Scattering of X-ray radiation by an electron. Diffuse scattering of X-rays. Addition of two harmonic oscillations with the same frequency, but different amplitude and initial phase
DIFFUSE SCATTERING OF X-RAYS- scattering x-rays substance in directions for which it is not fulfilled Bragg - Wolf condition.
In an ideal crystal, elastic scattering of waves by atoms located at periodic nodes. lattice, as a result, occurs only at a certain point. directions vector Q, coinciding with the directions of the reciprocal lattice vectors G: Q= k 2 -k 1 where k 1 and k 2 - wave vectors of the incident and scattered waves, respectively. The scattering intensity distribution in reciprocal lattice space is a set of d-shaped Laue-Bragg peaks at reciprocal lattice sites. Displacements of atoms from lattice sites disrupt the periodicity of the crystal, and interference. the picture is changing. In this case, in the scattering intensity distribution, along with the maxima (which remain if an averaged periodic lattice can be identified in a distorted crystal), a smooth component appears I 1 (Q), corresponding to D. r. R. l. on crystal imperfections.
Along with elastic scattering, D. r. R. l. may be due to inelastic processes accompanied by excitation of the electronic subsystem of the crystal, i.e., Compton scattering (see Compton effect) and scattering with plasma excitation (see. Solid state plasma). Using calculations or special experiments, these components can be excluded by highlighting D. r. R. l. on crystal imperfections.
In amorphous, liquid and gaseous substances, where there is no long-range order, scattering is only diffuse. Intensity distribution I 1 (Q Q) D. R. R. l. crystal in a wide range of values Intensity distribution, corresponding to the entire unit cell of the reciprocal lattice or several cells, contains detailed information about the characteristics of the crystal and its imperfections. Experimentally) can be obtained using a method using monochromatic. X-ray and allows you to rotate the crystal around different axes and change the directions of wave vectors Q over a wide range of values. Less detailed information can be obtained Debye - Scherrer method or Laue method.
In a perfect crystal D.r.r.l. caused only by thermal displacements and zero oscillations Q atoms of the lattice and can be associated with the processes of emission and absorption of one or more. . For small basic Single-phonon scattering plays a role, in which only phonons with q =Q-G G, Where Q-reciprocal lattice vector closest to . The intensity of such scattering I Q 1T (
) in the case of monatomic ideal crystals is determined by the f-loy Where N - number of elementary cells of the crystal, f -structural amplitude, - Debye-Waller factor, t
- atomic mass, -frequencies and . phonon vectors j th branch with wave vector q th branch with wave vector. At small th branch with wave vector frequency, i.e., when approaching the nodes of the reciprocal lattice, it increases as 1/ th branch with wave vector 2. Defining for vectors
, parallel or perpendicular to the directions , , in cubic crystals, where the oscillation frequencies for these directions are uniquely determined by considerations. . The intensity of such scattering 0 (Q In nonideal crystals, defects of finite sizes lead to a weakening of the intensities of correct reflections Intensity distribution)and to D.r.r.l. ) to static displacements and changes in structural amplitudes caused by defects ( 
s
- cell number near the defect, - type or orientation of the defect). In slightly distorted crystals with a low concentration of defects (the number of defects in the crystal) and
intensity D.r.r.l. where and are the Fourier components. Displacements decrease with distance where and are the Fourier components. r th branch with wave vector from defect as 1/ Intensity distribution 2, as a result of which at small th branch with wave vector and near reciprocal lattice nodes Intensity distribution)increases as 1/ 2. Angle addiction)qualitatively different for defects Intensity distribution different types Intensity distribution and symmetry, and the magnitude
) is determined by the amount of distortion around the defect. Distribution study ) in crystals containing point defects (for example, interstitial atoms and vacancies in irradiated materials, impurity atoms in weak solid solutions), makes it possible to obtain detailed information about the type of defects, their symmetry, position in the lattice, configuration of atoms forming the defect, tensors dipoles of forces with which defects act on the crystal. When combining point defects into groups, the intensity th branch with wave vector I 1 in the field of small increases strongly, but turns out to be concentrated in relatively small regions of reciprocal lattice space near its nodes, and at (
Study of areas of intensive D. r. R. l. makes it possible to study the size, shape and other characteristics of particles of the second phase in aging solutions. loops of small radius in irradiated or deformed. materials.
When means. concentrations of large defects, the crystal is strongly distorted not only locally near the defects, but also as a whole, so that in most of its volume. As a result, the Debye-Waller factor and the intensity of correct reflections I 0 decrease exponentially, and the distribution Intensity distribution) is qualitatively rearranged, forming broadened peaks slightly displaced from the reciprocal lattice nodes, the width of which depends on the size and concentration of defects. Experimentally, they are perceived as broadened Bragg peaks (quasi-lines on the Debye diagram), and in some cases diffraction patterns are observed. doublets consisting of pairs of peaks . The intensity of such scattering 0 and ) in crystals containing point defects (for example, interstitial atoms and vacancies in irradiated materials, impurity atoms in weak solid solutions), makes it possible to obtain detailed information about the type of defects, their symmetry, position in the lattice, configuration of atoms forming the defect, tensors dipoles of forces with which defects act on the crystal.. These effects appear in aging alloys and irradiated materials.
In concentrated solutions, single-component ordering crystals, ferroelectrics, non-ideality is not due separately. defects, and fluctuations. inhomogeneities of concentration and internal Intensity distribution parameters and th branch with wave vector) can be conveniently considered as scattering by th. fluctuation wave of these parameters ( q=Q-G)
) in the case of monatomic ideal crystals is determined by the f-loy - number of elementary cells of the crystal,. For example, in binary solutions A - B with one atom per cell, neglecting static scattering. displacements A and f B -atomic scattering factors of atoms A and B, With - concentration - correlation parameters, - probability of substitution of a pair of nodes separated by a lattice vector A Intensity distribution, atoms A. Having determined Intensity distribution) in the entire cell of the reciprocal lattice and by performing the Fourier transform f-tions can be found for decomp. coordination spheres Static scattering biases are excluded based on intensity data Intensity distribution) in several reciprocal lattice cells. - concentration - correlation parameters, - probability of substitution of a pair of nodes separated by a lattice vector Distributions ) can also be used directly. determination of solution ordering energies for different
in the model of pair interaction and its thermodynamic. characteristics. Features of D.r.r.l. metallic
Diffuse scattering of thermal neutrons by static. heterogeneities similar to D. r. R. l. and is described by similar phrases. The study of neutron scattering makes it possible to study also dynamic. characteristics of atomic vibrations and fluctuations. heterogeneities (see).
Inelastic neutron scattering Lit.:
James R., Optical principles of X-ray diffraction, trans. from English, M., 1950; Iveronova V.I., Revkevich G.P., Theory of X-ray scattering, 2nd ed., M., 1978;.
Iveronova V.I., Katsnelson A.A., Short-range order in solid solutions, M., 1977; Cowley J., Physics of Diffraction, trans. from English, M., 1979; Krivoglaz M A., Diffraction of X-rays and neutrons in nonideal crystals, K., 1983; by him, Diffuse scattering of X-rays and neutrons on fluctuation inhomogeneities in nonideal crystals, K., 1984.
M. A. Krivoglaz
Municipal educational institution secondary school No. 21
Abstract on physics
"SCATTERING OF X-RAYS
ON FULLEREN MOLECULES"
I've done the work
student of 11th grade
Lykov Vladimir Andreevich
Teacher:
Kharitonova Olga Alexandrovna
1. 3.5. Fraunhofer diffraction of x-rays on crystal atoms38 Goals of work
Computer modelling
X-ray scattering on fullerene molecules and fragments of fullerite crystals.
