Quantum systems and their properties.
Probability distribution over energies in space.
Boson statistics. Fermi-Einstein distribution.
fermion statistics. Fermi-Dirac distribution.
Quantum systems and their properties
In classical statistics, it is assumed that the particles that make up the system obey the laws classical mechanics. But for many phenomena, when describing micro-objects, it is necessary to use quantum mechanics. If the system consists of particles obeying quantum mechanics, we will call it a quantum system.
The fundamental differences between a classical system and a quantum one include:
1) Corpuscular-wave dualism of microparticles.
2) Discreteness physical quantities describing micro-objects.
3) Spin properties of microparticles.
The first implies the impossibility of accurately determining all the parameters of the system that determine its state from the classical point of view. This fact is reflected in the Heisandberg uncertainty relation:
In order to mathematically describe these features of microobjects in quantum physics, the quantity is assigned a linear Hermitian operator that acts on the wave function .
The eigenvalues of the operator determine the possible numerical values of this physical quantity, the average over which coincides with the value of the quantity itself.
Since the momenta and coefficients of the microparticles of the system cannot be measured simultaneously, the wave function is presented either as a function of coordinates:
Or, as a function of impulses:
The square of the modulus of the wave function determines the probability of detecting a microparticle per unit volume:
The wave function describing a particular system is found as an eigenfunction of the Hamelton operator:
Stationary Schrödinger equation.
Non-stationary Schrödinger equation.
The principle of indistinguishability of microparticles operates in the microworld.
If the wave function satisfies the Schrödinger equation, then the function also satisfies this equation. The state of the system will not change when 2 particles are swapped.
Let the first particle be in state a and the second particle be in state b.
The system state is described by:
If the particles are interchanged, then: since the movement of the particle should not affect the behavior of the system.
This equation has 2 solutions:
It turned out that the first function is realized for particles with integer spin, and the second for half-integer.
In the first case, 2 particles can be in the same state:
In the second case:
Particles of the first type are called spin integer bosons, particles of the second type are called femions (the Pauli principle is valid for them.)
Fermions: electrons, protons, neutrons...
Bosons: photons, deuterons...
Fermions and bosons obey non-classical statistics. To see the differences, let's count the number of possible states of a system consisting of two particles with the same energy over two cells in the phase space.
1) Classical particles are different. It is possible to trace each particle separately.
classical particles.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS The principle of size quantization The whole complex of phenomena usually understood by the words "electronic properties of low-dimensional electronic systems" is based on a fundamental physical fact: a change in the energy spectrum of electrons and holes in structures with very small sizes. Let us demonstrate the basic idea of size quantization using the example of electrons in a very thin metal or semiconductor film of thickness a.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Quantization principle The electrons in a film are in a potential well with a depth equal to the work function. The depth of the potential well can be considered infinitely large, since the work function exceeds the thermal energy of the carriers by several orders of magnitude. Typical work function values in most solids have a value of W = 4 -5 e. B, several orders of magnitude higher than the characteristic thermal energy of the carriers, which is of the order of magnitude k. T, equal at room temperature to 0.026 e. C. According to the laws of quantum mechanics, the energy of electrons in such a well is quantized, i.e., it can take only some discrete values En, where n can take integer values 1, 2, 3, …. These discrete energy values are called size quantization levels.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization For a free particle with an effective mass m*, whose motion in the crystal in the direction of the z axis is limited by impenetrable barriers (i.e., barriers with infinite potential energy), the energy of the ground state increases compared to the state without limitation This increase in energy is called the size quantization energy of the particle. Quantization energy is a consequence of the uncertainty principle in quantum mechanics. If the particle is limited in space along the z-axis within the distance a, the uncertainty of the z-component of its momentum increases by an amount of the order of ħ/a. Correspondingly, the kinetic energy of the particle increases by the value E 1. Therefore, the considered effect is often called the quantum size effect.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization The conclusion about the quantization of the energy of electronic motion refers only to motion across the potential well (along the z axis). The well potential does not affect the motion in the xy plane (parallel to the film boundaries). In this plane, the carriers move as free and are characterized, as in a bulk sample, by a continuous energy spectrum quadratic in momentum with an effective mass. The total energy of carriers in a quantum-well film has a mixed discretely continuous spectrum
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization In addition to increasing the minimum energy of a particle, the quantum-size effect also leads to quantization of the energies of its excited states. Energy spectrum of a quantum-dimensional film - the momentum of charge carriers in the plane of the film
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization Let the electrons in the system have energies less than E 2 and therefore belong to the lower level of size quantization. Then no elastic process (for example, scattering by impurities or acoustic phonons), as well as scattering of electrons by each other, can change quantum number n , transferring the electron to a higher level, since this would require additional energy costs. This means that during elastic scattering electrons can only change their momentum in the plane of the film, i.e., they behave like purely two-dimensional particles. Therefore, quantum-dimensional structures in which only one quantum level is filled are often called two-dimensional electronic structures.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Principle of size quantization There are other possible quantum structures where the movement of carriers is limited not in one, but in two directions, as in a microscopic wire or filament (quantum filaments or wires). In this case, the carriers can move freely only in one direction, along the thread (let's call it the x-axis). In the cross section (the yz plane), the energy is quantized and takes on discrete values Emn (like any two-dimensional motion, it is described by two quantum numbers, m and n). The full spectrum is also discrete-continuous, but with only one continuous degree of freedom:
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Quantization principle It is also possible to create quantum structures resembling artificial atoms, where the movement of carriers is limited in all three directions (quantum dots). In quantum dots, the energy spectrum no longer contains a continuous component, i.e., it does not consist of subbands, but is purely discrete. As in the atom, it is described by three discrete quantum numbers (not counting the spin) and can be written as E = Elmn , and, as in the atom, the energy levels can be degenerate and depend on only one or two numbers. A common feature of low-dimensional structures is the fact that if the motion of carriers along at least one direction is limited to a very small region comparable in size to the de Broglie wavelength of the carriers, their energy spectrum changes noticeably and becomes partially or completely discrete.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Definitions Quantum dots - quantum dots - structures whose dimensions in all three directions are several interatomic distances (zero-dimensional structures). Quantum wires (threads) - quantum wires - structures, in which the dimensions in two directions are equal to several interatomic distances, and in the third - to a macroscopic value (one-dimensional structures). Quantum wells - quantum wells - structures whose size in one direction is several interatomic distances (two-dimensional structures).
