Study of the spectrum of the hydrogen atom. Studying the spectrum of the hydrogen atom

A spectral line is emitted or absorbed as a result of a transition between two discrete energy levels. The formulas derived in the previous chapter allow us to get an idea of the spectra of the hydrogen atom and hydrogen-like ions.

14.1. Spectral series of the hydrogen atom

A spectral series is a set of transitions with a common lower level. For example, the Lyman series of the hydrogen atom and hydrogen-like ions consists of transitions to the first level: n→ 1, where the main quantum number of the upper level, or its number n, takes values 2, 3, 4, 5, etc., and the Balmer series - transitions n→ 2 for n> 2. Table 14.1.1 shows the names of the first few series of the hydrogen atom.

Table 1 4.1.1 Spectral series of the hydrogen atom

	Title of the series
n → 1	Lyman (Ly)
n → 2	Balmera (H)
n → 3	Pashena (P)
n → 4	Bracket (B)
n → 5	Pfunda (Pf)
n → 6	Humphrey
n → 7	Hansen–Strong

The Lyman series of the hydrogen atom falls entirely within the vacuum ultraviolet region. In the optical range there is the Balmer series, and in the near infrared region there is the Paschen series. The first few transitions of any series are numbered with letters of the Greek alphabet according to the scheme in Table 14.1.2:

Table 14.1.2 Designations of the first lines of the spectral series

Dn

As a result of a spontaneous transition from the upper level i to the bottom j an atom emits a quantum, energy Eij which is equal to the difference

During a radiative transition from j on i a quantum with the same energy is absorbed. In the planetary model of the hydrogen atom, the energy of the levels is calculated using formula (13.5.2), and the charge of the nucleus is equal to unity:

Dividing this formula by hc, we obtain the transition wave number:

The wavelength in vacuum is equal to the reciprocal of the wave number:

As the top level number increases i the transition wavelength decreases monotonically. In this case, the lines move closer together without limit. There is a lower limit to the wavelength of the series, corresponding to the ionization limit. It is usually indicated by the suffix "C" next to the series symbol. Figure 14.1.1 shows schematically

transitions, and in Fig. 14.1.2 - spectral lines of the Lyman series of the hydrogen atom.

The concentration of levels and lines near the ionization boundary is clearly visible.

Using formulas (1.3) and (1.4) with the Rydberg constant (13.6.4), we can calculate the wavelengths for any series of the hydrogen atom. Table 14.1.3 contains information about the first

Table 14.1.3. Lyman series of the hydrogen atom

n		E 12 eV	E 12 , Ry	Wavelength, Å
n		E 12 eV	E 12 , Ry	l exp.	l theory
	Ly a	10. 20	0.75	1215.67	1215.68
	Ly b	12.09	0.89	1025.72	1025.73
	Ly g	12.75	0.94	972.537	972.548
	Ly d	13.05	0.96	949.743	949.754
	LyC	13.60	1.00	______	911.763

lines of the Lyman series. The first column shows the number of the top level number n, in the second - the transition designation. The third and fourth contain the transition energy, respectively, in electronvolts and in Rydbergs. The fifth contains the measured wavelengths of the transitions, the sixth contains their theoretical values, calculated using the planetary model. Radiation with l<2000Å сильно поглощается в земной атмосфере, поэтому длины волн серии Лаймана приведены для вакуума.

Good agreement between theory and experiment indicates the reasonableness of the provisions underlying Bohr's theory. The discrepancy in hundredths of an angstrom is due to relativistic effects, which were mentioned in the previous section. We will look at them below.

Formula (1.4) gives the wavelength in vacuum λvac. . For the optical range (λ > 2000Å), the spectroscopic tables give wavelengths λ atm. , measured in the conditions of the earth's atmosphere. Transition to λ vac. performed by multiplying by the refractive index N:

(1.5) λ vac. = N·λ atm. .

For the refractive index of air at normal humidity, the following empirical formula is valid:

(1.6) N- 1 = 28.71·10 -5 (1+5.67·10 -3 λ 2 a tm.)

Here the atmospheric wavelength is expressed in microns. We can also substitute λvac into the right-hand side of (1.6). : a slight error in wavelength has little effect on the value N – 1.

Information about the Balmer series ( j= 2) are contained in Table 14.1.4. The experimental transition wavelengths in the fifth column are given for

Table 14.1.4 Balmer series of hydrogen

n	Line	Energy of transition		Wavelength . , Å
n	Line	eV		Measured in the atmosphere	Theoretical for vacuum	Theoretical for the atmosphere
	H a	1.89	0.14	6562.80	6564.70	6562.78
	H b	2.55	0.18	4861.32	4862.74	4861.27
	H g	2.86	0.21	4340.60	4341.73	4340.40
	H d	3.02	0.22	4101.73	4102.94	4101.66
		3.40	0.25	______	3647	3646

normal atmospheric conditions. Theoretical wavelengths, corrected refractions using formulas (1.5) and (1.6), are given in the last column. The spectral lines of the Balmer series can be schematically depicted in

Fig.14.1.3. The position of the line is marked with a colored line; above - wavelength in angstroms, below - the accepted designation of the transition. Head line H a is in the red range of the spectrum; it usually ends up being the strongest line of the series. The remaining transitions monotonically weaken as the principal quantum number of the upper number increases. Line H b is located in the blue-green region of the spectrum, and the rest are in the blue and violet regions.

The nature of the Balmer jump

A Balmer jump is a depression of radiation in the spectra of stars at wavelengths shorter than 3700 Å. Figure 14.1.4 shows the recording patterns of the spectra of two stars. red border

photoelectric effect due to the ionization of the hydrogen atom from the second level is marked with a red dotted line ( l=3646Å), and the actual Balmer jump is blue ( l=3700Å). In the lower spectrum it is clearly depression visible near blue lines. For comparison, above is the driving star spectrum, which does not have any features in the range of 3600< l < 3700 Å.

The noticeable discrepancy between the red and blue lines in Fig. 14.1.4 does not allow us to consider the photoelectric effect as the direct cause of the phenomenon under consideration. Here, an important role is played by the superposition of lines of the Balmer series at large values n. Let us calculate the difference in wavelengths ∆λ of two adjacent transitions: i→2 and ( i+1)→2. Let us use formulas (1.3), (1.4) twice for j= 2, replacing the index i on n: For n ? 1 can be neglected compared to n, as well as four compared to ( n+1) 2:

We have obtained a quantitative expression for the above-mentioned unlimited approach of the upper members of any hydrogen series. The last formula for n> 10 has an accuracy of no worse than 5%.

Absorption lines have a certain width, depending on the physical conditions in the star’s atmosphere. As a rough approximation, it can be taken to be 1Å. We will consider two lines indistinguishable if the width of each of them is equal to the distance between the lines. Then from (1.7) it turns out that the merging of lines should occur at n≈15. Approximately this picture is observed in the spectra of real stars. So, the Balmer jump is determined by the merger of the high members of the Balmer series. We will discuss this issue in more detail in chapter seventeen.