2. Study of rotational pseudosymmetry of the angular intensity distribution of scattered X-rays.
2. Theoretical part
2.1. Oscillations
2.1.1. One-dimensional oscillatory movements
Let us consider the one-dimensional periodic motion of a material point. Periodicity of motion means that the coordinate of point x is a periodic function of time t:
In other words, for any moment of time the equality
f(t + T) = f(t), (1.2)
where the constant value T is called the oscillation period. It is important that the coordinate can be not only Cartesian, but also an angle, etc. There are many types of periodic motion. For example, this is
uniform motion
material point along a circle.
liquid surface).
Fig.1.3. A ball suspended on a thread.
Fig.1.4. Float on the surface of a liquid. Fig.1.5. U-shaped tube containing liquid. Fig.1.6.
Electrical circuit
containing a capacitor with capacitance C and a coil with inductance L.
The simplest type of oscillations are harmonic. The coordinate of a material point changes over time during harmonic oscillations according to the law
x(t) =Acos(wt + j0) (1.3)
where A is the displacement amplitude (the maximum displacement of the point from the equilibrium position), w is the frequency associated with the period by the relation
w = 2p / T. (1.4)
The equilibrium position is the location of a material point in which the sum of the forces acting on it is equal to zero.
The cosine argument wt + j0 in function (1.3) is called the oscillation phase. It can be seen that the phase is a dimensionless quantity and linear function time. The constant value j0 is called the initial phase.
Oscillations of the physical systems shown in Fig. 1.1. – 1.6. would perform strictly harmonic oscillations at the following additional conditions:
System 1.1. – in the absence of air resistance, system 1.2. – in the absence of thorns, system 1.3. – at small angles and no air resistance, systems 1.4. and 1.5. – in the absence of liquid viscosity, system 1.6. – in the absence of active resistance of the coil and wires.
For simplicity, let us first consider one-dimensional harmonic oscillations, when a material point moves along one straight line.
Having calculated the derivative of function (1.3) with respect to time, we obtain the speed of the material point:
v(t) = -wAsin(wt+j0) (1.5)
It can be seen that speed is also a periodic function of time.
Now we take the derivative of function (1.5) with respect to time and obtain the acceleration of the material point.
a(t) = -w2 Acos(wt+j0) (1.6)
Comparing functions (1.3) and (1.6) we find that the coordinate and acceleration are related by the following expression
a(t) = -w2 x(t),(1.7)
which is executed at any time.
In other words, for any one-dimensional harmonic oscillations, the acceleration of a particle is directly proportional to its coordinate, and the proportionality coefficient is negative.
Fig.1.7. Time dependences of the coordinates (circles), speed (squares) and acceleration (triangles) of a particle performing one-dimensional harmonic oscillations. Amplitudes A=2, period T=5, initial phase j0=0.
As is known, the acceleration of a particle (according to the basic law of dynamics) is directly proportional to the force acting on the particle. Consequently, if the force is directly proportional to the coordinate with the opposite sign, then the particle will perform a harmonic oscillation. Such forces are called restoring forces.
An important example of a restoring force is the Hooke's force (elastic force). Thus, if on material point Hooke's force acts, then the point performs harmonic oscillations.
Since we are considering one-dimensional vibrations, to analyze the problem it is enough to project the Hooke force vector onto an axis parallel to this force. If the zero of the x coordinate is chosen at the point at which the restoring force is zero, then the projection of the force is equal to
where the coefficient k is called stiffness.
Comparing equations (1.7) and (1.8), and using Newton’s 2nd law, we obtain an important expression for the oscillation frequency:
This means that the oscillation frequency is described by the parameters of the physical system, and does not depend on the initial conditions. In particular, expression (1.9) determines the frequency harmonic vibrations systems shown in Fig. 1.1. and 1.2.
As an instructive example, consider the one-dimensional movements performed by weights attached to springs (see Fig. 1.8).
Fig.1.8. Weights on springs.
Let the masses of the springs be negligible compared to the masses of the loads.
Loads are considered as material points.
First, consider the system shown in Fig. 18. A. Let's assume that initially the load was shifted to the left and, as a result, the spring stretched. In this case, 3 forces act on the load (material point): the force of gravity mg, the elastic force F and the normal reaction force of the support N. We neglect friction in this problem (see Fig. 1.9).
Fig.1.9. Forces on a load resting on a smooth support when a spring is stretched.
Let's write down Newton's second law for the body shown in Fig. 1.9.
ma = mg + F + N(1.10)
The elastic force for small deformations of springs is described by Hooke's law
F = – kd(1.11)
where d is the spring deformation vector, k is the spring stiffness coefficient.
Note that when the load moves, the tension of the spring can be replaced by compression. In this case, the deformation vector d will change its direction to the opposite, therefore, the same will happen with the Hooke force (1.11). From this, in particular, it follows that during the initial compression of the spring, the vector equation of motion (1.10) will have the same form:
ma = mg – kd + N(1.12)
Let us choose the origin of coordinates at the point where the load is located with an undeformed spring. Let's direct the X axis horizontally, the Y axis vertically, i.e. perpendicular to the support (see Fig. 1.9).
Since the load moves horizontally along the support, the projection of acceleration onto the Y axis is zero. Then the force of gravity is completely compensated by the normal reaction of the support
N + mg = 0 (1.13)
Projecting the equation of motion (1.12) onto the X axis gives the scalar equation:
ma = – kd,(1.14)
where a is the horizontal projection of the acceleration of the load, d is the projection of the spring deformation vector.
In other words, the acceleration is directed along the horizontal X axis and is equal to
a = – (k/m) d(1.15)
Let us note once again that equation (1.15) is valid for both tension and compression of the spring.
Since the origin of coordinates is chosen so that it coincides with the end of the undeformed spring, the projection of deformation coincides with the value of the horizontal coordinate of the load x:
a = – (k/m) x (1.16)
By definition, the acceleration projection is equal to the second derivative of the corresponding coordinate with respect to time. Consequently, the one-dimensional equation of motion (1.16) can be rewritten in the form
In other words, the acceleration projection is directly proportional to the coordinate, and the proportionality coefficient has a negative sign.
Equation (1.17) is a second order differential equation; the general theory of solving such equations is studied in the course of mathematical analysis. However, it is easy to prove by direct substitution that the harmonic oscillation function (1.3) satisfies equation (1.17). As has already been proven earlier, the oscillation frequency is expressed by formula (1.9).
The amplitude A and the initial phase j0 of oscillations are determined from the initial conditions.
Let the load be initially displaced to the right from the equilibrium position by a distance d0, and the initial speed of the load be zero. Then, using functions (1.3) and (1.5), we write for the moment t=0 following equations:
d0 =Acos(j0) (1.18)
0 = -wAsin(j0) (1.19)
The solution to system (1.18) – (1.19) is the following values A = d0 and j0= 0.
For other initial conditions, the quantities A and j0 will naturally acquire different values.
Now consider the system shown in Fig. 1.8. b. In this case, only two forces act on the load: the force of gravity mg and the elastic force F (see Fig. 1.10). It is clear that in the equilibrium position these forces compensate each other, therefore, the spring is stretched.
Let the load move slightly vertically. Then the vector equation of motion will have a form similar to equation (1.12)
ma = mg – kd(1.20)
and regardless of the direction of vertical displacement (up or down).
All vectors in equation (1.20) are directed vertically, so it is advisable to project this equation onto the vertical coordinate axis. Let's direct the axis down, and choose the origin of coordinates at the point where the body is in a state of equilibrium (see Fig. 1.10).
Fig.1.10. Forces acting on a load hanging on a spring.
Projecting (1.18) onto the X axis we get:
a = g – (k/m) d(1.21)
where a is the projection of body acceleration, d is the projection of spring deformation.
To solve equation (1.21), it is useful to return to the equilibrium position of the load. Newton's equation for this position is:
0 = g – (k/m) d0(1.22)
where d0 is the deformation of the spring when the load is in equilibrium. Therefore, the vector d0 is equal to
It can be seen that in the equilibrium position of the body the spring is actually stretched, since the vector d0 is directed parallel to the vector g, i.e. down.
