Another atomic effect associated with specific properties of the nucleus is the splitting of atomic energy levels as a result of the interaction of electrons with the spin of the nucleus - called hyperfine level structure. Due to the weakness of this interaction, the intervals of this structure are very small, including in comparison with the intervals of the fine structure. Therefore, the hyperfine structure must be considered for each of the fine structure components separately.

We will denote the spin of the nucleus in this section (in accordance with the custom in atomic spectroscopy) by i, retaining the designation J for the total moment of the electron shell of the atom. We denote the total moment of the atom (together with the nucleus) as . Each component of the hyperfine structure is characterized by a certain value of this moment.

According to the general rules for adding moments, the quantum number F takes the values

so that each level with a given J is split into an (if ) or (if ) component.

Since the average distances of electrons in an atom are large compared to the radius R of the nucleus, the main role in hyperfine splitting is played by the interaction of electrons with the multipole moments of the nucleus of the lowest orders. These are the magnetic dipole and electric quadrupole moments (the average dipole moment is zero - see § 75).

The magnetic moment of the nucleus is of the order of magnitude where is the velocity of the nucleons in the nucleus. The energy of its interaction with the magnetic moment of the electron is of the order of

The quadrupole moment of a nucleus is the energy of interaction of the field it creates with the charge of an electron of the order of

Comparing (121.2) and (121.3), we see that the magnetic interaction (and therefore the level splitting it causes) is times greater than the quadrupole interaction; although the ratio is relatively small, the ratio is large.

The operator of the magnetic interaction of electrons with the nucleus has the form

(similar to electron spin-orbit interaction). The dependence of the level splitting it causes on F is therefore given by the expression

(121,5)

The operator of the quadrupole interaction of electrons with the nucleus is composed of the operator of the quadrupole momentum tensor of the nucleus and the components of the electron momentum vector J. It is proportional to the scalar composed of these operators

that is, it has the form

here it is taken into account that it is expressed through the nuclear spin operator by a formula of the form (75.2). Having calculated the eigenvalues ​​of the operator (121.6) (this is done exactly as in the calculations in Problem 1 § 84), we find that the dependence of the quadrupole hyperfine level splitting on the quantum number F is given by the expression

The effect of magnetic hyperfine splitting is especially noticeable for levels associated with an outer electron located in the -state, due to the relatively high probability of finding such an electron near the nucleus.

Let us calculate the hyperfine splitting for an atom containing one external electron (E. Fermi, 1930). This electron is described by a spherically symmetrical wave function of its motion in the self-consistent field of the remaining electrons and the nucleus.

We will look for the operator of interaction of an electron with a nucleus as an operator of energy - the magnetic moment of the nucleus in the magnetic field created (at the origin) by the electron. According to the well-known formula of electrodynamics, this field

where j is the operator of the current density created by the moving electron spin, and is the radius vector from the center to the element. According to (115.4), we have

( - Bohr magneton). Having written and performed integration, we find

Finally, for the interaction operator we have

If the total moment of the atom is, then hyperfine splitting leads to the appearance of a doublet; according to (121.5) and (121.9) we find for the distance between two levels of the doublet

Since the value is proportional (see § 71), the magnitude of this splitting increases in proportion to the atomic number.

Tasks

1. Calculate hyperfine splitting (associated with magnetic interaction) for an atom containing one electron with orbital momentum I beyond closed shells (E. Fermi, 1930).

Solution. The vector potential and the strength of the magnetic field created by the magnetic moment of the nucleus are equal

When studied using high-resolution spectral instruments, the lines of most elements reveal a complex structure, much narrower than the multiplet (fine) line structure. Its occurrence is associated with the interaction of the magnetic moments of nuclei with the electron shell, leading to hyperfine structure of levels and with isotopic shift of levels .

