Renovations! Please excuse the mess, but we are in the process of overhauling the site.

ELEMENTARY PROCESSES TO A PLASMA, say, "plasma" Edit, click, copy, click.

Another Gary Search Page, say, "HotBot", click,

Then, "HotBot", Search, click, paste for "plasma" , click, then Find

Searching Entries for Find (Edit, then Find)

Seven types of string searches are available, 1) search for an author, (your name) 2) search for a title (or part of a title), 3) search for a journal, 5) search by year, or 6) search by keyword.

55

The elementary processes in a plasma are essentially the processes of interaction between the component particles: atoms, ions and electrons. The fundamental theory and the cross section data for these elementary processes--appropriate for the low—temperature cesium plasma of the TIC--are outlined briefly in this chapter. A more complete account of the theory of collisions, of the methods for measuring cross sections and of the main experimental results can be found in various monographs

[1-5].

Consider the collision of particles with masses m_a and m_b and with velocities v_a and v_b. As a result of collisions, the particles acquire velocities v_a^’ and v_b^’. It follows from the laws of conservation of energy and of momentum that the velocity of the center of mass

Thus, the result of a collision is the rotation of the relative velocity vector by an angle q _o. This problem is equivalent mathematically to the problem of scattering a single particle with a velocity of g and reduced mass

The angle q _o between velocities g and g^' is determined by the impact parameter b, i.e., the distance from the scattering center to the projected line of the initial trajectory.

trajectories with impact parameters in the range b to b + db. The initial curvilinear sections of these trajectories intersect a plane perpendicular to g within the limits of the elemental area 2p bdb. The directions of vector g’ for these trajectories are located in the solid angle dW ₀ = 2p sinq _od q _o. The value of 2p bdb is called the scattering cross section for solid angle dW ₀. If this scattering cross section is divided by the value of solid angle dW ₀ = 2p sinq _od q _o, then we obtain the differential scattering cross section:

From this formula, knowing the dependence of q _o on b, the value of the differential scattering cross section can be obtained.

Q = p (r_a + r_b) for the collision of two rigid spheres with radii r_a and r_b.

The total cross section Q(g) is a measure of the elastic collision probability. For example, let there be contained in a unit volume the particle a (moving at a velocity of v_a = g) and a fixed particle b. Collision between them during a unit time interval occurs if particle a is in any cylinder of volume v_a Q. If N_b of the particles b are in a unit volume, the probability of collision within unit time is equal to N_b v_a Q. The reciprocal characterizes the average time between collisions and is called the mean free time t = 1/N_b v_a Q. Within time t , particle a goes the distance

We can calculate the changes in the projection of the momentum along the x-axis, D P_x, and the energy D E of a particle with mass m_a for the case where particle b is at rest before the collision (i.e., v_b = 0), and the velocity v_a = g is directed along the x-axis.

*Expressions (1.8) and (1.9) are also valid if particle b moves.

By multiplying D P_x and D E by N_bgs (g, q _o) 2p sin q _odq _o, which equals the probability of collision per unit time, and by integrating over all values of q _o, we find

Here t _P^ab and t E ^ab are the momentum and energy relaxation times:

where the value of Q_P(g)is calculated by the expression

Consider the collision of an electron with a fixed ion. The electron trajectory in the field of the ion located at point 0 is shown in Fig. 3.2. The equation for this trajectory in polar coordinates r and b (where the angle b is read from the straight line which intersects the origin and the electron trajectory at point r = r_min -- see Fig. 3.2) can be obtained from the laws of conservation of angular momentum and of total energy, whose values prior to collision are equal to mbv and mv²/2:

The first two terms in the right side of (2.2) correspond to the kinetic energy, and the value of e²/ r is equal to the electron potential energy in the field of the ion. By integrating (2.1) and (2.2), we find

The value of r_min is calculated using the condition dr/db = 0 or db /dr = ¥ which, according to (2.3), indicates that the expression under the radical approaches zero as r ® r_min.

