Stark–Zeeman and Blokhintsev Spectra of Rydberg Atoms

Abstract

The problem of Rydberg atomic spectra in crossed electric F and magnetic B fields (combined Stark–Zeeman effects), as well as an oscillating electric field are under consideration. The main problem is the great array of radiative transitions between the Rydberg atomic states. The different versions of semiclassical approaches are applied for the solution of the problem. New approximate selection rules are established to simplify the matrix elements of radiative transitions, making it possible to reduce the complex expressions of Gordon’s hypergeometric series to the trivial trigonometric functions. Simple analytical formulas for the dipole matrix elements of a Rydberg hydrogen-like atom in external crossed electric and magnetic fields for H_nα are obtained. The specific calculations are presented for n = 10 to 9 transition both for parallel and perpendicular orientations of F–B fields. The result of Blokhintsev for a single Stark component in oscillating field is generalized for the array of radiative transitions between Rydberg atomic states. Such spectra with large values of modulation index are compared with static ones.

1 INTRODUCTION

The problem of hydrogen atomic spectra in external fields, that are considered in this paper, has formally precise quantum mechanical solution. However, in application to the specific radiation transitions one faces with anomalous large numbers of matrix elements especially for highly excited (Rydberg) atomic states. The case of a constant electric field is deeply studied from different angles. General expressions for the energy shift and the radiative transitions intensities are well known both in spherical [1–3] and parabolic [4–6] coordinate systems. In the case of the spherical quantization the situation is simplified by the existence of the selection rules for the orbital momentum quantum numbers l, which makes it possible to perform fast calculations of the transition probabilities and investigation of their connection with the classical results [2]. Coming to the parabolic quantization in the presence of electric field, there are no selection rules for the parabolic quantum numbers n₁, n₂ which results in essential difficulties for radiative intensities calculation, especially for Rydberg atomic states, where the massive of radiative transitions grows sharply (proportionally to n⁴). The problem of radiative transitions between Rydberg atomic states is of interest both from fundamental point of view and applications in line shape calculations for laboratory [7–9] and especially astrophysical plasma [10]. It is the goal of the present paper to compare different approaches of calculating of the dipole matrix elements between nearby parabolic states, determine the approximate selection rules for the radiative transition probabilities between the Rydberg states in the presence of an electrical field. Using these results, we show how complicated formulas for intensity in crossed F and B fields can be simplified. In this paper, we consider comparatively strong magnetic fields. Thus, one can neglect the corrections connected with the spin–orbit interaction.

General approach for hydrogen-like atomic systems spectra calculations is based on the application of the Coulomb field symmetry properties. The atomic state evolution in external fields is determined by the additional angular momentum vectors J_1,2, being the combination of standard angular momentum L and specific constant of motion in the Coulomb field—the Runge–Lenz vector A [11]:

$${{{\mathbf{J}}}_{{{\mathbf{1}}{\mathbf{,2}}}}} = \frac{1}{2}({\mathbf{L}} \pm {\mathbf{A}}),$$

(1)

where 1 relates to + and 2 to –.

The atomic energy shift for hydrogen atom in a single F field is determined by the projections of the vectors (1) on the z axis (z coincides with the direction of the electric field),

$$\Delta {{E}^{{(1)}}} = \frac{3}{2}nF({{i}_{2}} - {{i}_{1}}),$$

(2)

where n is the principle quantum number and i_{1, 2} are the projections of the vectors (1) on the z axis.

In relation (2) and in every expression of this paper, we use atomic units. Moreover, we need only the first order of perturbation theory, so we will omit upper index “(1)” in formulas for energy shifts. There is the connection [1] between parabolic quantum numbers and i_{1, 2},

$$\left\{ \begin{gathered} {{i}_{2}} - {{i}_{1}} = {{n}_{1}} - {{n}_{2}} \hfill \\ {{i}_{2}} + {{i}_{1}} = m, \hfill \\ \end{gathered} \right.$$

(3)

where n₁, n₂ are the parabolic quantum numbers.

The atomic energy shift in crossed F and B is determined by projections of the vectors (1) on directions of ω_{1, 2}:

$${{\omega }_{{{\mathbf{1}}{\mathbf{,2}}}}} = \frac{1}{{2c}}{\mathbf{B}} \mp \frac{3}{2}n{\mathbf{F}}.$$

(4)

The reason for it is that we can rewrite the perturbation part of the Hamiltonian H₁ = $\frac{1}{{2c}}$BL + Fr in the following way:

$${{H}_{1}} = {{\omega }_{{\mathbf{1}}}}{{{\mathbf{J}}}_{{\mathbf{1}}}} + {{\omega }_{{\mathbf{2}}}}{{{\mathbf{J}}}_{{\mathbf{2}}}},$$

(5)

$$E = {{\omega }_{1}}n{\kern 1pt} '\, + {{\omega }_{2}}n{\kern 1pt} '',$$

(6)

where n' and n'' are the projections of vectors (1) on the vectors (4).

We can do this because in the Coulomb field there is connection between the Runge–Lenz vector and the coordinate:

$${\mathbf{A}} = - \frac{2}{{3n}}{\mathbf{r}}.$$

(7)

Using angular momentum properties of the vectors (1), we can express the wave functions in the representation of n, n', n'' in terms of parabolic states.

$${\text{|}}n,n{\kern 1pt} ',n{\kern 1pt} ''\rangle = \sum\limits_{{{i}_{1}} = - j}^j {\sum\limits_{{{i}_{2}} = - j}^j {d_{{{{i}_{1}}n'}}^{j}({{\alpha }_{1}})d_{{{{i}_{2}}n''}}^{j}({{\alpha }_{2}})} } {\text{|}}n{{i}_{1}}{{i}_{2}}\rangle ,$$

(8)

where $d_{{{{m}_{1}}{{m}_{2}}}}^{j}$(β) is the Wigner d-function.

$$j = \frac{{n - 1}}{2}.$$

(9)

In relation (8), α_{1, 2} are the angles between vectors J_{1, 2} and ω_{1, 2}. We choose the reference frame, in which direction of magnetic field coincides with the z axis,

$$\cos {{\alpha }_{{1,2}}} = \frac{{\frac{1}{{2c}}B \mp \frac{3}{2}nF\cos \theta }}{{{{\omega }_{{1,2}}}}},$$

(10)

where θ is an angle between electric F and magnetic fields B.

