Experience shows that the spectra of non-interacting atoms, as is the case for rarefied gases, consist of individual lines grouped in series. In Fig. Figure 5.3 shows the series lines of the spectrum of the hydrogen atom located in the visible region. The wavelength corresponding to the lines in this series, called Balmer series , is expressed by the formula

Where, n = 3, 4, 5, ...; - Rydberg constant.

Line corresponding n= 3, is the brightest and is called head , and the value n= ∞ corresponds to a line called border of the series .

A series of lines were also discovered in other regions of the spectrum (ultraviolet, infrared). All of them can be represented generalized Balmer-Rydberg formula

Where m- an integer constant for each series.

At m = 1; n = 2,3,4, ... - Lyman series . Observed in the ultraviolet region.
At m = 2; n = 3,4,5, ... - Balmer series - in the visible region.
At m = 3; n = 4,5,6, ... - Paschen series - in the infrared (IR) region.
At m = 4; n = 5,6,7, ... - Bracket series - also in the IR region, etc.

Discreteness in the structure of atomic spectra indicates the presence of discreteness in the structure of the atoms themselves. For the energy of radiation quanta of hydrogen atoms, we can write the following formula

When writing this expression, formulas (5.1), (3.21) and (5.8) were used. Formula (5.9) was obtained based on analysis of experimental data.

Bohr's postulates

First quantum theory The structure of the atom was proposed in 1913 by the Danish physicist Niels Bohr. It was based on the nuclear model of the atom, according to which an atom consists of a positively charged nucleus around which negatively charged electrons revolve.
Bohr's theory is based on two postulates.

I Bohr's postulate - postulate of stationary states. In an atom, there are stationary (not changing with time) states in which it does not emit energy. These stationary states correspond to stationary orbits along which electrons move. The movement of electrons in stationary orbits is not accompanied by energy emission.

II Bohr's postulate called the "frequency rule". When an electron moves from one stationary orbit to another, a quantum of energy is emitted (or absorbed) equal to the difference energies of stationary states

Where h- Planck's constant; v- frequency of energy emission (or absorption);
hv- energy of a quantum of radiation (or absorption);
E n And E m- energies of stationary states of the atom before and after radiation (absorption), respectively. At E m < E n a quantum of energy is emitted, and when E m > E n- absorption.

According to Bohr's theory, the energy value of an electron in a hydrogen atom is equal to

Where m e- electron mass, e- electron charge, ε e- electrical constant
,

h- Planck's constant,
n- integer, n = 1,2,3,...

Thus, the energy of an electron in an atom is discrete quantity, which can only change abruptly.

The set of possible discrete frequencies of quantum transitions determines line spectrum atom

The frequencies of spectral lines for the hydrogen atom calculated using this formula turned out to be in excellent agreement with experimental data. But the theory did not explain the spectra of other atoms (even helium, next to hydrogen). Therefore, Bohr's theory was only a transitional stage on the path to building a theory of atomic phenomena. She pointed out the inapplicability classical physics to intra-atomic phenomena and the overriding importance quantum laws in a microcosm.

By the end of the 19th century, for 150 years in European physical laboratories experiments were carried out to study the light radiation of various heated gases. Using various optical instruments It was experimentally established that the radiation of atoms that do not interact with each other consists of individual spectral lines. Lines in atomic spectra are not randomly located, but are combined into groups called spectral series. Line spectra of atoms have individual structure, however, general patterns were identified.

In 1885 the Swiss school teacher mathematician Johan Balmer discovered that the wavelengths of a series of lines of the hydrogen atom lying in the visible spectrum are related by the relation

n = R (1/n 2 – 1/m 2), R=3.29 10 15 Hz – Rydberg constant, n and m – integers. Based on the formula obtained, Balmer predicted the existence of spectral series of hydrogen in the ultraviolet and infrared regions, which were discovered 20 years later.

The line frequencies of other atoms can be represented as the difference of two terms, having more complex look than for hydrogen atoms.

Discovery of radioactivity

In the early years of the twentieth century, new types of radiation were discovered - radioactive , called a, b, and g radiation. The phenomenon of radioactivity was studied by Antoine Becquerel (1852-1908) and the spouses Pierre (1859-1906) and Marie 1867-1934 Curie.