Formula (2.4) determines the relationship of the angle q and the impact parameter b, which makes it possible to find the differential scattering cross section by using (1.5). This calculated cross section is given by Rutherford’s formula:

An important characteristic of the Coulomb interaction is that it leads to a divergent integral in expression (1.15) for the momentum transfer cross section. This divergence is related to the large contribution of remote collisions. However, the collisions are no longer regarded as binary when the impact parameter exceeds the average distance between the particles: b > n^-1/3. Because of the long-range effect of Coulomb forces, a charged particle scatters simultaneously from many charged particles. This scattering may also be regarded as the effect of electric field fluctuations. Theory shows that these fluctuations can be taken into account by introducing a "shielded" Coulomb potential (instead of the Coulomb potential) as the interaction potential between two charged particles [6]:

is the Debye radius, and T and n are the electron temperature and density.

corresponds to the maximum impact parameter L_D, according to (2.4). The cross section Q_P for a cut-off Coulomb potential can be obtained if the Rutherford formula (2.5) is used during the integration in (1.15) and if the value of q _min instead of 0 is selected as the lower integration limit. By leaving only the values which increase without limit as q _min ® 0, we find

where the value of ln((mv²/e²) L_D) = lnL is called the Coulomb logarithm. Expression (2.10) for the momentum transfer cross section was obtained within the limits of classical mechanics and is valid given the condition e²/hv « l.* This criterion is easily satisfied for conditions of the TIC.

Calculation of elastic electron-atom scattering cross sections requires the use of quantum mechanical scattering theory [9, 10]. In this theory, the electron flow is regarded as a plane monochromatic wave, while the electrons scattered from the atom are regarded as a combination of divergent spherical waves. Each of these scattered waves makes its contribution to the scattering cross section.

Consideration of S-scattering of slow electrons shows that resonance phenomena, which lead to a sharp drop in the scattering cross section, may occur with specific ratios between the electron wavelength and the dimensions of the atom. This phenomenon is called the Ramsauer effect [1]. It is especially pronounced for inert gases atoms (Xe, Kr, and Ar).

The most reliable measurements of the electron scattering cross sections for cesium, obtained by different methods [11-19], are pre-

*In the opposite case of e²/hv » 1, the quantum mechanical expression for the momentum transfer cross section [7] must be used, which in the Born approximation for a shielded potential (2.6) has the form [8]

presented in Fig. 3.3. The results of a theoretical calculation are also shown in the same figure [20]. It is obvious from the figure that the scattering cross section in the electron energy range of most interest for the TIC (E = 0.2 - 0.4 eV) hardly varies with energy and is close to the value of Q^ea = 3.5 x 10 ¹⁴ cm², which is the value used in this monograph. *

*We note that, according to [21, 22], there are Ramsauer type minima which are deeper than previously noted in the low energy range.

The most reliable values for the scattering (charge-exchange) cross sections of cesium ions in cesium for thermal energies (» 0.l eV) are presented in Table 3.1.

The value Q^ia = l x l0^-14 cm² is used here for plasma TIC calculations, where the ion temperature is l000 - 2000° K.

If electron impact leads to a transition in the atom from the excitation level k to the level l, then energy conservation leads to the following relation:

where E_k and E_l are the excitation energies of levels k and l, and v and v’ are the electron velocities before and after collision. The excitation process, when E_l > E_k, is called a collision of the first kind, while the de-excitation process, when E_l < E_k, is called a collision of the second kind.

If the collision of an electron with an atom at level k leads to its ionization, the law of conservation of energy has the form

where E_ion is the ionization energy of the atom in the ground state and v₂ is the velocity of the secondary electron. We must also add the principle of detailed balance [26] to our consideration of these inelastic processes, i.e., that the transition probability of the system going from state 1 to state 2 is equal to the probability of the reverse transition, from state 2 to state 1.