Now let’s place the origin of coordinates at the equilibrium point of the load on the spring, and then equation (1.21) will take the form:
a = g – (k/m) (x+ d0) (1.24)
where d0 is the modulus of the spring deformation vector d0.
Substituting the value d0 obtained from relation (1.23) into equation (1.24), we obtain:
a = g – (k/m) (x+ (m/k) g)
a = – (k/m) x (1.25)
The resulting equation completely coincides with equation (1.16). Thus, the body shown in Fig. 1.8. b, also performs harmonic oscillatory motion, described by function (1.3), like the load in the system shown in Fig. 1.8. A. Frequency of vibration The only difference is the direction of vibration (vertical instead of horizontal). But the oscillation frequency is still determined by the spring stiffness and the mass of the load by formula (1.9).
It is characteristic that the initial deformation of the spring in the system in Fig. 1.8. b does not affect the oscillation frequency.
2.1.3. Addition of vibrations
2.1.3.1. Addition of two harmonic oscillations with the same amplitudes and frequencies
Let's consider the example of sound waves, when two sources create waves with the same amplitudes A and frequencies ω. We will install a sensitive membrane at a distance from the sources. When the wave “travels” the distance from the source to the membrane, the membrane will begin to vibrate. The effect of each wave on the membrane can be described by the following relationships using oscillatory functions:
x1(t) = A cos(ωt + φ1),
x2(t) = A cos(ωt + φ2).
x(t) = x1 (t) + x2 (t) = A (1.27)
The expression in parentheses can be written differently using the trigonometric sum of cosines function:
In order to simplify function (1.28), we introduce new quantities A0 and φ0 that satisfy the condition:
A0 = φ0 = (1.29)
Substituting expressions (1.29) into function (1.28), we obtain
Thus, the sum of harmonic oscillations with the same frequencies ω is a harmonic oscillation of the same frequency ω. In this case, the amplitude of the total oscillation A0 and the initial phase φ0 are determined by relations (1.29).
2.1.3.2. Addition of two harmonic oscillations with the same frequency, but different amplitude and initial phase
Now consider the same situation, changing the oscillation amplitudes in function (1.26). For the function x1 (t) we replace the amplitude A with A1, and for the function x2 (t) A with A2. Then functions (1.26) will be written in the following form
x1 (t) = A1 cos(ωt + φ1), x2 (t) = A2 cos (ωt + φ2); (1.31)
Let us find the sum of harmonic functions (1.31)
x= x1 (t) + x2 (t) = A1 cos(ωt + φ1) + A2 cos (ωt + φ2) (1.32)
Expression (1.32) can be written differently, using the trigonometric sum cosine function:
x(t) = (A1cos(φ1) + A2cos(φ2)) cos(ωt) – (A1sin(φ1) + A2sin(φ2)) sin(ωt) (1.33)
In order to simplify function (1.33), we introduce new quantities A0 and φ0 that satisfy the condition:
Let us square each equation of system (1.34) and add the resulting equations. Then we get the following relation for the number A0:
Let's consider expression (1.35). Let us prove that the quantity under the root cannot be negative. Since cos(φ1 – φ2) ≥ –1, this means that this is the only quantity that can affect the sign of the number under the root (A12 > 0, A22 > 0 and 2A1A2 > 0 (from the definition of amplitude)). Let's consider the critical case (cosine is equal to minus one). Under the root is the formula for the square of the difference, which is always a positive quantity. If we begin to gradually increase the cosine, then the term containing the cosine will also begin to increase, then the value under the root will not change its sign.
Now let’s calculate the relationship for the value φ0 by dividing the second equation of system (1.34) by the first and calculating the arctangent:
Now let’s substitute the values from system (1.34) into function (1.33)
x = A0(cos(φ0) cosωt – sin(φ0) sinωt) (1.37)
Transforming the expression in parentheses using the cosine sum formula, we get:
x(t) = A0 cos(ωt + φ0) (1.38)
And again it turned out that the sum of two harmonic functions of the form (1.31) is also a harmonic function of the same type. More precisely, the addition of two harmonic oscillations with the same frequencies ω also represents a harmonic oscillation with the same frequency ω. In this case, the amplitude of the resulting oscillation is determined by relation (1.35), and the initial phase – by relation (1.36).
2.2. Waves
2.2.1. Propagation of vibrations in a material environment
Let us consider vibrations in the material environment. One example is the oscillation of a float on the surface of the water. If the role of an observer is a bird flying over the float, then it will notice that the float forms circles around itself, which, surprisingly, increases in radius over time as it moves away. But if the role of observer is a person standing on the shore, then he will see “humps” and “hollows”, which, alternating, approach the shore. This phenomenon is called a traveling wave.
In order to understand the properties of the wave, we neglect air resistance and the viscosity of water and air, i.e. dissipative forces. Then the mechanical energy of water droplets can be assumed to be conserved. In this case, the movement of the wave can be schematically depicted as shown in Figure 1, replacing the water droplets with numbered balls. Let's denote ball No. 1 as the float.
Rice. 2.1. Schematic representation of a transverse wave.
We see that the cause of the movement is ball No. 1, i.e. float. With the help of interaction, he engages ball No. 2 in movement, ball No. 2 engages ball No. 3, etc. But the interaction between particles does not occur instantly, so ball No. 2 will lag behind in time. You can also notice that ball #13 oscillates in the same way as #1. Then we can conclude that ball No. 2 will lag behind No. 1 by 1/12 of the period.
Hence, the wave period (T) can be called the period of oscillation of ball No. 1, the wave amplitude (A) is the maximum deviation of the ball from the horizontal axis, and the wave length (λ) is the minimum distance between the maxima of the nearest humps or the minima of the nearest troughs.
In the previously considered example, the wave propagated perpendicular to the oscillations of the source, in other words, it was considered transverse wave.
Longitudinal waves are waves that propagate parallel to the movement of the source. If we consider longitudinal waves schematically (Fig. 2.2), we can see that over time the source of oscillations (ball No. 1) oscillates left and right and involves other particles in the same oscillatory motion. Then, for a longitudinal wave, the definition of wave period described above will remain unchanged, but the definitions of wavelength and amplitude will look different. Generalized concepts will look like this: wavelength – the minimum distance between balls moving with the same phases; wave amplitude – maximum deviation from the equilibrium position.
2.2.2. Wave function
Let us consider a source that performs harmonic oscillations in a material medium with frequency w. Then its motion is described by a function of the form . Let the initial phase j0 be zero. Then the source coordinate is next function time.
x = Acos(wt) (2.1)
Due to particle interaction environment are involved in movements, which will also be harmonic oscillations. But interparticle interaction does not occur instantly, so oscillations of neighboring particles will occur with a time shift. Due to the finite and constant speed of interaction transmission, this time shift of oscillations is directly proportional to the distance of the next particle from the source.
From the previous examples it follows that as a result, disturbances called waves will propagate in the medium. When surface waves this disturbance represents the deflection of water particles from the surface in a calm state. In the case of sound waves, the disturbance is the deviation of air density from the average air density at rest. Regardless of the type of waves (longitudinal or transverse), this disturbance must be described by some function of time and coordinates.
At the source point, the disturbance is a function of time coinciding with (2.1)
y(0, t) = Acos(wt). (2.2)
Let us consider the propagation of a harmonic disturbance in the direction specified by the 0Z coordinate axis. According to the above, particles of the material environment located at a distance z from the source perform harmonic oscillations with a time delay (due to the finite speed of propagation of the interaction). Consequently, the disturbance at point z and at an arbitrary moment of time t coincides with the disturbance at point z = 0 of the source at some previous moment of time t¢
y(z, t) = y(0, t¢) (2.3)
The speed of propagation of a disturbance in a given medium is clearly expressed by the speed of movement of the hump (or depression) in surface waves or the speed of movement of compaction (or rarefaction) in a sound wave. This speed vf is called the phase speed of the wave. Thus, a hump, a depression, or any other type of disturbance in the medium travels a distance z in time z/vf.