Rutherford's experiments

In 1907, Ernst Rutherford (1871-1937), a professor of physics at the University of Manchester, who studied the problems of radioactivity, and his collaborators studied the passage of alpha particles through thin metal foil. a-particles were emitted by some radioactive substance, had a speed of the order of 10 9 cm/s and positive charge, equal to twice the electron value. When passing through the foils, most of the a-particles deviated from the original direction at some small angles. It turned out, however, that a certain number of a-particles are deflected at angles of the order of 180 0, which, according to classical theory scattering is possible only if inside the atom there is an extremely strong EM field, concentrated in a small volume and charge created large mass.

Example. Contradiction with Thomson's atomic model.

An atom is a positively charged ball containing an electron.

When an electron deviates from its equilibrium position, a quasi-elastic force arises, under the influence of which the electron will oscillate and emit elastic magnets. waves.

Based on experimental data, Rutherford proposed in 1911 nuclear model atom:

ü in the center of the atom there is a heavy positively charged nucleus with a charge Ze and dimensions not exceeding 10 -12 m;

ü there are Z electrons around the nucleus, distributed throughout the entire volume occupied by the atom, the dimensions of the atom are about

In Rutherford's experiments, the deflection of a-particles is due to the action of atomic nuclei on them.

The question of exactly how electrons are distributed around the nucleus remained open. Rutherford considered the possibility planetary model atom, according to which electrons rotate around atomic nucleus. nuclear model, however, it turned out to be in conflict with the laws classical mechanics and electrodynamics. Since the system stationary charges cannot be in a state stable equilibrium, Rutherford had to assume that electrons move around the nucleus along curvilinear trajectories. But in this case, the electron moves with acceleration, and according to the laws of classical electrodynamics, it must emit an electric magnet. waves, losing energy in the process, as a result of which it must ultimately fall onto the core.

Bohr's atomic model.

The young Danish student Niels Bohr, who arrived in Manchester in Rutherford's group, became interested in the planetary model of the atom. At the beginning of 1912, Bohr prepared for Rutherford a paper “On the structure of atoms and molecules”, in which he suggested that within the framework of the planetary model there could be some stationary orbits of electrons, which should somehow be related to the Planck-Einstein formula E=hn. A breakthrough was made when Bohr discovered Balmer's formula.

To resolve the contradictions that arose in 1913, Niels Bohr proposed two postulates :

1. From infinite number electron orbits allowed classical mechanics, in reality, only some discrete orbits satisfying certain quantum conditions are realized. An electron, being in such an orbit, does not emit EM waves.

2. Radiation is emitted or absorbed in the form light quantum energy during the transition of an electron from one stationary state to another. The magnitude of the energy quantum is equal to the difference in energies of stationary states

hn = E 1 – E 2

According to Bohr's postulate only those electron orbits, for which the angular momentum is a multiple of Planck's constant

L = mvR = n h/2p

(The first proposal to quantize angular momentum was published by Nicholson in 1912).

Using the classical description of electron motion as rotation in the Coulomb field of the nucleus, Bohr obtained analytical expressions for the radii of stationary orbits and energies of the corresponding states of the atom:

Where r 1 =0.53 A= 0.53 10 -10 m

, where Ry=-13.6 eV.

Bohr's theory made it possible to explain the spectra of the hydrogen atom. The theoretically calculated value of the Rydberg constant differed by only a few percent from that obtained by Balmer. Bohr's theory combined classical and quantum approaches to the description of atomic processes. It was a transitional stage on the path to creating quantum mechanics, currently has mainly historical significance.

A more thorough experimental study of the spectrum of the hydrogen atom showed the presence large number spectral lines, which were no longer described by Bohr's theory. Arnold Sommerfeld (1868-1951), theorist, professor from Munich, took into account the ellipticity of electron orbits, which made it possible to explain additional spectral lines and required the introduction of additional quantum number I (orbital quantum number). IN last decade In the 19th century, Dane Peter Zeeman (1865-1943) discovered that additional spectral lines appeared in the spectrum of excited hydrogen atoms placed in a magnetic field (the Zeeman effect). Sommerfeld suggested that the observed phenomenon of splitting of spectral lines in a magnetic field is associated with different orientations of the electron orbits relative to external field. Sommerfeld introduced one more thing into consideration - magnetic quantum number m.

More subtle experiments with magnetic field made it possible to detect additional spectral lines (anomalous Zeeman effect) that were not described by the Bohr-Sommerfeld theory. The Swiss theoretical physicist Wolfgang Pauli (1900-1958) became interested in the AEZ problem and accepted Bohr's invitation to work in Copenhagen in 1922-23. Reflections on the nature of AEZ led Pauli to the idea that the electron is characterized by some additional rotational process, which corresponds to an additional angular momentum. Pauli proposed introducing a fourth quantum number into the theory of the atom, which can take only two values. Pauli sought to understand physical essence phenomenon and was in no hurry to publish. At the same time, two young Dutch physicists Uhlenbeek and Goudsmit came up with the same idea. Their supervisor, Professor Paul Ehrenfest, forwarded their paper for publication. Subsequently, Uhlenbeck and Goudsmit received for this work Nobel Prize in physics.