Consider an atom at level k and an electron with velocity v placed in volume V. The flux of electrons, i.e., the number of electrons passing through 1 cm in 1 sec, is equal to v/V, and the probability of atomic excitation to level l is equal to (v/V)Q_kl(v), where Q_kl(v) is the excitation cross section for passing from level k to level l. The probability of ionization and the generation of a second electron in the velocity range of v₂ to v₂ + dv₂ is equal to (v/V)Q_ion(v,k ® v). The cross section Q_ion(v,k ® v₂) for this process, unlike those considered previously, has a dimension of [cm× sec]. It is obvious that the total ionization cross section from the k level, Q_{k ion}(v), is related to the ionization cross section Q_ion(v,k ® v₂) by the relation

(5.3)

The reverse process with respect to the excitation process is de-excitation to level k, the probability of which is (v’/V)Q_lk(v’) and the reverse process with respect to the ionization is the recombination process, whose probability is equal to (v’/V) (v₂/V)Q_rec(v’,v_2® k) where v’/V and v₂/V are the fluxes of electrons colliding with the ion and Q_rec(v’,v_2® k) is the recombination cross section.

However, the probabilities considered above are the transition probabilities not for one, but for a whole group of quantum-mechanical states. In order to use the principle of detailed balance, it is necessary to find the probability of transition for a single quantum-mechanical state, i.e., it is necessary to separate the probabilities obtained above according to the number of finite states, which is equal to the products of the number of possible finite states of the electrons participating in the process. The number of possible states per unit volume of space of free electron velocities is equal to 2Vm³/h³, and that in a spherical layer with radii of v and v + dv is equal to 2p n ²dn x 2Vm³/h³. The number of possible states for the valence electron of an atom at the k-level of excitation is given by the statistical weight g_k of the level.

where the expression n dn = n ’dn ’ has been used, which is based on the laws of conservation (5.1) and (5.2). Equations (5.6) and (5.7) make it possible to find the cross sections for collisions of the second kind and for recombination, if the cross sections for collisions of the first kind and for ionization are known.

It is easiest to analyze the cross sections for inelastic collisions of the first kind and for ionization using the classical Thomson theory [27]. In this theory the valence (optical) electron of the atom is regarded as being at rest and free at the moment of collision between it and the bombarding electron. Thus, the effect of the atomic core (i.e., the nucleus and remaining electrons of the atom) on the collision process and also the motion of the valence electron inside the atom are not taken into account in any way. It is assumed that the inelastic transition of the atom from level k to level 1 occurs if the energy D E acquired by the valence electron is in the range of E_l - E_k < D E < E _l + 1 - E_k. The cross section of these collisions is obtained by integration of the differential Coulomb scattering cross section (Rutherford’s formula (2.5)) over all angles q for which D E (see (1.10)) is located in the range indicated above.* As a result we obtain

In like fashion, a contribution to the ionization cross section Q_ion(n , k ® n 2 ) is made by collisions with energy transfer of

If the angle of deflection of electron velocity, q (corresponding to these values of D E, are calculated by using (1.10), then we obtain

Integrating over the range of n 2 (see (5.3)), we obtain for the total ionization cross section from the k-level

There are also other calculations of the differential cross sections of inelastic processes that utilize classical concepts. Grizinski [28], for example, unlike Thomson, also took into account the motion of the valence electron. The cross section of atomic ionization from level k, according to Grizinski, has the form

where x = mn 2 /2 ( E_ion – E_k).

In the quantum-mechanical theory of collisions [1-3], the expressions easiest to obtain for inelastic cross sections are those using the Born approximation. In this approximation, the interaction of the free and valence electrons is regarded as a slight perturbation on the free electron. For inelastic collision of the first kind, the total cross section in the Born approximation has the form

(y _k and y _l are the wave functions of the valence electron at the k-level and l-levels of excitation), and @P = m(@n - @n ’) the momentum transferred on collision.

If for the exponential in (5.13) the approximate expression exp(i@P@r/h) » 1 + i@P@r/h is substituted, then we obtain the Bethe approximation:

where g = (Ö 3/p )ln {(n + n ’)/(n - n ’)} is the Kramers-Gaunt factor, and @j_kl is the oscillator strength for transition between levels k and l, related to the matrix element of transition by relation (7.14) below.

mass system is described by the same equations as is the motion of an electron in a heavy ion field, but in the case of the collision of two electrons, the mass m/2 must be substituted into (2.5) instead of the mass of the electron.