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Minimum and maximum sizes The lower limit of size quantization is determined by the critical size Dmin, at which at least one electronic level exists in a quantum-size structure. Dmin depends on the conduction band break DEc in the corresponding heterojunction used to obtain quantum size structures. In a quantum well, at least one electronic level exists if DEc exceeds the value h - Planck's constant, me* - the effective mass of an electron, DE 1 QW - the first level in a rectangular quantum well with infinite walls.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Minimum and maximum dimensions If the distance between energy levels becomes comparable to thermal energy k. BT , then the population of high levels increases. For a quantum dot, the condition under which the population of higher levels can be neglected is written as E 1 QD, E 2 QD are the energies of the first and second size quantization levels, respectively. This means that the benefits of size quantization can be fully realized if This condition sets upper limits for size quantization. For Ga. As-Alx. Ga 1-x. As this value is 12 nm.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Along with its energy spectrum, an important characteristic of any electronic system is the density of states g(E) (the number of states per unit energy interval E). For three-dimensional crystals, the density of states is determined using the Born-Karman cyclic boundary conditions, from which it follows that the components of the electron wave vector do not change continuously, but take a number of discrete values, here ni = 0, ± 1, ± 2, ± 3, and are the dimensions crystal (in the form of a cube with side L). The volume of k-space per one quantum state is equal to (2)3/V, where V = L 3 is the volume of the crystal.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Thus, the number of electronic states per volume element dk = dkxdkydkz, calculated per unit volume, will be equal here, the factor 2 takes into account two possible spin orientations. The number of states per unit volume in the reciprocal space, i.e., the density of states) does not depend on the wave vector In other words, in the reciprocal space the allowed states are distributed with a constant density.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures It is practically impossible to calculate the function of the density of states with respect to energy in the general case, since isoenergetic surfaces can have a rather complex shape. In the simplest case of an isotropic parabolic dispersion law, which is valid for the edges of energy bands, one can find the number of quantum states per volume of a spherical layer enclosed between two close isoenergetic surfaces corresponding to energies E and E+d. E.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures The volume of a spherical layer in k-space. dk is the layer thickness. This volume will account for d. N states Taking into account the relationship between E and k according to the parabolic law, we obtain From here the density of states in energy will be equal to m * - the effective mass of the electron
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Thus, in three-dimensional crystals with a parabolic energy spectrum, as the energy increases, the density of allowed energy levels (density of states) will increase in proportion to the density of levels in the conduction band and in the valence band. The area of the shaded regions is proportional to the number of levels in the energy interval d. E
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Let us calculate the density of states for a two-dimensional system. The total energy of carriers for an isotropic parabolic dispersion law in a quantum-well film, as shown above, has a mixed discretely continuous spectrum. In a two-dimensional system, the states of a conduction electron are determined by three numbers (n, kx, ky). The energy spectrum is divided into separate two-dimensional En subbands corresponding to fixed values of n.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Curves of constant energy represent circles in reciprocal space. Each discrete quantum number n corresponds to the absolute value of the z-component of the wave vector. Therefore, the volume in the reciprocal space, bounded by a closed surface of a given energy E in the case of a two-dimensional system, is divided into a number of sections.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Let us determine the energy dependence of the density of states for a two-dimensional system. To do this, for a given n, we find the area S of the ring bounded by two isoenergetic surfaces corresponding to the energies E and E+d. E: Here The value of the two-dimensional wave vector corresponding to the given n and E; dkr is the width of the ring. Since one state in the plane (kxky) corresponds to the area where L 2 is the area of a two-dimensional film of thickness a, the number of electronic states in the ring, calculated per unit volume of the crystal, will be equal, taking into account the electron spin
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Since here is the energy corresponding to the bottom of the n-th subband. Thus, the density of states in a two-dimensional film is where Q(Y) is the unit Heaviside function, Q(Y) =1 for Y≥ 0, and Q(Y) =0 for Y
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS The distribution of quantum states in low-dimensional structures The density of states in a two-dimensional film can also be represented as - whole part, equal to the number subbands whose bottom is below the energy E. Thus, for two-dimensional films with a parabolic dispersion law, the density of states in any subband is constant and does not depend on energy. Each subband makes the same contribution to the total density of states. For a fixed film thickness, the density of states changes abruptly when it does not change by unity.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures. Dependence of the density of states of a two-dimensional film on energy (a) and thickness a (b).
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures In the case of an arbitrary dispersion law or with another type of potential well, the dependences of the density of state on energy and film thickness may differ from those given above, but the main feature, a nonmonotonic course, will remain.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Let us calculate the density of states for a one-dimensional structure - a quantum wire. The isotropic parabolic dispersion law in this case can be written as x is directed along the quantum filament, d is the thickness of the quantum filament along the y and z axes, kx is a one-dimensional wave vector. m, n are positive integers characterizing where the axis is quantum subbands. The energy spectrum of a quantum wire is thus divided into separate overlapping one-dimensional subbands (parabolas). The motion of electrons along the x axis turns out to be free (but with an effective mass), while the motion along the other two axes is limited.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Energy spectrum of electrons for a quantum wire
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire versus energy Number of quantum states per interval dkx , calculated per unit volume where is the energy corresponding to the bottom of the subband with given n and m.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire as a function of energy Thus Hence In deriving this formula, the spin degeneracy of states and the fact that one interval d. E corresponds to two intervals ±dkx of each subband, for which (E-En, m) > 0. The energy E is counted from the bottom of the conduction band of the bulk sample.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire on energy Dependence of the density of states of a quantum wire on energy. The numbers next to the curves show the quantum numbers n and m. The degeneracy factors of the subband levels are given in parentheses.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum wire as a function of energy Within a single subband, the density of states decreases with increasing energy. The total density of states is a superposition of identical decreasing functions (corresponding to individual subbands) shifted along the energy axis. For E = Em, n, the density of states is equal to infinity. The subbands with quantum numbers n m turn out to be doubly degenerate (only for Ly = Lz d).
