Scheme of the Davisson–Germer experiment (1927): K – nickel single crystal; A – source of electrons; B – electron receiver; θ – angle of deflection of electron beams.

A beam of electrons falls perpendicular to the polished plane of the crystal S. When the crystal is rotated around the O axis, the galvanometer connected to the receiver B gives periodically occurring maxima

Recording of diffraction maxima in the Davisson–Germer experiment on electron diffraction at different angles of rotation of the crystal φ for two values ​​of the electron deflection angle θ and two accelerating voltages V . The maxima correspond to reflection from various crystallographic planes, the indices of which are indicated in brackets

Double slit experiment in the case of light and electrons

Light or electrons

Intensity distribution on the screen

English physicist

Paul Adrien Maurice Dirac

(8.08.1902-1984)

7.2.3. Heisenberg Uncertainty Principle

Quantum mechanics (wave mechanics) –

a theory that establishes the method of description and laws of motion of microparticles in given external fields.

It is impossible to make a measurement without introducing some kind of disturbance, even a weak one, into the object being measured. The very act of observation introduces significant uncertainty into either the position or momentum of the electron. This is what it's all about uncertainty principle,

first formulated by Heisenberg in

Heisenberg inequalities

Dx Dp x ³ , Dy Dp y ³ , Dz Dp z ³

Dt × D(E′ - E ) ³

7.2.4. Wave functions II

IN In quantum mechanics, the amplitude of, say, an electron wave is calledwave function

And denoted by the Greek letter "psi": Ψ.

Thus, Ψ specifies the amplitude of a new type of field, which could be called a matter field or wave, as a function of time and position.

The physical meaning of the function Ψ is that the square of its modulus gives the probability density (probability per unit volume) of finding a particle in the corresponding place in space.

Diffraction of part c, scattering of microparticles (electrons, neutrons, atoms, etc.) by crystals or molecules of liquids and gases, in which additionally deflected beams of these particles arise from the initial beam of particles of a given type; The direction and intensity of such deflected beams depend on the structure of the scattering object.

Dynamic particles can only be understood on the basis of quantum theory. Diffraction is a wave phenomenon; it is observed during the propagation of waves of various natures: diffraction of light, sound waves, waves on the surface of a liquid, etc. Diffraction during particle scattering, from the point of view of classical physics, is impossible.

directed towards the propagation of the wave, or along the movement of the particle.

Thus, the wave vector of a monochromatic wave associated with a freely moving microparticle is proportional to its momentum or inversely proportional to the wavelength.

Since the kinetic energy of a relatively slowly moving particle E = mv 2/2, the wavelength can also be expressed in terms of energy:

When a particle interacts with some object - with a crystal, molecule, etc. - its energy changes: the potential energy of this interaction is added to it, which leads to a change in the motion of the particle. Accordingly, the nature of the propagation of the wave associated with the particle changes, and this occurs according to the principles common to all wave phenomena. Therefore, the basic geometric laws of dynamic particles are no different from the laws of diffraction of any waves (see. Diffraction waves). The general condition for the diffraction of waves of any nature is the commensurability of the length of the incident wave l with the distance d between scattering centers: l £ d.

Particle diffraction experiments and their quantum mechanical interpretation. The first experiment on quantum mechanics, which brilliantly confirmed the original idea of ​​quantum mechanics—wave-particle duality, was the experience of American physicists K. Davisson and L. Germera (1927) on electron diffraction on nickel single crystals ( rice. 2 ). If you accelerate electrons electric field with tension V, then they will acquire kinetic energy E = eV, (e- electron charge), which after substituting numerical values ​​into equality (4) gives

Here V expressed in V, and l - in A (1 A = 10 -8 cm). At voltages V about 100 V, which were used in these experiments, the so-called “slow” electrons with l of the order of 1 A are obtained. This value is close to the interatomic distances d in crystals that are several A or less, and the ratio l £ d, necessary for diffraction to occur, is fulfilled.