The Born approximation is sufficiently accurate if the free electron energy is much greater than the energy threshold of the process being considered (for collisions of the first kind this is the difference of the excitation energies for two levels). At the same time, when investigating a low—temperature plasma, the most important values of the cross section are those near this threshold. This is due to the rapid decrease of the number of electrons as energy increases. Therefore, the Born approximation is not always more satisfactory than, for example, Thomson’s or Grizinski’s classical formulas. The use of more exact approximations than Born’s usually requires numerical calculations. The result of such a calculation for cesium, carried out by Hansen [29], is shown in Fig. 3.4 (curve 2). The values of cross sections, calculated using Thomson’s theory (curve 4), are also shown in this figure for comparison.

In view of the complexity of exact calculations of inelastic cross sections, the use of semi-empirical formulas is quite common [10]. A simple theoretical formula (Thomson’s formula or one obtained in the Born approximation) is taken as a basis and various correcting factors are then introduced by comparison with experiment.

Thus, Seaton [31] used formula (5.14) (Bethe-Born approximation) to calculate inelastic cross sections, but he introduced a multiplier in order to bring the calculated values of the cross section into agreement with the experimental data on the cross section for excitation to the first level in hydrogen, helium, and sodium. A similar procedure was used in [30], in which the excitation cross sections near the threshold were reduced tenfold compared to the Bethe—Born formula (5.14). The cross section for excitation to the first level of cesium, given by this formula, is shown in Fig. 3.4 (curve 3) as an example. It is obvious from the figure that the approximate, classical formula (curve 4) for calculating the cross sections in the initial, linear segment of the curve Q(E ), which is the region of much interest for a TIC plasma, yields good agreement with experimental results [31] (curve 1) and fair agreement with the attempt at a more accurate calculation [29] (curve 2).

Some measurements of the cross section for ionization of cesium from the ground state [32-34], and also theoretical ionization cross

sections, calculated by Thomson’s (curve 5) and Grizinski’s (curve 4) formulas, are presented in Fig. 3.5. The experimental data indicate an essentially linear increase of Q_ion(E ) just above threshold, as was also found with excitation cross section.

The slope of the ionization cross section above threshold for other alkali metals (from data of [33]) is presented in Table 3.2.

There is one optical electron outside the filled electron shells in atoms of Cs and other alkali metals. The effective field in which this electron moves is spherically symmetric. The state of the electron in this field is characterized by the set of quantum numbers n, l, j, and m. The number n is called the principal quantum number, the number l is called the azimuth quantum number, and m is called the magnetic quantum number. The number l determines the value of the orbital angular momentum, which is equal to hÖ l(l + 1)?. The number j determines the total angular momentum, which is the vector sum of the orbital and spin moments of the electron. The total electron moment is equal to hÖ j(j + 1)?. The magnetic quantum number m determines the projection of the total angular momentum to some axis. The value of this projection is equal to hm?.

The quantum numbers may assume the following values: n = 1, 2, 3, …; l = 0, 1, 2, …, n – 1; j = l ± 1/2 (at l = 0, j assumes a single value equal to 1/2); m = -j, -(j - 1), …, (j - 1), j. The electron energy for such relatively heavy atoms as Cs is mainly determined by the quantum numbers n and l. The quantum number j determines the minor corrections in the energy level due to interaction of the spin and the orbital angular momentum of the electron. With regard to these corrections, each (n, l)-level, besides levels at l = 0 and j = 1/2, is split into two levels distinguished by the values of j(j₁ = l - 1/2, j₂ = l + 1/2). This splitting is called the fine structure of the levels. The energy difference of this doublet splitting of the levels decreases as the quantum numbers l and n increase.

