Scattering of X-ray radiation by an electron. X-ray absorption and scattering
The relationships we have considered reflect the quantitative aspect of the X-ray attenuation process. Let us dwell briefly on the quality side of the process, or on those physical processes that cause weakening. This is, firstly, absorption, i.e. conversion of X-ray energy into other types of energy and, secondly, scattering, i.e. changing the direction of propagation of radiation without changing the wavelength (classical Thompson scattering) and with changing the wavelength (quantum scattering or Compton effect).
1. Photoelectric absorption... X-ray quanta can snatch electrons from the electron shells of atoms of a substance. They are commonly referred to as photoelectrons. If the energy of the incident quanta is low, then they knock out electrons from the outer shells of the atom. Large kinetic energy is imparted to photoelectrons. With an increase in energy, X-ray quanta begin to interact with electrons located in the deeper shells of the atom, in which the binding energy with the nucleus is greater than that of the electrons of the outer shells. With such an interaction, almost all the energy of the incident X-ray quanta is absorbed, and part of the energy given to the photoelectrons is less than in the first case. In addition to the appearance of photoelectrons, quanta of characteristic radiation are emitted in this case due to the transition of electrons from higher levels to levels located closer to the nucleus.
Thus, as a result of photoelectric absorption, a characteristic spectrum of a given substance appears - a secondary characteristic radiation. If the electron is pulled out from the K-shell, then the entire line spectrum appears, which is characteristic of the irradiated substance.
Rice. 2.5. Spectral distribution of the absorption coefficient.
Let us consider the change in the mass absorption coefficient t / r caused by photoelectric absorption as a function of the wavelength l of the incident X-ray radiation (Fig. 2.5). The breaks in the curve are called absorption jumps, and the corresponding wavelength is called the absorption boundary. Each jump corresponds to a certain energy level of the atom K, L, M, etc. At l gr, the energy of the X-ray quantum turns out to be sufficient to knock out an electron from this level, as a result of which the absorption of X-ray quanta of a given wavelength increases sharply. The shortest wavelength jump corresponds to the removal of an electron from the K-level, the second from the L-level, etc. The complex structure of the L and M-boundaries is due to the presence of several sublevels in these shells. For x-rays with wavelengths somewhat longer than l gr, the energy of the quanta is insufficient to snatch an electron from the corresponding shell, the substance is relatively transparent in this spectral region.
Dependence of the absorption coefficient on l and Z with the photoeffect is defined as:
t / r = Сl 3 Z 3 (2.11)
where C is the coefficient of proportionality, Z Is the ordinal number of the irradiated element, t / r is the mass absorption coefficient, l is the wavelength of the incident X-ray radiation.
This dependence describes the portions of the curve in Fig. 2.5 between the absorption jumps.
2. Classical (coherent) scattering explains the wave theory of scattering. It takes place if the X-ray quantum interacts with the electron of the atom, and the energy of the quantum is insufficient to pull the electron out of the given level. In this case, according to the classical theory of scattering, X-rays cause forced vibrations of the bound electrons of the atoms. Oscillating electrons, like all oscillating electric charges, become a source of electromagnetic waves that propagate in all directions.
The interference of these spherical waves leads to the appearance of a diffraction pattern, naturally related to the structure of the crystal. Thus, it is precisely coherent scattering that makes it possible to obtain diffraction patterns, on the basis of which one can judge the structure of the scattering object. Classical scattering occurs when soft X-rays with wavelengths of more than 0.3 Å pass through a medium. The scattering power by one atom is equal to:
, (2.12)
and one gram of substance
where I 0 is the intensity of the incident X-ray beam, N is the Avogadro number, A is the atomic weight, Z- serial number of the substance.
From here we can find the mass coefficient of classical scattering s cells / r, since it is equal to P / I 0 or 
.
Substituting all the values, we get 
.
Since most of the elements Z/[email protected], 5 (except for hydrogen), then
those. the mass coefficient of classical scattering is approximately the same for all substances and does not depend on the wavelength of the incident X-ray radiation.