However, it remained unclear why all the electrons in multi-electron atoms do not go to the ground state. Pauli answered this question.

Pauli principle

So, the state of each electron in an atom is characterized by four quantum numbers:

main n (n=1, 2, …)

azimuthal l (l=1, 2, …, n-1)

magnetic m l (m l =-l,…,-1,0,+1,…,+l)

spin m s (m s =+1/2, -1/2)

In the normal (unexcited) state of an atom, electrons should be located at the lowest available to them energy levels. According to Pauli principle , in the same atom (or another quantum system) there cannot be two electrons that have the same set of quantum numbers.

In an atom, each n state can correspond to n 2 states differing ( n, l, m l ), ​​and in addition, the spin quantum number can take values ​​of ±1/2. Thus,

n=1 – 2 electrons,

n=2 – 8 electrons,

n=3 – 18 electrons, etc.

The collection of electrons having same values principal quantum number n, forms shell.

Value n 1 2 3 4 …

Shell designation K L M N …

The Pauli principle provides an explanation for the repeatability of the properties of atoms. Atoms with the same number of electrons in the outer shell have similar properties (a completely filled shell is characterized by the total orbital and spin moments being equal to zero) (see Fig. periodic table Mendeleev's elements: alkali metals, metals, halogens, inert gases).

Electron waves in an atom.

Bohr's quantum conditions received a simple explanation based on the wave-particle duality applied to electrons located in stationary orbits. Waves associated with electrons were considered as standing waves, similar to those that arise on a string fixed on both sides. Then the length of the orbit must contain an integer number of waves

Using the de Broglie relation, it is easy to obtain the condition for quantizing the angular momentum.

"Old" quantum theory, created by Planck, Einstein, de Broglie, Rutherford, Bohr, Sommerfeld, Pauli and others, was able to explain:

ü spectrum of the hydrogen atom;

ü quantization of energy in stationary states atom;

ü Mendeleev's periodic system.

The fundamental ideas of the new quantum mechanics were laid, but the semiclassical theory could not answer many important questions.

Regularities in atomic spectra

Material bodies are sources electromagnetic radiation, having a different nature. In the second half of the 19th century. Numerous studies have been carried out on the emission spectra of molecules and atoms. It turned out that the emission spectra of molecules consist of widely diffuse bands without sharp boundaries. Such spectra were called striped. The emission spectrum of atoms consists of individual spectral lines or groups of closely spaced lines. Therefore, the spectra of atoms were called line spectra. For each element there is a completely definite line spectrum emitted by it, the type of which does not depend on the method of excitation of the atom.

The simplest and most studied is the spectrum of the hydrogen atom. Analysis of empirical material has shown that individual lines in the spectrum can be combined into groups of lines, which are called series. In 1885, I. Balmer established that the frequencies of lines in the visible part of the spectrum of hydrogen can be represented in the form of a simple formula:

( 3, 4, 5, …), (7.42.1)

where 3.29∙10 15 s -1 is the Rydberg constant. Spectral lines that differ different meanings, form the Balmer series. Subsequently, several more series were discovered in the spectrum of the hydrogen atom:

Lyman series (located in the ultraviolet part of the spectrum):

( 2, 3, 4, …); (7.42.2)

Paschen series (lies in the infrared part of the spectrum):

( 4, 5, 6, …); (7.42.3)

Bracket series (lies in the infrared part of the spectrum):

( 5, 6, 7, …); (7.42.4)

Pfund series (lies in the infrared part of the spectrum):

( 6, 7, 8, …); (7.42.5)

Humphrey series (located in the infrared part of the spectrum):

( 7, 8, 9, …). (7.42.6)

The frequencies of all lines in the spectrum of the hydrogen atom can be described by one formula - the generalized Balmer formula:

, (7.42.7)

where 1, 2, 3, 4, etc. – defines a series (for example, for Balmer series 2), and defines a line in a series, taking integer values ​​starting from 1.