Electron energy is not dependent on the value of the magnetic quantum number m. The independence of energy from m(m-degeneracy) has a simple physical meaning: all directions in space are equivalent in a field which has central symmetry, and therefore, energy may not be dependent on the spatial orientation of the angular momentum vector.

The multiplicity of degeneracy (the statistical weight) of level i is

In spectroscopy, the accepted designation for the states corresponding to the values l = 0, 1, 2, ... are the letters S, P, D, ….The number which indicates the value of the principal quantum number n is placed before this letter. The quantum number j is indicated as a subscript. For example, a level at n = 3, l = 2 and j = 3/2 is denoted as 3D_3/2. A Grotrian diagram for the Cs levels is shown in Fig. 3.6.

The lowest energy state of an optical electron in a Cs atom is the state 6S_1/2. The lower states are occupied by the electrons of the closed shells. It is interesting to compare the Cs spectrum with that of the simplest atom - the hydrogen atom. The main difference of the H spectrum from that of Cs is in the fact that l-degeneracy, typical for the Coulomb field, occurs in the H spectrum: levels with different values of l, but identical values of n, have the same energy

where Ry = m⁴/2h? » 2.152 x 10^-11 erg is the so-called Rydberg unit of energy (the ionization energy of a hydrogen atom). Disregarding the fine structure, the multiplicity of degeneracy (i.e., the statistical weight) of a hydrogen level is

The effective field in atoms of alkali metals at great distances from the nucleus is essentially the Coulomb field of charge e, because the electrons of closed shells shield the field of the nucleus. There is no shielding at small distances from the nucleus. Therefore, the potential well for electrons in an atom of an alkali metal is deeper than that in a hydrogen atom, and accordingly, the levels of the atom of an alkali metal are shifted downward with respect to their corresponding levels in a hydrogen atom. As n and l increase, the distance of the valence electron from the closed shells increases, and the system of the levels approaches that of the hydrogen system. The foregoing is illustrated by Fig. 3.7, where the Cs and H spectra are compared. It is obvious that the S-, P- and D-levels of Cs are shifted downward with respect to the H levels, whereas the F—, G- etc. levels of Cs essentially coincide

where E_ion is the ionization energy of the atom and n_l is the so-called effective principal quantum number. It is obvious that n_l £ n. Therefore, n_l may be represented in the form of a difference

The value of D _l is called the quantum defect. The quantum defect is determined mainly by the orbital quantum number l and is weakly dependent on n, the quantum defect decreasing as l increases. The value D _l for a number of atoms of alkali metals are presented in Fig. 3.8. It is obvious that D _l increases as the charge of the nucleus Z increases.

Einstein’s coefficients. Absorption or radiation of light by atoms is accompanied by transitions of the atomic electron from one discrete level to another. Consider a lower level k and an upper level l. The number of transitions from level l to level k and from level k to level l, accompanied by radiation and absorption of a light quantum ?hw _lk = 2p ?hn _lk = hn _lk can be written in the following manner:

Here N_k and N_l are the concentrations of the excited atoms, A_lk + B_lkr (w _lk) and B_klr (w _lk) are the transition probabilities per unit time, having the dimension sec^-1, and r (w _lk) is the density of the electromagnetic radiation energy of frequency w _lk = (E_l – E_k)/h?. Formula (7.2) indicates that the probability of absorption of a light quantum ?hw _lk is proportional to the number of quanta in the electromagnetic field. At the same time it is obvious from formula (7.1) that the probability of emission of a quantum is the sum of two terms. The first term A_lk is the probability of spontaneous emission. The second term, B_lkr (w _lk), proportional to radiation density, is the probability of stimulated emission.

The values of A_lk, B_lk, and B_kl are called the Einstein coefficients

If the statistical weights of the levels are equal to g_k and g_l then

where W^spm_lm_k , W^stm_km_l, and Wm_km_l are the probabilities of corresponding transitions from one state m_l with energy E_l to one of the states m_k with energy E_k (by spontaneous or stimulated emission), and of re-turn by absorption.