The phase velocity allows us to relate the moments of time t¢ and t by the following relation
Using relations (2.2) – (2.4), we obtain an expression for the perturbation function in the following form:
The resulting expression is called a harmonic wave function or, in short, a harmonic wave.
In cases of homogeneous media and small disturbances, the phase velocity is a constant value.
Let us introduce a new quantity, called the wave number, with the following relation:
k = ω / vf(2.6)
Using the wave number, the harmonic wave function (2.5) can be written as:
y(z, t) = A cos(ωt – kz) (2.7)
Let's consider the quantity A. This quantity is the amplitude of the wave. As already mentioned, the amplitude of a wave is the maximum deviation of a particle from its equilibrium position. The amplitude of the wave can change over time (due to external forces).
The phase of the wave will be the quantity under the sign trigonometric function. Depending on the initial conditions, the phase of the wave function may contain a constant term j0 ¹ 0. The phase of the wave is a function of two arguments, time and coordinates.
Note that function (2.8) describes a wave process that is infinite in space and time.
Let us consider the physical meaning of the quantity k. Let's choose the moment of time t=0. Wave function (2.8) will take the form:
Function (2.9) can be interpreted as an instantaneous photograph of the wave process. It can be seen that this function is periodic in space.
According to the definition of the period, the following equality holds for any values of the z coordinate
A Cos(k (z + l)) = A Cos(k z)
The quantity l is called the wavelength. It represents the minimum distance between points with the same phase (humps, depressions, etc.).
If the cosines are equal, then the arguments differ by 2π
k (z+l) = kz +2π (2.9)
Through simple transformations we obtain the following expression:
λ = 2π/k(2.10)
It follows that the value of k is inversely proportional to the wavelength λ.
Let us consider a set of points in space at which the phase of the wave remains equal to zero.
wt – kz = 0(2.11)
The algebraic transformation gives:
The z/t ratio on the left above was defined as the phase velocity. According to (2.13), the phase velocity of a plane harmonic wave is equal to
From relation (2.15) it is also clear that for a harmonic traveling wave at a fixed moment in time, the rate of increase in phase per unit length is the value k (wave number) equal to
k = w/vF(2.14)
An example of harmonic waves was considered above. But in nature such waves are very rare. More often there are damped waves, i.e. waves in which the speed (due to air resistance, friction or other dissipative forces) becomes zero over time. The functions we obtained earlier are not valid for damped waves.
Above, we considered waves propagating along the interface between two media and waves propagating in volumes of matter. For example, only longitudinal sound waves can propagate in air, but both longitudinal and transverse ones can propagate in metal.
In addition, waves can be distinguished by the shape of the constant phase surface. Important special cases are plane and spherical waves.
2.2.3. Electromagnetic waves
It is known that a changing magnetic field generates an electric one. If we assume that a changing electric field generates a magnetic field, then we can assume, as Maxwell did, that this will produce an electromagnetic wave. And only then, in 1886, Hertz experimentally proved that Maxwell was right. Hertz, in his experiments, by reducing the number of coil turns and the area of the capacitor plates, as well as moving them apart, made the transition from a closed oscillatory circuit to an open oscillatory circuit (Hertz vibrator), which consists of two rods separated by a spark gap. If in a closed oscillatory circuit the alternating electric field is concentrated inside the capacitor, then in an open circuit it fills the space surrounding the circuit, which significantly increases the intensity of electromagnetic radiation. Oscillations in such a system are maintained due to e. from a source connected to the capacitor plates, and the spark gap is used to increase the potential difference to which the plates are initially charged. To excite electromagnetic waves, a Hertz vibrator 8 was connected to an inductor. When the voltage at the spark gap reached the breakdown value, a spark arose and free damped oscillations arose in the vibrator. When the spark disappeared, the circuit opened and the oscillations stopped. Then the inductor charged the capacitor again, a spark appeared, and oscillations were again observed in the circuit, etc. To record electromagnetic waves, Hertz used another vibrator, which had the same frequency of natural oscillations as the radiating vibrator, i.e. tuned in resonance with the vibrator. When electromagnetic waves reached the resonator, an electric spark jumped in its gap.
With the help of the described vibrator, Hertz reached frequencies of the order of 100 MHz and obtained waves whose length was approximately 3 m.P.N. Lebedev, using a miniature vibrator made of thin platinum rods, obtained millimeter electromagnetic waves with a wavelength λ = 6-4 mm. This is how electromagnetic waves were experimentally discovered. Hertz also proved that the speed of an electromagnetic wave is equal to the speed of light:
Then it was proven that electromagnetic waves are transverse. The source of electromagnetic waves are oscillating charges. A system of electric and magnetic fields arises in the space surrounding the charge. A “snapshot” of such a field system is shown in Fig. 2.3.
A qualitative characteristic of electromagnetic oscillations can be given both in the form of oscillation frequency, expressed in hertz, and in wavelengths. The higher the oscillation frequency, the shorter the propagated wavelength. The entire spectrum of these waves is conventionally divided into the following 16 ranges:
Wavelength | Name | Frequency |
| more than 100 km | Low frequency electrical vibrations | 0-3 kHz |
| 100 km - 1 mm | Radio waves | 3 kHz - 3 THz |
| 100-10 km | myriameter (very low frequencies) | 3 - 3 kHz |
| 10 - 1 km | kilometer (low frequencies) | 30 - 300 kHz |
| 1 km - 100 m | hectometric (mid frequencies) | 300 kHz - 3 MHz |
| 100 - 10 m | decameter (high frequencies) | 3 - 30 MHz |
| 10 - 1 m | meter (very high frequencies) | 30 - 300MHz |
| 1 m - 10 cm | decimeter (ultra-high) | 300 MHz - 3 GHz |
| 10 - 1 cm | centimeter (extra high) | 3 - 30 GHz |
| 1 cm - 1 mm | millimeter (extremely high) | 30 - 300 GHz |
| 1 - 0.1 mm | decimmillimeter (hyperhigh) | 300 GHz - 3 THz |
| 2 mm - 760 nm | Infrared radiation | 150 GHz - 400 THz |
| 760 - 380 nm | Visible radiation (optical spectrum) | 400 - 800 THz |
| 380 - 3 nm | Ultraviolet radiation | 800 THz - 100 PHz |
| 10 nm - 1 pm | X-ray radiation | 30 PHz - 300 EHz |
| <=10 пм | Gamma radiation | >=30 EHz |
One of the most common types of electromagnetic waves is light waves. But in our work we will consider another type of electromagnetic waves - x-rays.
2.2.4. X-rays
One of bright examples Electromagnetic waves can be considered x-rays.
In 1895 V.K. Roentgen (1845 – 1923) conducted research electric current in highly rarefied gases. To the electrodes, soldered into a glass tube, from which air was previously pumped out to a pressure of ~10–3 mm Hg. Art., a potential difference of several kilovolts was applied. It turned out that in this case the tube becomes a source of rays, which Roentgen called “X-rays”. The basic properties of X-rays were studied by Roentgen himself as a result of three years of work, for which he was awarded the Nobel Prize in 1901 - the first among physicists. The rays he discovered were subsequently rightly called X-rays.
Fig.2.3. Diagrams of X-ray tubes.
a) one of the first X-ray tubes, b) an X-ray tube from the end of the 20th century.
K – thermal cathode, A – high-voltage anode, T – heating of the thermal cathode, E – beams of accelerated electrons (dashed lines), P – fluxes of X-rays (dashed lines), O – windows in the tube body for the exit of X-rays.