Balmer series of deuterium

The nucleus of a heavy isotope of hydrogen - deuterium - consists of a proton and a neutron, and is approximately twice as heavy as the nucleus of a hydrogen atom - a proton. Rydberg constant for deuterium R D (13.6.5) is greater than that of hydrogen R H, so the deuterium lines are shifted to the blue side of the spectrum relative to the hydrogen lines. The wavelengths of the Balmer series of hydrogen and deuterium, expressed in angstroms, are given in Table. 14.1.5.

Table 14.1.5. Wavelengths of the Balmer series of hydrogen and deuterium.

		deuterium
	6562.78
	4861.27
	4340.40
	4101.66

The atomic weight of tritium is approximately three. Its lines also obey the law of the planetary model of the atom. They are blueshifted by approximately 0.6Å relative to the deuterium lines.

14.2. Transitions between highly excited states

Transitions between neighboring levels of a hydrogen atom with numbers n> 60 fall into the centimeter and longer wavelength ranges of the spectrum, which is why they are called “radio lines”. Frequencies of transitions between levels with numbers i And j are obtained from (1.3) if both sides of the formula are divided by Planck’s constant h:

The Rydberg constant, expressed in hertz, is equal to

A formula similar to (2.1) for states with n? 1 can be used not only in the case of hydrogen, but also for any atom. According to the material in the previous chapter, we can write

Where R(Hz) expressed in terms of R∞ (Hz) according to formula (13.8.1), as well as R through R ∞ .

Currently, radio links have become a powerful tool for studying interstellar gas. They are obtained as a result of recombination, that is, the formation of a hydrogen atom during the collision of a proton and an electron with the simultaneous emission of excess energy in the form of a light quantum. Hence their other name follows - recombination radio lines. They are emitted by diffuse and planetary nebulae, regions of neutral hydrogen around regions of ionized hydrogen, and supernova remnants. Emission of radio lines from space objects was detected in the wavelength range from 1 mm to 21 m.

The radio link designation system is similar to the optical transitions of hydrogen. The line is indicated by three symbols. First, the name of the chemical element is written down (in this case, hydrogen), then the number of the lower level and, finally, the Greek letter with which the difference is encrypted j - i:

Designation α β γ δ

Difference j - i 1 2 3 4

For example, H109α denotes the transition from the 110th to the 109th level of hydrogen, and H137β denotes the transition between its 139th and 137th levels. Let us give the frequencies and wavelengths of three transitions of the hydrogen atom, often found in astronomical literature:

Transition H66α H109α H137β

n(MHz)223645008.95005.03

l(cm)1.3405.98535.9900

The H109α and H137β lines are always seen separately, despite the fact that they are very close in the spectrum. This is a consequence of two reasons. Firstly, using radio astronomy methods, wavelengths are measured very accurately: with six and sometimes seven correct signs (in the optical range, no more than five correct signs are usually obtained). Secondly, the lines themselves in quiet regions of the interstellar medium are much narrower than the lines in stellar atmospheres. In rarefied interstellar gas, the only mechanism for line broadening remains the Doppler effect, while in dense stellar atmospheres pressure broadening plays an important role.

The Rydberg constant increases with increasing atomic weight of a chemical element. Therefore, the He109α line is shifted toward higher frequencies than the H109α line. For a similar reason, the frequency of the C109α transition is even higher.

This is illustrated in Fig. 14.2.1, which shows a section of the spectrum of a typical gas nebula (NGC 1795). The horizontal axis shows the frequency, measured in megahertz, and the vertical axis shows the brightness temperature in degrees Kelvin. The field of the figure shows the Doppler velocity of the nebula (–42.3 km/s), which slightly changes the wavelengths of the lines compared to their laboratory values.

14.3. Isoelectronic sequence of hydrogen

According to the definition given in the fourth section of the seventh chapter, ions consisting of a nucleus and one electron are called hydrogen-like. In other words, they are said to belong to the isoelectronic sequence of hydrogen. Their structure is qualitatively reminiscent of a hydrogen atom, and the position of the energy levels of ions whose nuclear charge is not too large ( Z < 10), может быть вычислено по простой формуле (13.5.2). Однако у многозарядных ионов (Z> 20) quantitative differences appear associated with relativistic effects: the dependence of the electron mass on speed and spin-orbit interaction.

Optical transitions of the HeII ion

The charge of the helium nucleus is equal to two, therefore the wavelengths of all spectral series of the HeII ion are four times less than those of similar transitions of the hydrogen atom: for example, the wavelength of the H line a equal to 1640Å.

The Lyman and Balmer HeII series lie in the ultraviolet part of the spectrum; and the Paschen (P) and Bracket (B) series partially fall into the optical range. The most interesting transitions are collected in Table 14.3.1. As with the Balmer series of hydrogen, “atmospheric” wavelengths are given.

Table 14.3.1. Wavelengths of the Paschen and Breckett series of the HeII ion


Designation	P a	P b	B g	B e
Wavelength, Å	4686	3202	5411	4541

The Rydberg constant for helium is:

Let us note an important feature of the HeII ion. From 13.5.2 it follows that the level energy Zn hydrogen-like ion with nuclear charge Z, equal to the level energy n hydrogen atom. Therefore, transitions between even levels are 2 n and 2 m HeII ion and transitions n → m hydrogen atoms have very similar wavelengths. The lack of complete agreement is mainly due to the difference in values R H and R He.

In Fig. Figure 14.3.1 compares the transition schemes of the hydrogen atom (left) and the HeII ion (right). The dotted line indicates HeII transitions that practically coincide with the Balmer lines of hydrogen. Solid lines mark the transitions B γ, B ε and B η, for which there is no pair among the hydrogen lines. The top line of Table 14.3.2 shows the wavelengths of the Bracket HeII series, and the bottom line shows the lines of the Balmer series of hydrogen. Bracket series lines are also called series

Table 14.3.2. Bracket series of the HeII ion and Balmer series of the hydrogen atom

HeII

6560

(6 → 4)

B b

5411

(7 → 4)

B g

4859

(8 → 4)

B d

4541

(9 → 4)

B ε

4339
(10→4)

B ζ

4200
(11 → 4)

Bη

4100

Bθ

B 13

6563

H a

_______

4861

H b

_______

4340

H g

_______

4102

H d

______

Pickering, named after the director of the Harvard Observatory, who first studied them in the spectra of hot stars in the southern sky. Note that the Pickering series was successfully explained precisely within the framework of the planetary model of the atom. Thus, she contributed to the establishment of modern views on the nature of the atom.

The reduced mass is higher for a heavier chemical element, so the level with number 2 m helium ion lies deeper than the level m hydrogen atom. Consequently, the lines of the Brackett HeII series are blueshifted relative to the corresponding transitions of the Balmer series. Relative amount of line shift Dl /l is determined in this case by the ratio of the Rydberg constants:

Absolute value Dl For l= 6560Å is approximately 3Å, in agreement with the data in table (14.3.2).