Expression of the coordinate matrix elements as representation of states (8) is complicated. It contains 4 sums with big number of terms (~n⁴), 4 Wigner functions (these objects are expressed in terms of Jacobi polynomials (see [12])), and the complicated Gordon’s matrix elements [4]. However, usage of the symmetry of the Coulomb field makes it possible to simplify this complex sums into comparatively short expression.

Calculation of the matrix elements of dipole radiative transition probabilities between the atomic wave functions corresponding to the projections of (1) on axes (4) is very cumbersome. However, when Δn ≪ n, we can use new selection rules for parabolic numbers and asymptotic expressions of the coordinate matrix elements to simplify formulas for intensity (the intensity of radiation in dipole approximation is proportional to square absolute value of coordinate matrix element).

Due to unique properties of the Coulomb field, classical results for radiative processes start working with good accuracy when n ~ 1. An accurate quantum expressions for dipole matrix elements were obtained by Gordon [4–6]. Connection between different expressions for Z-matrix elements were demonstrated in [13, 14]. In these works, the author showed that quantum result can be expressed in terms of Jacobi polynomials. Using this type of formula in the limit of large n, it can be reduced to the classical Born result. For nα series (Δn = 1), it is easy to repeat for the X‑matrix element (see the Appendix).

Calculation of the intensity of radiation of a hydrogen in an oscillating field is a topical problem in atomic physics. This situation can be realized for example in turbulent plasma. An oscillating field can also be generated by a laser. In [15], this problem was considered for the first time. Blokhintsev derived the expression for intensity of a single Stark component. It gave an impetus both to theoretical [16, 17] and experimental [18, 19] investigations. Detailed analysis of problems connected with oscillating electric field one can find in [20–23]. We will give brief information about this theory below.

As shown in [15], a single Stark-component splits into satellites separated by pω (p is integer number) from unperturbed frequency.

Let us consider a hydrogen-like atom in field

$${\mathbf{F}} = {{{\mathbf{F}}}_{{\mathbf{0}}}}\cos (\Omega t).$$

(11)

Therefore, the reduced frequency of one Stark-component equals

$$S\left( {\frac{{\Delta \omega }}{\Omega }} \right) = \sum\limits_{p = - \infty }^{ + \infty } {J_{p}^{2}\left( {\frac{{\Delta {{\omega }_{F}}}}{\Omega }} \right)\delta \left( {\frac{{\Delta \omega }}{\Omega } - p} \right)} ,$$

(12)

where J_p(z) is the Bessel function of order p, Δω_F = $\frac{3}{2}$F₀[n(n₁ – n₂) – $\bar {n}$(${{\bar {n}}_{1}}$ – ${{\bar {n}}_{2}}$).

In fact, number of indices p in Eq. (12) is finite. It is determined by the following relation:

$$\int {S(z)dz = 1} .$$

(13)

The problem of satellites number is discussed in [20].

When $\frac{{\Delta {{\omega }_{F}}}}{\Omega } \gg 1$, $J_{p}^{2}$(z) ($\frac{{\Delta {{\omega }_{F}}}}{\Omega }$ is called the modulation index) has the sharp maximum when z = p and it may be more convenient to use the following approximate intensity distribution instead of the squared Bessel functions:

$${{I}_{s}}(\Delta \omega ) = \frac{1}{\pi }\frac{1}{{\sqrt {{{{(\Delta {{\omega }_{F}})}}^{2}} - {{{(\Delta \omega )}}^{2}}} }}\theta {{(\Delta {{\omega }_{F}})}^{2}} - {{(\Delta \omega )}^{2}}),$$

(14)

where θ(z) is the Heaviside step function.

One of the most important properties of (14) is that full integral over all frequencies coincides with integration of the delta function distribution:

$$\int\limits_{ - \infty }^{ + \infty } {{{I}_{s}}(\omega )d\omega } = \int\limits_{ - \infty }^{ + \infty } {\delta (\omega )d\omega = 1} .$$

It shows the correspondence between cases of constant electric field and Ω → 0.

2 SELECTION RULES FOR PARABOLIC NUMBERS IN A RYDBERG HYDROGEN ATOM

The classical expressions for the coordinate matrix elements between parabolic states in hydrogen atom were obtained by Born and Kramers [24, 25].

$$\begin{gathered} z_{{{{n}_{1}}{{n}_{2}}m}}^{{{{{\bar {n}}}_{1}}{{{\bar {n}}}_{2}}m}} = ({{\sigma }_{2}}{{J}_{{\Delta {{n}_{1}}}}}(\Delta n{{\sigma }_{1}})J{{'}_{{\Delta {{n}_{2}}}}}(\Delta n{{\sigma }_{1}}) \\ - \,{{\sigma }_{1}}{{J}_{{\Delta {{n}_{2}}}}}(\Delta n{{\sigma }_{1}})J{{'}_{{\Delta {{n}_{1}}}}}(\Delta n{{\sigma }_{1}})){\text{/}}(n\Delta n), \\ \end{gathered} $$

(15)

$$\begin{gathered} x_{{{{n}_{1}}{{n}_{2}}m}}^{{{{{\bar {n}}}_{1}}{{{\bar {n}}}_{2}}m}} = ({{\sigma }_{3}}{{J}_{{\Delta n1}}}(\Delta n{{\sigma }_{1}}){{J}_{{\Delta {{n}_{2}}}}}(\Delta n{{\sigma }_{1}}) \\ - \,{{\sigma }_{4}}{{J}_{{\Delta {{n}_{2}} + 1}}}(\Delta n{{\sigma }_{1}}){{J}_{{\Delta {{n}_{1}} + 1}}}(\Delta n{{\sigma }_{1}})){\text{/}}(2n\Delta n), \\ \end{gathered} $$