3. Quantum (incoherent) scattering... When a substance interacts with hard X-ray radiation (wavelength less than 0.3 Å), quantum scattering begins to play an important role, when a change in the wavelength of the scattered radiation is observed. This phenomenon cannot be explained wave theory, but it is explained by quantum theory. According to quantum theory, such an interaction can be considered as a result of elastic collisions of X-ray quanta with free electrons (electrons of the outer shells). X-ray quanta give up part of their energy to these electrons and cause them to transition to other energy levels. The electrons that have received energy are called recoil electrons. X-ray quanta with energy hn 0 as a result of such a collision deviate from the initial direction by an angle y, and will have an energy hn 1 less than the energy of the incident quantum. The decrease in the frequency of the scattered radiation is determined by the ratio:
hn 1 = hn 0 - E dep, (2.15)
where Edet is the kinetic energy of the recoil electron.
Theory and experience show that the change in frequency or wavelength during quantum scattering does not depend on the serial number of the element Z, but depends on the scattering angle y. Wherein
l y - l 0 = l = × (1 - cos y) @ 0.024 (1 - cozy), (2.16)
where l 0 and l y are the wavelength of the X-ray quantum before and after scattering,
m 0 is the mass of an electron at rest, c Is the speed of light.
It can be seen from the formulas that as the scattering angle increases, l increases from 0 (at y = 0 °) to 0.048 Å (at y = 180 °). For soft rays with a wavelength of the order of 1 Å, this value is a small percentage of about 4–5%. But for hard rays (l = 0.05–0.01 Å), a change in wavelength by 0.05 Å means a change in l by half or even several times.
Due to the fact that quantum scattering is incoherent (l is different, the angle of propagation of the reflected quantum is different, there is no strict regularity in the propagation of scattered waves with respect to the crystal lattice), the order in the arrangement of atoms does not affect the nature of quantum scattering. These scattered X-rays are involved in creating the overall background on the X-ray. The dependence of the background intensity on the scattering angle can be theoretically calculated that practical application in X-ray diffraction analysis, it does not, because There are several reasons for the occurrence of the background, and its general significance cannot be easily calculated.
The processes of photoelectron absorption, coherent and incoherent scattering considered by us determine, mainly, the attenuation of X-rays. In addition to them, other processes are possible, for example, the formation of electron-positron pairs as a result of the interaction of X-rays with atomic nuclei. Under the influence of primary photoelectrons with high kinetic energy, as well as primary X-ray fluorescence, secondary, tertiary, etc. may arise. characteristic radiation and corresponding photoelectrons, but with lower energies. Finally, some of the photoelectrons (and partly of the recoil electrons) can overcome the potential barrier at the surface of the substance and fly out of it, i.e. external photoelectric effect may occur.
All the phenomena noted, however, have a much smaller effect on the value of the X-ray attenuation coefficient. For X-rays with wavelengths from tenths to a few angstroms, commonly used in structural analysis, all of these side effects it can be neglected and it can be assumed that the attenuation of the primary X-ray beam occurs on the one hand due to scattering and, on the other, as a result of absorption processes. Then the attenuation coefficient can be represented as the sum of two coefficients:
m / r = s / r + t / r, (2.17)
where s / r is the mass scattering coefficient, which takes into account energy losses due to coherent and incoherent scattering; t / r is the mass absorption coefficient, which mainly takes into account energy losses due to photoelectric absorption and excitation of characteristic rays.
The contributions of absorption and scattering to the attenuation of the X-ray beam are not equal. For X-rays used in structural analysis, incoherent scattering can be neglected. If we take into account in this case that the magnitude of coherent scattering is also small and approximately constant for all elements, then we can assume that
m / r "t / r, (2.18)
those. that the attenuation of the X-ray beam is mainly determined by the absorption. In this regard, for the mass attenuation coefficient, the regularities discussed above for the mass absorption coefficient in the photoelectric effect will be valid.
Selecting radiation ... The nature of the dependence of the absorption (attenuation) coefficient on the wavelength determines, to a certain extent, the choice of radiation in structural studies. Strong absorption in the crystal significantly reduces the intensity of the diffraction spots in the X-ray diffraction pattern. In addition, the fluorescence arising from strong absorption exposes the film. Therefore, it is unprofitable to work at wavelengths somewhat lower than the absorption limit of the substance under study. This can be easily understood from the diagram in Fig. 2.6.
