X-ray scattering intensity. Small angle x-ray scattering
Diffraction x-rays- scattering of X-rays, in which secondary deflected beams with the same wavelength arise from the initial beam of rays, which appeared as a result of the interaction of primary X-rays with the electrons of the substance. The direction and intensity of the secondary beams depend on the structure (structure) of the scattering object.
2.2.1 Scattering of X-rays by an electron
X-rays, which are an electromagnetic wave, directed at the object under study, act on any electron weakly bound to the nucleus, and bring it into oscillating motion. When a charged particle vibrates, electromagnetic waves are emitted. Their frequency is equal to the frequency of charge oscillations, and, consequently, the frequency of field oscillations in the beam of "primary" X-rays. This is coherent radiation. It plays a major role in the study of structure, since it is it that is involved in creating the interference pattern. So, under the influence of X-rays, an oscillating electron emits electromagnetic radiation, thus "scattering" the X-rays. This is the diffraction of X-rays. In this case, the electron absorbs part of the energy received from X-rays, and gives off part in the form of a scattered beam. These rays scattered by different electrons interfere with each other, that is, they interact, add up and can not only enhance, but also weaken each other, and also extinguish (the extinction laws play important role in x-ray analysis). It should be remembered that the rays that create the interference pattern and the X-rays are coherent, i.e. X-ray scattering occurs without changing the wavelength.
2.2.2 Scattering of X-rays by atoms
Scattering of X-rays by atoms differs from scattering by a free electron in that there can be Z-electrons on the outer shell of an atom, each of which, like a free electron, emits secondary coherent radiation. The radiation scattered by the electrons of atoms is defined as the superposition of these waves, i.e. intra-atomic interference occurs. The amplitude of X-rays scattered by one atom A a, having Z-electrons, is equal to
A a \u003d A e F (5)
where F is the structure factor.
The square of the structural amplitude indicates how many times the intensity of scattered radiation by an atom is greater than the intensity of scattered radiation by one electron:
The atomic amplitude I a is determined by the distribution of electrons in the atom of the substance, by analyzing the magnitude of the atomic amplitude, one can calculate the distribution of electrons in the atom.
2.2.3 Scattering of x-rays by a crystal lattice
It is of the greatest interest for practical work. The theory of X-ray interference was first substantiated by Laue. It made it possible to theoretically calculate the positions of interference maxima on X-ray patterns.
However, the wide practical application of the interference effect became possible only after the English physicists (Braggie's father and son) and, simultaneously with them, the Russian crystallographer G.V. Wulff created an extremely simple theory, discovering a simpler relationship between the location of the interference maxima on the X-ray pattern and the structure of the spatial lattice. At the same time, they considered the crystal not as a system of atoms, but as a system of atomic planes, assuming that X-rays experience mirror reflection from atomic planes.
Figure 11 shows the incident beam S 0 and the deflected plane (HKL) beam S HKL .
In accordance with the law of reflection, this plane must be perpendicular to the plane in which the rays S0 and SHKL lie, and divide the angle between them in half, i.e. the angle between the continuation of the incident beam and the deflected beam is 2q.
The spatial lattice is built from a number of planes P 1 , P 2 , P 3 ...
Consider the interaction of such a system of parallel; planes with a primary beam on the example of two adjacent planes P and P 1 (Fig. 12):
Rice. 12. To the derivation of the Wolf-Bragg formula
Parallel beams SO and S 1 O 1 fall at the points O and O 1 at an angle q to the planes P and P 1. Moreover, the wave hits the point O 1 with a delay equal to the difference in the path of the waves, which is equal to AO 1 \u003d d sinq, These rays will be mirrored from the planes P and P 1 at the same angle q, The path difference of the reflected waves is O 1 B \u003d d sinq . Cumulative path difference Dl=2d sinq. Rays reflected from both planes, propagating in the form of a plane wave, must interfere with each other.
The phase difference of both oscillations is equal to:
(7)
From equation (7) it follows that when the difference in the path of the rays is a multiple of an integer number of waves, Dl=nl=2d sinq, the phase difference will be a multiple of 2p, i.e. the oscillations will be in one phase, the "hump" of one wave coincides with the "hump" of the other, and the oscillations amplify each other. In this case, an interference peak will be observed on the radiograph. So, we get that the equality 2d sinq = nl (8) (where n is an integer, called the order of reflection and determined by the difference in the path of rays reflected by neighboring planes)
is a condition for obtaining an interference maximum. Equation (8) is called the Wulf-Bragg formula. This formula is the basis of X-ray diffraction analysis. It should be remembered that the introduced term "reflection from the atomic plane" is arbitrary.
It follows from the Wulf-Bragg formula that if an X-ray beam with a wavelength l falls on a family of plane-parallel planes, the distance between which is equal to d, then there will be no reflection (interference maximum) until the angle between the direction of the rays and the surface corresponds to this equation.
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 size, 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 by an individual electron occurs in a solid angle of 4p and has a significant anisotropy. If the primary X-ray radiation is not polarized, then the flux density of the scattered wave radiation is described by the following function
(3.18)where I0 is the primary radiation flux density, R is the distance from the scattering point to the place where the scattered radiation is detected, 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 radius of the electron.