(16)

$$\begin{gathered} \Delta n = n - \bar {n}, \\ \langle {{{\bar {n}}}_{1}}{{{\bar {n}}}_{2}}\bar {m}{\text{|}}x{\text{|}}{{n}_{1}}{{n}_{2}}m\rangle = {{( - 1)}^{{\Delta m}}}\langle {{{\bar {n}}}_{1}}{{{\bar {n}}}_{2}}\bar {m}{\text{|}}y{\text{|}}{{n}_{1}}{{n}_{2}}m\rangle , \\ \end{gathered} $$

(17)

where ${{J}_{{v}}}$(x), $J_{{v}}^{'}$(x) Bessel function and its derivative, correspondingly, and

$$\left\{ \begin{gathered} {{\sigma }_{1}} = \frac{1}{n}\sqrt {{{n}_{1}}({{n}_{1}} + m)} \hfill \\ {{\sigma }_{2}} = \frac{1}{n}\sqrt {{{n}_{2}}({{n}_{2}} + m)} \hfill \\ {{\sigma }_{3}} = \frac{1}{n}\sqrt {({{n}_{1}} + m)({{n}_{2}} + m)} \hfill \\ {{\sigma }_{4}} = \frac{1}{n}\sqrt {{{n}_{1}}{{n}_{2}}} . \hfill \\ \end{gathered} \right.$$

(18)

Relation (17) is general for all kinds of coordinate matrix elements.

An accurate quantum expression for dipole matrix elements contains hypergeometric series:

$${{\psi }_{m}}({{n}_{i}},{{\bar {n}}_{i}}) = F\left( { - {{n}_{i}}, - {{{\bar {n}}}_{i}},m + 1,\frac{{ - 4n\bar {n}}}{{{{{(n - \bar {n})}}^{2}}}}} \right),$$

(19)

where F(a, b, c, z) is the hypergeometric function, n and $\bar {n}$ are the principle quantum numbers of two different states.

The presence of these series makes calculations of the dipole-matrix elements very difficult for large n. Because of that, another asymptotic of matrix coordinate elements was obtained by Gulyaev [26, 27]. In these works, the author noticed that last term in (19) is much greater than others (Δn ≪ n). For the H_nα series, this approach leads to the following simple results:

$$X_{m}^{{m - 1}} = \frac{1}{4}b\sqrt {({{n}_{1}} + m)({{n}_{2}} + m)} \left[ {1 - \frac{{({{n}_{1}} + 1)({{n}_{2}} + 1)}}{{{{b}^{2}}}}} \right],$$

(20)

$$X_{m}^{{m + 1}} = \frac{1}{4}b\sqrt {{{n}_{1}}{{n}_{2}}} \left[ {1 - \frac{{({{n}_{1}} + m)({{n}_{2}} + m)}}{{{{b}^{2}}}}} \right],$$

(21)

$$Z_{m}^{m} = \frac{1}{4}b[\sqrt {({{n}_{1}} + m){{n}_{1}}} {{\delta }_{{K, + 1}}} + \sqrt {({{n}_{2}} + m){{n}_{2}}} {{\delta }_{{K, - 1}}}],$$

(22)

where b² = $\frac{{4n\bar {n}}}{{{{{(n - \bar {n})}}^{2}}}}$,

$$K = ({{n}_{1}} - {{n}_{2}}) - ({{\bar {n}}_{1}} - {{\bar {n}}_{2}}).$$

(23)

Here, values with bar relate to states with lower energy.

The final procedure that will be done for analysis of matrix elements is the comparison of Gulyaev’s and Born’s expressions for σ components (X element). The energy shift in this case equals

$$\frac{{\Delta \omega }}{{{{\omega }_{F}}}} = \bar {n}({{\bar {n}}_{1}} - {{\bar {n}}_{2}}) - n({{n}_{1}} - {{n}_{2}}),$$

(24)

where ω_F = $\frac{3}{2}$F.

First, this graphic (Fig. 1) shows that we can choose any previously mentioned semiclassical approximation. Second, this comparison is a good illustration of the correspondence principle in quantum mechanics. It is clear that the Rydberg atom results work even when n ~ 1.

Fig. 1.

Intensity of the sigma component (divided by the maximum of I) as a function of energy shift (reduced frequency–energy shift divided by ω_F = $\frac{3}{2}$F, where F is the absolute value of the electric field). Transition from n = 7 to n = 6.

One of the biggest advantages of spherical quantum numbers is selection rule for orbital number l. As we will see later, although investigation of Stark effect is easier with parabolic states, lack of selection rules for n₁ and n₂ remains a big problem for calculation of intensity in external electric and magnetic fields. However, this inconvenience can be eliminated for large n. In order to solve this problem, we use an approach from [26, 27]. At first, we should consider new quantum number K (see relation (23)).

The energy shift can be rewritten in the following way:

$$\frac{{\Delta \omega }}{{{{\omega }_{F}}}} = Kn + \Delta nk,$$

(25)

where k = ${{\bar {n}}_{1}}$ – ${{\bar {n}}_{2}}$ is electric quantum number.

For each Δn (spectral series), K can take a limited number of values. Moreover, in the semiclassical limit, the intensity of radiation is highly dependent on this number. For example, for nβ [27] absolute value of dipole matrix elements series with K = ±1 and K = ±2 is much greater than any other. Thus, with growth of n, we need to calculate only a small part of all matrix elements. For nα there are only three possibilities: K = +1, –1 and K = 0.