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot as a function of energy With a three-dimensional limitation of particle motion, we arrive at the problem of finding allowed states in a quantum dot or zero-dimensional system. Using the effective mass approximation and the parabolic dispersion law, for the edge of an isotropic energy band, the spectrum of allowed states of a quantum dot with the same dimensions d along all three coordinate axes will have the form n, m, l = 1, 2, 3 ... - positive numbers numbering the subbands. The energy spectrum of a quantum dot is a set of discrete allowed states corresponding to fixed n, m, l.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot as a function of energy The degeneracy of the levels is primarily determined by the symmetry of the problem. g is the level degeneracy factor
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL ELECTRONIC SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot versus energy Degeneracy of levels is primarily determined by the symmetry of the problem. For example, for the considered case of a quantum dot with the same dimensions in all three dimensions, the levels will be three times degenerate if two quantum numbers are equal to each other and not equal to the third, and six times degenerate if all quantum numbers are not equal to each other. A specific type of potential can also lead to an additional, so-called random degeneracy. For example, for the considered quantum dot, to a threefold degeneracy of the levels E(5, 1, 1); E(1, 5, 1); E(1, 1, 5), associated with the symmetry of the problem, a random degeneration E(3, 3, 3) is added (n 2+m 2+l 2=27 in both the first and second cases), associated with the form limiting potential (infinite rectangular potential well).
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Distribution of quantum states in low-dimensional structures Density of states in a quantum dot versus energy Distribution of the number of allowed states N in the conduction band for a quantum dot with the same dimensions in all three dimensions. The numbers represent quantum numbers; the level degeneracy factors are given in parentheses.
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Three-dimensional electronic systems The properties of equilibrium electrons in semiconductors depend on the Fermi distribution function, which determines the probability that an electron will be in a quantum state with energy E EF is the Fermi level or electrochemical potential, T - absolute temperature, k is the Boltzmann constant. The calculation of various statistical quantities is greatly simplified if the Fermi level lies in the energy band gap and is far from the bottom of the conduction band Ec (Ec – EF) > k. T. Then, in the Fermi-Dirac distribution, the unit in the denominator can be neglected and it passes into the Maxwell-Boltzmann distribution of classical statistics. This is the case of a non-degenerate semiconductor
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of carriers in low-dimensional structures Three-dimensional electron systems The distribution function of the density of states in the conduction band g(E), the Fermi-Dirac function for three temperatures, and the Maxwell-Boltzmann function for a three-dimensional electron gas. At T = 0, the Fermi-Dirac function has the form of a discontinuous function. For Е EF the function is equal to zero and the corresponding quantum states are completely free. For T > 0, the Fermi function. The Dirac smears in the vicinity of the Fermi energy, where it rapidly changes from 1 to 0 and this smearing is proportional to k. T, i.e., the more, the higher the temperature. (Fig. 1. 4. Edges)
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of Carriers in Low-Dimensional Structures Three-Dimensional Electronic Systems The electron density in the conduction band is found by summing over all states Note that we should take the energy of the upper edge of the conduction band as the upper limit in this integral. But since the Fermi-Dirac function for energies E >EF decreases exponentially rapidly with increasing energy, replacing the upper limit with infinity does not change the value of the integral. Substituting the values of the functions into the integral, we obtain the -effective density of states in the conduction band
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Carrier statistics in low-dimensional structures Two-dimensional electron systems Let us determine the charge carrier concentration in a two-dimensional electron gas. Since the density of states of a two-dimensional electron gas We obtain Here also the upper limit of integration is taken equal to infinity, taking into account the sharp dependence of the Fermi-Dirac distribution function on energy. Integrating where
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of carriers in low-dimensional structures Two-dimensional electron systems For a nondegenerate electron gas, when In the case of ultrathin films, when only the filling of the lower subband can be taken into account For a strong degeneracy of the electron gas, when where n 0 is an integer part
ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL SYSTEMS Statistics of carriers in low-dimensional structures It should be noted that in quantum-well systems, due to the lower density of states, the condition of complete degeneracy does not require extremely high concentrations or low temperatures and is quite often implemented in experiments. For example, in n-Ga. As at N 2 D = 1012 cm-2, degeneracy will already take place at room temperature. In quantum wires, the integral for calculation, in contrast to the two-dimensional and three-dimensional cases, is not calculated analytically by arbitrary degeneracy, and simple formulas can only be written in extreme cases. In a non-degenerate one-dimensional electron gas in the case of hyperthin filaments, when only the occupation of the lowest level with energy E 11 can be taken into account, the electron concentration is where the one-dimensional effective density of states is
Energy levels (atomic, molecular, nuclear)
1. Characteristics of the state of a quantum system
2. Energy levels of atoms
3. Energy levels of molecules
4. Energy levels of nuclei
Characteristics of the state of a quantum system
At the heart of the explanation of St. in atoms, molecules and atomic nuclei, i.e. phenomena occurring in volume elements with linear scales of 10 -6 -10 -13 cm lies quantum mechanics. According to quantum mechanics, any quantum system (ie, a system of microparticles, which obeys quantum laws) is characterized by a certain set of states. In general, this set of states can be either discrete (discrete spectrum of states) or continuous (continuous spectrum of states). Characteristics of the state of an isolated system yavl. internal energy system (everywhere below, just energy), total angular momentum (MKD) and parity.
System energy.
A quantum system, being in different states, generally speaking, has different energies. The energy of the bound system can take any value. This set of possible energy values is called. discrete energy spectrum, and energy is said to be quantized. An example would be energy. spectrum of an atom (see below). An unbound system of interacting particles has a continuous energy spectrum, and the energy can take arbitrary values. An example of such a system is free electron (E) in the Coulomb field of the atomic nucleus. The continuous energy spectrum can be represented as a set of infinite a large number discrete states, between to-rymi energetic. gaps are infinitely small.
The state, to-rum corresponds to the lowest energy possible for a given system, called. basic: all other states are called. excited. It is often convenient to use a conditional scale of energy, in which the energy is basic. state is considered the starting point, i.e. is assumed to be zero (in this conditional scale, everywhere below the energy is denoted by the letter E). If the system is in the state n(and the index n=1 is assigned to main. state), has energy E n, then the system is said to be at the energy level E n. Number n, numbering U.e., called. quantum number. In the general case, each U.e. can be characterized not by one quantum number, but by their combination; then the index n means the totality of these quantum numbers.