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 motion of electrons inside an atom complicates the anisotropy of the scattered radiation. The amplitude of an X-ray wave scattered by an atom is expressed as a function of the wavelength and the polar angle f(q, l), which is called the atomic amplitude.
Thus, the angular distribution of the intensity of an X-ray wave scattered by an atom is expressed by the formula
(3. 20)and has axial symmetry with respect to the direction of the wave vector of the primary wave k0. The square of the atomic amplitude f 2 is called the atomic factor.
As a rule, in experimental setups for X-ray diffraction and X-ray spectral studies, the detector of scattered X-rays is located at a distance R that is much larger than the dimensions of the scattering sample. In such cases, the entrance 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 scheme of X-ray scattering by atoms of sample 1 under Fraunhofer diffraction conditions.
2 – X-ray detector, k0 – wave vector of the primary X-ray wave, dashed arrows represent primary X-ray fluxes, dash-dotted arrows – scattered X-ray fluxes. The circles indicate the 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 entrance window of the detector.
Consequently, in this detection geometry, the detector perceives a stream 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 makes it possible to calculate the diffraction pattern (angular distribution of the scattered radiation intensity) with high accuracy. The proof is that the Fraunhofer diffraction approximation underlies X-ray diffraction methods for studying a substance, which make it possible to determine the parameters of the 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.
We introduce a rectangular coordinate system. Its beginning is compatible with the center of one of the atoms. The position of each atom center (scattering center) is given 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 the 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 both inelastic and elastic. Elastic scattering occurs without changing the X-ray wavelength. With inelastic scattering, the radiation wavelength increases, and the secondary waves are incoherent. In what follows, only the elastic scattering of x-rays by atoms is considered.
Let L be the distance from the origin of coordinates to the detector. Let us assume that the Fraunhofer diffraction conditions are satisfied. In particular, this 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
(3.21)If the primary wave is satisfactorily approximated by a plane harmonic, then all secondary (scattered by atoms) waves are coherent. The phase difference of the scattered waves depends on the difference between the paths of these waves.
Let us draw an auxiliary axis Or from the origin of coordinates to the point where the input window of the detector is located. 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.
The 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 the function (3.23) by a 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 wavenumber
k = 2π/l. (3.25)
To calculate the path difference of secondary waves (3.23), we first assume that the irradiated sample is a one-dimensional chain of atoms located along the coordinate axis Ox (see Fig. 3.9). The atomic coordinates are given 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 Or axis) is given 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 semiaxis Ox. (except for the atom located in the center of coordinates).
Since the Fraunhofer diffraction conditions are satisfied, the wave vectors of all waves scattered by the 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 the distance to the detector L – xisin(q). Therefore, the oscillation of the sensitive element of the detector, caused by the secondary wave emitted by the atom with coordinate xi, is described by the function
A1 f(q) cos(wt – k(L– xj sin(q)) + j0) (3.26)
The rest of the scattered waves that enter the window of the detector, which is in a given position, have a similar form.
The value of the initial phase j0 is determined, in essence, by the moment of the beginning of the time reference. Nothing prevents us from choosing j0 equal to –kL. Then the movement of the sensitive element of the detector will be represented by the sum
(3.27)This means that the path difference 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 wavelength l = Å, the frequency w is ~1019 s-1 in order of magnitude. Modern equipment cannot measure the instantaneous values of the electric and magnetic fields (1) with such rapid field changes, so all X-ray detectors record the average value of the square of the amplitude of electromagnetic oscillations.
At working at high voltage, as with normal voltage X-rays, it is necessary to use all known ways combating scattered X-rays.
Quantity scattered x-rays decreases with a decrease in the irradiation field, which is achieved by limiting the diameter of the working x-ray 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. Interchangeable diaphragms or tubes are still insufficiently used to limit the working beam of X-rays in the diameter.
To reduce the amount scattered x-rays compression should be applied where possible. With compression, the thickness of the object under study decreases and, of course, there are fewer centers of 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 often used.
Amount of scattered radiation decreases with increasing distance between the X-ray tube and the film. With an increase in this distance and an appropriate diaphragming, a less divergent working beam of x-rays is obtained. When increasing the distance between the X-ray tube and the film, it is necessary to reduce the irradiation field to the minimum possible size. In this case, the area under study should not be “cut off”.
To this end, in recent 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 being filmed to improve the quality of the X-ray image, but also to exclude excessive exposure to those parts of the human body that are not subject to radiography.
To reduce the amount scattered x-rays the part of the object under study should be as close as possible to the x-ray film. This does not apply to X-ray exposures with direct magnification of the X-ray image. In direct magnification radiography, the scattered study hardly reaches the x-ray film.
Sandbags used for commits of 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.
When radiography produced on a table without the use of a sifting grid, under the cassette or envelope with the film, a sheet of lead rubber should be placed as large as possible.
For absorption scattered x-rays screening x-ray gratings are used, which absorb these rays as they leave the human body.
Mastering technology production of x-rays at elevated voltages on the X-ray tube is exactly the path that brings us closer to the ideal X-ray image, i.e., such an image in which both bone and soft tissue are clearly visible in detail.