The HeII lines corresponding to transitions between levels with even numbers overlap with the hydrogen lines, since the line widths are much larger than the distance between them. Typically, hydrogen lines are much stronger than helium lines, but there is one exception - these are Wolf-Rayet type stars. The temperature of their atmospheres exceeds 30,000K, and the helium content in terms of the number of particles is ten times greater than hydrogen. Therefore, there are a lot of helium ions there, but, on the contrary, there is little neutral hydrogen. As a result, in the spectra of Wolf–Rayet stars, all hydrogen lines are observed only as weak additions to the HeII lines. The hydrogen content in stars of this type is estimated by comparing the line depths of the Breckt HeII series with even and odd numbers of the upper level: the first are somewhat larger due to the additional contribution of hydrogen.

In the spectra of normal stars, the strongest absorption lines always remain hydrogen lines if the atmospheric temperature is above 10,000K. In Fig. 14.3.2

The log record of a hot star of spectral class O3 is shown. The lines of the Pickering series and three Balmer lines are clearly visible in the figure.
Another example of the interaction of hydrogen and HeII lines is provided by the P α transition of the HeII ion with a wavelength λ=4686Å. This line in the spectra of stars can be observed as an emission line, while the next member of the Paschen series is l 3202Å - represents a conventional absorption line. The difference in the behavior of the lines is due to the fact that the population of the upper level ( n= 4) lines l 4686 can be significantly increased by absorbing the strong Ly line a hydrogen: the wavelengths of the 2→1 transitions of the hydrogen atom and the 4→2 transitions of the HeII ion are very close. This process does not affect the radiation in the line at all. l 3202Å, in which both levels have odd numbers (transition 5→3). The interaction effect is weakened if the lower level is located high enough, for example, l 5411 and l 4541. The latter is used in the spectral classification of stars as a temperature criterion.

Multiply charged ions

The planetary model, as we have seen, is a very effective tool for studying the hydrogen atom and hydrogen-like ions. However, it remains a very rough approximation to the real structure of atoms and, in particular, multiply charged ions. Table 14.3.3 compares the experimental and theoretical wavelengths of the resonant transition Ly a for several hydrogen-like ions of interest in astronomy. The first row of the table shows

Table 14.3.3. Wavelengths of resonant transitions of hydrogen-like ions

l theor , Å

l exp . , Å

303.78at i =2 and j= 1, and in the third - their experimental values. If, according to Table 14.1.3, the hydrogen atom has a discrepancy with experiment only in the sixth significant digit, then for HeII - in the fifth, for CVI and OVIII ions - in the fourth, and for FeXXVI - already in the third. These differences are due to relativistic effects, which we wrote about at the beginning of the chapter.

Based on (13.7.7), we calculate the difference between the energies of the second and first levels:

The factor in front of the left bracket is equal to the transition energy in the nonrelativistic approximation; it is obtained from (3.1a) at j= 1 and i = 2:

Value Δ E B corresponds to the theoretical wavelength from the second row of table (14.3.3). Now we can clarify the transition wavelength. To do this, compare the relative value of the relativistic correction

with relative difference

numbers from table (14.1.3). The calculation results are collected in table (14.3.4).

Table 14.3.4. Comparison of the relativistic correction with experiment

	HeII		OVIII	FeXXVI
dl	6.6(–5)	6.0(–4)	1.05(–3)	9.5(–3)
dR	6.6(–5)	6.0(–4)	1.06(–3)	1.1(–2)

A comparison of the second and third rows of the table shows that it is possible to obtain good agreement between theory and experiment, even while remaining within the framework of the semiclassical model of circular orbits.

Noticeable discrepancy between dR And dl present in the iron ion. Despite its small value, it cannot be eliminated within the framework of the applied model: calculations using formula (13.7.5) do not lead to an improvement in the result. The reason lies in the fundamental drawback of the planetary model with circular electron orbits: it relates the level energy to only one quantum number. In reality, the upper level of the resonant transition is split into two sublevels. This splitting is called fine structure level. It is this that introduces uncertainty into the transition wavelength. All hydrogen-like ions have a fine structure, and the amount of splitting increases rapidly as the nuclear charge increases. To explain the fine structure we will have to abandon the simple model of circular orbits. Remaining within the framework of semiclassical concepts, let us move on to the model of elliptical orbits, which is called the Bohr–Sommerfeld model.

LABORATORY WORK No. 10

BRIEF THEORY

The purpose of this work is to familiarize yourself with the spectrum of hydrogen and sodium. In the process of performing it, it is necessary to visually observe the visible part of the spectrum, measure the wavelengths and, based on the results of these measurements, determine the Rydberg constant.

The emission spectrum of the hydrogen atom consists of individual sharp lines and stands out for its simplicity. Balmer (1885), Rydberg (1890) and Ritz (1908) established empirically that the spectral lines of hydrogen can be grouped into series, and the wavelengths are expressed with high accuracy by the formula:

where is the wave number; l-wavelength, in vacuum; R= 109677.581 cm -1 - Rydberg constant; n = 1, 2, 3, ... - a natural number, constant for the lines of a given series, which can be considered as the series number; m = n + 1, n + 2, n + 3, ... - a natural number that “numbers” the lines of a given series.

The series with n = 1 (Lyman series) lies entirely in the ultraviolet part of the spectrum. The series corresponding to n = 2 (Balmer series) has the first four lines in the visible region. Series with n = 3 (Paschen), n = 4 (Brackett), n = 5 (Pfund) and so on are in the infrared range.

High-resolution spectroscopy shows that the serial lines (I) have a fine structure; each line consists of several closely spaced components at a distance of hundredths of an angstrom for the visible part of the spectrum.

Bohr's theory. Numerous attempts to explain the line structure of atomic spectra, in particular formula (1), from the point of view of classical physics were unsuccessful. In 1911, Rutherford's experiments established the nuclear model of the atom, which from the point of view of classical mechanics should be considered as a collection of electrons moving around the nucleus. According to the laws of classical electrodynamics, such an atomic model is unstable, since due to the acceleration necessary for curvilinear motion in orbits, electrons must emit energy in the form of electromagnetic waves and, as a result, quickly fall onto the nucleus. In 1913, Bohr, abandoning classical ideas, constructed a theory that was compatible with the nuclear model of the atom and explained the basic patterns in the spectrum of the hydrogen atom and similar atomic systems.

Bohr's theory is based on the following postulates:

1. An atomic system has discrete stable stationary states with a certain energy, which can be treated using ordinary mechanics, but in which the system does not radiate, even if it should radiate according to classical electrodynamics.

2. Radiation occurs during the transition from one stationary state to another in the form of an energy quantum hv monochromatic light (here v– radiation frequency; h= 6.62 10 -27 erg.sec - Planck’s constant).

3. In the special case of motion in circular orbits, only those orbits in which the angular momentum P of the electron is a multiple of h/2p:

Where n = 1, 2, 3,...; m e- electron mass, r n- radius n th orbit; Vn- electron speed per n th orbit.

In accordance with the law of conservation of energy and Bohr's first two postulates, the energy of a radiation quantum during the transition between stationary states with energies E" And E"" equal to

hv= E" - E"" . (3)

If we compare formulas (1) and (3), it is easy to see that the energy of stationary states of the hydrogen atom takes, up to sign, a discrete quantum series of values:

Where c- speed of light.