1. If the anode, consisting of the same atoms as the substance under study, will emit, then we get that the absorption limit, for example
Figure 2.6. The change in the intensity of X-ray radiation when passing through a substance.
The K-edge of absorption of the crystal (Fig. 2.6, curve 1) will be slightly shifted relative to its characteristic radiation to the short-wavelength region of the spectrum. This shift is of the order of 0.01–0.02 Å relative to the lines of the edge of the line spectrum. It always takes place in the spectral position of emission and absorption of the same element. Since the absorption jump corresponds to the energy that must be expended to remove an electron from the level outside the atom, the hardest K-series line corresponds to the transition to the K-level from the farthest level of the atom. It is clear that the energy E required to pull an electron out of the atom is always somewhat higher than that released when an electron moves from the most distant level to the same K-level. Fig. 2.6 (curve 1) it follows that if the anode and the crystal under study are one substance, then the most intense characteristic radiation, especially the K a and K b lines, lies in the region of weak absorption of the crystal with respect to the absorption boundary. Therefore, the absorption of such radiation by the crystal is small, and the fluorescence is weak.
2. If we take an anode whose atomic number is Z 1 more than the crystal under study, then the radiation of this anode, according to Moseley's law, will slightly shift to the short-wavelength region and will be located relative to the absorption boundary of the same test substance as shown in Fig. 2.6, curve 2. Here the K b - line is absorbed, due to which fluorescence appears, which can interfere with shooting.
3. If the difference in atomic numbers is 2-3 units Z, then the radiation spectrum of such an anode will shift even further to the short-wavelength region (Fig. 2.6, curve 3). This case is even more disadvantageous, since, firstly, X-rays are greatly attenuated and, secondly, strong fluorescence illuminates the film during shooting.
The most suitable, therefore, is the anode, the characteristic radiation of which lies in the region of weak absorption by the sample under study.
Filters. The selective absorption effect considered by us is widely used to attenuate the short-wavelength part of the spectrum. For this, a foil with a thickness of several hundredths is placed in the path of the rays mm. The foil is made of a substance whose serial number is 1–2 units less than Z anode. In this case, according to Fig.2.6 (curve 2), the edge of the foil absorption band lies between the K a - and K b - emission lines and the K b line, as well as the continuous spectrum, will be strongly weakened. The attenuation of K b in comparison with K a radiation is of the order of 600. Thus, we have filtered b-radiation from a-radiation, which hardly changes in intensity. The filter can be a foil made of a material whose serial number is 1–2 units less Z anode. For example, when working on molybdenum radiation ( Z= 42), zirconium ( Z= 40) and niobium ( Z= 41). In the series Mn ( Z= 25), Fe ( Z= 26), Co ( Z= 27) each of the previous elements can serve as a filter for the next one.
It is clear that the filter should be located outside the camera in which the crystal is shot so that the film is not exposed to fluorescence rays.
ATOMIC SCATTERING FACTOR
Scattering of X-rays by electrons in
atoms
K
S
E S Ee S f S Ee S f,
1/2
K0
r (r)
e 2 1 1 cos 2 2
Ee E0 2
mc
R
2
f,
r (r) is the distribution of the electron
density in the atom
S = K - K0
2
s - s0
For simplicity of calculations, we will
count the distribution of electrons
in an atom spherically symmetric
function. Then you can write.
E S
Ee S
Atomic scattering factor
r r
z r r dr
0
Here z is the number of electrons in the atom
We assume that a plane wave is incident on an atom
1
K
S
s
E
A0
K0
C
Aj
i t
Let at the origin, i.e.