MOU secondary school №21
Physics abstract
"SCATTERING OF X-RAY RAYS
ON FULLERENE MOLECULES»
I've done the work
student 11 "G" class
Lykov Vladimir Andreevich
Teacher:
Kharitonova Olga Alexandrovna
3.5. Fraunhofer diffraction of X-rays by crystal atoms38
Work goals
1. Computer modelling X-ray scattering on fullerene molecules and fragments of fullerite crystals.
2. Investigation of the rotational pseudosymmetry of the angular distribution of the scattered X-ray intensity.
2. Theoretical part
2.1. fluctuations
2.1.1. One-dimensional oscillatory motions
Consider the one-dimensional periodic motion of a material point. The periodicity of the movement means that the coordinate of the 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 essential that the coordinate can be not only Cartesian, but also an angle, etc.
There are many types of periodic motion. For example, such is uniform motion material point around the circle.
liquid surface).
Fig.1.3. A ball suspended from a thread.
Fig.1.4. float on the surface of a liquid.
Fig.1.5. U-shaped tube with liquid.
Fig.1.6. An electrical circuit containing a capacitor of capacitance C and a coil of inductance L.
In example 1.3. the deflection angle changes periodically. Finally, in example 1.6. the charge of the capacitor and the current in the coil change periodically. However, all these physical processes are described by the same mathematical functions.
2.1.2. harmonic vibrations
The simplest type of oscillations are harmonic. The coordinate of a material point over time with harmonic oscillations changes 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 related to the period by the relation
w = 2p / T. (1.4)
The position of equilibrium 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 a linear function of time. The constant j0 is called the initial phase.
Oscillations of physical systems shown in Fig.1.1. – 1.6. would make strictly harmonic oscillations under 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 absence of 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 is displaced along one straight line.
Calculating the derivative of function (1.3) with respect to time, we obtain the velocity of the material point:
v(t) = -wAsin(wt+j0) (1.5)
It can be seen that the speed is also a periodic function of time.
Now we take the derivative of the function (1.5) with respect to time and obtain the acceleration of the material point.
a(t) = -w2 Acos(wt+j0) (1.6)
Comparing the functions (1.3) and (1.6) we get 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 coefficient of proportionality is negative.
Fig.1.7. Time dependences of the coordinates (circles), velocity (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. Therefore, 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.
An important example of a restoring force is the Hooke force (elastic force). Thus, if on material point the Hooke force acts, then the point makes harmonic oscillations.
Since we are considering one-dimensional oscillations, to analyze the problem, it suffices to project the Hooke force vector onto an axis parallel to this force. If the x-coordinate reference zero is chosen at the point where the restoring force is zero, then the projection of the force is
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 of harmonic oscillations of the systems shown in Fig. 1.1. and 1.2.
As an instructive example, let's consider one-dimensional movements performed by weights attached to springs (see Fig. 1.8).
Fig.1.8. Spring weights.
Let the masses of the springs be negligible compared to the masses of the weights.
Loads are considered as material points.
First, consider the system depicted in Fig.18. A. Suppose that the load was initially shifted to the left and, as a result, the spring was stretched. At the same time, 3 forces act on the load (material point): gravity mg, elastic force F and the force of the normal reaction of the support N. We neglect friction in this problem (see Fig. 1.9).
Fig.1.9. Forces on a load lying on a smooth support, when the spring is stretched.
Let's write Newton's second law for the body shown in Fig. 1.9.
ma = mg + F + N(1.10)
The elastic force at small deformations of the springs is described by Hooke's law
F = –kd(1.11)
where d is the spring deformation vector, k is the spring stiffness factor.
Note that when the load is moving, 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 with the initial compression of the spring, the vector equation of motion (1.10) will have the same form:
ma = mg – kd + N(1.12)
We choose the origin of coordinates at the point where the load is located with an undeformed spring. We 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 the acceleration on the Y axis is zero. Then the force of gravity is fully 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 load acceleration, 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)
Once again, we note that equation (1.15) is valid both for 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 deformation projection 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 second-order differential, general theory solutions of such equations are studied in the course 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 proved 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 of the equilibrium position by a distance d0, and the initial speed of the load be equal to zero. Then, using the functions (1.3) and (1.5), we write the following equations for the time t=0:
d0 =Acos(j0) (1.18)
0 = -wAsin(j0) (1.19)
The solution of the system (1.18) - (1. 19) are the following values A = d0 and j0= 0.
For other initial conditions, the quantities A and j0 will naturally acquire other values.
Now consider the system shown in Figure 1.8. b. In this case, only two forces act on the load: the force of gravity mg and the force of elasticity 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 shift slightly vertically. Then the vector equation of motion will look like, similar to equation (1.12)
ma = mg - kd(1. 20)
and regardless of the direction of the 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 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 the acceleration of the body, d is the projection of the deformation of the spring.
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 are the deformations of the spring at load equilibrium. Therefore, the vector d0 is equal to
It can be seen that in the equilibrium position of the body, the spring is indeed stretched, since the vector d0 is directed parallel to the vector g, i.e. down.
Now we 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 deformation vector of the spring d0.