The magnetic moments of nuclei are associated with the presence of their mechanical angular momentum (spins). Nuclear spin is quantized according to the general rules of quantization of mechanical moments. If the mass number of the nucleus A is even, the spin quantum number I is an integer; if A is odd, the number I is a half-integer. A large group of so-called even-even nuclei, which have an even number of both protons and neutrons, have zero spin and zero magnetic moment. The spectral lines of even-even isotopes do not have a hyperfine structure. The remaining isotopes have non-zero mechanical and magnetic moments.

By analogy with the magnetic moments created in atoms by electrons and , the magnetic moment of the nucleus can be represented in the form

where is the proton mass, the so-called nuclear factor, which takes into account the structure of nuclear shells (in order of magnitude it is equal to unity). The unit of measurement for nuclear moments is the nuclear magneton:

The nuclear magneton is =1836 times smaller than the Bohr magneton. The small value of the magnetic moments of nuclei compared to the magnetic moments of electrons in an atom explains the narrowness of the hyperfine structure of spectral lines, which is an order of magnitude from multiplet splitting.

The energy of interaction of the magnetic moment of the nucleus with the electrons of the atom is equal to

where is the strength of the magnetic field created by electrons at the point where the nucleus is located.

Calculations lead to the formula

Here A is some constant value for a given level, F is the quantum number of the total angular momentum of the nucleus and electron shell

which takes values

F=J+I, J+I-1,…, |J-I|. (7.6)

Hyperfine splitting increases with increasing nuclear charge Z, as well as with increasing degree of ionization of the atom, approximately proportional to , where is the charge of the atomic residue. If for light elements the hyperfine structure is extremely narrow (on the order of hundredths), then for heavy elements such as Hg, T1, Pb, Bi, it reaches a value in the case of neutral atoms and several in the case of ions.

As an example in Fig. Figure 7.1 shows a diagram of the hyperfine splitting of levels and lines of the sodium resonance doublet (transition). Sodium (Z=11) has the only stable isotope with mass number A=23. The nucleus belongs to the group of odd-even nuclei and has spin I=3/2. The magnetic moment of the nucleus is 2.217. The common lower level of both components of the doublet is split into two ultrafine levels with F=1 and 2. The level into four sublevels (F=0, 1, 2, 3). The level splitting value is 0.095. The splitting of the upper levels is much smaller: for the level it is equal to 0.006, the full splitting for the level is 0.0035.

Studies of the hyperfine structure of spectral lines make it possible to determine such important quantities as the mechanical and magnetic moments of nuclei.

An example of determining the nuclear spin value The nuclear moment of thallium and the structure of the line with = 535.046 nm can be calculated directly from the number of components. The complete picture of level splitting is presented in Fig. 7.2. Thallium has two isotopes: and , the percentage of which in the natural mixture is: –29.50% and – 70.50%. The lines of both thallium isotopes experience an isotopic shift equal to nm, respectively. For both isotopes, the nuclear spin is I=1/2. According to the splitting scheme, one should expect that the thallium line with nm, which appears during the transition from level to level, consists of three hyperfine splitting components with an intensity ratio of 2:5:1, since the level consists of two sublevels with a distance between the sublevels, and the level also splits into two sublevels. The distance between sublevels is negligible, so spectroscopic observations reveal only two hyperfine splitting components for each isotope separately, located at a distance of nm (). The number of components shows that the spin of the thallium nucleus is I =1/2, since at J = 1/2 the number of components is 2I+1 =2. Quadrupole moment Q = 0. This indicates that the splitting of the term is very small and cannot be resolved spectroscopically. The anomalously narrow splitting of the term is explained by the fact that it is perturbed by the configuration. The total number of components of this line is four. Components A and B belong to a more common isotope, and components B belong to a rarer one. Both groups of components are shifted relative to each other by , with the heavier isotope corresponding to a shift to the violet side of the spectrum. Measuring the intensity ratio of components A: or B: b allows one to determine the content of isotopes in a natural mixture.