Crystals have a high degree of order. The atoms in them are located in a three-dimensional periodic crystal lattice, that is, they form a spatial diffraction grating for the corresponding wavelengths. Diffraction of waves on such a grating occurs as a result of scattering on systems of parallel crystallographic planes, on which scattering centers are located in a strict order. The condition for observing the diffraction maximum upon reflection from the crystal is Bragg-Wolff condition :

2d sin J = n l , (6)

here J is the angle at which the electron beam falls on a given crystallographic plane (grazing angle), and d- the distance between the corresponding crystallographic planes.

In the experiment of Davisson and Germer, when electrons were “reflected” from the surface of a nickel crystal at certain angles of reflection, maxima appeared ( rice. 3 ). These maxima of reflected electron beams corresponded to formula (6), and their appearance could not be explained in any other way except on the basis of ideas about waves and their diffraction; Thus, the wave properties of particles - electrons - were proven by experiment.

At higher accelerating electrical voltages (tens kv) electrons acquire sufficient kinetic energy to penetrate thin films of matter (thickness about 10 -5 cm, i.e. thousands A). Then the so-called diffraction of fast electrons by transmission occurs, which was first studied on polycrystalline films of aluminum and gold by the English scientist J.J. Thomson and Soviet physicist P. S. Tartakovsky.

Soon after this, it was possible to observe the phenomena of diffraction of atoms and molecules. Atoms with mass M in a gaseous state in a vessel at absolute temperature T, corresponds, according to formula (4), wavelength

The scattering ability of an atom is characterized quantitatively by a quantity called the atomic scattering amplitude f(J), where J is the scattering angle, and is determined by the potential energy of interaction of particles of a given type with atoms of the scattering substance. The particle scattering intensity is proportional to f 2(J).

If the atomic amplitude is known, then, knowing the relative position of the scattering centers - the atoms of the substance in the sample (i.e., knowing the structure of the scattering sample), it is possible to calculate the overall diffraction pattern (which is formed as a result of the interference of secondary waves emanating from the scattering centers).

Theoretical calculation, confirmed by experimental measurements, shows that the atomic amplitude of electron scattering f e is maximum at J = 0 and decreases with increasing J. Magnitude f e also depends on the charge of the nucleus (atomic number) Z and from the building electron shells atom, increasing on average with increasing Z approximately like Z 1/3 for small J and how Z 2/3 at large values ​​of J, but exhibiting oscillations associated with the periodic nature of the filling of electronic shells.

Atomic neutron scattering amplitude f H for thermal neutrons (neutrons with energy in hundredths ev) does not depend on the scattering angle, i.e., the scattering of such neutrons by the nucleus is the same in all directions (spherically symmetrical). This is explained by the fact that an atomic nucleus with a radius of about 10 -13 cm is a “point” for thermal neutrons, the wavelength of which is 10 -8 cm. In addition, there is no obvious dependence on the nuclear charge for neutron scattering Z. Due to the presence of so-called resonance levels in some nuclei with energy close to the energy of thermal neutrons, f H for such nuclei are negative.

An atom scatters electrons much more strongly than X-rays and neutrons: absolute values ​​of the electron scattering amplitude f e sub>- these are values ​​of the order of 10 -8 cm, x-rays - f p ~ 10 -11 cm, neutrons - f H ~ 10 -12 cm. Since the scattering intensity is proportional to the square of the scattering amplitude, electrons interact with matter (scatter) approximately a million times stronger than X-rays (and even more so neutrons). Therefore, samples for observing electron diffraction are usually thin films with a thickness of 10 -6 -10 -5 cm, whereas to observe diffraction of X-rays and neutrons you need to have samples several thick mm.

Diffraction by any system of atoms (molecule, crystal, etc.) can be calculated by knowing the coordinates of their centers r i and atomic amplitudes f i for a given type of particle.

The effects of dynamic particles are most clearly revealed by diffraction from crystals. However, the thermal motion of atoms in the crystal somewhat changes the diffraction conditions, and the intensity of the diffracted beams decreases with increasing angle J in formula (6). When D. ch. liquids, amorphous bodies or gas molecules whose ordering is significantly lower than crystalline, several blurred diffraction maxima are usually observed.

Dynamic particle, which at one time played such a large role in establishing the dual nature of matter—particle-wave dualism (and thereby served as an experimental basis for quantum mechanics), has long become one of the main working methods for studying the structure of matter. Two important modern methods for analyzing the atomic structure of matter are based on dynamic particles - electronography And neutronography .