If the states n 1, n 2, n 3,..., nk corresponds to the same energy, i.e. one U.e., then this level is called degenerate, and the number k- multiplicity of degeneration.
During any transformations of a closed system (as well as a system in a constant external field), its total energy, energy, remains unchanged. Therefore, energy refers to the so-called. conserved values. The law of conservation of energy follows from the homogeneity of time.
Total angular momentum.
This value is yavl. vector and is obtained by adding the MCD of all particles in the system. Each particle has both its own MCD - spin, and orbital moment, due to the motion of the particle relative to the common center of mass of the system. The quantization of the MCD leads to the fact that its abs. magnitude J takes strictly defined values: , where j- quantum number, which can take on non-negative integer and half-integer values (the quantum number of an orbital MCD is always an integer). The projection of the MKD on the c.-l. axis name magn. quantum number and can take 2j+1 values: m j =j, j-1,...,-j. If k.-l. moment J
yavl. the sum of two other moments , then, according to the rules for adding moments in quantum mechanics, the quantum number j can take the following values: j=|j 1 -j 2 |, |j 1 -j 2 -1|, ...., |j 1 +j 2 -1|, j 1 +j 2 , a . Similarly, the summation more moments. It is customary for brevity to talk about the MCD system j, implying the moment, abs. the value of which is ; about magn. The quantum number is simply spoken of as the projection of the momentum.
During various transformations of a system in a centrally symmetric field, the total MCD is conserved, i.e., like energy, it is a conserved quantity. The MKD conservation law follows from the isotropy of space. In an axially symmetric field, only the projection of the full MCD onto the axis of symmetry is preserved.
State parity.
In quantum mechanics, the states of a system are described by the so-called. wave functions. Parity characterizes the change in the wave function of the system during the operation of spatial inversion, i.e. change of signs of the coordinates of all particles. In such an operation, the energy does not change, while the wave function can either remain unchanged (even state) or change its sign to the opposite (odd state). Parity P takes two values, respectively. If nuclear or el.-magnets operate in the system. forces, parity is preserved in atomic, molecular and nuclear transformations, i.e. this quantity also applies to conserved quantities. Parity conservation law yavl. a consequence of the symmetry of space with respect to mirror reflections and is violated in those processes in which weak interactions are involved.
Quantum transitions
- transitions of the system from one quantum state to another. Such transitions can lead both to a change in energy. the state of the system, and to its qualities. changes. These are bound-bound, freely-bound, free-free transitions (see Interaction of radiation with matter), for example, excitation, deactivation, ionization, dissociation, recombination. It is also a chem. and nuclear reactions. Transitions can occur under the influence of radiation - radiative (or radiative) transitions, or when a given system collides with a c.-l. other system or particle - non-radiative transitions. An important characteristic of the quantum transition yavl. its probability in units. time, indicating how often this transition will occur. This value is measured in s -1 . Radiation probabilities. transitions between levels m And n (m>n) with the emission or absorption of a photon, the energy of which is equal to, are determined by the coefficient. Einstein A mn , B mn And B nm. Level transition m to the level n may occur spontaneously. Probability of emitting a photon Bmn in this case equals Amn. Type transitions under the action of radiation (induced transitions) are characterized by the probabilities of photon emission and photon absorption , where is the energy density of radiation with frequency .
The possibility of implementing a quantum transition from a given R.e. on k.-l. another w.e. means that the characteristic cf. time , during which the system can be at this UE, of course. It is defined as the reciprocal of the total decay probability of a given level, i.e. the sum of the probabilities of all possible transitions from the considered level to all others. For the radiation transitions, the total probability is , and . The finiteness of time , according to the uncertainty relation , means that the level energy cannot be determined absolutely exactly, i.e. U.e. has a certain width. Therefore, the emission or absorption of photons during a quantum transition does not occur at a strictly defined frequency , but within a certain frequency interval lying in the vicinity of the value . The intensity distribution within this interval is given by the spectral line profile , which determines the probability that the frequency of a photon emitted or absorbed in a given transition is equal to:
(1)
where is the half-width of the line profile. If the broadening of W.e. and spectral lines is caused only by spontaneous transitions, then such a broadening is called. natural. If collisions of the system with other particles play a certain role in the broadening, then the broadening has a combined character and the quantity must be replaced by the sum , where is calculated similarly to , but the radiat. transition probabilities should be replaced by collision probabilities.
Transitions in quantum systems obey certain selection rules, i.e. rules that establish how the quantum numbers characterizing the state of the system (MKD, parity, etc.) can change during the transition. The most simple selection rules are formulated for radiats. transitions. In this case, they are determined by the properties of the initial and final states, as well as the quantum characteristics of the emitted or absorbed photon, in particular its MCD and parity. The so-called. electric dipole transitions. These transitions are carried out between levels of opposite parity, the complete MCD to-rykh differ by an amount (the transition is impossible). In the framework of the current terminology, these transitions are called. permitted. All other types of transitions (magnetic dipole, electric quadrupole, etc.) are called. prohibited. The meaning of this term is only that their probabilities turn out to be much less than the probabilities of electric dipole transitions. However, they are not yavl. absolutely prohibited.
Quantum systems of identical particles
Quantum features of the behavior of microparticles, which distinguish them from the properties of macroscopic objects, appear not only when considering the motion of a single particle, but also when analyzing the behavior systems microparticles . This is most clearly seen in the example of physical systems consisting of identical particles - systems of electrons, protons, neutrons, etc.
For a system from N particles with masses T 01 , T 02 , … T 0 i , … m 0 N, having coordinates ( x i , y i , z i) , the wave function can be represented as
Ψ (x 1 , y 1 , z 1 , … x i , y i , z i , … x N , y N , z N , t) .
For elementary volume
dV i = dx i . dy i . dz i
magnitude
w =
determines the probability that one particle is in the volume dV 1 , another in volume dV 2 etc.
Thus, knowing the wave function of a system of particles, one can find the probability of any spatial configuration of a system of microparticles, as well as the probability of any mechanical quantity, both for the system as a whole and for an individual particle, and also calculate the average value of the mechanical quantity.