7.4. Description of installation.

HFS of spectral lines can only be observed when using high-resolution instruments, for example, a Fabry-Perot interferometer (FPI). An FPI is a device with a narrow spectral interval (for example, the free spectral interval for λ = 500 nm in an FPI with a distance between mirrors t = 5 mm is Δλ = 0.025 nm, within this interval Δλ it is possible to study the fine and ultrafine structure). As a rule, FPI is used in combination with a spectral device for preliminary monochromatization. This monochromatization can be carried out either before the light flux enters the interferometer, or after passing through the interferometer.

The optical scheme for studying the HFS of spectral lines is shown in Fig. 7.3.

Light source 1 (high-frequency electrodeless VSB lamp with metal vapors) is projected by lens 2 (F = 75 mm) onto the FPI (3). The interference pattern, localized at infinity, in the form of rings is projected by an achromatic condenser 4 (F=150mm) into the plane of the entrance slit 5 of the spectrograph (6,7,8 collimator, Cornu prism, chamber lens of the spectrograph). The central part of the concentric rings is cut out by the slit (5) of the spectrograph and the image of the picture is transferred to the focal plane 9, where it is recorded on a photographic plate. In the case of a line spectrum, the picture will consist of spectral lines crossed in height by interference maxima and minima. This picture can be observed visually from the cassette part through a magnifying glass. With proper adjustment of the IT, the picture has a symmetrical appearance (Fig. 7.4.).

ULTRA-FINE STRUCTURE(hyperfine splitting) of energy levels - splitting the energy levels of an atom, molecule or crystal into several. sublevels, due to the interaction of magnetic. moment of the core with magnetic field created by ch. arr. electrons, as well as interaction with inhomogeneous intra-atomic electric. field. Due to hyperfine level splitting in optical. in the spectra of atoms and molecules, instead of one spectral line, a group of very close lines appears - S. s. spectral lines.

If the nucleus of an atom or one of the atomic nuclei of a molecule has a spin I, then each sublevel of S. s. characterized by total moment F = J+ 7, where J is the vector sum of the total electron momentum and the momentum of the orbital motion of the nuclei. F full moment values ​​run through F = |J - I|, |J - I| + 1,..., J+I (J And I- quantum numbers of complete mechanical electronic and nuclear spin moments). When the number of sublevels is 2I + 1, and when J< I it is equal 2J+ 1. The energy of the sublevel is written as:

where is the energy of the level neglecting the S. s., is the energy of the magnet. dipole-dipole interaction, - electrical energy. quadrupole interaction.

In atoms and ions basic. Magnet plays a role. interaction, energy of which

constant A(Hz) is determined by averaging over the state with the total moment F of the magnetic operator. interaction of electrons with nuclear moment The magnitude of the interaction is proportional. nuclear magneton" , where is the Bohr magneton, T- electron mass and m р - proton mass. The distance between the sublevels of S. s. in an atom is approximately 1000 times less than the distance between components fine structure. Characteristic values ​​of hyperfine splitting for the order of one or several. GHz. The hyperfine splitting of excited energy levels decreases proportionally. binding energy of an excited electron to the power of 3/2 and rapidly decreases with increasing orbital momentum of the electron. In the case of hydrogen-like atoms (H, He +, etc.)

Where - Rydberg constant, - fine structure constant, Z- nuclear charge (in electron units), P And l- principal and orbital quantum numbers, gI- nuclear Lande multiplier.Electric. quadrupole interaction exists for nonspherical. cores s. It gives corrections to the energy of the sublevels of the atom

Constant IN is determined by averaging over the state with the total moment F of the quadrupole interaction operator

Where i, k = 1, 2, 3, - Kronecker symbol.Usually the quadrupole interaction constant IN one to one and a half orders of magnitude less than the constant A. Quadrupole interaction leads to violation of the Lande interval rule.

For dipole transitions between sublevels of the system. different levels are performed selection rules:. Between the sublevels of S. s. Magnets are allowed at the same level. dipole transitions with the above selection rules, as well as electrical quadrupole transitions with selection rules.