According to modern scientific research, X-rays are electromagnetic radiation invisible to the eye with a wavelength belonging to the approximate range of 10–2 - 10 nanometers.
X-rays are emitted when fast electrons in a substance are decelerated (in this case they form continuous spectrum) and during electron transitions from the outer electron shells of the atom to the inner ones (and give a line spectrum).
The most important properties of X-rays are the following:
The rays pass through all materials, including those opaque to visible light. The intensity of transmitted rays I decreases exponentially with thickness x of the layer of matter
I(x) = I0 exp(–m/x),(2.16)
where I0 is the intensity of the rays incident on the layer of irradiated material.
The coefficient m characterizes the attenuation of the X-ray flux by the substance and depends on the density of the material r and its chemical composition. Numerous experiments have shown that, to a first approximation, there is a dependence
Streams of X-rays pass through thick boards, metal sheets, human body etc. The significant penetrating power of X-rays is currently widely used in flaw detection and medicine.
X-rays cause luminescence of certain chemical compounds. For example, a screen coated with BaPt(CN) 4 salt glows yellow-green when hit by X-rays.
X-rays hitting photographic emulsions cause them to turn black.
X-rays ionize air and other gases, making them electrically conductive. This property is used in detectors that detect invisible X-rays and measure their intensity.
X-rays have a strong physiological effect. Long-term irradiation of living organisms with intense fluxes of X-rays leads to the emergence of specific diseases (the so-called “radiation sickness”) and even to fatal outcome.
As mentioned earlier, X-rays are emitted during the deceleration of fast electrons in a substance and during electron transitions from the outer electron shells of an atom to the inner ones (and give a line spectrum). Detectors that record X-rays are based on the properties of X-rays. Therefore, the most often used detectors are: photographic emulsions on film and plates, fluorescent screens, gas-filled and semiconductor detectors.
2.3. Wave diffraction
2.3.1. Diffraction and wave interference
Typical wave effects are the phenomena of interference and diffraction.
Initially, diffraction was the deviation of the propagation of light from the rectilinear direction. This discovery was made in 1665 by Abbot Francesco Grimaldi and served as the basis for the development of the wave theory of light. Diffraction of light was the bending of light around the contours of opaque objects and, as a consequence, the penetration of light into the region of geometric shadow.
After the creation of the wave theory, it turned out that the diffraction of light is a consequence of the phenomenon of interference of waves emitted by coherent sources located at different points in space.
Waves are said to be coherent if their phase difference remains constant over time. The sources of coherent waves are coherent oscillations of wave sources. Sine waves, whose frequencies do not change over time, are always coherent.
Coherent waves emitted by sources located at different points propagate in space without interaction and form a total wave field. Strictly speaking, the waves themselves do not “add.” But if a recording device is located at any point in space, then its sensitive element will be set into oscillatory motion under the influence of waves. Each wave acts independently of the others, and the movement of the sensing element is the sum of the oscillations. In other words, in this process there are no
waves, but vibrations caused by coherent waves.
Rice. 3.1. Dual source and detector system. L – distance from the first source to the detector, L’ – distance from the second source to the detector, d – distance between the sources.
As a basic example, consider the interference of waves emitted by two coherent point sources (see Fig. 3.1). The frequencies and initial phases of source oscillations coincide. The sources are located at a certain distance d from each other. The detector that records the intensity of the generated wave field is located at a distance L from the first source. The type of interference pattern depends on the geometric parameters of the sources of coherent waves, on the dimension of the space in which the waves propagate, etc.
Let us consider the functions of waves that are a consequence of oscillations emitted by two point coherent sources. To do this, let's set the z axis as shown in Fig. 3.1. Then the wave functions will look like this:
Let us introduce the concept of wave path difference. To do this, consider the distances from the sources to the recording detector L and L’. The distance between the first source and the detector L differs from the distance between the second source and the detector L’ by the amount t. In order to find t, consider right triangle, containing the values t and d. Then you can easily find t using the sine function:
This quantity will be called the wave path difference. Now let's multiply this value by the wave number k and get a value called the phase difference. Let's denote it as ∆φ
When two waves “reach” the detector, functions (3.1) take the form:
In order to simplify the law according to which the detector will oscillate, we set the value (–kL + j1) to zero in the function x1(t). Let us write the value of L’ in the function x2(t) using function (3.4). By simple transformations we obtain that
It can be noted that relations (3.3) and (3.6) are the same. Previously, this quantity was defined as the phase difference. Based on what was said earlier, Relationship (3.6) can be rewritten as follows:
Now let's add functions (3.5).
(3.8)
Using the method of complex amplitudes, we obtain the relation for the amplitude of the total oscillation:
where φ0 is determined by relation (3.3).
After the amplitude of the total oscillation has been found, the intensity of the total oscillation can be found as the square of the amplitude:
(3.10)
Let us consider the graph of the intensity of the total oscillation at different parameters. The angle θ varies in the interval (this can be seen from Figure 3.1), the wavelength varies from 1 to 5.
Let's consider the special case when L>>d. This case usually occurs in X-ray scattering experiments. In these experiments, the scattered radiation detector is usually located at a distance much greater than the size of the sample being studied. In these cases, secondary waves enter the detector, which can be approximately assumed to be plane with sufficient accuracy. In this case, the wave vectors of individual waves of secondary waves emitted by different centers of scattered radiation are parallel. It is believed that in this case the Fraunhofer diffraction conditions are satisfied.
2.3.2. X-ray diffraction
X-ray diffraction is a process that occurs during elastic scattering of X-ray radiation and consists of the appearance of deflected (diffracted) rays propagating at certain angles to the primary beam. X-ray diffraction is caused by the spatial coherence of secondary waves that arise when primary radiation is scattered by electrons that make up the atoms. In some directions, determined by the relationship between the wavelength of the radiation and the interatomic distances in the substance, the secondary waves add up while being in the same phase, resulting in the creation of an intense diffraction beam. In other words, under the influence of the electromagnetic field of the incident wave, the charged particles present in each atom become sources of secondary (scattered) spherical waves. Individual secondary waves interfere with each other, forming both amplified and weakened beams of radiation propagating in different directions.
We can assume that scattering is not accompanied by dispersion, and, therefore, the frequency of the scattered waves coincides with the frequency of the primary wave. If scattering is elastic, then the modulus of the wave vector also does not change.
Let us consider the result of the interference of secondary waves at a point distant from all scattering centers at a distance much greater than the interatomic distances in the sample under study (irradiated). Let there be a detector at this point and the oscillations caused by the scattered waves arriving at this point are added up. Since the distance from the scatterer to the detector significantly exceeds the wavelength of the scattered radiation, the sections of secondary waves arriving at the detector can be considered with a sufficient degree of accuracy as flat, and their wave vectors as parallel. Thus, the physical pattern of X-ray scattering, by analogy with optics, can be called Fraunhofer diffraction.
Depending on the scattering angle q (the angle between the wave vector of the primary wave and the vector connecting the crystal and the detector), the amplitude of the total oscillation will reach a minimum or maximum. The radiation intensity recorded by the detector is proportional to the square of the total amplitude. Consequently, the intensity depends on the direction of propagation of scattered waves reaching the detector, on the amplitude and wavelength of the primary radiation, and on the number and coordinates of scattering centers. In addition, the amplitude of the secondary wave formed by an individual atom (and hence the total intensity) is determined by the atomic factor - a decreasing function of the scattering angle q, which depends on the electron density of the atoms.
Let us consider the distribution of radiation intensity created by n coherent point sources of monochromatic waves. The geometry of a system consisting of n coherent point sources of monochromatic waves and a detector that can move along a straight line is shown in Fig. 5.1.
Fig.3.3. Geometry of a system of n sources.
The numbers 1,2,3,4,…,n indicate the positions of point sources.