The wave function of a system of particles is found from the Schrödinger equation
, Where
Hamilton function operator for a system of particles
+
.
force function for i-
th particle in an external field, and
Interaction energy i- oh and j- oh particles.
The indistinguishability of identical particles in the quantum
mechanics
Particles that have the same mass, electric charge, spin, etc. will behave in exactly the same way under the same conditions.
The Hamiltonian of such a system of particles with the same masses m oi and the same force functions U i can be written as above.
If the system changes i- oh and j- th particle, then, due to the identity of identical particles, the state of the system should not change. The total energy of the system, as well as all physical quantities characterizing its state, will remain unchanged.
The principle of identity of identical particles: in a system of identical particles, only such states are realized that do not change when the particles are rearranged.
Symmetric and antisymmetric states
Let us introduce the particle permutation operator in the system under consideration - . The effect of this operator is that it swaps i- wow Andj- th particle of the system.
The principle of identity of identical particles in quantum mechanics leads to the fact that all possible states of a system formed by identical particles are divided into two types:
symmetrical, for which
antisymmetric, for which
(x 1 , y 1 ,z 1 … x N , y N , z N , t) = - Ψ A ( x 1 , y 1 ,z 1 … x N , y N , z N , t).
If the wave function describing the state of the system is symmetric (antisymmetric) at some point in time, then this type of symmetry persists at any other point in time.
Bosons and fermions
Particles whose states are described by symmetric wave functions are called bosons Bose–Einstein statistics . Bosons are photons, π- And To- mesons, phonons solid body, excitons in semiconductors and dielectrics. All bosons havezero or integer spin .
Particles whose states are described by antisymmetric wave functions are called fermions . Systems consisting of such particles obey Fermi–Dirac statistics . Fermions include electrons, protons, neutrons, neutrinos and all elementary particles and antiparticles withhalf back.
The connection between the particle spin and the type of statistics remains valid in the case of complex particles consisting of elementary ones. If the total spin of a complex particle is equal to an integer or zero, then this particle is a boson, and if it is equal to a half-integer, then the particle is a fermion.
Example: α-particle() consists of two protons and two neutrons i.e. four fermions with spins +. Therefore, the spin of the nucleus is 2 and this nucleus is a boson.
The nucleus of a light isotope consists of two protons and one neutron (three fermions). The spin of this nucleus is . Hence the core is a fermion.
Pauli principle (Pauli prohibition)
In the system of identicalfermions no two particles can be in the same quantum state.
As for the system consisting of bosons, the principle of symmetry of wave functions does not impose any restrictions on the states of the system. can be in the same state any number of identical bosons.
Periodic system of elements
At first glance, it seems that in an atom, all electrons should fill the level with the lowest possible energy. Experience shows that this is not so.
Indeed, in accordance with the Pauli principle, in the atom there cannot be electrons with the same values of all four quantum numbers.
Each value of the principal quantum number P corresponds 2 P 2 states that differ from each other by the values of quantum numbers l , m And m S .
The set of electrons of an atom with the same values of the quantum number P forms the so-called shell. according to the number P
Shells are divided into subshells, differing in quantum number l . The number of states in a subshell is 2(2 l + 1).
Different states in a subshell differ in their quantum numbers T And m S .
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Bohr's model of the atom was an attempt to reconcile the ideas of classical physics with the emerging laws of the quantum world.
E. Rutherford, 1936:
How are the electrons arranged in the outer part of the atom? I regard Bohr's original quantum theory of the spectrum as one of the most revolutionary that has ever been made in science; and I don't know of any other theory that has more success. He was at that time in Manchester and, firmly believing in the nuclear structure of the atom, which became clear in experiments on scattering, he tried to understand how the electrons should be arranged in order to obtain the known spectra of atoms. The basis of his success lies in the introduction of completely new ideas into the theory. He introduced into our ideas the idea of a quantum of action, as well as an idea that is alien to classical physics, that an electron can orbit around the nucleus without emitting radiation. When putting forward the theory of the nuclear structure of the atom, I was fully aware that, according to the classical theory, electrons should fall on the nucleus, and Bohr postulated that for some unknown reason this does not happen, and on the basis of this assumption, as you know, he was able to explain the origin of the spectra. Using quite reasonable assumptions, he solved step by step the problem of the arrangement of electrons in all atoms of the periodic table. There were many difficulties here, since the distribution had to correspond to the optical and x-ray spectra of the elements, but in the end Bohr managed to propose an arrangement of electrons that showed the meaning of the periodic law.
As a result of further improvements, mainly introduced by Bohr himself, and modifications made by Heisenberg, Schrödinger and Dirac, the whole mathematical theory was changed and the ideas of wave mechanics were introduced. Quite apart from these further improvements, I regard Bohr's work as the greatest triumph of human thought.
To realize the significance of his work, one should only consider the extraordinary complexity of the spectra of the elements and imagine that within 10 years all the main characteristics of these spectra have been understood and explained, so that now the theory of optical spectra is so complete that many consider this an exhausted question, similar to how it was a few years ago with sound.
By the middle of the 1920s, it became obvious that N. Bohr's semiclassical theory of the atom could not give an adequate description of the properties of the atom. In 1925–1926 In the works of W. Heisenberg and E. Schrödinger, a general approach was developed for describing quantum phenomena - quantum theory.
The quantum physics |
|---|
(x,y,z,p x ,p y ,p z)
=∂H/∂p, = -∂H/∂t,
x, y, z, p x , p y , p z
ΔхΔp x ~
∆y∆p y ~
∆z∆p z ~
Determinism
|(x,y,z)| 2
The state of a classical particle at any moment of time is described by setting its coordinates and momenta (x,y,z,p x ,p y ,p z ,t). Knowing these values at the time t, it is possible to determine the evolution of the system under the action of known forces at all subsequent moments of time. The coordinates and momenta of the particles are themselves quantities that can be directly measured experimentally. In quantum physics, the state of a system is described by the wave function ψ(x, y, z, t).
Because for a quantum particle, it is impossible to accurately determine the values of its coordinates and momentum at the same time, then it makes no sense to talk about the movement of the particle along a certain trajectory, you can only determine the probability of the particle being at a given point at a given time, which is determined by the square of the modulus of the wave function W ~ |ψ( x,y,z)| 2.