Almost all molecules in the ground electronic state have a total mechanical the moment of electrons is zero and magnetic. S. s. oscillatory-rotate. energy levels ch. arr. associated with the rotation of the molecule. In the case of diatomic, linear polyatomic molecules and molecules of the symmetrical top type (see. Molecule), containing one nucleus with spin I on the axis of the molecule,

Where J and K- quantum numbers of the total rotation. moment and its projection onto the top axis, respectively. Magn. splitting ranges from 1-100 kHz. If several people have spin. nuclei of the molecule, then due to the magnetic interaction of nuclear moments arise complementary. splitting order several. kHz. Magnetic S. s. The energy levels of molecules with an electronic moment are of the same order as for atoms.

If the molecule in its state contains a nucleus c on its axis, Ch. quadrupole splitting plays a role:

where (Hz) is a constant characteristic of the data level TO And J. The magnitudes of quadrupole splittings are tens and hundreds of MHz.

In solutions, glasses and crystals of S. s. may, for example, have energy levels of impurity ions, free radicals, electrons localized on lattice defects.

Diff. chemical isotopes elements have different nuclear spin values, and their lines are isotopic. shift. Therefore, the spectra of different isotopes and synthetic substances often overlap. spectral lines is further complicated.

Lit.: Townes Ch., Shavlov A., Radiospectroscopy, trans. from English, M., 1959; Sobelman I.I., Introduction to the theory of atomic spectra, Moscow, 1977; Armstrong L. jr., Theory of the hyperfine structure of free atoms, N.Y.-, 1971; P a d ts i g A. A., S M i r n o v B. M., Parameters of atoms and atomic ions. Directory, 2nd ed., M., 1986. E. A. Yukov.

9. Compare the obtained value with the theoretical one, calculated through universal constants.

The report must contain:

1. Optical design of the spectrometer with a prism and a rotating prism;

2. Table of measurements of deviation angles of lines - mercury reference points and their average values;

3. Table of measurements of deviation angles of hydrogen lines and their average values;

4. The values ​​of the found frequencies of hydrogen lines and the interpolation formulas used for the calculations;

5. Systems of equations used to determine the Rydberg constant using the least squares method;

6. The obtained value of the Rydberg constant and its value calculated from the universal constants.

3.5.2. Spectroscopic determination of nuclear moments

3.5.2.1. Experimental determination of parameters of hyperfine splitting of spectral lines.

To measure the ultrafine structure of spectral lines, it is necessary to use spectral instruments with high resolving power, therefore in this work we use a spectral instrument with crossed dispersion, in which a Fabry-Perot interferometer is placed inside a prism spectrograph (see Fig. 3.5.1 and section 2.4.3.2,

rice. 2.4.11).

The dispersion of a prism spectrograph is sufficient to separate spectral emission lines caused by transitions of the valence electron in an alkali metal atom, but is completely insufficient to resolve the hyperfine structure of each of these lines. Therefore, if we used only a prism spectrograph, we would obtain an ordinary emission spectrum on a photographic plate, in which the components of the hyperfine structure would merge into one line, the spectral width of which is determined only by the resolution of the ICP51.

The Fabry-Perot interferometer makes it possible to obtain an interference pattern within each spectral line, which is a sequence of interference rings. The angular diameter of these rings θ, as is known from the theory of the Fabry-Perot interferometer, is determined by the ratio of the thickness of the standard air layer t and the wavelength λ:

		θ k = k

where k is the interference order for a given ring.

Thus, each spectral line is not just a geometric image of the entrance slit, constructed by the optical system of the spectrograph in the plane of the photographic plate, each of these images now turns out to be intersected by segments of interference rings. If there is no hyperfine splitting, then within a given spectral line one system of rings corresponding to different orders of interference will be observed.

If within a given spectral line there are two components with different wavelengths (hyperfine splitting), then the interference pattern will be two systems of rings for wavelengths λ and λ ", shown in Fig. 3.5.2 with solid and dotted lines, respectively.