The X axis is directed along the line of movement of the detector. Where Z1,Z2, Z3, Z4,…, Zn, are the distances from the first, second, third,…, nth sources to the receiver, along the axis X the vibration intensities are added, L– distance from axis X to the line connecting the sources.
In order to find the intensity of n sources, we use relation (3.10). Let's add the amplitudes using a vector method. Then for n sources function (3.10) will take the form:
This is the equation for calculating the radiation intensity of n sources, where
Here it can be calculated as follows:
Substituting (3.12), (3.13) and (3.14) into (3.11) we get:
2.3.4. Atomic factor
The atomic factor is a quantity that characterizes the ability of an isolated atom or ion to coherently scatter X-rays, electrons or neutrons (the X-ray, electron or neutron atomic factor is distinguished accordingly). The atomic factor determines the intensity of radiation scattered by an atom in a certain direction.
Let us consider the interaction of an X-ray wave with a single atom. The electric field of the wave generates periodic forces acting on all charged particles that make up the atom - electrons and the nucleus. The acceleration that a particle receives is inversely proportional to the mass of the particle. Each particle becomes a source of a secondary (i.e. scattered) wave. The radiation intensity is proportional to the square of the acceleration, so scattered radiation is generated almost exclusively by electrons, therefore the X-ray atomic factor depends on the distribution of electron density in the atom.
Electrons are dispersed inside the atom, and the size of the atom is comparable to the x-ray wavelength. Therefore, secondary waves created by individual electrons of an atom have a phase difference. This phase shift Dφ depends on the direction of propagation of the scattered wave relative to the direction of the wave vector of the primary wave. Consequently, the amplitude of the radiation scattered by an atom depends on the scattering angle.
The atomic factor f (or atomic scattering function) is defined as the ratio of the amplitude of the wave scattered by one atom to the amplitude of the wave scattered by one free electron. The magnitude of the atomic factor depends on the scattering angle q and the radiation wavelength l. The value g = sin(q) / l is used as an argument for the atomic factor function in X-ray diffraction studies.
If the polar angle q = 0, then the value of the atomic factor is equal to the number of electrons in the atom (in other words, the atomic number chemical element in the periodic table). As the scattering angle q increases, the atomic factor f(g) monotonically decreases to zero. A typical form of the atomic scattering function is shown in Fig. 3.4.
3.5. Fraunhofer diffraction of X-rays by crystal atoms
Let a flux of X-rays with a certain wavelength l be directed at a crystalline sample. In physical studies (when deciphering the atomic structure by X-ray diffraction, X-ray spectral elemental analysis, etc.), a geometric experimental design with the following geometric features is usually implemented (see Fig. 1).
Fig.3.5. Geometric diagram of irradiation of a small sample with a narrow beam of X-rays.
1 – X-ray generator (for example, X-ray tube), 2 – collimator, 3 – sample under study. Dashed arrows represent X-ray fluxes.
Using a collimator, a narrow beam of X-rays is formed. The irradiated crystalline sample is located from the exit from the collimator at a distance significantly greater than the size of the sample. In X-ray diffraction studies, samples are prepared smaller in size cross section beam. As they say, the sample is “bathed” in a beam of incident X-rays (see callout in Fig. 3.5).
Then we can assume with good accuracy that a plane electromagnetic wave with a length l is incident on the sample under study. In other words, all atoms of the sample are exposed to coherent plane waves with parallel wave vectors k0.
X-rays are electromagnetic waves that are transverse. If the coordinate axis Z is directed along the wave vector k0, then the components of the electric and magnetic fields of a plane electromagnetic wave can be written in the following form:
EX = EX0 cos(wt – k0 z + j0) EY = EY0 cos(wt – k0 z + j0)
BX = BX0 cos(wt – k0 z + j0) BY = BY0 cos(wt – k0 z + j0)
where t is time, w is the frequency of electromagnetic radiation, k0 is the wave number, j0 is the initial phase. The wave number is the modulus of the wave vector and is inversely proportional to the wavelength k0 = 2π/l. The numerical value of the initial phase depends on the choice of the initial time t0=0. The quantities EX0, EY0, BX0, BY0 are the amplitudes of the corresponding components (3.16) of the electric and magnetic fields of the wave.
Thus, all components (3.16) of a plane electromagnetic wave are described by elementary harmonic functions of the form:
Y = A0 cos(wt – kz+ j0) (3.17)
Let us consider the scattering of a plane monochromatic X-ray wave on a set of atoms of the sample under study (on a molecule, a crystal of finite dimensions, etc.). The interaction of an electromagnetic wave with the electrons of atoms leads to the generation of secondary (scattered) electromagnetic waves. According to classical electrodynamics, scattering from an individual electron occurs at a solid angle of 4p and has significant anisotropy. If the primary X-ray radiation is not polarized, then the flux density of the scattered radiation of the wave is described by the following function
where I0 is the primary radiation flux density, R is the distance from the scattering point to the place of registration of scattered radiation, q is the polar scattering angle, which is measured from the direction of the wave vector of the plane primary wave k0 (see Fig. 3.6). Parameter
» 2.818×10-6 nm(3.19)
historically called the classical electron radius.
Fig.3.6. Polar scattering angle q of a plane primary wave on a small Cr sample under study.
A certain angle q defines a conical surface in space. The correlated movement of electrons within an atom complicates the anisotropy of scattered radiation. The amplitude of an X-ray wave scattered by an atom is expressed using a function of wavelength and polar angle f(q, l), which is called the atomic amplitude.
Thus, the angular distribution of the intensity of the X-ray wave scattered by an atom is expressed by the formula
and has axial symmetry relative to the direction of the wave vector of the primary wave k0. The square of the atomic amplitude f 2 is usually called the atomic factor.
As a rule, in experimental installations for X-ray diffraction and X-ray spectral studies, the detector of scattered X-rays is located at a distance R significantly greater than the dimensions of the scattering sample. In such cases, the input window of the detector cuts out an element from the surface of the constant phase of the scattered wave, which can be assumed to be flat with high accuracy.
Fig.3.8. Geometric diagram of X-ray scattering on atoms of sample 1 under Fraunhofer diffraction conditions.
2 – X-ray detector, k0 – wave vector of the primary X-ray wave, dashed arrows depict the fluxes of primary X-rays, dash-dotted ones – fluxes of scattered X-rays. Circles indicate atoms of the sample under study.
In addition, the distances between neighboring atoms of the irradiated sample are several orders of magnitude smaller than the diameter of the detector entrance window.
Consequently, in this registration geometry, the detector perceives a flow of plane waves scattered by individual atoms, and the wave vectors of all scattered waves can be assumed to be parallel with high accuracy.
The above features of X-ray scattering and their registration have historically been called Fraunhofer diffraction. This approximate description of the process of x-ray scattering on atomic structures allows one to calculate the diffraction pattern (angular distribution of the intensity of the scattered radiation) with high accuracy. The proof is that the Fraunhofer diffraction approximation underlies X-ray diffraction methods for studying matter, which make it possible to determine the parameters of unit cells of crystals, calculate the coordinates of atoms, establish the presence of various phases in a sample, determine the characteristics of crystal defects, etc.
Consider a small crystalline sample containing a finite number N of atoms with a certain chemical number.
Let us introduce a rectangular coordinate system. Its origin is compatible with the center of one of the atoms. The position of each atomic center (scattering center) is specified by three coordinates. xj, yj, zj, where j is the atomic number.
Let the sample under study be exposed to a plane primary X-ray wave with a wave vector k0 directed parallel to the Oz axis of the selected coordinate system. In this case, the primary wave is represented by a function of the form (3.17).
Scattering of X-rays by atoms can be either inelastic or elastic. Elastic scattering occurs without changing the wavelength of X-ray radiation. With inelastic scattering, the radiation wavelength increases, and the secondary waves are incoherent. Below, only elastic scattering of X-rays on atoms is considered.