For K = 0, the magnetic quantum number has to be changed. For the parabolic states, the selection rules for m remain the same. Formulas for the dipole matrix elements are expressed in terms of absolute value of m; therefore, here and below, we will use notation |m| = m,

$${{n}_{1}} = {{\bar {n}}_{1}},\quad {{n}_{2}} = {{\bar {n}}_{2}}$$

(26)

if Δm = –1 and

$${{n}_{1}} = {{\bar {n}}_{1}} + 1,\quad {{n}_{2}} = {{\bar {n}}_{2}} + 1$$

(27)

if Δm = +1. For K = +1, we have

$${{n}_{1}} = {{\bar {n}}_{1}} + 1,\quad {{n}_{2}} = {{\bar {n}}_{2}}$$

(28)

while for K = –1, we obtain

$${{n}_{1}} = {{\bar {n}}_{1}},\quad {{n}_{2}} = {{\bar {n}}_{2}} + 1$$

(29)

Because of the selection rules for m, the case of K = 0 corresponds to the X-dipole elements and the case of K = ±1 conforms to Z.

We will show in detail how to get the selection rules for coordinate matrix elements for nα. Let us start with π components (Z-matrix element):

$$\left\{ \begin{gathered} ({{i}_{2}} - {{i}_{1}}) - ({{{\bar {i}}}_{2}} - {{{\bar {i}}}_{1}}) = \pm 1 \hfill \\ {{i}_{2}} + {{i}_{1}} = {{{\bar {i}}}_{2}} + {{{\bar {i}}}_{1}}. \hfill \\ \end{gathered} \right.$$

(30)

The first equation in (30) corresponds to the definition of K (see (23)) and the fact that K for the Z‑matrix element for nα can be only ±1. The second equation expresses the selection rule for the magnetic quantum number. The solution to Eq. (30) is

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} \pm \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} \mp \frac{1}{2}. \hfill \\ \end{gathered} \right.$$

(31)

For the σ-components (X-matrix elements), the situation is more complicated. The main reason for it is that (20) and (21) are written for the absolute value of Δm. In fact, here Δm = |$\bar {m}$ – m|. So equations for the X-dipole elements will be

$$\left\{ \begin{gathered} ({{i}_{2}} - {{i}_{1}}) - ({{{\bar {i}}}_{2}} - {{{\bar {i}}}_{1}}) = 0 \hfill \\ {\text{|}}{{i}_{2}} + {{i}_{1}}{\text{|}} = {\text{|}}{{{\bar {i}}}_{1}} + {{{\bar {i}}}_{2}}{\text{|}} \pm 1. \hfill \\ \end{gathered} \right.$$

(32)

The solution of Eqs. (32) is presented in the Appendix,

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} \pm \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} \pm \frac{1}{2}. \hfill \\ \end{gathered} \right.$$

(33)

Here, sign “+” corresponds to m, $\bar {m}$ ≥ 0 and “—” to m, $\bar {m}$ ≤ 0. Here, m is not the absolute value! In other words, upper sign equation in (33) is for an increase in the magnetic quantum number and lower is for a decrease.

3 DIPOLE MATRIX ELEMENTS IN EXTERNAL ELECTRIC AND MAGNETIC FIELDS FOR A RYDBERG HYDROGEN-LIKE ATOM

In [11], a hydrogen atom in crossed electric and magnetic fields was considered in the framework of quantum mechanics for the first time. These ideas were used in [8] to calculate spectra lines profiles. According to relation (8), the coordinate matrix element can be expressed in terms of the Wigner d-functions:

$$\begin{gathered} a_{{nn'n''}}^{{\bar {n}\bar {n}'\bar {n}''}} = \sum\limits_{{{{\bar {i}}}_{1}} = - \bar {j}}^{\bar {j}} {\sum\limits_{{{{\bar {i}}}_{2}} = - \bar {j}}^{\bar {j}} {\sum\limits_{{{i}_{1}} = - j}^j {\sum\limits_{{{i}_{2}} = - j}^j {d_{{{{{\bar {i}}}_{1}}\bar {n}'}}^{{\bar {j}}}({{{\bar {\alpha }}}_{1}})} } } } \\ \, \times d_{{{{{\bar {i}}}_{2}}\bar {n}''}}^{{\bar {j}}}({{{\bar {\alpha }}}_{2}})d_{{{{i}_{1}}n'}}^{j}({{\alpha }_{1}})d_{{{{i}_{2}}n''}}^{j}({{\alpha }_{2}})a_{{n{{i}_{1}}{{i}_{2}}}}^{{\bar {n}{{{\bar {i}}}_{1}}{{{\bar {i}}}_{2}}}}, \\ \end{gathered} $$

(34)

where a = X, Y, Z.

We will show how to do the simplification when Δn = 1. In order to do this we will use the results of Gulayev. Let us start with the π components. At first, we will consider the first square root in (22),

$$\sqrt {({{n}_{1}} + m){{n}_{1}}} .$$

There is the connection between the parabolic quantum numbers and the principal number n:

$$n = {{n}_{1}} + {{n}_{2}} + \,{\text{|}}m{\text{|}} + 1.$$

(35)

Using relations (3) and (35), we can obtain

$$\left\{ \begin{gathered} {{n}_{1}} = \frac{{n - \,{\text{|}}{{i}_{1}} + {{i}_{2}}{\text{|}} + {{i}_{2}} - {{i}_{1}} - 1}}{2} \hfill \\ {{n}_{2}} = \frac{{n - \,{\text{|}}{{i}_{1}} + {{i}_{2}}{\text{|}} + {{i}_{1}} - {{i}_{2}} - 1}}{2}. \hfill \\ \end{gathered} \right.$$

(36)

Here, Δn ≪ n, so (i₁ + i₂ $\simeq$ ${{\bar {i}}_{1}}$ + ${{\bar {i}}_{2}}$) and (i₁ – i₂ $\simeq$ ${{\bar {i}}_{1}}$ ‒ ${{\bar {i}}_{2}}$). The second expression here is right because of the selection rules for nα.