The transition probabilities in (7.3) are averaged for all the initial states of the electron and are added over all the final states. In this case, all possible electron transitions between the states at levels k and l are taken into account. Wm_km_l and W^stm_lm_k are the probabilities of forward of reverse transitions of the atomic electron , the latter occurring under the effects of an electromagnetic field. Therefore, according to the principle of detailed balance,

From (7.3) and (7.4), we obtain the relationship between coefficients B_lk and B_kl.

(see §1, Chapter 5), the number of radiation events should be equal to the number of absorption events. By setting (7.1) and (7.2) equal, and using (7.5), (7.6), and (7.7), we obtain the following relationship between A_lk and B_lk:

The relationships (7.5) and (7.8) between the Einstein coefficients permit calculation of all the coefficients, if the value of at least one of them is known. We also note that if the radiation density is close to equilibrium and the distance between the atomic levels is ?hw _lk » kT, it is sufficient to take into account only the spontaneous emission processes.

The probabilities of optical transitions. Oscillator strengths. The interaction of the atom with the electromagnetic field must be examined in order to calculate the Einstein coefficient and (using methods of quantum mechanics [35]) the transition probabilities of the valence (optical) electron. Accordingly, using equations (7.3) and (7.8), coefficients B_kl, B_lk and A_lk are calculated. The Einstein coefficients are expressed in terms of the matrix elements of the dipole moment of the optical electron

where g_k = 2j_k +1 and g_l = 2j_l +1 are the statistical weights of the levels.

Matrix elements ?Dm_km_l are distinct from zero only for those transitions for which the following conditions are fulfilled:

The corresponding transitions are shown in Fig. 3.6. Relations (7.12) are called the selection rules for dipole radiation. Those transitions that do not satisfy relations (7.12) are forbidden in the dipole approximation. Some forbidden transitions are also depicted in Fig. 3.6. Summation over m_k and m_l in (7.l0)-(7.ll) denotes summation over the values of the magnetic quantum number at levels k and l. The terms which satisfy the selection rules (7.12) are non-zero for sums over m_k and m_l.

The expression for the probability of spontaneous emission A_lk can be written in the following form:

where 2w _lk²e²/mc³ is three times the probability (relative rate) of oscillator radiation at frequency w _lk in classical electrodynamics (see, for example, [36], and ?f_kl is the mean oscillator strength for absorption. It follows from (7.11) and (7.13) that

The oscillator strengths for different transitions in cesium vapor were calculated in [37]. The calculated oscillator strengths agree quite satisfactorily with the experimental values that were obtained from measurements in an equilibrium cesium and argon-cesium plasma [38, 39].

Radiation of the continuous spectrum. Radiation of the continuous spectrum by free electrons may be caused by accelerations due to electrons interacting with electrons, ions, or atoms, and also by recombination processes. In a TIC plasma, the dominant mechanism is recombination radiation, i.e., radiation from electron transitions from a state in the continuum to discrete levels. This transition is accompanied by radiation of a light quantum with energy ?hw , equal to the sum of the kinetic energy E = mv²/2 of the free electron and of its binding energy E_nion = ?hw _n = E_ion - E_n(Fig. 3.9):

Since free electrons have a continuum of energies, the photons emitted during the recombination process have a continuous spectrum, beginning at the boundaries of the series. The energy emitted per unit volume of plasma per unit time in the frequency range of dw is given by the expression

where D S(w ) is the spectral intensity of radiation, n(E)dE is the number of free electrons with energy from E to E + dE, n(E) is the electron energy distribution function, n_i is the ion density, v = Ö 2E/m_ is the electron velocity, and Q_red^(ph)(E, w )? is the recombination cross section. If the free electrons have a Maxwell-Boltzmann distribution, and n(E) = Nm(E) (see (5.2.14)), then, from (7.17) and (7.18), we obtain

and n = n_e = n_i is the charged particle density in the plasma.