at point A0 the phase of the wave is zero
0 0
Each point of the atom (i.e., each
s0
rj
B
2
E E0 e
electron) under the action of the wave E
begins to emit spherical
wave. Electron located A0
emits a wave
E 0 i t
E A0
e
R
Here R is the distance from point A0 to observation point M in the direction
vector s (lines 1 and 2). The primary plane will reach the point Aj having a phase
j k s0, rj
Then the secondary spherical wave 2 emitted by an electron located
at the point Aj will have the form
1 M
K
s
E
A0
B
C
Aj
2
We will assume that A0M >> ІrjІ
S
Wave 2 will reach the observation point M c
additional phase due to the segment
path AjC = (s, rj). Hence
the additional phase will be equal to k (s, rj)
K0
Then the total phase of wave 2 reached
point M will have the form
s0
rj
EAj
E0 i t k s0, rj
e
R
k s, rj k s0, rj rjK rjK 0
K - K 0, rj S, rj
EM
Aj
E0 i t k s-s0, rj E0 i t i Srj
e
e e
R
R Let the falling beam
directed along the X axis
Let's calculate the intensity
scattered element
volume dv
dv d dr
r d rsin d dr An atom can be approximately considered as a volume with a continuous
charge distribution. Let us select in the volume of an atom a volume element dv
at a distance r from the center of the atom. The electron density at this point
denote by r (r). Amplitude of the wave scattered by the element
volume dv can be written as. (For simplicity, omit R)
dE Ee r r e
ik s s0, r
dv Ee r r e
ik S, r
dv
Let us substitute the volume element in this ratio in an explicit form. Then
the total amplitude scattered by all electrons of the atom will be
equal to the integral over the entire volume
E Ee r r e
iSr cos
dv
V
Ee d r r r 2 dr eiS cos sin d
r Remembering the definition of the atomic scattering factor
E S Ee S f,
f S f,
E S
Ee S
the above expression can be rewritten as
f S
2
0
0
0
2
iS cos
d
r
r
r
dr
e
sin d
ia cos x
sin x dx we are already familiar from the previous section
Integral of type e
ia cos x
e
sin x dx
sin ax
ax
Integration over, and r leads to the expression f sin /
0
sin (Sr)
2
4 r r (r)
dr
Sr
This is the atomic scattering factor.
It depends on the distribution
electron density inside the atom.
Let us examine the behavior of the function f (S). If
the function argument goes to zero,
integral fraction
tends to unity and therefore Let us examine the behavior of the function f (S). If the function argument tends to
zero, the fraction under the integral tends to unity and
hence f (S) approaches the value Z /
s 0
sin (Sr)
1
Sr
f sin / 4 r 2 r (r) dr z
0
f sin / Z
If the argument S grows, the function f (S) decreases and tends to zero
S 4
sin
sin (Sr)
0
Sr
f sin / 0
The form of the dependence of the atomic scattering function
on sin / for neutral Zn and Al atoms.
(Z for Zn = 40 and for Al = 13).
10.
The estimates made above are carried out on the condition that the electrons inatom are practically free and the equation of motion of an electron can be
write as mr eE. Real situation more difficult - electrons in
atoms move in their orbits and have natural frequencies
fluctuations and, therefore, it is necessary to consider the problem
the motion of a bound electron under the influence of an external periodic
perturbing force when the electron moves i.e. mr kr 2r eE. And this
0
not all. It is also necessary to take into account the damping during movement.
electrons. Then complete equation movement will look like
mr kr 0 2r eE
In this case, the amplitude of the wave scattered by the bound electron is
can be written as
2
E E 2
0 2 ik
e
or for everyone
electrons in an atom
2
E E 2
2
n 0 n ik
e
From the written relation it is seen that, firstly, the amplitude
scattering appears complex number and therefore
additional absorption appears near the intrinsic
resonance frequencies, and, secondly, the amplitude strongly depends on
frequency of the incident wave, i.e. there is variance. Correct accounting of these
corrections were carried out in the works of Lorentz.
11.
.If the wavelength of the incident radiation is far enough from
the edge of the absorption band, the atomic factor is simply f0.
However, as the wavelength of the incident radiation approaches
at the edge of the absorption band, the atomic factor becomes
complex value and it should be written as
f f 0 f i f
where f0 is the atomic scattering function,
obtained under the assumption of free electrons of the atom, and f "and
f "- dispersion corrections, the first of which takes into account
additional scattering for the case of bound electrons, and
the second is additional absorption near natural frequencies
vibrations of electrons in an atom. Dispersion corrections depend
on the wavelength and practically do not depend on sin. And since f0
decreases with increasing scattering angle, dispersion corrections
begin to play an increasing role at large angles
scattering.