Let us denote L as the distance from the origin to the detector. Let us assume that the Fraunhofer diffraction conditions are satisfied. This, in particular, means that the maximum distance between the atoms of the irradiated sample is several orders of magnitude smaller than the distance L. In this case, the sensitive element of the detector is exposed to plane waves with parallel wave vectors k. The moduli of all vectors are equal to the modulus of the wave vector k0 = 2π/l.
Each plane wave causes a harmonic oscillation with a frequency
If the primary wave is satisfactorily approximated by a plane harmonic wave, then all secondary (scattered by atoms) waves are coherent. The phase difference of the scattered waves depends on the difference in the path of these waves.
Let us draw an auxiliary axis Or from the origin of coordinates to the location of the detector input window. Then each secondary propagating in the direction of this axis can be described by the function
y = A1 fcos(wt– kr+ j0) (3.22)
where the amplitude A1 depends on the amplitude of the primary wave A0, and the initial phase j0 is the same for all secondary waves.
A secondary wave emitted by an atom located at the origin of coordinates will create an oscillation of the sensitive element of the detector, described by the function
A1 f(q) cos(wt – kL+ j0) (3.23)
Other secondary waves will create oscillations with the same frequency (3.21), but differing from function (3.23) in phase shift, which in turn depends on the difference in the path of the secondary waves.
For a system of plane coherent monochromatic waves moving in a certain direction, the relative phase shift Dj is directly proportional to the path difference DL
Dj = k×DL(3.24)
where k is the wave number
k = 2π/l. (3.25)
To calculate the difference in the path of secondary waves (3.23), we first assume that the irradiated sample is a one-dimensional chain of atoms located along the Ox coordinate axis (see Fig. 3.9). The coordinates of the atoms are specified by the numbers xi, (j = 0, 1, …, N–1), where x0 = 0. The surface of the constant phase of the primary plane wave is parallel to the chain of atoms, and the wave vector k0 is perpendicular to it.
We will calculate a flat diffraction pattern, i.e. angular distribution of scattered radiation intensity in the plane shown in Fig. 3.9. In this case, the orientation of the detector location (in other words, the direction of the auxiliary axis Or) is specified by the scattering angle, which is measured from the Oz axis, i.e. on the direction of the wave vector k0 of the primary wave.
Fig.3.9. Geometric scheme of Fraunhofer diffraction in a given plane on a rectilinear chain of atoms
Without loss of generality of reasoning, we can assume that all atoms are located on the right Ox semi-axis. (except for the atom located at the center of coordinates).
Since the Fraunhofer diffraction conditions are satisfied, the wave vectors of all waves scattered by atoms arrive at the input window of the detector with parallel wave vectors k.
From Fig. 3.9 it follows that the wave emitted by an atom with coordinate xi travels a distance to the detector L – xisin(q). Consequently, the oscillation of the sensitive element of the detector caused by a secondary wave emitted by an atom with coordinate xi is described by the function
A1 f(q) cos(wt – k(L– xj sin(q)) + j0) (3.26)
The remaining scattered waves entering the window of the detector located in a given position have a similar appearance.
The value of the initial phase j0 is determined, in essence, by the moment when time begins to count. Nothing prevents you from choosing the value of j0 equal to –kL. Then the movement of the sensitive element of the detector will be represented by the sum
This means that the difference in the paths of waves scattered by atoms with coordinates xi and x0 is –xisin(q), and the corresponding phase difference is equal to kxisin(q).
The frequency w of oscillations of electromagnetic waves in the X-ray range is very high. For X-rays with a wavelength l = Å, the frequency w in order of magnitude is ~1019 sec-1. Modern equipment cannot measure the instantaneous values of the electric and magnetic field strengths (1) with such rapid field changes, therefore all X-ray detectors record the average value of the square of the amplitude of electromagnetic oscillations.
The recorded intensity of X-rays scattered by atoms of the irradiated sample is the square of the amplitude of the total vibration (11). To calculate this value, it is advisable to use the complex amplitude method. We write each term of sum (11) in complex form
A1 fexp (3.28)
where i is the imaginary unit, Djj is the phase shift, equal to kxjsin(q) in the physical picture under consideration.
We rewrite expression (12) in the form
A1 feiwte–iDjj (3.29)
The time-dependent factor describes oscillations of the electromagnetic field with frequency w. The modulus of this quantity is equal to unity. As a consequence, the complex amplitude of the electromagnetic oscillation expressed by function (12) has the form:
A1 fexp [–iDjj] (3.30)
The complex amplitude of the total oscillation recorded by the detector is equal to the sum of values (3.30), and the summation is carried out over all scattering centers - i.e. over all atoms of the irradiated sample. The square of the real part of the specified sum determines the recorded intensity of the scattered X-ray radiation
accurate to the hardware coefficient (a factor determined by the characteristics of the recording equipment).
Intensity (3.31) is a function of the polar angle q and describes in the xoz plane the angular distribution of X-rays scattered by a chain of atoms located along the ox axis.
Now let us consider the scattering of X-rays on a finite set of atoms located in the same plane. Let a plane X-ray wave with a wave vector k0 perpendicular to the plane of the atoms fall on this system of atoms.
Let us associate the Cartesian coordinate axes with this physical system. The oz axis will be directed along the vector k0, and the ox and oY axes will be located in the atomic plane. The position of each atom is specified by two coordinates xj and yj, where j = 0, ... N – 1. Let the origin of coordinates be combined with the center of one of the atoms, which has number j = 0.
Let us consider the scattering of X-rays into a half-space z > 0. In this case, we can assume that the detector moves along a hemisphere of a certain radius R, which is much larger than the size of the irradiated sample. The direction to the detector under Fraunhofer diffraction conditions coincides with the wave vectors k of scattered waves arriving at the input window of the detector. This direction is characterized by two angles: polar q, which is measured from the oz axis (as in Fig. 3.9 and 3.10), and azimuth Ф, which is measured from the ox axis in the xoY plane (see Fig. 3.10). In other words, q is the angle between the wave vectors of the primary k0 and scattered k waves. Azimuth Ф is the angle between the OX axis and the projection of the vector k onto the XOY plane.
As in the previous case of a one-dimensional chain of atoms, the amplitude of the total vibration recorded by the detector is determined by the relative phase shifts of coherent waves scattered by individual atoms. The phase shift of scattered waves is related to the path difference by relation (3.24), as in the case considered above.
Let's find the path difference between waves scattered by atoms with coordinates (x0=0, y0=0) and (x, y) in the direction specified by the wave vector k (i.e., certain angles q and Ф). Let's draw the auxiliary axis OU along the projection of the vector k onto the XOY plane (see Fig. 3.10).
Fig.3.10. Toward the calculation of the path difference of secondary waves scattered on a plane system of atoms under Fraunhofer diffraction conditions.
Point F on the OU axis is the projection of the center of the jth atom. The length of the segment OF is equal to xcos(Ф) + ysin(Ф), which can be obtained by transforming coordinates or geometric construction. The projection of the segment OF onto the direction of the wave vector k gives the desired path difference - the length of the segment OG, equal to
Dl = sin(q). (3.32)
Consequently, the phase shift of secondary waves scattered by atoms with coordinates (x0=0, y0=0) and (xj, yj) in the direction specified by certain angles q and Ф is equal to
Djj = k sin(q). (3.33)
The recorded intensity of scattered X-ray radiation is expressed by a formula similar to (3.31):
Finally, consider Fraunhofer diffraction of X-rays by a three-dimensional object. Let's use the Cartesian coordinate system used in the previous problem. The only difference between the physical picture and the previous one is that the centers of some atoms have coordinates zj¹ 0.