The evolution of a quantum system in the nonrelativistic case is described by a wave function that satisfies the Schrödinger equation
where is the Hamilton operator (the operator of the total energy of the system).
In the nonrelativistic case − 2 /2m + (r), where t
is the mass of the particle, is the momentum operator, (x,y,z) is the operator of the potential energy of the particle. To set the law of motion of a particle in quantum mechanics means to determine the value of the wave function at every moment of time at every point in space. In the stationary state, the wave function ψ(x, y, z) is a solution to the stationary Schrödinger equation ψ = Eψ. Like any bound system in quantum physics, the nucleus has a discrete spectrum of energy eigenvalues.
The state with the highest binding energy of the nucleus, i.e., with the lowest total energy E, is called the ground state. States with higher total energy are excited states. The lowest energy state is assigned a zero index and the energy E 0
=
0.
E0 → Mc 2 = (Zm p + Nm n)c 2 − W 0 ;
W 0 is the binding energy of the nucleus in the ground state.
Energies E i (i = 1, 2, ...) of excited states are measured from the ground state.
Scheme of the lower levels of the 24 Mg nucleus.
The lower levels of the kernel are discrete. As the excitation energy increases, the average distance between the levels decreases.
An increase in the level density with increasing energy is a characteristic property of many-particle systems. It is explained by the fact that with an increase in the energy of such systems, the number of different ways of distributing energy between nucleons rapidly increases.
quantum numbers- integer or fractional numbers that determine the possible values of physical quantities characterizing a quantum system - an atom, an atomic nucleus. Quantum numbers reflect the discreteness (quantization) of physical quantities characterizing the microsystem. A set of quantum numbers that exhaustively describe a microsystem is called complete. So the state of the nucleon in the nucleus is determined by four quantum numbers: the main quantum number n (can take values 1, 2, 3, ...), which determines the energy E n of the nucleon; orbital quantum number l = 0, 1, 2, …, n, which determines the value L
the orbital angular momentum of the nucleon (L = ћ 1/2); the quantum number m ≤ ±l, which determines the direction of the orbital momentum vector; and the quantum number m s = ±1/2, which determines the direction of the nucleon spin vector.
quantum numbers
| n | Principal quantum number: n = 1, 2, … ∞. |
| j | The quantum number of the total angular momentum. j is never negative and can be integer (including zero) or half-integer depending on the properties of the system in question. The value of the total angular momentum of the system J is related to j by the relation J 2 = ћ 2 j(j+1). = + where and are the orbital and spin angular momentum vectors. |
| l | Quantum number of orbital angular momentum. l can only take integer values: l= 0, 1, 2, … ∞, The value of the orbital angular momentum of the system L is related to l relation L 2 = ћ 2 l(l+1). |
| m | The projection of the total, orbital, or spin angular momentum onto a preferred axis (usually the z-axis) is equal to mћ. For the total moment m j = j, j-1, j-2, …, -(j-1), -j. For the orbital moment m l = l, l-1, l-2, …, -(l-1), -l. For the spin moment of an electron, proton, neutron, quark m s = ±1/2 |
| s | Quantum number of spin angular momentum. s can be either integer or half-integer. s is a constant characteristic of the particle, determined by its properties. The value of the spin moment S is related to s by the relation S 2 = ћ 2 s(s+1) |
| P | Spatial parity. It is equal to either +1 or -1 and characterizes the behavior of the system under mirror reflection P = (-1) l . |
Along with this set of quantum numbers, the state of the nucleon in the nucleus can also be characterized by another set of quantum numbers n, l, j, jz . The choice of a set of quantum numbers is determined by the convenience of describing a quantum system.
The existence of conserved (invariant in time) physical quantities for a given system is closely related to the symmetry properties of this system. So, if an isolated system does not change during arbitrary rotations, then it retains the orbital angular momentum. This is the case for the hydrogen atom, in which the electron moves in the spherically symmetric Coulomb potential of the nucleus and is therefore characterized by a constant quantum number l. An external perturbation can break the symmetry of the system, which leads to a change in the quantum numbers themselves. A photon absorbed by a hydrogen atom can transfer an electron to another state with different values of quantum numbers. The table lists some quantum numbers used to describe atomic and nuclear states.
In addition to quantum numbers, which reflect the space-time symmetry of the microsystem, the so-called internal quantum numbers of particles play an important role. Some of them, such as spin and electric charge, are conserved in all interactions, others are not conserved in some interactions. So the strangeness quantum number, which is conserved in the strong and electromagnetic interactions, is not conserved in the weak interaction, which reflects the different nature of these interactions.
The atomic nucleus in each state is characterized by the total angular momentum. This moment in the rest frame of the nucleus is called nuclear spin.
The following rules apply to the kernel:
a) A is even J = n (n = 0, 1, 2, 3,...), i.e. an integer;
b) A is odd J = n + 1/2, i.e. half-integer.
In addition, one more rule has been experimentally established: for even-even nuclei in the ground state Jgs
= 0. This indicates mutual compensation of nucleon moments in the ground state of the nucleus – special property internucleon interaction.
The invariance of the system (hamiltonian) with respect to spatial reflection - inversion (replacement → -) leads to the parity conservation law and the quantum number parity R. This means that the nuclear Hamiltonian has the corresponding symmetry. Indeed, the nucleus exists due to the strong interaction between nucleons. In addition, an essential role in nuclei is also played by electromagnetic interaction. Both of these types of interactions are invariant to spatial inversion. This means that nuclear states must be characterized by a certain parity value P, i.e., be either even (P = +1) or odd (P = -1).
However, weak forces that do not preserve parity also act between nucleons in the nucleus. The consequence of this is that a (usually insignificant) admixture of a state with the opposite parity is added to the state with a given parity. The typical value of such an impurity in nuclear states is only 10 -6 -10 -7 and in most cases can be ignored.
The parity of the nucleus P as a system of nucleons can be represented as the product of the parities of individual nucleons p i:
P \u003d p 1 p 2 ... p A ,
moreover, the parity of the nucleon p i in the central field depends on the orbital moment of the nucleon , where π i is the internal parity of the nucleon, equal to +1. Therefore, the parity of a nucleus in a spherically symmetric state can be represented as the product of the orbital parities of nucleons in this state:
Nuclear level diagrams usually indicate the energy, spin, and parity of each level. The spin is indicated by a number, and the parity is indicated by a plus sign for even levels and a minus sign for odd levels. This sign is placed to the right of the top of the number indicating the spin. For example, the symbol 1/2 + denotes an even level with spin 1/2, and the symbol 3 - denotes an odd level with spin 3.