From formulas (7.42.1) – (7.42.7) it is clear that each of the frequencies in the spectrum of the hydrogen atom is the difference between two quantities of the form depending on an integer. Expressions of the form where 1, 2, 3, 4, etc. are called spectral terms. According to the Ritz combination principle, all emitted frequencies can be represented as combinations of two spectral terms:

(7.42.8)

and always >

Study of spectra more complex atoms showed that the frequencies of their emission lines can also be represented as the difference between two spectral terms, but their formulas are more complicated than for the hydrogen atom.

The experimentally established patterns of atomic radiation are in conflict with classical electrodynamics, according to which electromagnetic waves emits an accelerating charge. Therefore, atoms contain electric charges, moving with acceleration in a limited volume of an atom. When radiating, the charge loses energy in the form of electromagnetic radiation. This means that the stationary existence of atoms is impossible. However, the established patterns indicated that spectral radiation atoms is the result of as yet unknown processes inside the atom.

Material bodies are sources of electromagnetic radiation of different nature. In the second half of the 19th century. Numerous studies have been carried out on the emission spectra of molecules and atoms. It turned out that the emission spectra of molecules consist of widely diffuse bands without sharp boundaries. Such spectra were called striped. The emission spectrum of atoms consists of individual spectral lines or groups of closely spaced lines. Therefore, the spectra of atoms were called line spectra. For each element there is a completely definite line spectrum emitted by it, the type of which does not depend on the method of excitation of the atom.

The simplest and most studied is the spectrum of the hydrogen atom. Analysis of empirical material has shown that individual lines in the spectrum can be combined into groups of lines, which are called series. In 1885, I. Balmer established that the frequencies of lines in the visible part of the spectrum of hydrogen can be represented in the form of a simple formula:

( 3, 4, 5, …), (7.42.1)

where 3.29∙10 15 s -1 is the Rydberg constant. Spectral lines with different values ​​form the Balmer series. Subsequently, several more series were discovered in the spectrum of the hydrogen atom:

Lyman series (located in the ultraviolet part of the spectrum):

( 2, 3, 4, …); (7.42.2)

Paschen series (lies in the infrared part of the spectrum):

( 4, 5, 6, …); (7.42.3)

Bracket series (lies in the infrared part of the spectrum):

( 5, 6, 7, …); (7.42.4)

Pfund series (lies in the infrared part of the spectrum):

( 6, 7, 8, …); (7.42.5)

Humphrey series (located in the infrared part of the spectrum):

( 7, 8, 9, …). (7.42.6)

The frequencies of all lines in the spectrum of the hydrogen atom can be described by one formula - the generalized Balmer formula:

, (7.42.7)

where 1, 2, 3, 4, etc. – defines a series (for example, for Balmer series 2), and defines a line in a series, taking integer values ​​starting from 1.

From formulas (7.42.1) – (7.42.7) it is clear that each of the frequencies in the spectrum of the hydrogen atom is the difference between two quantities of the form depending on an integer. Expressions of the form where 1, 2, 3, 4, etc. are called spectral terms. According to the Ritz combination principle, all emitted frequencies can be represented as combinations of two spectral terms:

(7.42.8)

and always >

A study of the spectra of more complex atoms showed that the frequencies of their emission lines can also be represented as the difference between two spectral terms, but their formulas are more complicated than for the hydrogen atom.

The experimentally established patterns of atomic radiation are in conflict with classical electrodynamics, according to which electromagnetic waves are emitted by an accelerating charge. Consequently, atoms contain electric charges that move with acceleration in a limited volume of the atom. When radiating, the charge loses energy in the form of electromagnetic radiation. This means that the stationary existence of atoms is impossible. Nevertheless, the established patterns indicated that the spectral radiation of atoms is the result of as yet unknown processes inside the atom.

Spectral analysis of the radiation emitted by atoms provides extensive information about their structure and properties. Usually observed emission of light from hot monatomic gases (or low density vapors) or when electrical discharge in gases.

The emission spectrum of atoms consists of individual discrete lines, which are characterized by wavelength or frequency v = c/X. Along with emission spectra, there are absorption spectra, which are observed when radiation with a continuous spectrum (“white” light) is passed through cold vapors. Absorption lines are characterized by the same wavelength as emission lines. Therefore they say that the emission and absorption lines of atoms mutually invertible ( Kirchhoff, 1859).

In spectroscopy, it is more convenient to use not the radiation wavelength, but reciprocal v = l/X, which is called spectroscopic wave number, or simply wave number (Stoney, 1871). This value shows how many wavelengths fit per unit length.