Lit.: Blokhintsev D.I., Fundamentals of quantum mechanics, 4th ed., M., 1963, ch. 1, § 7, 8; Pinsker Z.G., Electron Diffraction, M. - L., 1949; Vainshtein B.K., Structural electron diffraction, M., 1956; Bacon J., Neutron Diffraction, trans. from English, M., 1957; Ramsey N., Molecular beams, trans. from English, M., 1960.

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Example 4.1.(C4). Soap film is a thin layer of water, on the surface of which there is a layer of soap molecules, which provides mechanical stability and does not affect the optical properties of the film. The soap film is stretched over a square frame, two sides of which are horizontal and the other two are vertical. Under the influence of gravity, the film took the shape of a wedge (see figure), the thickness of which at the bottom turned out to be greater than at the top. When a square is illuminated by a parallel beam of laser light with a wavelength of 666 nm (in air), incident perpendicular to the film, part of the light is reflected from it, forming an interference pattern on its surface consisting of 20 horizontal stripes. How much greater is the thickness of the soap film at the base of the wedge than at the top if the refractive index of water is equal to ?

Solution. The number of stripes on the film is determined by the difference in the path of the light wave in its lower and upper parts: Δ = Nλ"/2, where λ"/2 = λ/2n is the number of half-waves in a substance with refractive index n, N is the number of stripes, and Δ difference in film thickness in the lower and upper parts of the wedge.

From here we get the relationship between the wavelength laser radiation in the air λ and the parameters of the soap film, from which the answer follows: Δ = Nλ/2n.

Example 4.2.(C5). When studying the structure of a crystal lattice, a beam of electrons having the same speed is directed perpendicular to the crystal surface along the Oz axis, as shown in the figure. After interacting with the crystal, the electrons reflected from the upper layer are distributed throughout space so that diffraction maxima are observed in some directions. There is such a first-order maximum in the Ozx plane. What is the angle between the direction of this maximum and the Oz axis if the kinetic energy of the electrons is 50 eV and the period of the crystal structure of the atomic lattice along the Ox axis is 0.215 nm?

Solution. The momentum p of an electron with kinetic energy E and mass m is equal to p = . The de Broglie wavelength is related to the momentum λ = = . The first diffraction maximum for a grating with a period d is observed at an angle α satisfying the condition sin α = .

Answer: sin α = ≈ 0.8, α = 53 o.

Example 4.3.(C5). When studying the structure of a monomolecular layer of a substance, a beam of electrons having the same speed is directed perpendicular to the layer under study. As a result of diffraction on molecules that form a periodic lattice, some electrons are deflected at certain angles, forming diffraction maxima. At what speed do electrons move if the first diffraction maximum corresponds to the deviation of electrons by an angle α=50° from the original direction, and the period of the molecular lattice is 0.215 nm?

Solution. The momentum p of an electron is related to its speed p = mv. The de Broglie wavelength is determined by the electron momentum λ = = . The first diffraction maximum for a grating with a period d is observed at an angle α satisfying the condition sin α = = . v = .

Example 4.4. (C5). A photon with a wavelength corresponding to the red limit of the photoelectric effect knocks an electron out of a metal plate (cathode) in a vessel from which air has been evacuated and a small amount of hydrogen has been introduced. The electron is accelerated by a constant electric field to an energy equal to the ionization energy of the hydrogen atom W = 13.6 eV, and ionizes the atom. The resulting proton is accelerated by the existing electric field and hits the cathode. How many times is the momentum p m transferred to the plate by the proton greater than the maximum momentum p e of the electron that ionized the atom? The initial velocity of the proton is assumed to be zero, and the impact is considered absolutely inelastic.

Solution. The energy E e acquired by an electron in an electric field is equal to the energy E p acquired by a proton and is equal to the ionization energy: E e = E p = W. Expressions for momentum:

proton: p p = m n v n or p p = ;

electron: p e = m e v e or p e = ; from here .