Substituting into equation (1.24) the value d0 obtained from relation (1.23), 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 depicted in Fig. 1.8. b, also performs a harmonic oscillatory motion, described by the function (1.3), as well as the load in the system shown in Fig. 1.8. A. Oscillation frequency The only difference is in the direction of oscillation (vertical instead of horizontal). But the oscillation frequency is still determined by the stiffness of the spring 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
Consider the example of sound waves, when two sources create waves with the same amplitudes A and frequencies ω. Install a sensitive membrane at a distance from the sources. When the wave "passes" the distance from the source to the membrane, the membrane will begin to oscillate. The impact of each of the waves on the membrane can be described by the following relations, using the oscillatory functions:
x1(t) = Acos(ωt + φ1),
x2(t) = Acos(ωt + φ2).
x(t) = x1 (t) + x2 (t) = A (1.27)
The expression in brackets can be written differently using the trigonometric function of the sum of cosines:
To simplify the function (1.28), we introduce new quantities A0 and φ0 satisfying the condition:
A0 = φ0 = (1.29)
We substitute 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 let's consider the same situation, changing the oscillation amplitudes in the 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 the functions (1.26) can 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 function of the cosine of the sum:
x(t) = (A1cos(φ1) + A2cos(φ2)) cos(ωt) – (A1sin(φ1) + A2sin(φ2)) sin(ωt) (1.33)
In order to simplify the function (1.33), we introduce new quantities A0 and φ0 satisfying the condition:
Let us square each equation of the system (1.34) and add the resulting equations. Then we get the following relation for the number A0:
Consider expression (1.35). Let us prove that the value under the root cannot be negative. Since cos(φ1 - φ2) ≥ -1, this means that this is the only value that can affect the sign of the number under the root (A12 > 0, A22 > 0 and 2A1A2 > 0 (from the amplitude definition)). 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 positive. If we begin to gradually increase the cosine, then the term containing the cosine will also begin to grow, then the value under the root will not change its sign.
Now we calculate the ratio for the value φ0 by dividing the second equation of the system (1.34) by the first and calculating the arc tangent:
And now let's substitute into the function (1.33) the values from the system (1.34)
x = A0(cos(φ0) cosωt – sin(φ0) sinωt) (1.37)
Transforming the expression in brackets according to 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 frequency ω is also 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 the material environment
Consider fluctuations in the material environment. One example is the oscillation of a float on the surface of the water. If a bird flying over a float acts as an observer, then it will notice that the float forms circles around itself, which, surprisingly, with the passage of time, moving away, increases the radius. But if the role of the 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 the air resistance, the viscosity of water and air, i.e. dissipative forces. Then the mechanical energy of water droplets can be considered conserved. In this case, the wave motion can be schematically depicted as shown in Figure 1, replacing water droplets with numbered balls. Let's denote the float ball number 1.
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, it involves the ball #2 in motion, the ball #2 involves the #3 ball, and so on. But the interaction between the particles does not occur instantly, so the ball number 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 #2 will lag behind #1 by 1/12 of the period.
Hence, the period of the wave (T) can be called the oscillation period of the ball No. 1, the amplitude of the wave (A) is the maximum deviation of the ball from the horizontal axis, and the wavelength (λ) is the minimum distance between the maxima of the nearest humps or the minima of the nearest depressions.
In the previously considered example, the wave propagated perpendicular to the oscillations of the source, in other words, a transverse wave was considered.
Longitudinal waves are waves that propagate parallel to the movement of the source. If we consider longitudinal waves schematically (Fig. 2.2), then we can see that over time, the source of oscillations (ball No. 1) oscillates left-right and involves other particles in the same oscillatory movement. Then, for a longitudinal wave, the definition of the 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 the balls moving with the same phases; wave amplitude is the maximum deviation from the equilibrium position.
2.2.2. wave function
Consider a source that performs harmonic oscillations in a material medium with a frequency w. Then its motion is described by a function of the form . Let the initial phase j0 be equal to zero. Then the source coordinate is the next function of 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 the vibrations of neighboring particles will occur with a shift in time. Due to the finite and constant rate of transmission of interaction, this shift of oscillations in time is directly proportional to the distance of the next particle from the source.
It follows from the previous examples that, as a result, disturbances called wave disturbances will propagate in the medium. In the case of surface waves, this perturbation is the deflection of water particles from the surface in a quiescent state. In the case of sound waves, the perturbation is the deviation of the air density from the mean air density at rest. Regardless of the type of waves (longitudinal or transverse), this perturbation must be described by some function of time and coordinates.
At the source point, the perturbation 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 given by the coordinate axis 0Z. According to the foregoing, the particles of the material medium, located at a distance z from the source, perform harmonic oscillations with a delay in time (due to the finite speed of propagation of the interaction). Consequently, the perturbation at the point z and at an arbitrary time t coincides with the perturbation at the point z = 0 of the source at some previous time t¢
y(z, t) = y(0, t¢) (2.3)
The propagation velocity of a disturbance in a given medium is clearly expressed by the velocity of the hump (or depression) of surface waves or the velocity of compaction (or rarefaction) of a sound wave. This speed vf is called the phase velocity of the wave. Thus, a hump, depression or any other type of disturbance of the medium runs the distance z in the time z/vf.