Atomic scattering functions for the case of free electrons in an atom in
dependence on sin / and the corresponding dispersion corrections in
depending on the wavelength for all elements of the periodic table
are usually given in the form of tables. The most accurate values of these quantities are given
in international tables. (International Tables for X-Ray Crystallography, vol. 14, Birmingam, IDC, 1980)
12.
Amplitude of atomic scattering of electronsIn diffraction experiments, along with X-ray
radiation uses electrons with energies from tens to hundreds
keV (electrons with an energy of 50 keV have a wavelength of 0.037 Å). By
simple calculations can show that the amplitude of the atomic
scattering for electrons is related to the atomic scattering amplitude
X-rays by the following expression
Analysis of the written expression shows that at large angles
scattering, where fx is small, fe> Z and decreases inversely proportional to
(sin /) 2. In electron diffraction and electron microscopy, usually
a multiple of the atomic scattering amplitude is used and
included in the first Born approximation of scattering theory
electrons, namely
13.
The form of the atomic scattering functions of the hydrogen atom forX-rays and electrons, calculated in
first Born approximation.
25.0
20.0
15.0
10.0
5.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
14.
The estimates of the amplitudes of atomic scattering of electrons made above arelead to important features in the application of scattering
electrons versus X-rays. With one
On the other hand, a higher electron scattering amplitude (by two or three orders of magnitude) noticeably increases the luminosity of the diffraction pattern and
along with the possibility of focusing the incident electron beam
allows one to study very small crystals in
polycrystalline systems. On the other hand, a noticeable
absorption of electrons with energies of the order of several tens of keV
opens up an advantageous opportunity to study the structure of thin
surface layers 10-6-10-7cm thick. For comparison in
radiography under optimal conditions, a layer is recorded
about 10-2-10-4cm.
Weaker dependence of the atomic scattering amplitude
electrons versus X-rays from atomic
rooms allows structural studies for the lungs
atoms.
The presence of spin and magnetic moment of electrons opens
additional opportunities for studying the magnetic structure
materials.
15.
Atomic scattering functions for the casefree electrons in an atom depending on
values sin / and the corresponding
dispersion corrections versus length
waves for all elements of the periodic table
are usually given in the form of tables. Most
the exact values of these quantities are given in
international tables. (International Tables
forX-Ray Crystallography, vol. 1-4, Birmingam, IDC,
X-ray diffraction is the scattering of X-rays by crystals or molecules of liquids and gases, in which secondary deflected beams (diffracted beams) of the same wavelength arise from the initial beam of rays, resulting from the interaction of primary X-rays with the electrons of the substance. The direction and intensity of the secondary beams depend on the structure of the scattering object. Diffracted beams form part of the total X-ray radiation scattered by matter. Along with scattering without a change in wavelength, scattering with a change in wavelength is observed - the so-called Compton scattering. The phenomenon of X-ray diffraction, which proves their wave nature, was first experimentally discovered on crystals by the German physicists M. Laue, W. Friedrich, P. Knipping in 1912.
A crystal is a natural three-dimensional diffraction grating for X-rays, since the distance between scattering centers (atoms) in a crystal is of the same order of magnitude as the wavelength of X-rays (~ 1Å = 10-8 cm). X-ray diffraction by crystals can be considered as the selective reflection of X-rays from systems of atomic planes crystal lattice... The direction of the diffraction maxima simultaneously satisfies three conditions determined by the Laue equations.
The diffraction pattern is obtained from a stationary crystal using continuous-spectrum X-rays (the so-called Lauegram) or from a rotating or vibrating crystal illuminated by monochromatic X-rays or from a polycrystal illuminated by monochromatic radiation. The intensity of the diffracted beam depends on the structural factor, which is determined by the atomic factors of the atoms of the crystal, their location inside the unit cell of the crystal, and the nature of the thermal vibrations of the atoms. The structural factor depends on the symmetry of the arrangement of atoms in the unit cell. The intensity of the diffracted beam depends on the size and shape of the object, on the perfection of the crystal.