According to [40], the recombination cross section for hydrogen is given by the expression

where a₀ - ?h²/me² = 0.529x10^-8 cm is the Bohr radius, a = e2/?hc = 1/137 is the fine structure constant, w _n = w ₀/n² is the threshold frequency of the recombination continuum (w ₀ = Ry/?h) , and n is the principal quantum number.

Expression (7.21) was obtained in classical approximation and is applicable for transitions to hydrogen levels with a large quantum number n. However, comparison with precise formulas indicates that (7.21) also determines with sufficient accuracy the transition cross section to lower levels, i.e., levels with small values of n. Formula (7.21) may also be used in calculating the recombination cross section to levels of hydrogen - like atoms -cesium, for example - if the effective quantum number n_l is substituted into the formula instead of n. It is obvious from (7.l9)-(7.2l) that the value of C is not dependent on w , and therefore, the intensity of the recombination continuum decreases exponentially away from the threshold, exp[-?h(w - w _n)/kT_e]. The recombination spectra in cesium vapor are easily observed near the boundaries of the 6S, 6P, and 5D series.

is expanded as a Fourier integral, then, instead of a single frequency w _lk, a range of frequencies near the value of w _lk is obtained. The energy distribution over the spectrum, L_lk(w ), is proportional to the square of the Fourier component of this expansion.

The line contour L_lk(w ), obtained during damped oscillations,

has the so-called Lorentz form [36]:

Here L_lk(w ) is normalized to unity, i.e., ¦ ?L_lk(w )dw = 1. The emission intensity in the line at w - w _lk = g ₀/2, is half the maximum value. Therefore, g ₀ is the half-width of the emission line. The emission intensity in the line is D S_lk(w ) = D S_lkI_lk(w ), where D S_lk is the total emission intensity in the line, calculated above (see (7.15)).

Line broadening, when it is related to the finite duration of the radiation process itself, is called natural broadening. The half-width of the line, in this case, is obtained from the value of the Einstein coefficient for spontaneous emission: g _{o =} A_lk . If the emission corresponds to atomic transitions between two excited states, then the natural half-width of the line is obtained by adding the probabilities for all spontaneous transitions from each of the two levels. Since A_lk = 10⁸ sec^-1, the natural line width is very narrow, and the line width in a plasma is usually determined by other processes, those related to the motion of the atoms and to the interaction between the atoms and charged particles.

Broadening by charged particles (Stark broadening). In a TIC plasma, the main mechanism of line broadening (excluding resonance lines) is the interaction between the excited atom and charged particles: the electrons and ions. The moving charged particles apply a transient electric field E(t) at the location of the radiating atom. Unlike the electromagnetic radiation field, the field E(t) hardly changes during a

time the order of w _lk^-1. This field does not induce transitions of the atomic electron between states l and k, but it leads to an instantaneous shift and splitting of the atomic levels, i.e., to the Stark effect. Therefore, line broadening due to interaction with the electrons and ions is called Stark broadening.

The shift of the energy level, h D w , is dependent on the instantaneous value of field intensity E(t) created at the location of the radiating atom by the charged particle. It is known (see, for example, [36]) that the linear Stark effect (D w ~ E) occurs in a hydrogen atom because of the removal of l-degeneracy by the electric field, whereas the quadratic Stark effect (D w ~ E ²) occurs for other atoms, including cesium. Since E(t) ~ 1/r²(t), where r(t) is the distance between the atom and the charged particle, the instantaneous shift of the vibration frequency is

where n = 2 for the linear Stark effect and n = 4 for the quadratic Stark effect, and C_n is some constant..

electron moving at velocity v is equal to r(t) = C_n[b² + v²(t – t₀)²]^1/2(where b is the impact parameter and t₀ is the time at the moment of closest approach), the vibration frequency shift of the oscillator is

This frequency shift leads, during time dt, to a change in the oscillation phase of D w (t)dt. The total phase shift D c due to collision is calculated to be

where G (z) = @x^z-1exp(-x)dx is the Gamma function. Assume that the vibration coherence is disrupted if the phase shift exceeds some specific value c ₀. Collisions must then be understood as occurring for those impact parameters b £ b, where D c (b₀) = c ₀. In this case, from (8.3), we obtain

Usually c ₀ is assumed equal to unity.