(a) ${{\bar {i}}_{1}}$ + ${{\bar {i}}_{2}}$ ≥ 0,

$$\left\{ \begin{gathered} {{n}_{1}} \simeq \frac{n}{2} - {{{\bar {i}}}_{1}} \hfill \\ {{n}_{2}} + m \simeq \frac{n}{2} + {{{\bar {i}}}_{2}}; \hfill \\ \end{gathered} \right.$$

(b) ${{\bar {i}}_{1}}$ + ${{\bar {i}}_{2}}$ ≤ 0,

$$\left\{ \begin{gathered} {{n}_{1}} \simeq \frac{n}{2} + {{{\bar {i}}}_{2}} \hfill \\ {{n}_{2}} + m \simeq \frac{n}{2} - {{{\bar {i}}}_{1}}, \hfill \\ \end{gathered} \right.$$

$$({{n}_{1}} + m){{n}_{1}} = \left( {\frac{n}{2} - {{{\bar {i}}}_{1}}} \right)\left( {\frac{n}{2} + {{{\bar {i}}}_{2}}} \right).$$

(37)

Using the same manipulations, we can obtain the expression for the second term in (22):

$$({{n}_{2}} + m){{n}_{2}} = \left( {\frac{n}{2} + {{{\bar {i}}}_{1}}} \right)\left( {\frac{n}{2} - {{{\bar {i}}}_{2}}} \right).$$

(38)

From relation (10), we can see that α_{1, 2} $\simeq$ ${{\bar {\alpha }}_{{1,2}}}$ when Δn ≤ n.

Then we should use the selection rules for i₁, i₂. It allows us to reduce relation (34) to a double sum. Substitution of relations (37) and (38) into (34) for a = z splits the expression for the matrix element into two independent sums:

$$Z_{{nn'n''}}^{{\bar {n}\bar {n}'\bar {n}''}} = Z_{{1nn'n''}}^{{\bar {n}\bar {n}'\bar {n}''}} + Z_{{2nn'n''}}^{{\bar {n},\bar {n}',\bar {n}''}},$$

(39)

$$\begin{gathered} Z_{{1,2nn'n''}}^{{\bar {n}\bar {n}'\bar {n}''}} = \sum\limits_{{{{\bar {i}}}_{1}} = - \bar {j}}^{\bar {j}} {\sum\limits_{{{{\bar {i}}}_{2}} = - \bar {j}}^{\bar {j}} {d_{{{{{\bar {i}}}_{1}}\bar {n}'}}^{{\bar {j}}}({{\alpha }_{1}})} } \\ \, \times d_{{{{{\bar {i}}}_{2}}\bar {n}''}}^{{\bar {j}}}({{\alpha }_{2}})d_{{{{{\bar {i}}}_{1}} \pm \frac{1}{2}n'}}^{j}({{\alpha }_{1}})d_{{{{{\bar {i}}}_{2}} \mp \frac{1}{2}n''}}^{j}({{\alpha }_{2}}){{F}_{{1,2}}}({{{\bar {i}}}_{1}},{{{\bar {i}}}_{2}}), \\ \end{gathered} $$

(40)

where

$${{F}_{1}} = \left( {\frac{n}{2} - {{{\bar {i}}}_{1}}} \right)\left( {\frac{n}{2} + {{{\bar {i}}}_{2}}} \right),$$

$${{F}_{2}} = \left( {\frac{n}{2} + {{{\bar {i}}}_{1}}} \right)\left( {\frac{n}{2} - {{{\bar {i}}}_{2}}} \right).$$

The next step of the simplification is using the recurrence relations for the d-functions [12]

$$\begin{gathered} d_{{{{m}_{1}},{{m}_{2}}}}^{j}(\beta ) = \sqrt {\frac{{j - {{m}_{2}}}}{{j - {{m}_{1}}}}} \cos \left( {\frac{\beta }{2}} \right)d_{{{{m}_{1}} + \frac{1}{2},{{m}_{2}} + \frac{1}{2}}}^{{j - \frac{1}{2}}}(\beta ) \\ - \,\sqrt {\frac{{j + {{m}_{2}}}}{{j - {{m}_{1}}}}} \sin \left( {\frac{\beta }{2}} \right)d_{{{{m}_{1}} + \frac{1}{2},{{m}_{2}} - \frac{1}{2}}}^{{j - \frac{1}{2}}}(\beta ), \\ \end{gathered} $$

(41)

$$\begin{gathered} d_{{{{m}_{1}},{{m}_{2}}}}^{j}(\beta ) = \sqrt {\frac{{j - {{m}_{2}}}}{{j - {{m}_{1}}}}} \sin \left( {\frac{\beta }{2}} \right)d_{{{{m}_{1}} - \frac{1}{2},{{m}_{2}} + \frac{1}{2}}}^{{j - \frac{1}{2}}}(\beta ) \\ - \,\sqrt {\frac{{j + {{m}_{2}}}}{{j - {{m}_{1}}}}} \cos \left( {\frac{\beta }{2}} \right)d_{{{{m}_{1}} - \frac{1}{2},{{m}_{2}} - \frac{1}{2}}}^{{j - \frac{1}{2}}}(\beta ). \\ \end{gathered} $$

(42)

Here, j ≠ m₁, but we can neglect these terms when n ≫ 1. For proper usage of relations (41) and (42), we also have to switch low indices of the d-functions. In order to do that, we will use the symmetry property of the Wigner function:

$$d_{{{{m}_{1}},{{m}_{2}}}}^{j}(\beta ) = {{( - 1)}^{{{{m}_{1}} - {{m}_{2}}}}}d_{{{{m}_{2}},{{m}_{1}}}}^{j}(\beta ).$$

(43)