Example 4.5. (C6). To accelerate spacecraft in outer space and correct their orbits, it is proposed to use a solar sail - a lightweight, large-area screen made of a thin film attached to the apparatus, which specularly reflects sunlight. The mass of the spacecraft (including the sail) m = 500 kg. By how many m/s will the speed of a spacecraft in Mars orbit change in 24 hours after deploying the sail, if the sail has dimensions of 100 m x 100 m, and the power W of solar radiation incident on 1 m 2 of surface perpendicular to the sun's rays is approximately Earth 1370 W? Assume that Mars is 1.5 times farther from the Sun than the Earth.

Solution. Formula for calculating the pressure of light during its specular reflection: p = . Pressure force: F = . Dependence of radiation power on distance to the Sun: ( . Applying Newton's second law: F = m A, we get the answer: Δv = .

DEFINITION

Electron diffraction call the process of scattering of these elementary particles on systems of particles of matter. In this case, the electron exhibits wave properties.

In the first half of the 20th century, L. de Broglie presented a hypothesis about the wave-particle duality of various forms of matter. The scientist believed that electrons, along with photons and other particles, have both corpuscular and wave properties. The corpuscular characteristics of a particle include: its energy (E), momentum (), wave parameters include: frequency () and wavelength (). In this case, the wave and corpuscular parameters of small particles are related by the formulas:

where h is Planck's constant.

Each particle of mass, in accordance with de Broglie’s idea, is associated with a wave having a length of:

For the relativistic case:

Electron diffraction by crystals

The first empirical evidence that confirmed de Broglie's hypothesis was an experiment by American scientists K. Davisson and L. Germer. They found that if a beam of electrons is scattered on a nickel crystal, a clear diffraction pattern is obtained, which is similar to the pattern of X-ray scattering on this crystal. The atomic planes of the crystal played the role of a diffraction grating. This became possible because at a potential difference of 100 V, the De Broglie wavelength for an electron is approximately m, this distance is comparable to the distance between the atomic planes of the crystal used.

The diffraction of electrons by crystals is similar to the diffraction of X-rays. The diffraction maximum of the reflected wave appears at values ​​of the Bragg angle () if it satisfies the condition:

where d is the crystal lattice constant (the distance between the reflection planes); - order of reflection. Expression (4) means that the diffraction maximum occurs when the difference in the paths of the waves reflected from neighboring atomic planes is equal to an integer number of De Broglie wavelengths.

G. Thomson observed the pattern of electron diffraction on thin gold foil. On the photographic plate, which was located behind the foil, concentric light and dark rings were obtained. The radius of the rings depended on the speed of electron movement, which, according to De Broglie, is related to the wavelength. To establish the nature of the diffracted particles in this experiment, a magnetic field was created in the space between the foil and the photographic plate. The magnetic field must distort the diffraction pattern if the diffraction pattern is created by electrons. And so it happened.

Diffraction of a beam of monoenergetic electrons on a narrow slit, with normal incidence of the beam, can be characterized by the expression (condition for the occurrence of main intensity minima):

where is the angle between the normal to the grating and the direction of propagation of diffracted rays; a is the width of the slot; k is the order of minimum diffraction; is the de Broglie wavelength for the electron.

In the middle of the 20th century, an experiment was carried out in the USSR on diffraction on a thin film of single electrons that flew in turns.

Since diffraction effects for electrons are observed only if the wavelength associated with an elementary particle is of the same order as the distance between atoms in a substance, the electronography method, based on the phenomenon of electron diffraction, is used to study the structure of a substance. Electron diffraction is used to study the structures of body surfaces, since the penetrating ability of electrons is low.

Using the phenomenon of electron diffraction, the distances between atoms in a molecule of gases that are adsorbed on the surface of a solid are found.

Examples of problem solving

EXAMPLE 1

Exercise	A beam of electrons having the same energies falls on a crystal having a period of nm. What is the electron velocity (v) if the first order Bragg reflection appears if the grazing angle is ?
Solution	As a basis for solving the problem, we will take the condition for the occurrence of a maximum of diffraction of the reflected wave: where by condition. According to de Broglie's hypothesis, the electron wavelength is (for the relativistic case): Let's substitute the right side of expression (1.2) into the formula: From (1.3) we express the required speed: where kg is the mass of the electron; Js is Planck's constant. Let's calculate the electron speed:
Answer

EXAMPLE 2