Rice. 3.5.2. Interference structure of a spectral line consisting of two close components.

The linear diameter of the interference rings d in the small angle approximation is related to the angular diameter θ by the relation:

d = θ×F 2,

where F 2 is the focal length of the spectrograph camera lens.

Let us obtain expressions relating the angular and linear diameters of the interference rings to the wavelength of the radiation that forms the interference pattern in the Fabry-Perot interferometer.

In the small angle approximation cos θ 2 k ≈ 1− θ 8 k and for two lengths

waves λ and λ "the conditions for the interference maximum of the kth order will be written accordingly:

						4λ"
θk = 8	−k			θ" k = 8	−k

From here, for the difference between the wavelengths of the two components, we obtain:

d λ = λ" −λ =			(θ k 2		− θ" k 2 )
The angular diameter (k +1) of the 1st order of wavelength is determined by
ratio:
	8 − (k +1)
k+ 1

From (3.5.9) and (3.5.11) we obtain:

		= θ2			− θ2
					k+ 1
Excluding t						from (3.5.10)-(3.5.12) we obtain:
d λ =				θk 2 − θ" k 2
			k θ2 − θ2
						k+ 1

At small angles, the order of interference is given by the relation

k = 2 λ t (see (3.5.8)), so equality (3.5.13) takes the form:

d λ =				θk 2 − θ" k 2
	2 t θ 2				− θ2
					k+ 1
Moving on to wave numbers ν =								We get:
		1 d k 2 − d "k 2
d ν =
				− d 2
				k+ 1

Now, to determine d ~ ν, we need to measure the linear diameters of two systems of interference rings for two components of the hyperfine structure inside the spectral line under study. To increase the accuracy of determining d ~ ν, it makes sense to measure the diameters of the rings, starting from the second and ending with the fifth. Further rings are located close to each other and the error in determining the difference in the squares of the diameters of the rings grows very quickly. You can average the entire right-hand side (3.5.16), or separately the numerator and denominator.

3.5.2.2. Determination of nuclear magnetic moment

In this work, it is proposed to determine the values ​​of the splitting of the ground state 52 S 1 2 of the stable isotope Rb 87 by super-

Although we have completed the task of finding the energy levels of the ground state of hydrogen, we will still continue to study this interesting system. To say something else about it, for example to calculate the rate at which a hydrogen atom absorbs or emits radio waves of length 21 cm, you need to know what happens to him when he is outraged. We need to do what we did with the ammonia molecule - after we found the energy levels, we went further and found out what happens when the molecule is in an electrical field. And after this it was not difficult to imagine the influence of the electric field of a radio wave. In the case of the hydrogen atom, the electric field does nothing with the levels, except that it shifts them all by some constant value proportional to the square of the field, and this is not interesting to us, because it does not change differences energies. This time it's important magnetnew field. This means that the next step is to write the Hamiltonian for the more complex case when the atom sits in an external magnetic field.

What is this Hamiltonian? We will simply tell you the answer, because we cannot give any “proof”, except to say that this is exactly how the atom is structured.

The Hamiltonian has the form

Now it consists of three parts. First member A(σ e ·σ p) represents the magnetic interaction between electron and proton; it is the same as if there was no magnetic field. The influence of the external magnetic field is manifested in the remaining two terms. Second term (— μ e σ e· B) is the energy that an electron would have in a magnetic field if it were alone there. In the same way, the last term (- μ р σ р ·В) would be the energy of a single proton. According to classical physics, the energy of both of them together would be the sum of their energies; According to quantum mechanics, this is also correct. The interaction energy arising due to the presence of a magnetic field is simply the sum of the energies of interaction of an electron with a magnetic field and a proton with the same field, expressed through sigma operators. In quantum mechanics, these terms are not actually energies, but referring to the classical formulas for energy helps to remember the rules for writing the Hamiltonian. Be that as it may, (10.27) is the correct Hamiltonian.

