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S. Persheyev, D.A. Rogatkin
90
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
''photon'' travels from left to right. Thus, initial ''photons'' penetrating Δ
x
through the
left boundary have the state
i
= 1, backscattered from
Δx
''photons'' have the state
i
= 2
and transmitted by Δ
x
— the state
i
= 2
N
+
1. Strictly speaking, for this new
enumerating scheme for every even state
i
, excluding
i
= 2, only states
i
−
1 or
i
−
2 are
permitted for a transition with probabilities
R
and 1 −
R
respectively. For every
uneven state
i
, excluding
i
= 2
N
+
1, the permitted states are
i
+
1 and
i
+
2. Any
transitions from
i
= 2 or
i
= 2
N
+
1 states to any other states are not permitted (these
transition probabilities equal to zero). After creating such a statistical scheme, one can
start the general calculation.
To determine a backscattered radiation inside Δ
x
at multiple pure scattering, the
total probability
P
s
(
N
) of a ''photon'' transition from the state
i
= 1 to the state
i
= 2
through any
s
steps (
s
=1, 2, …,
) should be derived as a function of a number of
heterogeneities
N
. It means that we must find the unlimited sum:
12
1
( )
( ).
N
s
s
P N p s
(31)
The matrix of all one-step transitions
1
, for example, in the case of
N
= 2, is the
matrix
2
( 1)
N
ik
s
p
with the dimension of 6×6:
2
1
0
1
0 0 0
0 0 0 0 0 0
0 0 0
1 0
(1)
.
0 1
0 0 0
0 0 0 0 0 0
0 0 0 1
0
ik
R R
R R
R R
R R
p
(32)
The probabilities of reaching the state
i
= 2 from the state
i
= 1 with the use of any
s
steps can now be calculated by multiplication of corresponding matrixes (32). For
example, if
N
= 2, then the corresponding probabilities are:
s
= 1:
p
12
(1) =
R
;
s
= 2:
p
12
(2) = 0;
s
= 3:
p
12
(3) =
R
(1−
R
)
2
;
s
= 4:
p
12
(4) = 0;
s
= 5:
p
12
(5) =
R
3
(1
−R
)
2
;
s
= 6:
p
12
(6) = 0;
s
= 7:
p
12
(7) =
R
5
(1
−R
)
2
, etc.
Thus, the sum of them is a series:
2
2 3
2
12
1
(2)
( )
(1 )
(1 ) ...
s
s
P
p s R R R R R
(33)