The last property of the d-functions which will be used here is the orthogonality relations:

$$\sum\limits_{{{m}_{3}} = - j}^j {{{{( - 1)}}^{{{{m}_{3}} - {{m}_{2}}}}}d_{{{{m}_{2}},{{m}_{3}}}}^{j}(\beta )d_{{{{m}_{3}},{{m}_{1}}}}^{j}(\beta ) = {{\delta }_{{{{m}_{1}},{{m}_{2}}}}}} .$$

(44)

Here, j $\simeq$ n/2. After the substitution of relations (41), (42) into (40), we can observe a lucky circumstance. Factors F_{1, 2} coincide with the denominators in square roots in the recurrence relations. Because of this, the coefficients in series (40) are independent of the summation indices. So, we can now freely use relation (44) and get

$$Z_{{nn'n''}}^{{\bar {n}\bar {n}'\bar {n}''}} = {{( - 1)}^{{\Delta \bar {n}' + \Delta \bar {n}''}}}\frac{{{{a}_{0}}}}{4}b[Z_{{nn'n''}}^{{(1)\bar {n}\bar {n}'\bar {n}''}} + Z_{{nn'n''}}^{{(2)\bar {n}\bar {n}'\bar {n}''}}],$$

(45)

$$\begin{gathered} Z_{{nn'n''}}^{{(1)\bar {n}\bar {n}'\bar {n}''}} = \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} '} \cos \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' + 1/2}}}} \right. \\ \left. { - \sqrt {\frac{n}{2} + n{\kern 1pt} '} \sin \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' - 1/2}}}} \right) \\ \times \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} ''} \sin \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' + 1/2}}}} \right. \\ \left. { + \sqrt {\frac{n}{2} + n{\kern 1pt} ''} \cos \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' - 1/2}}}} \right), \\ \end{gathered} $$

$$\begin{gathered} Z_{{nn'n''}}^{{(2)\bar {n}\bar {n}'\bar {n}''}} = \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} '} \sin \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' + 1/2}}}} \right. \\ \left. { - \sqrt {\frac{n}{2} + n{\kern 1pt} '} \cos \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' - 1/2}}}} \right) \\ \times \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} ''} \cos \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' + 1/2}}}} \right. \\ \left. { + \sqrt {\frac{n}{2} + n{\kern 1pt} ''} \sin \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' - 1/2}}}} \right). \\ \end{gathered} $$

The same manipulations for X give us

$$X_{{nn'n''}}^{{\bar {n}\bar {n}'\bar {n}''}} = {{( - 1)}^{{\Delta \bar {n}' + \Delta \bar {n}'' - 1}}}\frac{{{{a}_{0}}}}{4}b[X_{{nn'n''}}^{{(1)\bar {n}\bar {n}'\bar {n}''}} - X_{{nn'n''}}^{{(2)\bar {n}\bar {n}'\bar {n}''}}],$$

(46)

$$\begin{gathered} X_{{nn'n''}}^{{(1)\bar {n}\bar {n}'\bar {n}''}} = \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} '} \sin \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' + 1/2}}}} \right. \\ \left. { - \sqrt {\frac{n}{2} + n{\kern 1pt} '} \cos \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' - 1/2}}}} \right) \\ \times \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} ''} \sin \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' + 1/2}}}} \right. \\ \left. { + \sqrt {\frac{n}{2} + n{\kern 1pt} ''} \cos \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' - 1/2}}}} \right), \\ \end{gathered} $$

$$\begin{gathered} X_{{nn'n''}}^{{(2)\bar {n}\bar {n}'\bar {n}''}} = \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} '} \cos \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' + 1/2}}}} \right. \\ \left. { - \sqrt {\frac{n}{2} + n{\kern 1pt} '} \sin \frac{{{{\alpha }_{1}}}}{2}{{\delta }_{{\bar {n}',n' - 1/2}}}} \right) \\ \times \left( {\sqrt {\frac{n}{2} - n{\kern 1pt} ''} \cos \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' + 1/2}}}} \right. \\ \left. { + \sqrt {\frac{n}{2} + n{\kern 1pt} ''} \sin \frac{{{{\alpha }_{2}}}}{2}{{\delta }_{{\bar {n}'',n'' - 1/2}}}} \right). \\ \end{gathered} $$

According to relations (17) and (37), we have

$$Y_{{nn'n''}}^{{\bar {n}\bar {n}'\bar {n}''}} = {{( - 1)}^{{\Delta \bar {n}' + \Delta \bar {n}'' - 1}}}\frac{{{{a}_{0}}}}{4}b[X_{{nn'n''}}^{{(1)\bar {n}\bar {n}'\bar {n}''}} + X_{{nn'n''}}^{{(2)\bar {n}\bar {n}'\bar {n}''}}].$$

(47)

As we can see from relations (45)–(47) that for large n, there are the selection rules for n' and n'' (because of the presence of delta symbols). This is a consequence of the angular momentum properties of vectors (1).

In order to show the connection the between of expressions (45)–(47) and the Stark and Paschen–Back effects, we will calculate the intensity function of ω with fixed angle θ between B and F and different values of parameter u:

$$u = \frac{B}{{3cnE}},$$

(48)

$$\omega = ({{\bar {\omega }}_{1}}\bar {n}{\kern 1pt} '\, + {{\bar {\omega }}_{2}}\bar {n}{\kern 1pt} ''\, - {{\omega }_{1}}n{\kern 1pt} '\, - {{\omega }_{2}}n{\kern 1pt} ''){\text{/}}{{\Omega }_{{FB}}},$$

(49)

$${{\Omega }_{{FB}}} = \frac{1}{{2c}}B + \frac{3}{2}nF.$$

(50)

The intensity of radiation as a function of energy shift (49) in the external parallel F and B fields is presented in Fig. 2. Zero value of the magnetic field corresponds to u = 0 (Fig. 2a). As we can see, formulas (49)–(51) lead us to the ordinary Stark effect. This result is in agreement with [26]. Moreover, the substitution of θ = 0 and B = 0 in relation (10) returns the obtained expressions to (20)–(22). With growth of u, matrix element X remains equal to Y (Figs. 2b, 2c). This is clearly seen from careful consideration of relations (45)–(48). Physical reason for this is that here z is the symmetry axis. When B ≫ F, the obtained expressions for the dipole matrix elements describe the Paschen–Back effect (Fig. 2d).