The phase velocity makes it possible to relate the moments of time t¢ and t by the following relationship
Using relations (2.2) - (2.4), we obtain an expression for the perturbation function in the following form:
The resulting expression is called the harmonic wave function or, in short, the harmonic wave.
In the cases of homogeneous media and small perturbations, the phase velocity is a constant value.
We introduce a new quantity, called the wavenumber, by 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) = Acos(ωt – kz) (2.7)
Consider the quantity A. This quantity is the amplitude of the wave. As already mentioned, the amplitude of the wave is the maximum deviation of the particle from the equilibrium position. The wave amplitude can change over time (due to external forces).
The phase of the wave will be called 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 of time and position.
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) takes the form:
Function (2.9) can be interpreted as an instant 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 value 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)
By 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.
Consider the set of points in space at which the phase of the wave remains equal to zero.
wt – kz = 0(2.11)
Algebraic transformation gives:
The ratio z/t, standing on the left, was defined above 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 seen that for a harmonic traveling wave at a fixed moment of time, the rate of phase increase per unit length is the value k (wave number) equal to
k = w / vF(2.14)
The example of harmonic waves was considered above. But in nature, such waves are very rare. Damped waves are more common, i.e. waves whose velocity (due to air resistance, friction, or other dissipative forces) vanishes 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, while in metal both longitudinal and transverse ones can propagate.
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 field. If we assume that a changing electric field generates a magnetic field, then we can assume, as Maxwell did, that an electromagnetic wave will form because of this. And only later, in 1886, Hertz experimentally proved that Maxwell was right. Hertz in his experiments, reducing the number of turns of the coil and the area of the capacitor plates, as well as pushing them apart, made the transition from a closed oscillatory circuit to an open oscillatory circuit (Hertz vibrator), which is two rods separated by a spark gap. If in a closed oscillatory circuit an 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 supported by 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, the Hertz vibrator 8 was connected to an inductor. When the voltage across the spark gap reached the breakdown value, a spark appeared, and free damped oscillations arose in the vibrator. When the spark disappeared, the circuit opened and the oscillations stopped. After that, the inductor charged the capacitor again, a spark appeared, and oscillations were again observed in the circuit, etc. To register electromagnetic waves, Hertz used another vibrator having the same frequency of natural oscillations as the radiating vibrator, i.e. tuned to resonance with the vibrator. When electromagnetic waves reached the resonator, an electric spark jumped in its gap.
Using the described vibrator, Hertz achieved frequencies of the order of 100 MHz and received 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 of 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 proved 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 system of fields is shown in Figure 2.3.
A qualitative characteristic of electromagnetic oscillations can be given both in the form of an oscillation frequency, expressed in hertz, and in wavelengths. The higher the oscillation frequency, the shorter the wavelength of the propagated wave. The entire spectrum of these waves is conditionally divided into the following 16 ranges:
Wavelength | Name | Frequency |
over 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 | hectometer (medium 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 | decimillimeter (hyper-high) | 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 - 1pm | x-ray radiation | 30Phz - 300Ehz |
<=10 пм | Gamma radiation | >=30 Hz |
One of the most common types of electromagnetic waves are light waves. But in our work we will consider another type of electromagnetic waves - x-rays.
2.2.4. X-rays
One of clear examples electromagnetic waves can be considered x-rays.
In 1895 V.K. Roentgen (1845 - 1923) researched electric current in highly rarefied gases. To 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 main properties of "X-rays" were studied by Roentgen himself as a result of three years of work, for which in 1901 he was awarded Nobel Prize- the first among physicists. The rays discovered by him were subsequently rightly called x-rays.
Fig.2.3. Schemes of x-ray tubes.
a) one of the first Roentgen tubes, b) X-ray tube of the end of the 20th century.
K is a thermal cathode, A is a high-voltage anode, T is the hot cathode incandescence, E are accelerated electron beams (dashed-dotted lines), P are X-ray fluxes (dashed lines), O are windows in the tube body for X-ray output.
According to modern scientific research, X-rays are electromagnetic radiation invisible to the eye with a wavelength belonging to the range with approximate boundaries of 10-2 - 10 nanometers.
X-rays are emitted during deceleration of fast electrons in matter (they form a continuous spectrum) and during transitions of electrons from the outer electron shells of the atom to the inner ones (and give a line spectrum).
The most important properties X-rays are the following properties:
The rays pass through all materials, including those that are opaque to visible light. The intensity of the transmitted rays I decreases exponentially with the thickness x of the substance layer
I(x) = I0exp(–m/x),(2.16)
where I0 is the intensity of the rays incident on the layer of the 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, in the first approximation, there is a dependence
X-ray beams pass through thick boards, metal sheets, the human body, etc. Significant penetrating power of X-rays is currently widely used in flaw detection and medicine.
X-rays cause luminescence of some chemical compounds. For example, a screen coated with salt BaPt(CN) 4 glows yellow-green when hit by X-rays.
X-rays, falling on photographic emulsions, cause them to blacken.
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. Prolonged irradiation of living organisms with intense X-ray fluxes leads to the occurrence of specific diseases (the so-called "radiation sickness") and even to death.