Isospin of atomic nuclei. Another characteristic of nuclear states is isospin I. Core (A, Z) consists of A nucleons and has a charge Ze, which can be represented as the sum of nucleon charges q i , expressed in terms of projections of their isospins (I i) 3
is the projection of the isospin of the nucleus onto axis 3 of the isospin space.
Total isospin of the nucleon system A
All states of the nucleus have the value of the isospin projection I 3 = (Z - N)/2. In a nucleus consisting of A nucleons, each of which has isospin 1/2, isospin values are possible from |N - Z|/2 to A/2
|N - Z|/2 ≤ I ≤ A/2.
The minimum value I = |I 3 |. The maximum value of I is equal to A/2 and corresponds to all i directed in the same direction. It has been experimentally established that the higher the excitation energy of the nuclear state, the greater the value of isospin. Therefore, the isospin of the nucleus in the ground and low-excited states has a minimum value
I gs = |I 3 | = |Z - N|/2.
The electromagnetic interaction breaks the isotropy of the isospin space. The interaction energy of a system of charged particles changes during rotations in isospace, since during rotations the charges of particles change and in the nucleus part of the protons passes into neutrons or vice versa. Therefore, the actual isospin symmetry is not exact, but approximate.
Potential well. The concept of a potential well is often used to describe the bound states of particles. Potential well - a limited region of space with a reduced potential energy of a particle. The potential well usually corresponds to the forces of attraction. In the area of action of these forces, the potential is negative, outside - zero.
The particle energy E is the sum of its kinetic energy T ≥ 0 and potential energy U (it can be both positive and negative). If the particle is inside the well, then its kinetic energy T 1 is less than the depth of the well U 0, the particle energy E 1 = T 1 + U 1 = T 1 - U 0 In quantum mechanics, the energy of a particle in a bound state can take only certain discrete values, i.e. there are discrete levels of energy. In this case, the lowest (main) level always lies above the bottom of the potential well. In order of magnitude, the distance Δ E between the levels of a particle of mass m in a deep well of width a is given by
ΔE ≈ ћ 2 / ma 2.
An example of a potential well is the potential well of an atomic nucleus with a depth of 40-50 MeV and a width of 10 -13 -10 -12 cm, in which nucleons with an average kinetic energy of ≈ 20 MeV are located at different levels.
Under such boundary conditions, the particle, being inside the potential well 0< x < l, не может выйти за ее пределы, т. е.
ψ(x) = 0, x ≤ 0, x ≥ L.
Using the stationary Schrödinger equation for the region where U = 0,
we obtain the position and energy spectrum of the particle inside the potential well.
For an infinite one-dimensional potential well, we have the following:
The wave function of a particle in an infinite rectangular well (a), the square of the modulus of the wave function (b) determines the probability of finding a particle at various points in the potential well.
The Schrödinger equation plays the same role in quantum mechanics as Newton's second law plays in classical mechanics.
The most striking feature of quantum physics turned out to be its probabilistic nature.
The probabilistic nature of the processes occurring in the microworld is a fundamental property of the microworld.
E. Schrödinger:
“The usual quantization rules can be replaced by other provisions that no longer introduce any “whole numbers”. Integrity is obtained in this case in a natural way by itself, just as the integer number of knots is obtained by itself when considering a vibrating string. This new representation can be generalized and, I think, is closely related to the true nature of quantization.
It is quite natural to associate the function ψ with some oscillatory process in the atom, in which the reality of electronic trajectories has recently been repeatedly questioned. At first, I also wanted to substantiate the new understanding of quantum rules using the indicated comparatively clear way, but then I preferred a purely mathematical method, since it makes it possible to better clarify all the essential aspects of the issue. It seems to me essential that quantum rules are no longer introduced as a mysterious " integer requirement”, but are determined by the need for the boundedness and uniqueness of some specific spatial function.
I do not consider it possible, until more are successfully calculated in a new way. challenging tasks, consider in more detail the interpretation of the introduced oscillatory process. It is possible that such calculations will lead to a simple coincidence with the conclusions of conventional quantum theory. For example, when considering the relativistic Kepler problem according to the above method, if we act according to the rules indicated at the beginning, a remarkable result is obtained: half-integer quantum numbers(radial and azimuth)…
First of all, it is impossible not to mention that the main initial impetus that led to the appearance of the arguments presented here was de Broglie's dissertation, which contains many deep ideas, as well as reflections on the spatial distribution of "phase waves", which, as shown by de Broglie, each time corresponds to periodic or quasi-periodic motion of an electron, if only these waves fit on the trajectories integer once. The main difference from de Broglie's theory, which speaks of a rectilinearly propagating wave, lies here in the fact that we are considering, if we use the wave interpretation, standing natural oscillations.
M. Laue:
“The achievements of quantum theory accumulated very quickly. It had a particularly striking success in its application to radioactive decay by the emission of α-rays. According to this theory, there is a "tunnel effect", i.e. penetration through a potential barrier of a particle whose energy, according to the requirements of classical mechanics, is insufficient to pass through it.
G. Gamov gave in 1928 an explanation of the emission of α-particles, based on this tunnel effect. According to Gamow's theory, the atomic nucleus is surrounded by a potential barrier, but α-particles have a certain probability of "stepping over" it. Empirically found by Geiger and Nettol, the relationship between the radius of action of an α-particle and the half-period of decay was satisfactorily explained on the basis of Gamow's theory.
Statistics. Pauli principle. The properties of quantum mechanical systems consisting of many particles are determined by the statistics of these particles. Classical systems consisting of identical but distinguishable particles obey the Boltzmann distribution
In a system of quantum particles of the same type, new features of behavior appear that have no analogues in classical physics. Unlike particles in classical physics, quantum particles are not only the same, but also indistinguishable - identical. One reason is that, in quantum mechanics, particles are described in terms of wave functions, which allow us to calculate only the probability of finding a particle at any point in space. If the wave functions of several identical particles overlap, then it is impossible to determine which of the particles is at a given point. Since only the square of the modulus of the wave function has physical meaning, it follows from the particle identity principle that when two identical particles are interchanged, the wave function either changes sign ( antisymmetric state), or does not change sign ( symmetrical state).
