
u
d wd vd u d
n n
( ) ( )
( )
( )
∂ ( )
∂
00 01
γ
γ
τ τ
τ
, , ,
, , ,
, , ,
0
0
0
2
1
2
L
L
x y z
n nC
n
z
y
L L L
c x e t
∫
∫ ∑
− ( ) ( ) ×
=
∞
π
F c y e c u s v c w
C u v w
P u v w T
C
nC n nC n n n
( ) −
( ) ( ) ( ) ( )
( )
( )
∂
τ
τ
γ
γ
00 01
, , ,
, , ,
u
u v w
v
d wd vd u d
L
L
L
t z
y
x
, , ,τ
τ
( )
∂
×
∫
∫
∫
∫ 0
0
0
0
n c z
L L L
n F c x c y c z e t e c u c v
n
x y z
nC n n n nC nC n n
( )− ( ) ( ) ( ) ( ) −
( ) ( ) ( )
2
2
π
τ s
s w
n
L
L
L
t
n
z
y
x
( ) ×
∫
∫
∫
∫
∑
=
∞
0
0
0
0
1
C u v w
P u v w T
C u v w
w
d wd vd u d
00 01
γ
γ
τ τ
τ
, , ,
, , ,
, , ,
( )
( )
∂ ( )
∂
;

24. Pankratov: On approach to increase integration rate
AJMS/Jul-Sep-2020/Vol 4/Issue 3 82
C x y z t
L L L
n F c x c y c z e t e s u
x y z
nC n n n nC nC n
11 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
c v c w
n n
L
L
L
t
n
z
y
x
0
0
0
0
1
g u v w T
C u v w
u
d wd vd u d
L L L
n F c x c y
L
x y z
nC n n
, , ,
, , ,
( )
∂ ( )
∂
− ( ) ( )
01
2
2
τ
τ
π
c
c z e t
n nC
n
( ) ( ) ×
=
∞
∑
1
e c u s v c w g u v w T
C u v w
v
d wd vd u d
nC n n n L
L
−
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
τ
τ
τ
, , ,
, , ,
01
0
z
z
y
x
L
L
t
x y z
L L L
∫
∫
∫
∫ − ×
0
0
0
2
2π
n e t e c u c v s w g u v w T
C u v w
w
d wd
nC nC n n n L
( ) −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂


, , ,
, , ,
01
v
vd u d
L
L
L
t
n
z
y
x

0
0
0
0
1
∫
∫
∫
∫
∑
=
∞
×
F c x c y c z
L L L
F c x c y c z e t e
nC n n n
x y z
nC n n n nC nC
( ) ( ) ( )− ( ) ( ) ( ) ( ) −
(
2
2
π
τ )
) ( ) ( ) ×
∫
∫
∫
∑
=
∞
s u c v
n n
L
L
t
n
y
x
0
0
0
1
n c w
C u v w
P u v w T
C u v w
u
d wd vd u d
n
Lz
( )
( )
( )
∂ ( )
∂
−
∫
00 10
0
γ
γ
τ τ
τ
, , ,
, , ,
, , , 2
2
2
1
π
L L L
n F c x c y
x y z
nC n n
n
( ) ( )×
=
∞
∑
c z e t e c u s v c w
C u v w
P u v w T
n nC nC n n n
( ) ( ) −
( ) ( ) ( ) ( )
( )
( )
∂
τ
τ
γ
γ
00 , , ,
, , ,
C
C u v w
v
d wd vd u d
L
L
L
t z
y
x
10
0
0
0
0
, , ,τ
τ
( )
∂
−
∫
∫
∫
∫
2
2
0
π
τ
L L L
n F c x c y c z e t e c u c v s w
C
x y z
nC n n n nC nC n n n
( ) ( ) ( ) ( ) −
( ) ( ) ( ) ( ) 0
0
0
0
0
0
1
γ
γ
τ
u v w
P u v w T
L
L
L
t
n
z
y
x
, , ,
, , ,
( )
( )
×
∫
∫
∫
∫
∑
=
∞
∂ ( )
∂
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