Now you need to go back to the beginning and solve the whole problem again. But most of the work has already been done; we just need to add the effects caused by the new members. Let us assume that the magnetic field B is constant and directed along z. Then to our old Hamiltonian operator N you need to add two new pieces; let's designate them N′:

Look how convenient it is! The operator H′, acting on each state, simply gives a number multiplied by the same state. In the matrix<¡|H′| j>there is therefore only diagonal elements, and one can simply add the coefficients from (10.28) to the corresponding diagonal terms in (10.13), so that the Hamiltonian equations (10.14) become

The form of the equations has not changed, only the coefficients have changed. And bye IN does not change over time, you can do everything the same as before.
Substituting WITH= a l e-(¡/h)Et, we get

Fortunately, the first and fourth equations are still independent of the others, so the same technique will be used again. One solution is the state |/>, for which

The other two equations require more work because the coefficients of a 2 and a 3 are no longer equal to each other. But they are very similar to the pair of equations that we wrote for the ammonia molecule. Looking back at equations (7.20) and (7.21), the following analogy can be drawn (remember that the subscripts 1 and 2 there correspond to subscripts 2 and 3 here):

Previously, energies were given by formula (7.25), which had the form

In Chapter 7 we used to call these energies E I and E II, now we will designate them E III And E IV

So, we have found the energies of four stationary states of the hydrogen atom in a constant magnetic field. Let's check our calculations, for which we'll direct IN to zero and see if we get the same energies as in the previous paragraph. You see that everything is fine. At B=0 energy E I, E II And E III contact +A, a E IV - V - 3A. Even our numbering of states is consistent with the previous one. But when we turn on the magnetic field, each energy will begin to change in its own way. Let's see how this happens.

First, recall that the electron μ e negative and almost 1000 times greater μ p, which is positive. This means that μ e +μ р and μ e -μ р are both negative and almost equal to each other. Let's denote them -μ and -μ′:

(AND μ , and μ′ are positive and almost coincide in value with μ e, which is approximately equal to one Bohr magneton.) Our quartet of energies will then turn into

Energy E I initially equal to A and increases linearly with growth IN with speed μ. Energy E II is also equal at first A, but with growth IN linear decreases the slope of its curve is - μ . Changing these levels from IN shown in Fig. 10.3. The figure also shows energy graphs E III And E IV. Their dependence on IN different. At small IN they depend on IN quadratic; At first their slope is zero, and then they begin to bend and when large B approach straight lines with a slope ± μ ′ close to the slope E I And E II.

The shift in atomic energy levels caused by the action of a magnetic field is called Zeeman effect. We say that the curves in FIG. 10.3 show Zeeman splitting ground state of hydrogen. When there is no magnetic field, one simply gets one spectral line from the hyperfine structure of hydrogen. State transitions | IV> and any of the other three occur with the absorption or emission of a photon whose frequency is 1420 MHz:1/h, multiplied by the energy difference 4A. But when the atom is in a magnetic field B, then there are much more lines. Transitions can occur between any two of the four states. This means that if we have atoms in all four states, then energy can be absorbed (or emitted) in any of the six transitions shown in Fig. 10.4 with vertical arrows. Many of these transitions can be observed using the Rabi molecular beam technique, which we described in Chap. 35, § 3 (issue 7).

What causes the transitions? They arise if, along with a strong constant field IN apply a small disturbing magnetic field that varies with time. We observed the same thing under the action of an alternating electric field on an ammonia molecule. Only here the culprit of the transitions is the magnetic field acting on the magnetic moments. But the theoretical calculations are the same as in the case of ammonia. The easiest way to obtain them is to take a disturbing magnetic field rotating in a plane hu, although the same will happen from any oscillating horizontal field. If you insert this perturbing field as an additional term into the Hamiltonian, you get solutions in which the amplitudes change with time, as was the case with the ammonia molecule. This means you can easily and accurately calculate the probability of transition from one state to another. And you will find that all this is consistent with experience.