The constant phase surface of the primary plane monochromatic wave reaches scattering centers with different coordinates z¹ 0 at different times. As a consequence, the initial phase of the wave scattered by an atom with coordinate z¹ 0 will lag behind the phase of the wave scattered by an atom with coordinate z = 0 by the amount wDt, where Dt = z / v, v is the speed of wave propagation. Frequency and wavelength are related by
w = 2pv / l(3.35)
therefore, the phase shift of the scattered wave is equal to -2pz/l or -kz.
On the other hand, if the coordinate of the jth atom is zj¹ 0, the path difference relative to the “zero” scattered wave additionally increases by the value zcos(q). As a result, the phase shift of a wave scattered by an atom with arbitrary coordinates (xj, yj, zj) in the direction specified by the angles q and Ф is equal to
Djj = k ( sin(q) + zjcos(q) -zj). (3.36)
The intensity of scattered x-rays recorded by the detector is expressed by the following formula:
3. Practical part
3.1. Pseudosymmetry
3.1.1. Rotational pseudosymmetry of diffraction patterns
Symmetry is the invariance of a physical or geometric system with respect to various types of transformations.
Various types symmetries are determined by transformations with respect to which the this system. There is translational symmetry, rotational symmetry, similarity symmetry, etc.
Symmetry is one of the fundamental properties of the Universe. Even the basic laws of physics: conservation of energy, momentum and angular momentum are associated with certain symmetric transformations of the space-time continuum.
The specific transformation with respect to which a given system is invariant is called a symmetry operation. The set of points that remain fixed during a symmetric transformation form an element of symmetry. For example, if the symmetry operation is rotation, then the corresponding symmetry element will be the axis around which the rotation is performed.
The symmetry of finite physical systems whose symmetry elements intersect at least at one point is called point symmetry. Point symmetry includes invariance with respect to rotation through a certain angle (rotational symmetry), invariance with respect to reflection in a certain plane (mirror symmetry), and invariance with respect to inversion at a given point (inversion symmetry).
The symmetry of the vast majority of physical objects is not absolute. This means that the physical or geometric system is not completely invariant under the transformation in question.
To quantitatively describe deviations from exact symmetry, a functional called the degree of invariance or pseudosymmetry coefficient is used.
Let any physical characteristic of the object under study be described by a point function. This function can be mass density, temperature electric potential, density electric charge etc. We are the symmetry of a given object with respect to a transformation, which is specified by some operation. Then the degree of invariance is determined by the following formula (4.1), where V is the volume of the object. Under the integral in the numerator is the product of the function and the function of the same object subjected to transformation. The numerator is called the convolution of the function with respect to the operation. The denominator is definite integral by the volume of the object from the square of the function.
The denominator of formula (4.1) serves as a normalization, so the value of the functional can vary from 0 to 1. If the physical system under consideration is completely invariant with respect to the operation, then the pseudosymmetry coefficient is equal to one. The value = 0 corresponds to the case when the symmetry of the system with respect to the operation is completely absent.
The concept of the degree of invariance can be extended to describe the symmetry of the angular distribution of the intensity of scattered X-rays. First of all, we are interested in the invariance of diffraction patterns with respect to rotation through a certain azimuthal angle around the point corresponding to the polar angle q = 0. In other words, the goal of the study is the rotational symmetry of the angular distribution of the intensity of scattered X-rays, and the rotation is carried out around the wave vector k0 of the primary radiation.
To study the features of rotational symmetry of diffraction patterns, one can adapt a general functional (1). The function under study in this case is the angular distribution of the intensity of scattered x-rays I(q, Ф), and the symmetry operation is the rotation of the diffraction pattern by an azimuthal angle a around the central point of the pattern with a polar angle q = 0. Thus, a quantitative characteristic of the rotational symmetry of the diffraction pattern is the following functionality:
Internal integrals are taken over the range of the azimuthal angle ФО, and external integrals over the range of the polar angle qО.
It is worth paying attention to some important features of all diffraction patterns. In Fig. 4.1. It can be seen that in the center of the polar diagram there is a central maximum of the intensity of the scattered radiation. This maximum has high symmetry, close to the symmetry of the limit group C¥. In the angular distribution of scattered radiation, the central maximum occupies a certain interval of polar angles qО. The half-width of the central maximum depends significantly on the X-ray wavelength l and the number of scattering atoms.
It is also very important that the intensity of the central maximum significantly exceeds the intensity of all other points in the two-dimensional angular distribution of scattered X-ray radiation. On the contrary, as the polar angle increases, the intensity of the scattered radiation on average drops sharply. This means that the peripheral region of the diffraction pattern (the region of polar angles exceeding a certain value qM) has virtually no effect on the value of the rotational pseudosymmetry coefficient (4.2).
As a consequence, the main contribution to the degree of invariance (4.2) comes from the central maximum. In other words, the high symmetry of the central maximum suppresses the symmetry features of all other characteristic features of the diffraction pattern.
For a detailed study of the rotational pseudosymmetry of the angular distribution of scattered X-ray radiation, it is advisable to calculate functionals of the following form:
External integrals over the polar angle have limits that can be set by the researcher, which makes it possible to study rotational pseudosymmetry in different intervals of the polar angle. In other words, quantities like (4.3) give quantitative estimates of the rotational pseudosymmetry of the diffraction pattern inside the ring, specified by a pair of polar angles q1 and q2. (see Fig. 4.1).
It is natural to divide the range of polar angles into subranges of a certain width dq = q2 -q1 and calculate the pseudosymmetry coefficients for all such subranges.
Fig.4.1. A ring in the polar diagram of a diffraction pattern that limits the range of polar angles.
It was indicated above that in computer modeling of the angular distribution of the intensity of scattered X-rays, the function I(q, Ф) is represented by a two-dimensional set of numerical values I(ql, jm) for finite discrete sets of angles ql = lDq, l=1,…nq; Фm = mDF, m =1,…nФ. Consequently, when calculating the pseudosymmetry coefficients ha from the results of calculating the angular distribution of the intensity of scattered X-rays, the double integrals in expression (4.2) turn into double sums
If we are interested in the averaged rotational pseudosymmetry of the entire diffraction pattern, then the degree of invariance is given by the following formula:
If we want to study rotational pseudosymmetry in various subranges of the polar angle (see Fig. 4.1), then it is necessary to calculate the ratio of the sums for the corresponding intervals of type (4.3). Then the pseudosymmetry coefficients will be presented in the form:
where the indices l1 and l2 correspond to the values of the polar angles q1 and q2
q1 = l1 Dq, q2 = l2 Dq. (4.6)
By specifying certain values of the rotation angle a, it is possible to calculate the pseudosymmetry coefficients ha of diffraction patterns for rotations of various orders. If we are interested in rotational pseudosymmetry of the nth order, then the angle of rotation a is expressed by the relation.
an = 2p/n. (4.7)
3.1.2. Computer simulation of X-ray scattering on molecules and fragments of crystal structures
In this work, we calculated the characteristics of X-ray radiation scattered by a finite set of atoms under Fraunhofer diffraction conditions. Primary X-ray radiation was represented by a plane monochromatic wave with a certain wave vector k0 and wavelength l.
The angular distribution of the intensity of X-rays scattered on a finite set of atoms is represented by a function I(q, Ф), depending on two angles - polar q and azimuthal Ф. Angles q and Ф determine the direction of the scattered X-ray detector, which coincides under Fraunhofer diffraction conditions with wave vector k of the scattered X-ray wave.
The polar angle q is measured from the direction of the wave vector k0 of the primary X-ray wave. The azimuthal angle Ф is plotted in a plane perpendicular to the vector k0. Azimuth Ф is the angle between the projection of the wave vector k of the scattered wave onto this plane and an arbitrarily chosen azimuthal axis.
The set of values of the function I(q, Ф) for all possible values of the arguments q and Ф is often called the diffraction pattern.
Our problem considers the scattering of X-rays into the “front” hemisphere. Therefore, the polar angle q belongs to the range ,. The azimuthal angle Ф takes values in the interval )