The diffraction of X-rays from polycrystalline bodies leads to the appearance of cones of the secondary rays. The axis of the cone is the primary ray, and the opening angle of the cone is 4J (J is the angle between the reflecting plane and the incident ray). Each cone corresponds to a specific family of crystal planes. All crystals participate in the creation of the cone, the family of planes of which is located at an angle J to the incident ray. If the crystals are small and they are very a large number of per unit volume, then the cone of rays will be solid. In the case of texture, that is, the presence of a preferred orientation of the crystals, the diffraction pattern (X-ray diffraction pattern) will consist of unevenly blackened rings.
Dedicated to the 100th anniversary of the discovery of X-ray diffraction
BACKSCATTERING OF X-RAY RAYS (DIFFRACTION BY THE BRAGG ANGLE n / 2)
© 2012 V. V. Leader
Institute of Crystallography RAS, Moscow E-mail: [email protected] Received September 29, 2011
The possibilities of using X-ray backscattering in X-ray optics and metrology, as well as for structural characterization of crystalline objects are considered. varying degrees perfection.
Introduction
1. Features of X-ray backscattering
2. Experimental implementation of backscattering
3. High-resolution X-ray optics based on backscattering
3.1. Monochromators
3.2. Analyzers
3.3. Crystal cavity
3.3.1. Crystalline cavity for the formation of a coherent beam
3.3.2. Crystalline cavity for time-resolving experiments
3.3.3. Crystal cavity for X-ray free electron laser
3.3.4. Fabry-Perot X-ray resonator
3.3.4.1. Resonator theory
3.3.4.2. Resonator implementation
3.3.4.3. Possibilities of using the resonator
4. Materials for monochromators and crystal mirrors
5. Using backscattering for structural characterization of crystals
5.1. Precise determination of crystal lattice parameters and wavelengths of γ-radiation sources
5.2. Using OR to study imperfect (mosaic) crystals
Conclusion
INTRODUCTION
It is known from the dynamic theory of X-ray scattering (X-ray scattering) that the width of the X-ray diffraction reflection (DRR) curve from a perfect crystal is given by the formula
u = 2C |% Ar | / d1 / 281P20. (1)
Here 0 is the Bragg angle,% br is the real part of the Fourier component of the crystal polarizability, the polarization factor C = 1 for the wave field components polarized perpendicular to the scattering plane (cp-polarization) and C = eo820 for the components polarized in this plane (n- polarization); B = y (/ ye is the asymmetry coefficient of the Bragg reflection, y;, ye are the direction cosines of the incident and diffracted radar lines, respectively, (y = 8m (0 - φ), ye = (0 + φ), φ is the angle of inclination of the reflecting planes to the surface of the crystal, which can be both positive and negative; in the Bragg geometry | f |< 0, а в случае Лауэ |ф| > 0).
Since Xng ^ 10-5, X-ray diffraction occurs in a very narrow angular interval, not exceeding a few arc seconds. This fact, as well as the dependence of the DRW width on the asymmetry coefficient, are widely used to create multicomponent X-ray optical systems for the formation of X-ray beams (using both laboratory radiation sources and synchrotron radiation (SR)) with specified parameters. One of the main parameters is the spectral divergence of the beam. Known multichip monochromator circuits that use antiparallel geometry of diffraction of at least two optical elements and provide a bandwidth equal to several millielectronvolts. Such high degree the monochromaticity of the beam is necessary, for example, for carrying out experiments on inelastic and nuclear resonance scattering. However, the applied dispersive diffraction scheme leads to a significant loss of the intensity of the X-ray beam at the output of the monochromator, which can complicate the experiment.
Backscattering (OR) was first considered from the point of view of the dynamic theory
Rice. 1. DiMond diagram for the area 0 "n / 2; -the receiving angle of the crystal.