Consider an atom consisting of a nucleus with a charge Z e and one electron. For hydrogen Z= 1, for singly ionized helium (He+) Z= 2, for doubly ionized lithium (Li++) Z= 3, etc. The strength of the Coulomb interaction between the nucleus and the electron will be equal to:

Where r- the distance between the nucleus and the electron. Under the influence of this force, the electron moves around the nucleus in an elliptical orbit, in particular, in a circle. If we count potential energy U from its value for an electron at infinity, then

When moving in a circle, the centripetal force is equal to

where does kinetic energy come from?

Total Energy

From relations (2) and (7) we find for the radius of a circular stationary orbit

Equality (10) shows that stationary orbits are circles whose radii increase in proportion to the square of the orbit number.

Substituting (10) into (9), we obtain the energy in stationary states (Fig. 2):

Expression (11) coincides with (4), if we put

Value (12) is somewhat different from the value of the Rydberg constant found from spectroscopic measurements. The fact is that when deriving formula (11), we assumed the nucleus to be motionless, whereas, due to the finiteness of its mass, it, together with the electron, moves around their common center of inertia. To take this circumstance into account, it is enough to introduce the reduced mass of the electron and nucleus instead of the electron mass:

Where M- core mass.

Replacing in (12) m e on m, we obtain in the case of the hydrogen atom ( M = Mp):

which is in excellent agreement with experiment. Here R corresponds to an infinitely large mass of the nucleus and coincides with (12).

Expression (14) shows that the Rydberg constant for hydrogen isotopes (deuterium with M d = 2M p and tritium M T = 3M p), due to the difference in the reduced masses, differs from the Rydberg constant Rp for light hydrogen. This is in good agreement with the observed line shift in the spectra of deuterium and tritium compared to the spectrum of hydrogen (isotopic shift).

To describe more subtle effects, for example, the splitting of spectral lines emitted by atoms in an external field, it is not enough to consider only circular orbits. More general stationarity conditions than (2), suitable for elliptical orbits, were given by Sommerfeld in the following form: if a mechanical system with i th degrees of freedom is described by generalized coordinates q i and corresponding generalized impulses p i = ¶T/¶q i, then only those states of the system are stationary for which

Where n i- integer quantum numbers, and integration extends to the entire range of changes q i. In the case of an ellipse described by polar coordinates r And j, we have

Where n j And n r- azimuthal and radial quantum numbers. Due to the constancy of the angular momentum p j= const = p condition (16) gives, as in the case of a circular orbit,

The corresponding calculation shows that the electron energy depends on the amount n j +n r = n according to formula (11). n called the principal quantum number. Because n j = 1, 2, ...n, for a given n, available n elliptical orbits with the same energy (11) and with different angular momentum (18). If we consider the third degree of freedom, then the quantization condition (15) for it leads to the fact that each orbit can be oriented in space not in an arbitrary way, but only in such a way that the projection of the angular momentum onto any fixed direction OZ can take 2 n+ 1 values, multiples h/(2p) :

m = - n j , - n j + 1, . . . . . n j- 1 , n j . (20)

The Bohr–Sommerfeld theory clearly demonstrated the inapplicability of classical physics and the primacy of quantum laws for microscopic systems. She explained the basic patterns in the spectra of hydrogen-like ions, alkali metals, and X-ray spectra. Within its framework, the regularities of the periodic system of elements were explained for the first time. On the other hand, the theory did not provide a consistent explanation for the intensity and polarization of spectral lines. Attempts to construct a theory of the simplest two-electron system—the helium atom—failed. The shortcomings of Bohr's theory are a consequence of its internal inconsistency. Indeed, on the one hand, it attracts ideas of quantization that are alien to classical physics, and on the other hand, it uses classical mechanics to describe stationary states. The most correct picture of intra-atomic physical phenomena was given by a consistent quantum theory - quantum mechanics, in relation to which Bohr's theory was the most important transition stage.

Quantum mechanical description of stationary states. The main difference between quantum mechanics and Bohr's theory is the rejection of the idea of electron motion along a classically defined orbit. In relation to a microparticle, we can talk not about its place on the trajectory, but only about the probability dW find this particle in volume dV, equal

dW = | Y (x, y, z)| 2 dx dy dz, (21)

where Y (x, y, z) is a wave function that obeys the equation of motion of quantum mechanics. In the simplest case, the equation obtained by Schrödinger for stationary states has the form

Where E And U- total and potential energy of a particle with mass m e.

Probability of electron presence in unit volume Y |(x, y, z)| 2, calculated for each point, creates an idea of the electron cloud as a certain statistical distribution of electron charge in space. Each stationary state is characterized by its own electron density distribution, and the transition from one stationary state to another is accompanied by a change in the size and configuration of the electron cloud.

Electron cloud density is a function of distance from the nucleus r. It is interesting to note, for comparison with Bohr's theory, that the maximum radial density of the ground state of the hydrogen atom corresponds to the point r, determined by formula (10), i.e., the greatest probable distance of the electron from the nucleus is exactly equal to the radius of the first orbit in Bohr’s theory (Fig. 1).

As the size of the electron cloud increases, its energy usually increases. E n, characterized by the principal quantum number n. The shape of the electron cloud determines the “orbital” angular momentum р l, characterized by quantum number l.

Rice. 1. Probability distribution for an electron in states:

1 - n = 1, l= 0 and 2 - n = 2, l = 0

The orientation of the cloud determines the projection of the moment p lz in space, characterized by the quantum number m l. In addition to the orbital momentum, the electron has its own angular momentum - spin r s, which can have two orientations in space, which is characterized by two values of the quantum number m s= - 1/2, + 1/2. One can imagine that the spin momentum is caused by the rotation of the electron around its axis (similar to how the Earth rotates around its axis while moving in orbit around the Sun). This simple picture is useful as a visual geometric representation of the possible origin of spin. Only quantum theory can give a strict definition of spin.

According to quantum mechanics, angular momentum and their projections are determined by the following relations:

Note that the Bohr–Sommerfeld quantization rules (18), (19) are an approximation to (23), (24) for large l.

Thus, in order to unambiguously determine the state of an electron in an atom, four physical quantities can be specified E n , p l , p lz , p sl , or, what is the same thing, the quadruple of quantum numbers m, l, m l, m s. The values of these quantum numbers are limited by formulas (23) - (26).

n = 1, 2, 3, 4, ... ; (27)

l = 1, 2, 3, 4, ..., n - 1 ; m l = - l, - l+ 1, ..., 0, ..., l- 1, l;

m s = -1/2 , +1/2 .

Orbital number l= 0, 1, 2, 3, 4, etc. usually denoted by letters s, p, d, f, q and further alphabetically.

By changing the quadruple of quantum numbers, all possible states of the atom can be obtained. The sequence of filling these electronic states is determined by two principles: the Pauli principle and the principle of least energy.