As mentioned earlier, X-rays are emitted during the deceleration of fast electrons in matter and during the transition of electrons from the outer electron shells of the atom to the inner ones (and give a line spectrum). X-ray detectors are based on the properties of X-rays. Therefore, the following are most often used as detectors: photographic emulsions on film and plates, luminescent screens, gas-filled and semiconductor detectors.
2.3. Wave diffraction
2.3.1. Wave Diffraction and Interference
Typical wave effects are the phenomena of interference and diffraction.
Initially, diffraction was called the deviation of the propagation of light from a 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. The diffraction of light was the light bending around the contours of opaque objects and, as a result, 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. Sources of coherent waves are coherent oscillations of wave sources. Sinusoidal 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 up". But if a recording device is located at any point in space, then its sensitive element will be brought into oscillatory motion under the action 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, not
waves, but oscillations caused by coherent waves.
Rice. 3.1. System of two sources and a detector. L is the distance from the first source to the detector, L' is the distance from the second source to the detector, d is the 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 the source oscillations coincide. The sources are at a certain distance d from each other. The detector that registers 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.
Consider the functions of waves that are the result of oscillations emitted by two point coherent sources. To do this, let's start the z-axis as shown in Figure 3.1. Then wave functions will look like this:
Let us introduce the concept of the 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 value t. To find t consider right triangle, containing the values t and d. Then you can easily find t using the sine function:
This value will be called the difference in the path of the waves. And now we 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 of function (3.1), they will take the form:
In order to simplify the law according to which the detector will oscillate, we set the value (–kL + j1) in the function x1(t) to zero. Let us write the value L' in the function x2(t) according to the function (3.4). By simple transformations, we get that
It can be seen that relations (3.3) and (3.6) are the same. Previously, this value was defined as the phase difference. Proceeding from the above, Relation (3.6) can be rewritten as follows:
Now we add the 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)
Consider the graph of the intensity of the total fluctuation for different parameters. The angle θ varies in the interval (this can be seen from Figure 3.1), the wavelength varies from 1 to 5.
Consider special case when L>>d. Usually such a case occurs in X-ray scattering experiments. In these experiments, the scattered radiation detector is usually located at a distance much greater than the dimensions of the sample under study. In these cases, secondary waves enter the detector, which can be approximately assumed to be plane waves 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 assumed that the Fraunhofer diffraction conditions are satisfied in this case.
2.3.2. X-ray diffraction
X-ray diffraction is a process that occurs during elastic scattering of X-rays and consists in the appearance of deflected (diffracted) rays propagating at certain angles to the primary beam. The diffraction of X-rays is due to the spatial coherence of the secondary waves that arise when the primary radiation is scattered by the electrons that make up the atoms. In some directions, determined by the ratio between the wavelength of the radiation and the interatomic distances in the substance, the secondary waves are added, being in the same phase, resulting in an intense diffraction beam. In other words, under the influence electromagnetic field charged particles present in each atom become sources of secondary (scattered) spherical waves. Separate secondary waves interfere with each other, forming both amplified and attenuated beams of radiation propagating in different directions.
We can assume that scattering is not accompanied by dispersion, and, consequently, the frequency of the scattered waves coincides with the frequency of the primary wave. If the scattering is elastic, then the modulus of the wave vector does not change either.
Let us consider the result of the interference of secondary waves at a point remote from all scattering centers at a distance much greater than the interatomic distances in the investigated (irradiated) sample. Let the detector be located at this point and the oscillations caused by the scattered waves that have arrived at this point are added. Since the distance from the scatterer to the detector significantly exceeds the wavelength of the scattered radiation, the sections of the secondary waves arriving at the detector can be considered flat with a sufficient degree of accuracy, and their wave vectors are parallel. Thus, the physical picture 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 the 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, which is a decreasing function of the scattering angle q, which depends on the electron density of the atoms.
Let us consider the intensity distribution of radiation produced 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 denote the positions of point sources.
The X axis is directed along the line of detector movement. Where Z1 ,Z2, Z3, Z4 ,…, Zn, are the distances from the first, second, third, …, nth sources to the receiver, along the axis X there is an addition of the intensities of oscillations, L- distance from the axis X to the line connecting the sources.
In order to find the intensity of n sources, we use relation (3.10). We add the amplitudes in a vector way. Then for n sources the function (3.10) takes 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 (respectively, the X-ray, electron or neutron atomic factor is distinguished). 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 an individual atom. The electric field of the wave generates periodic forces that act on all charged particles that make up the atom - on electrons and on the nucleus. The acceleration 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 the scattered radiation is generated practically only by electrons, so the X-ray atomic factor depends on the electron density distribution in the atom.
The electrons are dispersed inside the atom, and the size of the atom is commensurate with the length of the x-ray wave. Therefore, the secondary waves created by the individual electrons of the 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 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 wave amplitude scattered by one atom to the wave amplitude scattered by one free electron. The value of the atomic factor depends on the scattering angle q and the radiation wavelength l. The quantity g = sin(q) / l is used as an argument of 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) decreases monotonically 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 an X-ray beam with a certain wavelength l be directed onto a crystalline sample. In physical research (when deciphering the atomic structure by the X-ray diffraction method, X-ray spectral elemental analysis, etc.), a geometric scheme of the experiment is usually implemented with the following geometric features (see Fig. 1).