Fig. 2.

Transition from n = 10 to 9. Intensity (divided by the sum of all components) as a function of the energy shift (48), (49) with θ = 0: u = (a) 0, (b) 1, (c) 10, (d) 1000; u = $\frac{B}{{3cnE}}$.

The case of perpendicular fields shows lower degree of the symmetry. With a growth of u, there may be unequal couples of matrix elements (components) as in Fig. 2b. When magnetic field is zero, F is directed along the x axis. This results in replacing X with Z (Fig. 2a). For large u, we again come to pure Zeeman components (Fig. 3d).

Fig. 3.

Same as in Fig. 2, but θ = π/2.

Figures 2 and 3 present graphics of intensity divided by the sum of all components. Due to the fact that all three matrix elements were calculated, there is the non-conservation of the full intensity integral.

4 BLOKHINTSEV SPECTRA OF A RYDBERG ATOM

In order to calculate the intensity of a Rydberg atom in field (11), we use relations (12), (13) and the matrix elements from [26, 27]. Let us consider the following parameter:

$$r = \frac{3}{2}\frac{F}{\Omega }.$$

(51)

Practice shows that in real physical situations, always r $\gtrsim$ 1 or even r ≫ 1 (, for example, [28]). In the case of a Rydberg atom with strong modulation, there is a large number of satellites. Also, if spectral series have the central component (such as H_nα) and r $\gtrsim$ 1, the intensity distribution will have asymptotic behavior of the delta function. The lateral components will be low as compared to the central component. This circumstance is discussed in detail in [21, 23].

In order to show interesting effects, we focus on series without a central component. We will calculate the intensity profiles of H_nβ and π component of H_nα in the case of large n and r. We will compare them with profile in the constant field.

The distribution of intensity of all Stark components equals

$$I(\Delta \omega ) = \frac{1}{\Omega }\sum\limits_\tau ^{} {a_{\tau }^{2}J_{{\frac{{\Delta \omega }}{\Omega }}}^{2}\left( {\frac{{\Delta {{\omega }_{\tau }}}}{\Omega }} \right)} \,,$$

(52)

where τ is the full set of quantum numbers, a_τ is the coordinate matrix element.

Using the results from [27], relations (3), (9), and setting n $\gtrsim$ 1, we can obtain the matrix elements for H_nβ:

$$\begin{gathered} {{(X_{m}^{{m - 1}})}^{2}} + {{(X_{m}^{{m + 1}})}^{2}} = (j \mp {{i}_{1}})(j \mp {{i}_{2}}) \\ \, \times [(j - {{i}_{1}})(j - {{i}_{2}}) + (j + {{i}_{1}})(j + {{i}_{2}})], \\ \end{gathered} $$

(53)

where K = +1 for upper sign and K = –1 for lower one. This is σ_±1,

$${{(Z_{m}^{m})}^{2}} = {{(j \mp {{i}_{1}})}^{2}}{{(j \pm {{i}_{2}})}^{2}},$$

(54)

where K = +2 for upper sign and K = –2 for lower one. This is π_±2.

Energy shift (23) depends only on two quantum numbers i₁ and i₂ for fixed K (in fact, on ${{\bar {i}}_{1}}$ and ${{\bar {i}}_{2}}$, but it is no matter for a Rydberg atom). So, in relation (52), τ = i₁, i₂. There is a degeneracy, and the summation over the left quantum number will give only a mutual coefficient. It does not affect the profile shape.

As we have already said, we need series only with K = ±1 and K = ±2. The intensity of all others series can be neglected. Using results for the matrix elements (for H_nα see relations (20)–(23)) and (51), we can calculate the intensity distribution.

As we can see in Fig. 4, the satellites that correspond to every Stark component form the curve that is similar to the stationary profile. It is the result of specific behavior of satellites. As we can see from (14), when Δω_F → Δω, the distribution has the peak. However, it is obvious (especially in Fig. 4b) that the satellites come from “wings” to point Δω = 0 and the form central component. For H_nα, it leads to the anomalous growth of the central component (see Fig. 5a). The expression for the intensity profile in a stationary field one can find in [26, 27].

Fig. 4.

Intensity (divided by the sum of all components) as a function of the energy shift (reduced frequency–energy shift divided by $\frac{3}{2}$F₀; $\frac{3}{2}\frac{F}{\Omega }$ = 1000; (a) transition from n = 10 to 9 (only the π component!); (b) transition from n = 10 to 8.

Fig. 5.

Comparison of the semiclassical expression, accurate quantum results [20], and the stationary case. Intensity (divided by the sum of all components) as a function of the energy shift (reduced frequency–energy shift divided by $\frac{3}{2}$F₀. (a) Transition from n = 6 to 5; $\frac{3}{2}\frac{F}{\Omega }$ = 8. (b) Transition from n = 4 to 2; $\frac{3}{2}\frac{F}{\Omega }$ = 3.

The comparison of the accurate quantum results (see [20]) and the semiclassical approximation is presented in Fig. 5b. Even for transition 4–2, one can see the correspondence between two approaches. In fact, condition Δn $\lesssim$ n doesn’t work for this situation. However, the location of all peaks is the same. Also, there is a comparatively big area where classical calculations coincide with the quantum.