According to the Pauli principle, an atom cannot have two electrons with the same set of quantum numbers. According to the principle of lowest energy, the filling of electronic states occurs from low energy values to higher ones in the sequence

1s < 2s < 2p < 3s < 3p . (28)

In accordance with the Pauli principle and restrictions (27) in states with given n And l cannot be more than 2(2 l+ 1) electrons. Therefore in s-state ( l= 0) there can be no more than two electrons in p-state ( l= 1) – no more than six electrons and so on. In a state with a given quantum principal number n There can be no more than electrons.

A set of states with a given n called the electron shell, a set of states with a given pair of numbers n And l called a subshell. The distribution of electrons in an atom across subshells is called the electronic configuration. For example, electronic configurations of the ground states of hydrogen, lithium, helium, sodium atoms, etc. have the form:

1s 1 (H)

1s 2 (He)

1s 2 2s 1 (Li)

1s 2 2s 2 2p 6 3s 1(Na)

where the superscripts indicate the number of electrons in the corresponding subshells, and the numbers in the row indicate the value of the principal quantum number n. Let us explain the rule for writing electronic configurations using the example of the sodium atom Z= 11. Knowing the maximum number of electrons in states s And p(2 and 6, respectively), we place 11 electrons, following inequality (28) from left to right, then we get 1s 2 2s 2 2p 6 3s 1. The electronic configuration of other atoms is obtained in a similar way.

Rice. 2. Diagram of energy levels and radiative transitions of the hydrogen atom

Wavelengths in the emission spectrum of mercury

PROCEDURE FOR PERFORMANCE OF THE WORK

1. Turn on the power supply of the UM-2 monochromator and the mercury lamp.

2. Using the table, calibrate the monochromator (build a graph).

3. Turn on the gas-discharge tube with sodium and determine the wavelengths in the visible part of the spectrum using the graph.

4. Determine the Rydberg constant for each line and find the average value.

5. Determine the ionization potential of the sodium atom.

TEST QUESTIONS AND TASKS

1. Tell us about the theory of atomic structure created by Bohr.

2. How does Bohr’s theory differ from quantum mechanical theory?

3. What quantum numbers do you know? What is the Pauli principle?

4. Write the Schrödinger equation for a hydrogen-like atom.

5. How is the spectroscopic charge of an electron determined?

6. What is the generalized Balmer formula?

7. Explain the diagrams of energy levels and radiative transitions of the hydrogen and sodium atoms.

Literature

1. Zherebtsov I.P. Basics of Electronics. Leningrad, 1990.

2. Koshkin N.I., Shirkevich M.G. Handbook of elementary physics. –M., 1988.

3. Mirdel K. Electrophysics. – M. 1972

4. Optics and atomic physics: Laboratory workshop in physics / Ed. R.I. Soloukhina. 1976.

5. Pestrov E.G., Lapshin G.M. Quantum electronics. –M. 1988.

6. Workshop on spectroscopy / Ed. L.V. Levshite, –M, 1976.

7. Savelyev I.V. General physics course. –M., T.-2, 3., 1971.

8. Sivukhin D.V. General physics course. T-3, – M., 1990.

9. Trofimova T.I. Physics course. –M., Nauka, 1990.

10. FanoU., Fano L. Physics of atoms and molecules. – M., 1980.

11. Sheftel I.T. Thermistors. – M., 1972

12. Shpolsky E.V. Atomic physics. – M. 1990

13. Yavorsky B.M., Seleznev Yu.A. Physics Reference Guide. –M., 1989.

Educational edition

Alekseev Vadim Petrovich

Paporkov Vladimir Arkadevich

Rybnikova Elena Vladimirovna

Laboratory workshop

Group student

1. Purpose of work 2

2. Description of the setup and methodology of experiment 2

3. Results of work and their analysis 3

4. Conclusions 6

Answers to security questions 7

List of used literature 10

Appendix A 11

1. Purpose of the work

The purpose of the work is to study the emission spectrum of hydrogen atoms and experimentally determine the Rydberg constant.

2. Description of the setup and experimental technique

To study the spectrum of the hydrogen atom, a spectroscope based on a UM-2 prism monochromator is used. The experimental setup diagram is shown in Figure 2.1.

1 - laser; 2 - slot; 3 - screen with millimeter scale

Figure 2.1 – Schematic diagram for observing Fraunhofer diffraction using a laser

Light from source 1 through the entrance slit 2 and lens 3 falls in a parallel beam onto a spectral prism with a high 4. By the prism, the light is decomposed into a spectrum and through lens 6 is directed to the eyepiece 8. When the prism is rotated, different parts of the spectrum appear in the center of the field of view. The prism is rotated using drum 5, on which a scale in degrees is printed. By rotating the drum, the spectral line is brought to the pointer arrow 7 located in the eyepiece, and the reading on the drum scale is recorded.

The light source in this work is a gas-discharge hydrogen tube and a high-pressure mercury lamp DRSh-250-3.

3. Results of work and their analysis

Table 3.1 – Spectroscope calibration data for the spectrum of mercury*

*Wavelengths of mercury spectral lines taken from Table 5.1 on page 8 of the manual.

Figure 3.1 – Calibration graph

The wavelength values λ of the hydrogen spectral lines are determined from a calibration graph: the ϕ values are plotted on the Y axis, and the corresponding values on the X axis are selected so that the point coincides with the line.

Table 3.2 – Experimental data on the spectrum of the hydrogen atom

Table 3.3 – Reciprocal values of the wavelengths of the spectral lines of hydrogen, principal quantum numbers.

To check the validity of the Balmer formula, a graph of the dependence 1/n/(1/n 2) is plotted.

Figure 3.2 – Graph of linear dependence 1/l(1/n 2)

From the graph we determine the Rydberg constant as the angular coefficient of the linear dependence 1/l/(1/) according to formula (3.1).

Parameters of line 1 in Figure 3.2

The absolute value of the slope K of the straight line is the Rydberg constant R = |K| = 1.108E+07

Absolute error of the found Rydberg constant s(R) = s(K) = 1.057E+05

Table value of the Rydberg constant: 1.097E+07

The difference between the found and tabulated values of the Rydberg constant |1 - R/ |Х100% = 0.98%

In accordance with §8 on page 8 p. the result is recorded with a guarantee.

R = (1.108 ± 0.01) ;

Here e(R) is the relative error, which is calculated using f. (1.2) on page 2 p.

Using the wavelength values obtained from experiment, we will construct a fragment of the energy spectrum of the hydrogen atom.

Transitions observed in experiment: 6s → 2p, 5s → 2p, 4s → 2p, 3s → 2p.

4. Conclusions

During laboratory work, the emission spectrum of atoms was studied

hydrogen. A graph of the linear relationship (1/l)/(1/) was constructed, from which it was possible to determine the Rydberg constant:

R = (1.108 ± 0.01) ;

The error in determining the Rydberg constant was 0.9%.

The results obtained are consistent with theoretical data.

Answers to security questions

1. Explain the principle of operation of a prism spectroscope.

The operating principle of a prism spectroscope is based on the phenomenon of light dispersion. Disintegration of the input light flux into different spectral components.

2. What is the calibration of a spectroscope?

The angle of deflection of rays of monochromatic light by a prism is not proportional to either the wavelength or its frequency. Therefore, dispersive spectral devices must be pre-calibrated using standard light sources. In this laboratory work, the reference light source was a mercury lamp.