Fig.3.5. Geometric scheme of irradiating a small sample with a narrow X-ray beam.
1 – X-ray generator (for example, X-ray tube), 2 – collimator, 3 – test sample. Dashed arrows depict X-ray fluxes.
A narrow beam of X-rays is formed using a collimator. The irradiated crystalline sample is located from the collimator exit at a distance much greater than the sample size. In x-ray diffraction studies, samples are prepared with a size smaller than the beam cross section. The sample is said to be "bathed" in the beam of incident X-rays (see callout in Figure 3.5).
Then it can be assumed with good accuracy that a plane electromagnetic wave with length l is incident on the sample under study. In other words, all sample atoms 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 size, 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 by an individual electron occurs in a solid angle of 4p and has a significant anisotropy. If the primary X-ray radiation is not polarized, then the flux density of the scattered wave radiation 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 where the scattered radiation is detected, 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 radius of the electron.
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 motion of electrons inside an atom complicates the anisotropy of the scattered radiation. The amplitude of an X-ray wave scattered by an atom is expressed as a function of the wavelength and the polar angle f(q, l), which is called the atomic amplitude.
Thus, the angular distribution of the intensity of an X-ray wave scattered by an atom is expressed by the formula
and has axial symmetry with respect to the direction of the wave vector of the primary wave k0. The square of the atomic amplitude f 2 is called the atomic factor.
As a rule, in experimental setups for X-ray diffraction and X-ray spectral studies, the detector of scattered X-rays is located at a distance R that is much larger than the dimensions of the scattering sample. In such cases, the entrance 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 scheme of X-ray scattering by atoms of sample 1 under Fraunhofer diffraction conditions.
2 – X-ray detector, k0 – wave vector of the primary X-ray wave, dashed arrows represent primary X-ray fluxes, dash-dotted arrows – scattered X-ray fluxes. The circles indicate the 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 entrance window of the detector.
Consequently, in this detection geometry, the detector perceives a stream 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 makes it possible to calculate the diffraction pattern (angular distribution of the scattered radiation intensity) with high accuracy. The proof is that the Fraunhofer diffraction approximation underlies X-ray diffraction methods for studying a substance, which make it possible to determine the parameters of the 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.
We introduce a rectangular coordinate system. Its beginning is compatible with the center of one of the atoms. The position of each atom center (scattering center) is given 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 the 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 both inelastic and elastic. Elastic scattering occurs without changing the X-ray wavelength. With inelastic scattering, the radiation wavelength increases, and the secondary waves are incoherent. In what follows, only the elastic scattering of x-rays by atoms is considered.
Let L be the distance from the origin of coordinates to the detector. Let us assume that the Fraunhofer diffraction conditions are satisfied. In particular, this 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, then all secondary (scattered by atoms) waves are coherent. The phase difference of the scattered waves depends on the difference between the paths of these waves.
Let us draw an auxiliary axis Or from the origin of coordinates to the point where the input window of the detector is located. 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.
The 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 the function (3.23) by a 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 wavenumber
k = 2π/l. (3.25)
To calculate the path difference of secondary waves (3.23), we first assume that the irradiated sample is a one-dimensional chain of atoms located along the coordinate axis Ox (see Fig. 3.9). The atomic coordinates are given 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 Or axis) is given 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 semiaxis Ox. (except for the atom located in the center of coordinates).
Since the Fraunhofer diffraction conditions are satisfied, the wave vectors of all waves scattered by the 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 the distance to the detector L – xisin(q). Therefore, the oscillation of the sensitive element of the detector, caused by the secondary wave emitted by the atom with coordinate xi, is described by the function
A1 f(q) cos(wt – k(L– xj sin(q)) + j0) (3.26)
The rest of the scattered waves that enter the window of the detector, which is in a given position, have a similar form.
The value of the initial phase j0 is determined, in essence, by the moment of the beginning of the time reference. Nothing prevents us from choosing j0 equal to –kL. Then the movement of the sensitive element of the detector will be represented by the sum
This means that the path difference 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 wavelength l = Å, the frequency w is ~1019 s-1 in order of magnitude. Modern equipment cannot measure the instantaneous values of the electric and magnetic fields (1) with such rapid field changes, so 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 the 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 method of complex amplitudes. We write each term in the sum (11) in the complex form
A1 fexp (3.28)
where i is the imaginary unit, Djj is the phase shift equal to kxjsin(q) in the considered physical picture.
Expression (12) can be rewritten in the form
A1 feiwte–iDjj (3.29)
The time-dependent factor describes the oscillations of the electromagnetic field with the frequency w. The modulus of this quantity is equal to one. 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 quantities (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 this sum determines the detected intensity of the scattered X-ray radiation
up to the instrumental coefficient (factor determined by the characteristics of the recording equipment).
The 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 consider the scattering of x-rays by a finite set of atoms in the same plane. Let a plane X-ray wave with wave vector k0 perpendicular to the atomic plane be incident on this system of atoms.
Associate with this physical system axes of Cartesian coordinates. Let us direct the oz axis along the vector k0, and place the ox and oY axes in the atomic plane. The position of each atom is given by two coordinates xj and yj, where j = 0, … N – 1. Let the origin of coordinates be aligned with the center of one of the atoms, which has the number j = 0.