X-ray diffraction by a perfect crystal by Kora and Matsushita in 1972. In this work, two interesting features of the OR were noted: as the Bragg angle approaches 90 °, the spectral transmission band of the crystal sharply decreases, while its DRC sharply increases. Thus, the opportunity has opened up to create X-ray high-aperture optics with high energy resolution on the basis of OR. In the 80s. there was a sharp surge of interest in OR. Subsequently, a large number of publications appeared on the use of X-ray backscattering X-ray optics. high resolution, metrology, as well as for the structural characterization of various crystalline objects. Works on the theory of OR and Fabry-Perot resonators, experimental use of monochromators and spherical analyzers, precision determination of the crystal lattice parameters and wavelengths of several γ-radiation sources are considered in the book by Yu.V. Shvidko, and his dissertation. Investigations of the near-surface region of crystals using the method of standing X-ray waves (STW) in the OD geometry were combined by D.P. Woodruff in reviews.
The purpose of this work is to attempt to describe various possibilities of using X-ray backscattering, based both on and on publications that were not included in them and appeared after 2004.
1. FEATURES OF BACKSCATTERING OF X-RAY RAYS
Taking into account the X-ray refraction, the "traditional" form of writing the Wolfe-Bragg equation (k = 2dsin0, where k is the X-ray wavelength, d is the interplanar distance of the crystal) will change
k (1 + w) = 2d sin 0, (2)
where w = - X0r (d / k) 2 (1 + 1 / b) (X0r is a negative value).
Two parameters characterizing an X-ray optical crystal element are the energy (spectral) resolution (AE) k / E and the extinction length L:
(AE) k / E = w ctg e = C | xJ / b1 / 2sin2e, (3)
L = MY / Ye) 1/2 / lxJ. (4)
For OR e «n / 2, therefore, C« 1, b «1, (Y / Ye) 1/2 ~ cosph. Then (2) - (4) will take the form:
X (1 + w) «2d (1 - s2 / 2), (5)
(AE) k / E «N, (6)
where в is the half angle between the incident and diffracted X-ray beams: в =
Combining (6) and (7) and assuming that X «2d, we get:
(AE) k / E «d / nL = 1 / nNd, (8)
where Nd is the number of reflecting planes that fit into the extinction length.
Thus, the energy resolution is inversely proportional to the effective number of reflecting planes that form the diffraction pattern. Since the presence of a deformation gradient in a crystal leads to a decrease in the extinction length, the deviation of the energy resolution from its tabular (theoretical) value can be used to judge the degree of crystal imperfection.
With an increase in the XRL energy, the extinction length increases, and, as a consequence, the energy resolution decreases. For E «14 keV, the extinction length is 10-100 microns, therefore (AE) k / E« 10-6-10-7, which corresponds to (AE) to «« 1-10 meV (Table 1).
The expression for the receiving angle (width of the DRC) can be obtained using (5), (6) and Fig. 1:
10 = 2 (lXhrl) 1/2. (nine)
(A rigorous conclusion (9) based on the dynamic theory of X-ray scattering can be found in).
In the experimental observation of X-ray backscattering for the (620) reflection of a germanium crystal and Co ^ a1 radiation, the measured DRW width was 35 arcsec. min, which is about 3 orders of magnitude higher than ω / for e< < п/2. Формулы (6), (9) справедливы при отклонении угла Брэгга от 90° на величину, не превышающую (2|xJ)1/2 или даже (|Xhrl)1/2 , т.е. равную сотым долям градуса.
2. EXPERIMENTAL REALIZATION OF BACKSCATTERING
The small angular distance between the primary and diffracted beams creates the problem of registering the latter, since its trajectory
Analyzer (s) 81 ^ 13 13) Detector
Double-crystal premonochromator 81 (111)
Monochromator 81 (13 13 13)
Monochromator Ionization Sample (g) chamber
Solid state
detector detector
Rice. 2. Diagrams of experimental stations for studying the OR (a, c, d), determining the lattice parameter of Ge (b) and sapphire (e), studying the wave field of the SRW in the OR condition (f), using different ways registration of the PR; b: 1 - premonochromator, 2 - plane-parallel deflector, 2 - wedge-shaped deflector, 3 - thermostatically controlled sample, 4 - detector; e: M - premonochromator, E - Fe57 foil, B - transparent time-resolving detector; f: 1 - pre-monochromator, 2 - first crystal reflector, 3 - second (thermostated) reflector, which is both an analyzer and a CCD detector, 4 - photographic film, 5 - detector. For clarity, the primary and scattered beams are separated (c, d).
can be blocked by an X-ray source (pre-monochromator) or a detector. There are several ways to solve the problem.