The graduation was as follows:

Place a mercury lamp in front of the entrance slit of the spectroscope at a distance of 30-40 cm. Turn on the mercury lamp using the “NETWORK” and “DRSH LAMP” toggle switches. Light the mercury lamp by pressing the “START” button several times and let the lamp warm up for 3-5 minutes. By changing the width of the entrance slit and moving the eyepiece, ensure that the spectral lines visible through the eyepiece are thin and sharp.

Measure the angle of rotation of the drum for various lines of the spectrum of mercury, aligning the lines in sequence with the pointer arrow in the eyepiece. Lines should be drawn to the indicator only on one side in order to reduce the error due to the backlash of the drum.

3. How is the state of an electron in a hydrogen atom determined in quantum mechanics?

Eigenfunctions corresponding to energies En

define the stationary states of the electron in the hydrogen atom and depend on the quantum numbers n, l and m.

The orbital quantum number l for a certain n can take the values l=0, 1, 2, …, n-1. The magnetic quantum number for a given l takes on the values .

4. What is the meaning of the squared modulus of the wave function?

In accordance with the interpretation of the wave function, the square of the modulus of the wave function gives the probability density of finding an electron at various points in space.

5. Write the stationary Schrödinger equation for an electron in a hydrogen atom.

Rnl(r) – radial part of the wave function;

Ylm(u, q) – angular part of the wave function;

n – principal quantum number;

l – orbital quantum number;

m – magnetic quantum number.

6. Give possible states for an electron in a hydrogen atom with n = 3.

For n = 3, the possible states of the electron in the hydrogen atom are: s, p, d.

7. What is the ionization energy of a hydrogen atom called?

The 1s state of an atom is called the ground state. It corresponds to the lowest energy level E1 = -13.6 eV, also called the ground level. All other states and energy levels are called excited. Quantity |E1| is the ionization energy of the hydrogen atom.

8. Prove that the probability density of finding an electron at a distance equal to the Bohr radius is maximum.

The probability of detecting an electron in a spherical layer from r to r+dr is equal to the volume of this layer multiplied by . Probability density of detecting an electron at a distance r from the nucleus

reaches its maximum at r=r0.

The quantity r0, which has the dimension of length, coincides with the radius of the first Bohr orbit. Therefore, in quantum mechanics, the radius of the first Bohr orbit is interpreted as the distance from the nucleus at which the probability of finding an electron is maximum.

9. What selection rule does the orbital quantum number obey and why?

From the law of conservation of angular momentum during the emission and absorption of light by an atom, a selection rule arises for the orbital quantum number l.

10. Indicate the types of transitions for the Lyman and Paschen series.

For the Lyman series: np → 1s (n = 2, 3...).

For the Paschen series: np → 3s, ns → 3p, nd → 3p, np → 3d, nf → 3d (n = 4, 5 ...)

11. Find the short-wave and long-wave boundaries (l1 and l∞) for the Lyman, Balmer, Paschen series.

For the Lyman series: m = 1, n = 2, 3, … ∞.

R = 1.097 ∙ 107 (m-1)

for n = ∞. , l1 = 1/(1.097 ∙ 107) ∙ 109 = 91.2 (nm)

L∞ = 1/(1.097 ∙ 107 ∙ 3/4) ∙ 109 = 121.5 (nm)

For the Balmer series: m = 2, n = 3, 4 … ∞.

R = 1.097 ∙ 107 (m-1)

for n = ∞. , l1 = 1/(1.097 ∙ 107 ∙ 1/4) ∙ 109 = 364.6 (nm)

L∞ = 1/(1.097 ∙ 107 ∙ 0.1389) ∙ 109 = 656.3 (nm)

For the Paschen series: m = 3, n = 4, 5 ... ∞.

R = 1.097 ∙ 107 (m-1)

for n = ∞. , l1 = 1/(1.097 ∙ 107 ∙ 1/9) ∙ 109 = 820.4 (nm)

L∞ = 1/(1.097 ∙ 107 ∙ 0.04861) ∙ 109 = 1875.3 (nm)

Bibliography

, Kirillov spectrum of the hydrogen atom. Guide to laboratory work for students of all specialties. – Tomsk: TUSUR, 2005. – 10 p. Ripp of measurement errors. Guidelines for a laboratory workshop on a physics course for students of all specialties. – Tomsk: FDO, TUSUR, 2006. – 13 p.

Appendix A

The report file is accompanied by a registration file with the results of the experiments phyLab7.reg.

1 In Excel, the parameters of a straight line constructed from given points can be obtained using the LINEST() function, which implements the least squares method (LSM). In the manual, MNC is described on pp. 12–13 f. (10.2)–(10.5).

Send your good work in the knowledge base is simple. Use the form below

Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.

LABORATORYJOB

STUDYING THE SPECTRUM OF THE HYDROGEN ATOM

1. TARGETWORKS

1.1 Study the spectrum of atomic hydrogen in the visible region of the spectrum and measure the wavelengths of hydrogen lines N b, N V, N G, N d .

1.2 Calculate the value of the Rydberg constant.

1.3 According to the found value R calculate Planck's constant h.

2. RANGEHYDROGENANDENERGYLEVELS

2.1 ExperimentsRutherford.Structureatom

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In 1910, Rutherford and his collaborators conducted a series of experiments to observe the scattering of alpha particles as they passed through thin metal foil. The experiment was carried out as follows (Fig. 1). A beam of alpha particles emitted by a radioactive source released through a narrow hole in a container AND, fell on a thin metal foil F. When passing through the foil, alpha particles deviated from their original direction of motion at various angles. Scattered alpha particles hit the screen E, coated with zinc sulphide, and the scintillations (flashes of light) they caused were observed under a microscope M. The microscope and screen could be rotated around an axis passing through the center of the foil, and thus installed at any angle. The entire apparatus was placed in a vacuum chamber to eliminate the scattering of alpha particles by collision with air molecules.

Observations have shown that the bulk of alpha particles deviate from the original direction only by small angles, but at the same time, the scattering angle of a small number of alpha particles turns out to be significantly large and can even reach 180 o. After analyzing the results of the experiment, Rutherford came to the conclusion that such a strong deviation of alpha particles from the original direction is possible only if there is an extremely strong electric field inside the atom, which is created by a charge associated with a large mass. The small fraction of particles scattered at large angles indicates that the positive charge and associated mass are concentrated in a very small volume and the likelihood of a direct hit is low. Based on this conclusion, Rutherford proposed a nuclear model of the atom in 1911. According to Rutherford, an atom is a system of charges, in the center of which there is a heavy positively charged nucleus, having dimensions not exceeding 10 -12 cm, and negatively charged electrons rotate around the nucleus (so as not to fall on the nucleus), the total charge of which is equal in magnitude to the charge of the nucleus . Almost all the mass of an atom is concentrated in the nucleus.