Let us consider X-ray scattering 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 the scattered waves arriving at the entrance window of the detector. This direction is characterized by two angles: polar q, which is plotted from the oz axis (as in Figures 3.9 and 3.10), and azimuth Ф, which is measured from the ox axis in the xoY plane (see Figure 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 the scattered waves is related to the path difference by relation (3.24), as in the case considered above.
Let us find the path difference between the waves scattered by atoms with coordinates (x0=0, y0=0) and (x, y) in the direction given by the wave vector k (ie, 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. On the calculation of the path difference of secondary waves scattered on a planar 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 OF segment is xcos(Ф) + ysin(Ф), which can be obtained by coordinate transformation or geometric construction. The projection of the segment OF onto the direction of the wave vector k gives the required path difference - the length of the segment OG, equal to
Dl = sin(q). (3.32)
Therefore, the phase shift of secondary waves scattered by atoms with coordinates (x0=0, y0=0) and (xj, yj) in the direction given by certain angles q and Ф equals
Djj = k sin(q). (3.33)
The registered intensity of the scattered X-ray radiation is expressed by a formula similar to (3.31):
Finally, consider the Fraunhofer diffraction of X-rays by a three-dimensional object. Let's use the Cartesian coordinate system used in the previous problem. The physical picture differs from the previous one only in that the centers of some atoms have coordinates zj¹ 0.
The surface of the constant phase 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 the atom with coordinate z¹ 0 will lag behind the phase of the wave scattered by the atom with coordinate z = 0 by wDt, where Dt = z / v, v is the wave propagation velocity. The frequency and wavelength are related by the relationship
w = 2pv / l(3.35)
therefore, the phase shift of the scattered wave is -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 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 the 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 kinds of transformations.
The different types of symmetry are determined by the transformations under which the given system is invariant. 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: the conservation of energy, momentum and angular momentum are associated with certain symmetrical transformations of the space-time continuum.
A particular transformation under which a given system is invariant is called a symmetry operation. The set of points that remain fixed under 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), invariance with respect to inversion in 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 considered transformation.
For a quantitative description of deviations from exact symmetry, a functional called the degree of invariance or the coefficient of pseudosymmetry is used.
Let any physical characteristic of the object under study be described by a function of a point. This function can be mass density, temperature electric potential, density electric charge etc. Us the symmetry of a given object with respect to a transformation that is given 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 considered physical system 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 also be extended to describe the symmetry of the angular distribution of the scattered X-ray intensity. First of all, we are interested in the invariance of diffraction patterns with respect to rotation by a certain azimuthal angle around the point corresponding to the polar angle q = 0. In other words, the aim 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 the rotational symmetry of diffraction patterns, one can adapt the functional general view(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 the 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:
The internal integrals are taken over the range of the azimuthal angle ФО , and the external integrals are taken over the range of the polar angle qО .
Attention should be paid to some important features of all diffraction patterns. In Fig.4.1. It can be seen that the central maximum of the intensity of the scattered radiation is located at the center of the polar diagram. This maximum has a high symmetry, close to the symmetry of the limit group С¥. In the angular distribution of scattered radiation, the central maximum occupies a certain range of polar angles qÎ . The half-width of the central maximum essentially depends 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-rays. On the contrary, as the polar angle increases, the intensity of the scattered radiation sharply decreases on average. This means that the peripheral region of the diffraction pattern (the region of polar angles exceeding a certain value qM) has practically no effect on the value of the coefficient of rotational pseudosymmetry (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 expedient to calculate the 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 the rotational pseudosymmetry in various intervals of the polar angle. In other words, quantities of the type (4.3) give quantitative estimates of the rotational pseudosymmetry of the diffraction pattern inside a ring given 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 mentioned above that in computer simulation of the angular distribution of the intensity of scattered X-rays, the function I(q, Ф) is represented by a two-dimensional set 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 scattered X-ray intensity, the double integrals in expression (4.2) turn into double sums
If we are interested in the average rotational pseudosymmetry of the entire diffraction pattern, then the degree of invariance is given by the following formula:
If we want to investigate the rotational pseudosymmetry in different 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 the type (4.3). Then the pseudosymmetry coefficients are 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 setting certain values of the rotation angle a, one can calculate the pseudosymmetry coefficients ha of diffraction patterns for rotations of different orders. If we are interested in rotational pseudosymmetry of the nth order, then the rotation angle 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 crystalline structures
In the present work, we calculated the characteristics of X-ray radiation scattered by a finite set of atoms under Fraunhofer diffraction conditions. The primary X-ray radiation was represented as a plane monochromatic wave with a certain wave vector k0 and wavelength l.
The angular distribution of the intensity of X-rays scattered by a finite set of atoms is represented by a function I(q, Ф), which depends on two angles - polar q and azimuthal Ф. The angles q and Ф determine the direction to the detector of scattered X-rays, which coincides with the 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 a diffraction pattern.
In our problem, X-ray scattering into the "forward" hemisphere is considered. Therefore, the polar angle q belongs to the range ,. Azimuth angle Ф takes values in the interval )