The first consists in increasing the distance between the nodes of the experimental station (for example, between the optical element, providing
and a detector). One of such stations of the European Synchrotron Center (ESRF) is described in. Due to the large distance between the 81 (111) preliminary monochromator and 81 (13 13 13) monochromator (Fig. 2a), it was possible to obtain a Bragg angle of 89.98 ° for E = 25.7 keV.
<111> ■■-
Rice. 3. Path of rays in a monoblock monochromator.
With the distance between the monochromator arms
197 mm, for the 81 (777) reflection and E = 13.84 keV, the limiting Bragg angle is 89.9 °.
For laboratory experimental facilities, increasing the distance between optical elements is often difficult. Therefore, another possibility of realizing X-ray backscattering is to "separate" the primary and diffracted beams. In the left fig. 2b shows a schematic diagram of an experiment to determine the lattice parameter of germanium. Here, deflector 2, which is a thin plane-parallel crystal plate, reflects a pre-monochromatized X-ray beam onto sample 3, but at 2e> udef (udef is the receiving angle of the deflector) it turns out to be transparent for the diffracted beam. In this case, for detector 4, the region of angles 2e< юдеф является "dead zone". In order for scattered X-rays to be recorded by the detector at e = 0, it was proposed to use wedge-shaped crystal 2 as a deflector (right side of Fig. 2b). Then, due to the correction for X-ray refraction, the Bragg angles for different sides of the deflector (which in this scheme can also serve as an analyzer), according to (2),
A. E. BLAGOV, M. V. Kovalchuk, V. G. Kon, Yu. V. Pisarevsky, P. A. Prosekov - 2010
Irzhak D. V., Roshchupkin D. V., Snigirev A. A., Snigireva I. I. - 2011
A.E. BLAGOV, M.V. KOVALCHUK, V.G. KON, E.KH. MUKHAMEDZHANOV - 2011
At work at increased voltages As in the case of radiography at conventional voltages, it is necessary to use all known methods of dealing with scattered X-ray radiation.
Quantity scattered x-rays decreases with a decrease in the irradiation field, which is achieved by limiting the X-ray beam across the working beam. With a decrease in the irradiation field, in turn, the resolution of the X-ray image improves, i.e., the minimum size of the detail determined by the eye decreases. Replaceable diaphragms or tubes are far from being used enough to restrict the working beam of X-rays across the cross-section.
To reduce the amount scattered x-rays compression should be used where possible. During compression, the thickness of the object under study decreases and, of course, there are fewer centers for the formation of scattered X-ray radiation. For compression, special compression belts are used, which are included in the set of X-ray diagnostic devices, but they are not used often enough.
The amount of scattered radiation decreases with increasing distance between the X-ray tube and the film. With an increase in this distance and a corresponding aperture, a less diverging working X-ray beam is obtained. With an increase in the distance between the X-ray tube and the film, it is necessary to reduce the irradiation field to a minimum possible sizes... In this case, the investigated area should not be "cut off".
To this end, in the last structures X-ray diagnostic devices are provided with a pyramidal tube with a light centralizer. With its help, it is possible not only to limit the area to be removed to improve the quality of the X-ray image, but also excludes unnecessary irradiation of those parts of the human body that are not subject to X-ray.
To reduce the amount scattered x-rays the investigated detail of the object should be as close as possible to the X-ray film. This does not apply to direct magnification X-ray imaging. In direct magnification X-ray, diffuse scattering hardly reaches the X-ray film.
Sandbags used for fixing the object under study should be located farther from the cassette, since sand is a good medium for the formation of scattered X-ray radiation.
With radiography, produced on a table without using a screening grid, a sheet of leaded rubber of the largest possible size should be placed under the cassette or envelope with the film.
For absorption scattered x-rays X-ray screening gratings are used, which absorb these rays as they leave the human body.
Mastering the technique X-ray production at increased stresses on the X-ray tube, this is exactly the way that brings us closer to the ideal X-ray picture, that is, a picture in which both bone and soft tissues are clearly visible in detail.
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