However, the nuclear model turned out to be in conflict with the laws of classical mechanics and electrodynamics. The essence of the contradiction is as follows: an electron, moving along a curved path, must have centripetal acceleration. According to the laws of classical electrodynamics, a charge moving with acceleration must continuously emit electromagnetic waves. The process of radiation is accompanied by a loss of energy, so that the electron (if you follow the classical laws) should gradually descend, moving in a spiral and, ultimately, fall onto the nucleus. Estimates have shown that the time after which an electron must fall onto the nucleus should be approximately 10 -8 s. At the same time, continuously changing the radius of its orbit, it should emit a continuous spectrum, while in experiments with rarefied gases it was established that the spectra of atoms are lined. Thus, a contradiction arose between the ideas about the atom resulting from the results of Rutherford’s experiments and the laws of classical physics, according to which an atom having the indicated structure must be unstable, and its radiation spectrum must be continuous

2.2 PostulatesBora.ElementaryBorovskayatheoryhydrogendnogoatom

A way out of the contradiction that arose between the laws of classical physics and the conclusions arising from the results of Rutherford's experiments was proposed by Niels Bohr, who in 1913 formulated the following postulates: Postulate - a statement accepted without proof, as an axiom. The validity of a particular postulate can be judged by comparing the results obtained when using a particular postulate with experiment. :

1) Of the infinite number of electron orbits possible for an electron in an atom from the point of view of classical mechanics, only a few, called stationary. While on stationary orbit electron Not emits energy (Em waves) Although And moves With acceleration. For a stationary orbit, the angular momentum of the electron must be an integer multiple of the constant value

(-Dirac constant).

Those. the following ratio must be satisfied:

Where m e- electron mass, v-electron speed, r - electron orbit radius, n- an integer that can take the values 1, 2, 3, 4... and is called the principal quantum number.

2) Radiation is emitted or absorbed by an atom in the form of a light quantum of energy during the transition of an electron from one stationary (stable) state to another. The magnitude of the light quantum is equal to the difference in energies of those stationary states E n 1 And E n 2 , between which a quantum jump of an electron occurs:

The same relationship is valid for the case of absorption. Relationship (2) is called rulefrequenciesBora.

2.3 ModelBoraatomhydrogen

Bohr based the model of the hydrogen atom on the planetary model of the Rutherford atom and the postulates already mentioned above. From Bohr's first postulate it follows that only such orbits of motion of an electron around a nucleus are possible for which the angular momentum of the electron is equal to an integer multiple of the Dirac constant (see (1)). Bohr then applied the laws of classical physics. In accordance with Newton's second law, for an electron rotating around a nucleus, the Coulomb force plays the role of a centripetal force and the following relation must be satisfied:

excluding the speed from equations (1) and (3), an expression was obtained for the radii of permissible orbits:

Here n - principal quantum number ( n = 1,2,3…

The radius of the first orbit of a hydrogen atom is called Borovskyfor the sake ofatsom and is equal

The internal energy of an atom is equal to the sum of the kinetic energy of the electron and the potential energy of interaction between the electron and the nucleus (the nucleus, due to its large mass, is considered to be motionless to a first approximation).

So as (see formula (3))

Substituting into (6) the expression r n from (4), we find the allowed values of the internal energy of the atom:

Where n = 1, 2, 3, 4…

When a hydrogen atom transitions from the state n 1 in a state n 2 a photon is emitted.

The inverse wavelength of the emitted light can be calculated using the formula:

2.4 PatternsVatomicspectra

When conducting experimental studies of the emission spectra of hydrogen, Balmer found that hydrogen atoms (like atoms of other elements) emit electromagnetic waves of strictly defined frequencies. Moreover, it turned out that the reciprocal of the wavelength of the spectral line can be calculated as the difference of some two quantities, which are called spectral terms, i.e. the following ratio is valid:

Quantitative processing of experimentally obtained hydrogen spectra showed that the terms can be written as follows:

Where R is the Rydberg constant, and n is an integer that can take a number of integer values 1,2,3... The value of the Rydberg constant obtained experimentally was:

Taking into account the above, the wavelength of any spectral line of hydrogen can be calculated from generalizedformulaBalmera:

where are the numbers n 1 And n 2 can take values: n 1 = 1,2,3...; n 2 = n 1 , n 1 +1, n 1 +2 …

The wavelengths calculated using formula (15) very accurately coincided with the experimentally measured wavelengths in the hydrogen emission spectrum.

Comparing formulas (11) and (15), we can conclude that formula (11) is the same generalized Balmer formula, but obtained theoretically. Therefore, the value of the Rydberg constant can be calculated using the formula:

Numbers n 1 , n 2 - these are quantum numbers, which are the numbers of stationary orbits between which a quantum jump of an electron occurs. If you measure the value of the Rydberg constant experimentally, then using relation (16) you can calculate the Planck constant h.

atomic hydrogen boron rydberg

3. METHODOLOGYPERFORMANCEWORKS

3.1 Workersformulas

Rangeradiation is an important characteristic of a substance, which makes it possible to establish its composition, some characteristics of its structure, and the properties of atoms and molecules.

Gases in the atomic state emit line spectra, which can be divided into spectral series.A spectral series is a set of spectral lines for which the quantum number n 1 (the number of the level to which transitions are made from all higher levels) has the same meaning. The simplest spectrum is that of the hydrogen atom. The wavelengths of its spectral lines are determined by the Balmer formula (15) or (11).

Each series of the spectrum of a hydrogen atom has its own specific value. n 1 . Values n 2 represent a sequential series of integers from n 1 +1 to?. Number n 1 represents the number of the energy level of the atom to which the electron transitions after radiation; n 2 - the number of the level from which an electron passes when an atom emits electromagnetic energy.

According to formula (15 ), The hydrogen emission spectrum can be represented in the form of the following series (see Fig. 2):

Series Lyman (n 1 =1) - ultraviolet part of the spectrum:

Series Balmera (n 1 = 2) - visible part of the spectrum:

Fig. 2. Series of the spectrum of the hydrogen atom

a) energy diagram, b) transition diagram, c) wavelength scale.

Series Pashen (n 1 = 3) - infrared part of the spectrum:

Series Bracket (n 1 = 4) - infrared part of the spectrum:

Series Pfunda(n 1 = 5) - infrared part of the spectrum:

In this paper, we study the first four lines of the Balmer series, corresponding to transitions to the level n 1 = 2. Magnitude n 2 for the first four lines of this series, lying in the visible region, takes values 3, 4, 5, 6. These lines have the following designations:

H b- Red line ( n 2 = 3),

H V- green-blue ( n 2 = 4),

H n- blue( n 2 = 5),

H d- purple ( n 2 = 6).

The experimental determination of the Rydberg constant using lines of the Balmer series can be carried out using the formula obtained on the basis of (15):

The expression for calculating Planck's constant can be obtained by transforming formula (16):

Where m = 9.1 ? 10 -31 kg,e - 1.6 ? 10 -19 Kl,C - 3 ? 10 8 m/With,e 0 =8.8 ? 10 -12 f/ m.

3.2 Conclusionformulascalculationerrors

The expression for calculating the absolute measurement error of the Rydberg constant DR can be obtained by differentiating formula (17). It should be taken into account that the values of quantum numbers n 1 , n 2 are exact and their differentials are zero.