Symmetric wave functions describe particles with integer spin - bosons (pions, photons, alpha particles ...). Bosons obey Bose-Einstein statistics
An unlimited number of identical bosons can be in one quantum state at the same time.
Antisymmetric wave functions describe particles with half-integer spin - fermions (protons, neutrons, electrons, neutrinos). Fermions obey Fermi-Dirac statistics
The relationship between the symmetry of the wave function and spin was first pointed out by W. Pauli.
For fermions, the Pauli principle is valid - two identical fermions cannot simultaneously be in the same quantum state.
The Pauli principle determines the structure of the electron shells of atoms, the filling of nucleon states in nuclei, and other features of the behavior of quantum systems.
With the creation of the proton-neutron model of the atomic nucleus, the first stage in the development of nuclear physics can be considered completed, in which the basic facts of the structure of the atomic nucleus were established. The first stage began in the fundamental concept of Democritus about the existence of atoms - indivisible particles of matter. The establishment of the periodic law by Mendeleev made it possible to systematize atoms and raised the question of the reasons underlying this systematics. The discovery of electrons in 1897 by J. J. Thomson destroyed the concept of the indivisibility of atoms. According to Thomson's model, electrons are the building blocks of all atoms. The discovery by A. Becquerel in 1896 of the phenomenon of uranium radioactivity and the subsequent discovery by P. Curie and M. Sklodowska-Curie of the radioactivity of thorium, polonium and radium showed for the first time that chemical elements are not eternal formations, they can spontaneously decay, turn into other chemical elements . In 1899, E. Rutherford found that as a result of radioactive decay, atoms can eject α-particles from their composition - ionized helium atoms and electrons. In 1911, E. Rutherford, generalizing the results of the experiment of Geiger and Marsden, developed a planetary model of the atom. According to this model, atoms consist of a positively charged atomic nucleus with a radius of ~10 -12 cm, in which the entire mass of the atom and negative electrons rotating around it is concentrated. The size of the electron shells of an atom is ~10 -8 cm. In 1913, N. Bohr developed a representation of the planetary model of the atom based on quantum theory. In 1919, E. Rutherford proved that protons are part of the atomic nucleus. In 1932, J. Chadwick discovered the neutron and showed that neutrons are part of the atomic nucleus. The creation in 1932 by D. Ivanenko and W. Heisenberg of the proton-neutron model of the atomic nucleus completed the first stage in the development of nuclear physics. All the constituent elements of the atom and the atomic nucleus have been established.
1869 Periodic system of elements D.I. Mendeleev
By the second half of the 19th century, chemists had accumulated extensive information on the behavior of chemical elements in various chemical reactions. It was found that only certain combinations of chemical elements form a given substance. Some chemical elements have been found to have roughly the same properties while their atomic weights vary widely. D. I. Mendeleev analyzed the relationship between chemical properties elements and their atomic weight and showed that the chemical properties of elements arranged as atomic weights increase are repeated. This formed the basis of his periodic system elements. When compiling the table, Mendeleev found that the atomic weights of some chemical elements fell out of the regularity he had obtained, and pointed out that the atomic weights of these elements were determined inaccurately. Later precise experiments showed that the originally determined weights were indeed incorrect and the new results corresponded to Mendeleev's predictions. Leaving some places blank in the table, Mendeleev pointed out that there should be new yet undiscovered chemical elements and predicted their chemical properties. Thus, gallium (Z = 31), scandium (Z = 21) and germanium (Z = 32) were predicted and then discovered. Mendeleev left the task of explaining the periodic properties of chemical elements to his descendants. The theoretical explanation of Mendeleev's periodic system of elements, given by N. Bohr in 1922, was one of the convincing proofs of the correctness of the emerging quantum theory.
Atomic nucleus and the periodic system of elements
The basis for the successful construction of the periodic system of elements by Mendeleev and Logar Meyer was the idea that atomic weight can serve as a suitable constant for the systematic classification of elements. Modern atomic theory, however, has approached the interpretation of the periodic system without touching upon atomic weight at all. The place number of any element in this system and, at the same time, its chemical properties are uniquely determined by the positive charge of the atomic nucleus, or, what is the same, by the number of negative electrons located around it. The mass and structure of the atomic nucleus play no part in this; thus, at the present time, we know that there are elements, or rather types of atoms, which, with the same number and arrangement of outer electrons, have vastly different atomic weights. Such elements are called isotopes. So, for example, in a galaxy of zinc isotopes, the atomic weight is distributed from 112 to 124. On the contrary, there are elements with significantly different chemical properties that exhibit the same atomic weight; they are called isobars. An example is the atomic weight of 124 found for zinc, tellurium and xenon.
For determining chemical element one constant is enough, namely, the number of negative electrons located around the nucleus, since all chemical processes flow among these electrons.
Number of protons n 2
, located in the atomic nucleus, determine its positive charge Z, and thereby the number of external electrons that determine the chemical properties of this element; some number of neutrons n 1
enclosed in the same core, in total with n 2
gives its atomic weight
A=n 1
+n 2
. Conversely, the serial number Z gives the number of protons contained in the atomic nucleus, and from the difference between the atomic weight and the charge of the nucleus A - Z, the number of nuclear neutrons is obtained.
With the discovery of the neutron, the periodic system received some replenishment in the region of small serial numbers, since the neutron can be considered an element with an ordinal number equal to zero. In the area of high ordinal numbers, namely from Z = 84 to Z = 92, all atomic nuclei unstable, spontaneously radioactive; therefore, it can be assumed that an atom with a nuclear charge even higher than that of uranium, if it can only be obtained, should also be unstable. Fermi and his collaborators recently reported on their experiments, in which, when uranium was bombarded with neutrons, the appearance of a radioactive element with the serial number 93 or 94 was observed. It is quite possible that the periodic system has a continuation in this region as well. It only remains to add that Mendeleev's ingenious foresight provided for the framework of the periodic system so broadly that each new discovery, remaining within its scope, further strengthens it.
