Russian Federation
Russian Federation
Using two-dimensional recurrence relations, a description of dynamical X-ray diffraction in crystals is presented. It is shown that this approach makes it possible to calculate Xray fields inside the crystal and reciprocal space maps.
dynamical X-ray diffraction, rocking curve, reciprocal space map
Introduction
To describe diffraction in lateral crystals of a rectangular
cross section, two-dimensional recurrent relations were
obtained [1, 2], which differ from the one-dimensional Darwin
algebraic equations. On the other hand, diffraction in perfect
crystals of a rectangular cross section can also be considered
using the two-dimensional Takagi-Taupin equations [3].
On the example of a cylindrical crystal, it was shown that twodimensional
recurrence relations can be used to calculate reciprocal
space maps (RSMs) for crystals of arbitrary shape
[4]. It should be noted that numerical integration based on
the Takagi-Taupin equations is not always stable, while calculations
based on two-dimensional recurrence relations are
always stable. In this paper, we show that the diffraction of
spatially restricted X-ray beams in periodic structures can be
described using two-dimensional recurrence relations. Using
these recurrence relations, it is possible to calculate the
x-ray fields inside the crystal and RSMs.
Recurrent relations
Two-dimensional recurrent relations can be written in the
following form
Tm
n = a Tm−1
n−1 + b1 Sm−1
n−1 ,
Sm
n = a Sm−1
n+1 + b2 Tm−1
n+1 ,
(1)
where Tm
n and Sm
n are the amplitudes of the transmitted and
diffracted wave, a = (1 − i q0) exp
i 2πd
λ sin θB
, b1 =
−i ¯q exp
i
2πd
λ sin θB
, b2 = −i q exp
i
2πd
λ sin θB
,
q0 = − πd
λ sin θB
χ0, ¯q = − πd
λ sin θB
χ−g, q =
− πd
λ sin θB
χg, χ0, χh and χ−h — are the Fourier components
of the X-ray polarizability, d =
λ
2 sin θB
— is the interplanar
distance, λ is the X-ray wave length, θB is the Bragg angle for
reflective lattice planes. The distance between nodes along
the x axis is есть Δx = d/ tg θB.
Figure 1a shows the directions of the transmitted and
diffracted X-ray beam, taking into account the dynamical interaction
of X-ray waves. The horizontal lines represent the
crystal planes, which are numbered from top to bottom. Arrows
directed down correspond to transmitted waves, arrows
directed up refer to diffracted waves.
According to Figure 1a, the magnitude of the amplitude
Tm
n of the transmitted beam at the node (m; n)
consists of the contributions of the transmitted Tm−1
n−1
and reflected downward Sm−1
n−1 waves at the node (m −
1; n − 1). This physical process is described by the
first equation of recurrent relations (1). The diffraction
wave Sm
n at the node (m; n) is formed as a result
of the upward passage of the wave Sm−1
n+1 and the
88
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n + 1
n
n − 1
x
d
d
m − 2 m − 1 m
B
Tm−1
n−1
Sm−1
n−1
Tm−1
n+1
Sm−1
n+1
Tm
n
Sm
n
x
z
Lz
1
2
l
(in)
x
l(ex)
x
(a) (b)
Figure 1. (a) Scheme of two-dimensional X-ray diffraction on discrete crystal lattice planes. The vertical coordinate (see (b)) is defined as z = nd, where n
is the number of the crystal plane. The horizontal coordinate is x = mΔx, where Δx is the lateral distance between the nodes, m is the node number.
(b) Geometry of X-ray diffraction. The incident restricted beam illuminates the surface of the crystal l(in)
x . The exited beam size is l(ex)
x .
Рисунок 1. (a) Схема двумерной дифракции рентгеновских лучей на кристаллических плоскостях. Вертикальная координата определена как z = nd,
где n — номер кристаллической плоскости. Горизонтальная координата: x = mΔx, гдеΔx — расстояние между узлами в латеральном направлении,
m— номер узла. (b) Геометрия дифракции рентгеновских лучей. Ширина падающего на кристалл пучка l(in)
x . Ширина выходящего пучка l(ex)
x .
diffraction of the wave Tm−1
n+1 at the node (m − 1; n + 1).
The exponential factor exp
i
2πd
λ sin θB
, which is included
in the coefficients of the recurrence relations, takes into account
the phase variation during the passage of X-ray waves
from one crystal plane to the neighboring plane.
To solve the diffraction problem, recurrent relations (1)
must be supplemented with boundary conditions (Figure 1b).