Using experimental data, the Swiss physicist Ritz in 1908 found an empirical rule called combinational principle , according to which there is a system spectral terms, or simply terms, T p And T, the difference between which determines the spectroscopic wave number of a certain spectral line:

Therms are considered positive. Their value should decrease as the number increases P(and l,). Since the number of emission lines is infinite, the number of terms is also infinite. Let's fix an integer P. If we consider the number n as a variable with the values ​​n+ 1, n + 2, n + 3,..., then, according to formula (1.8), a series of numbers arises, which correspond to a system of spectral lines called spectral series. A spectral series is a set of spectral lines located in a certain regular sequence, and the intensity of which also varies according to a certain law. At l,-o term T->0. Corresponding wave number v n = T p called border of this series. When approaching the boundary, the spectral lines become denser, i.e., the difference in wavelengths between them tends to zero. The intensity of the lines also decreases. The series boundary follows continuous spectrum. The totality of all spectral series forms the spectrum of the atom under consideration.

The combination principle (1.8) also has a different form. If yaya =T-T And y = T-T - wave numbers of two spectra

LL| P L| PP 2 P *

tral lines of the same series of some atom, then the difference of these wave numbers (for l, > l 2):

represents the wave number of a spectral line of some other series of the same atom. At the same time, not every possible combination line is actually observed in experiment.

The combination principle was completely incomprehensible at one time and was considered fun game numbers. Only Niels Bohr in 1913 saw in this “game” a manifestation of deep internal patterns atom. For most atoms, analytical expressions for terms are unknown. Approximate formulas were selected by analyzing experimental data. For the hydrogen atom, such formulas turned out to be accurate. In 1885, Balmer showed that the wavelengths of the four visible lines observed in the spectrum of the hydrogen atom are

H Q, Нр, Н у, H ft (Fig. 1.6), which were first measured by Angstrom (1868), with to a large extent accuracy can be calculated using the formula

where number l = 3,4, 5, 6,.... Constant B= 3645.6-10 8 cm was determined empirically. For the wave number, the formula follows from (1.10)

Where R- empirical Rydberg constant (1890), R = 4/B. For the hydrogen atom the Rydberg constant is equal to

From formula (1.11) it is clear that the term for the hydrogen atom has a simple expression:

Consequently, for the wave numbers of the spectral series of the hydrogen atom, generalized Balter formula:

This formula correctly describes the spectral series of the hydrogen atom discovered in the experiment:

Balter series(l = 2, l, = 3, 4, 5, ...) - in the visible and near ultraviolet parts of the spectrum X = (6562...3646)* 10" 8 cm:

Lyman series(1914) (l = 1, l, = 2, 3, 4, ...) - in the ultraviolet part of the spectrum A = (1216...913)-10“ 8 cm:

Paschen series(1908) (l = 3, l, =4, 5, 6,...) - in the infrared part of the spectrum X = 1.88...0.82 microns:

series Brackett(1922) (l = 4, l, = 5, 6, 7, ...) - in the far infrared part of the spectrum X. = 4.05 ... 1.46 μm:

Pfund series(1924) (l = 5, l, =6, 7, 8,...) - in the far infrared part of the spectrum X = 7.5...2.28 microns:

Humphrey series(1952) (l = 6, l, = 7, 8,...) - in the far infrared part of the spectrum X = 12.5...3.3 µm:

The boundary of each series is determined by l, the head line of this series.

1. Find the limiting wavelengths of the spectral series of the hydrogen atom.

Answer. X t = n 1 /R. f/

2. Determine the head lines of the spectral series.

Answer. X^ =l 2 (l + 1) 2 /i (2l + 1).

3. Determine the limiting wavelengths between which the spectral lines of the Balmer series are located.

ANSWER: Xf = 3647-10" 8 cm, X^ = 6565-10' 8 cm.

4. Determine the classical spectrum of the hydrogen atom.

Solution. An electron together with a nucleus can be considered as electric dipole, whose radius vector changes periodically. Projections of the radius vector of the electron onto the Cartesian axes are also periodic functions, which, in general, can be represented as series

Fourier: *(/)= ^2 , y(t)= I^e^ , where A s, B s- constants;

co is the frequency of electron revolution around the nucleus, determined by Kepler's third law. Average over the period 7'=2l/o) dipole radiation intensity

is determined by the formula: I =----(x 2 +y 2 where x 2 = - G dtx2. From here barely

6L? 0 C 3 V >T.J.

blows: / = ---((/I 2 + 5 2)w 4 + (l 2 + B)(2В)(3ш) 4 +...) Evil 0 s 3

Thus, the spectrum contains the frequency o and its harmonics 2o), 30,... and represents a series equally spaced lines. This contradicts experiment.