Fig.3. Finding the error DC according to the calibration schedule

The magnitude of the absolute error in determining the wavelength l can be found using a calibration graph of wavelength versus drum division l (ts) (see Fig. 2) . To do this, it is necessary to estimate the error in taking a reading on the drum DC and, as shown in Fig. 3, find the corresponding error Dl at a given wavelength.

However, due to the fact that the values ? are very small, then with the existing scale of the graph l = f(ts) it is not possible to determine the value Dl. That's why Dl is determined with sufficient accuracy using formula (24).

To determine Planck's constant, tabular values of quantities are used m e, e, e 0, C, which are known with an accuracy significantly exceeding the accuracy of determining the Rydberg constant, therefore the relative error in determining h will be equal to:

Where DR- error in determining the Rydberg constant.

3.3 Descriptionlaboratoryinstallations

The light source, in the visible part of the spectrum of which the lines of atomic hydrogen predominate, is an H-shaped glow discharge lamp, powered by a high-voltage rectifier 12. The highest brightness of the spectrum is achieved when the end of the horizontal part of the tube (capillary) serves as the light source.

To measure the wavelengths of spectral lines, a prism monochromator UM-2 is used in this work (Fig. 4). In front of the entrance slit of the monochromator, a hydrogen lamp S and a condenser K move on the optical rail on riders; the condenser serves to concentrate light on the entrance slit of the monochromator (1).

The entrance slot 1 is equipped with a micrometric screw 9, which allows you to open the slot to the desired width. The collimator lens 2 forms a parallel beam of light, which falls further onto the dispersing prism 3. The micrometric screw 8 allows you to move the lens 2 relative to the slit 1 and serves to focus the monochromator.

Fig. 4. Laboratory setup diagram.

Prism 3 is installed on a rotary table 6, which rotates around a vertical axis using a screw 7 with a counting drum. A helical track with degree divisions is applied to the drum. The direction indicator of the drum 11 slides along the track. As the drum rotates, the prism rotates, and in the center of the field of view of the telescope, consisting of a lens 4 and an eyepiece 5, various parts of the spectrum appear. Lens 4 produces an image of entrance slit 1 in its focal plane.

Pointer 10 is located in this plane. To change the brightness of the pointer illumination, there is a regulator and a switch on the monochromator.

The images of the slit produced by different wavelengths of light are spectral lines.

4. ORDERPERFORMANCEWORKS

After reading the description of the laboratory installation, turn it on in the following order:

4.1. Turn the handle "PREPARATION" clockwise until it stops, without applying excessive force.

4.2. Click the button "ON"HIGH." At this point the light will light up NET", instrument arrow "CURRENTDISCHARGE" deviate by 6...8 divisions, a hydrogen lamp discharge will occur.

4.3. Using the condenser adjustment screws, focus the light spot from the hydrogen lamp onto the crosshairs of the cap at the collimator inlet, then remove the cap.

4.4. Find the red, green-blue, blue and violet lines in the spectrum of hydrogen. This region of the spectrum is located approximately in the range of 750...3000 drum divisions. The violet line has weak intensity. Along with the lines of atomic hydrogen, lines of molecular hydrogen are observed in the spectrum of the hydrogen tube in the form of weak red-yellow, green and blue bands. They should not be confused with the clear lines of atomic hydrogen.

Rotating drum 7, align each of the lines with the eyepiece indicator and take the drum count according to indicator 11.

4.5. Repeat this operation three times for each of the four lines of the spectrum, bringing it to the eyepiece pointer from different sides. Record the measurement results (N 1 ... N 3) in table 1.

4.6. After 10 minutes the device will turn off, indicating the shutdown with a bell. If it is necessary to turn it on again, repeat the operations in paragraphs 4.1 and 4.2. To turn off the unit in an emergency, turn the knob "PREPARATION" counterclock-wise. Calculate the tabular values of drum counts for each line using formulas (21…24)

Table 1

CalculationsByresultsmeasurementsare being doneoncomputer

Calculate the tabular values of drum counts for each line using formulas (21…24)

The magnitude of the absolute error that occurs when measuring the number of divisions of the drum is determined by the formula:

The wavelength of each spectrum line can be determined from the monochromator calibration graph. However, it is easier to do this using the interpolation formula:

410.2+5.5493*10 -2 (N avg -753.3)2.060510 -7 (N avg - 753.3) 2 +

1.5700 *10 -8 (N avg -753.3) 3 (23)

The absolute error in determining each of the wavelengths can be calculated using the interpolation formula, having previously differentiated it by N CP:

d = 5.5493-10 -2 dNav- 4.121? 10 -7 (N avg - 753.3) dN avg +

4.7112?10 -8 (N c p - 753.3) 3 dN avg (24)

Now we can begin to calculate the Rydberg and Planck constants using formulas (17) and (18), respectively. The magnitude of the absolute error in determining the Rydberg constant is calculated using formula (19), and then the relative error in determining the Planck constant is calculated using formula (20).

Thus, for each of the spectral lines we obtain our own values of the Rydberg and Planck constants, which, strictly speaking, should be the same for all these lines. However, as a result of errors in wavelength measurements, these values differ slightly from each other.

To obtain a final answer about the value of the constants being determined, it is advisable to proceed as follows. Take their average value as the value of the Rydberg and Planck constants, and take the maximum of the errors as the value of the absolute error in their determination. You just need to remember that the error value is rounded to the first significant digit. The value of the constants is rounded to a figure of the same order as the error. Enter the calculation results in Table 2.

Table 2.

At the end of the calculations, write down the results of the work performed in the form:

R = (R avg ± R)?10 7 1/m

h = (h avg ± h)?10 -34 J s

5. CONTROLQUESTIONS

5.1. What experimental facts is the Bohr model of the hydrogen atom based on?

5.2. State Bohr's postulates.

5.3. What is Balmer's formula?

5.4. What is Rydberg's constant?

5.5. What is the essence of Bohr's theory of the hydrogen atom? Derive a formula for the radius of the first and subsequent Bohr orbits of an electron in a hydrogen atom.

5.6. Derive a formula for the position of electron energy levels in a hydrogen atom.

5.7. What is the energy spectrum of a hydrogen atom? Name the series of spectral lines of the hydrogen atom. What does a particular series of spectral lines of a hydrogen atom represent?

LITERATURE

I.V. Savelyev. General physics course T.3. Ed. M. “Science” 1988.

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Study of the spectrum of the hydrogen atom. Studying the spectrum of the hydrogen atom

14.1. Spectral series of the hydrogen atom

The nature of the Balmer jump

Balmer series of deuterium

14.2. Transitions between highly excited states

14.3. Isoelectronic sequence of hydrogen

Optical transitions of the HeII ion

Multiply charged ions

1. Purpose of the work

2. Description of the setup and experimental technique

3. Results of work and their analysis

4. Conclusions

Answers to security questions

Bibliography

Appendix A

Send your good work in the knowledge base is simple. Use the form below

STUDYING THE SPECTRUM OF THE HYDROGEN ATOM

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