The collision cross section for disrupting the vibration coherence can now be calculated as Q₀(v) = p b₀²(v). The average duration of unperturbed vibration t ₀ is calculated by the expression 1/t ₀ = n_e< vQ₀(v) >,where the symbol <...> denotes averaging with the electron distribution function (see §1, Chapter 4).

Thus, the radiating atom is made to correspond to an oscillator with a limited vibration duration. The probability W(t ) that the unperturbed vibrations of the atomic oscillator will continue for time segment t , is equal to W(t ) = exp(-t /t ₀).

The expression obtained determines the shape of the oscillator line for a duration of t oscillations. To find the shape of the emission line L_lk(w ), it is necessary to average L_lk(w ,t ) for all values of t :

By substituting (8.5) into (8.6) and by normalizing the result obtained to unity, we obtain the Lorentz line contour (8.1), whose half-width is g _o = 2/t ₀.

broadening [41-45]. Since, in impact theory, the time of unperturbed vibration t ~ (D w )^-1 should considerably exceed the duration of impact t ~ b₀/v, the condition of applicability of impact theory is the inequality D w < < ? v/b₀. By using (8.4), the latter inequality can be written in the form

Since D w ~ g _o, it is then necessary that g _o < < ? v^n/n-1C_n^-1/(n-1) for the applicability of impact theory. In the opposite limiting case, the charged particles do not manage to move during the radiation time t . In this case, one should assume that each charge particle creates a static electric field at the location of the atom, which leads to a corresponding change in the radiation frequency given by (8.2). Statistical averaging of the shifts of (8.2) for all positions of the interacting particles leads to broadening of the line and to a shift of its maximum. This contains the main idea of the so-called static theory of line broadening, which is used when calculating line broadening by ions [46,47]. We note that, according to (8.7), the remote wings of the line which correspond to large values of D w , in a number of cases should be calculated by the statistical theory even if the shape of the line close to the center is calculated by collision broadening.

A generalized theory of spectral line broadening has recently been developed [40, 48-50] in which the broadening caused by interaction with both ions and electrons is taken into account simultaneously.

where L_lk(w and L_lk(w are the line shapes during electron and ion broadening, respectively. The experimental line shapes for cesium are presented in [51], and calculated line shapes in [51,52].

Broadening by uncharged particles. A cause of spectral line broadening may also be the interaction of the radiating atom with neutral gas atoms [53]. In this case, the following mechanisms of broadening are distinguished: van der Waals broadening, related to interaction with other neutral atoms, and resonance broadening, related to interaction with identical atoms. For the cesium plasma of the TIC, resonance broadening is of greatest interest, because it is this mechanism that causes broadening of the resonance line (transitions 6P_1/2 ® 6S_1/2 and 6P_3/2 ® 6S_1/2) of the cesium atom, and it is in these lines that the greatest energy loss occurs. The main mechanism of resonance line broadening of alkali metals in a weakly ionized plasma is the transfer of resonance excitation from one atom to another during collisions. The lifetime of the atom in an excited state is inversely proportional to the concentration of atoms N_a. Therefore, the half-width g _o is proportional to E_a By denoting the proportionality constant by 2p h we find

The values of h for the resonance lines of a number of alkali metals (transitions h P_j ® S_1/2) are presented in Table 3.3 from the data of [54].

where v_z is the component of velocity in the direction of radiation propagation (the z—axis) and c is the speed of light. This is the Doppler effect. If the velocities of the various radiating atoms are different, the emission line is broadened. With a Maxwellian velocity distribution, the number of atoms in which the velocity component v_z is included in the range from v_z to v_z + dv_z is proportional to exp(-Mv_z²/2kT). Therefore, during Doppler broadening

Doppler broadening is usually small in a dense low temperature cesium plasma. Thus, at T = 3000° K and for g o = 7000 @, we have for the half-width D g _D = 2.4 x 10^-2 @.