Let the X-ray beam fall on the crystal surface at an angle θ1
, which, in the general case, may differ from the Bragg angle
θB . We consider the case when an incident X-ray wave illuminates
the crystal surface l(in)
x , then the boundary condition
has the form
Tm
0 = exp
i
2π
λ
cos θ1 · mΔx
, mΔx ⩽ l(in)
x ,
Tm
0 = 0, mΔx > l(in)
x .
(2)
It follows from relations (2) that the modulus of the amplitude
of the incident X-ray beam on the crystal surface is
equal to unity. The exponential factor takes into account the
change in the phase of the incident wave amplitude along the
x axis (Figure 1b).
The calculation of the amplitudes of X-ray fields is performed
on the basis of recurrence relations (1) taking into account
the boundary conditions (2) for all nodes of a rectangular
network (m; n), where 0 ⩽ m ⩽ Mx = l(in)
x /Δx and
0 ⩽ n ⩽ Nz = Lz/d, Lz is the thickness of the crystal.
In a triple-axis diffraction scheme, the exited beam is registered,
for example, at a different angle θ2. In this case, it
is necessary to take into account additional phase variations
φm = −(2π/λ)mΔx cos θ2 for the reflected X-ray wave
at the front of the exited beam, which is shown by the dotted
line in Figure 1b.
The intensity of the diffracted X-ray wave is found from
the relation:
Ih(qx, qz) =
XMx
m=0
Sm
0 exp(iφm)
2
(3)
where qx =
2π sin θB
λ
(Δθ1 − Δθ2), qx =
−2π cos θB
λ
(Δθ1 + Δθ2), Δθ1 = θ1 − θB, Δθ2 =
θ2 − θB.
Calculated results
Numerical calculations of X-ray diffraction in a perfect silicon
crystal are performed for symmetric (333) reflection of
σ–polarized X-ray CuKα1 radiation. The calculation results
are presented taking into account the shift of the coordinate
system by the angular distance associated with the X-ray refraction,
which is proportional to the real part of the coefficient
a0 in the diffraction equations (1). The length of the
primary Bragg extinction for (333) reflection from silicon is
lext = λ| sin θB|/(Cπ|χh|) = 8.03 μm. The Bragg angle
for the selected reflection is 47.476 arc. deg. The interplanar
distance of the reflecting planes is d = 0.1045 nm.
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89
(a) (b)
Figure 2. (a) The calculated distribution of the diffraction intensity inside the crystal at a deviation from the Bragg angle by 2 arc. sec. (b) RSM on a logarithmic
scale, calculated on the basis of two-dimensional recurrence relations. The width of the incident and diffraction beams is 110 μm.
Рисунок 2. (a) Распределение интенсивности дифрагированной волны внутри кристалла при отклонении от угла Брегга на 2 угловые сек. (b) Карта
интенсивности в логарифмическом масштабе, рассчитанная методом рекуррентных соотношений. Ширина падающего и отраженного пучков 110 μm.
Using recurrent relations (1), the field of the diffracted
wave inside the crystal was calculated for an arbitrary angle
of incidence, which differs slightly from the Bragg angle.
Figure 2a shows the scattering intensity distribution inside the
crystal when the incident beam deviates from the Bragg angle
by 2 arc.sec.
Solution (3) makes it possible to calculate RSMs for various
sizes of X-ray beams. Figure 2b shows the calculated
RSM on a logarithmic scale, which completely coincides with
calculations based on differential diffraction equations. The
width of the incident beam is 110 μm. The effective depth of
penetration of X-ray wave into a crystal is 82 μm.
The study was supported by the Russian Science Foundation,
grant No. 23-22-00062.
1. Punegov, V.I. Darwin’s approach to X-ray diffraction on lateral crystalline structures / V.I. Punegov , S.I. Kolosov, K.M. Pavlov // Acta Cryst. - 2014. - Vol. A70. - P. 64-71.
2. Punegov, V.I. Bragg-Laue X-ray dynamical diffraction on perfect and deformed lateral crystalline structures / V.I. Punegov , S.I. Kolosov, K.M. Pavlov // J. Appl. Cryst. - 2016. - Vol. 49. - P. 1190-1202.
3. Authier, A. Dynamical theory of X-ray diffraction / A. Authier. New York: Oxford University Press, 2001.
4. Punegov, V.I. Simulation of X-ray diffraction in a cylindrical crystal /V.I. Punegov , S.I. Kolosov // J. Appl. Cryst. - 2020. - Vol. 53. - P. 1203-1211.