5 CONCLUSIONS

The detailed investigation of Rydberg atomic spectra in crossed electric and magnetic fields, which is closely related to intensity representation in the parabolic coordinate system, was made by using different semiclassical methods. Using the connection between the Jacobi polynomials and the hypergeometric function, we will reduce the Gordon formulas [5] to the classical Born result not only for Z (as in [13]), but also for the X (sigma) components. The comparison of the classical Born results and he semiclassical representation [26] for the sigma components is demonstrated in Fig. 1, making it possible to apply the representation for a further simplification of matrix elements. The semiclassical representation makes it possible to formulate the approximate selection rules for the parabolic quantum numbers and to reach essential simplification for the radiative transition probabilities in terms of elementary algebraic functions (square roots and trigonometry functions). That is the transition of complicated expressions (34) to simple formulas (45)–(48). It opens a simple way for the calculation of atomic Rydberg spectra. All these results show classical properties and the symmetry of the Coulomb field. To generalize the method to the n > 1 case, it is necessary to use recurrence relations (41), (42) several times.

The results obtained by Blokhintsev in [15] are generalized for the array of radiative transitions. This can be realized in a turbulent plasma or generated by a laser field. The Rydberg atom approach can be useful there, especially for hydrogen-like ions, because the frequency of radiation emitted by an atom or ion is proportional to Z ² (Z is the charge of a nucleus). In order to be within the visible spectrum, one needs to investigate transitions with large n; therefore, expressions for the semiclassical limit would simplify calculations. If a spectral series has no central component, the envelope of satellites (see expression (53)) form the curve of a stationary profile (Fig. 4). Figure 5b demonstrates classical properties of the Coulomb field. Even for the transition 4–2, there is the correspondence between classical and quantum results. For more excited levels, one can freely use Gulyaev’s expressions instead of complicated Gordons’s formulas.

Appendices

APPENDIX A

1.1 X-Matrix Element in Terms of Jacobi Polynomials

There is a connection between the hypergeomtric function and the Jacobi polynomials:

$$\begin{gathered} F\left( { - k, - k - \alpha ,1 + \beta , - {{{\cot }}^{2}}\frac{\varphi }{2}} \right) = \frac{{k!\Gamma (1 + \beta )}}{{k + 1 + \beta }} \\ \times {{\left( { - \frac{1}{{{{{\sin }}^{2}}\frac{\varphi }{2}}}} \right)}^{k}}P_{k}^{{(\alpha ,\beta )}}(\cos \varphi ), \\ \end{gathered} $$

(55)

where $P_{k}^{{(\alpha ,\beta )}}$(z) is a Jacobi polynomial and Γ(z) is the gamma function. Using formula (55), we can obtain the result for Gordon’s X-matrix element in terms of Jacobi polynomials:

$$X_{{{{n}_{1}},{{n}_{2}},m - 1}}^{{{{{\bar {n}}}_{1}}{{{\bar {n}}}_{2}},m}} = {{\Pi }_{x}}[\Upsilon _{{{{x}_{1}}}}^{J} - \Upsilon _{{{{x}_{2}}}}^{J}],$$

(56)

where

$$\begin{gathered} {{\Pi }_{x}} = {{( - 1)}^{{{{{\bar {n}}}_{1}} + {{{\bar {n}}}_{2}}}}}\frac{{{{a}_{0}}}}{{4{{{((m - 1)!)}}^{2}}}} \\ \times \sqrt {\frac{{({{n}_{1}} + m)!({{n}_{2}} + m)!({{{\bar {n}}}_{1}} + m - 1)({{{\bar {n}}}_{2}} + m - 1)!}}{{{{n}_{1}}!{{n}_{2}}!{{{\bar {n}}}_{1}}!{{{\bar {n}}}_{2}}!}}} \\ \times {{\left( {\frac{{4n\bar {n}}}{{{{{(n - \bar {n})}}^{2}}}}} \right)}^{{m + 1}}}{{\left( {\frac{{n - \bar {n}}}{{n + \bar {n}}}} \right)}^{{n + \bar {n}}}}, \\ \end{gathered} $$

$$\begin{gathered} \Upsilon _{{{{x}_{1}}}}^{J} = \frac{{{{N}_{1}}!{{N}_{2}}!{{{[(m - 1)!]}}^{2}}}}{{({{N}_{1}} + m)!({{N}_{2}} + m)}}{{\left( { - \frac{1}{{{{{\sin }}^{2}}\frac{\varphi }{2}}}} \right)}^{{{{N}_{1}} + {{N}_{2}}}}} \\ \, \times P_{{{{N}_{1}}}}^{{({{\Delta }_{1}},m - 1)}}(\cos \varphi )P_{{{{N}_{2}}}}^{{({{\Delta }_{2}},m - 1)}}(\cos \varphi ), \\ \end{gathered} $$

$$\begin{gathered} \Upsilon _{{{{x}_{2}}}}^{J} = {{\left( {\frac{{\Delta n}}{{n + \bar {n}}}} \right)}^{2}}\frac{{{{{\bar {N}}}_{1}}!{{{\bar {N}}}_{2}}!{{{[(m - 1)!]}}^{2}}}}{{({{{\bar {N}}}_{1}} + m - 1)!({{{\bar {N}}}_{2}} + m - 1)!}} \\ \, \times P_{{{{{\bar {N}}}_{1}}}}^{{({{{\bar {\Delta }}}_{1}},m - 1)}}(\cos \varphi )P_{{{{{\bar {N}}}_{2}}}}^{{({{{\bar {\Delta }}}_{2}},m - 1)}}(\cos \varphi ), \\ \end{gathered} $$

$$\begin{gathered} {{\Delta }_{1}} = {\text{|}}{{n}_{1}} - {{{\bar {n}}}_{1}}{\text{|}},\quad {{\Delta }_{2}} = {\text{|}}{{n}_{2}} - {{{\bar {n}}}_{2}}{\text{|}}, \\ {{{\bar {\Delta }}}_{1}} = {\text{|}}{{n}_{1}} - {{{\bar {n}}}_{1}} - 1{\text{|}},\quad {{{\bar {\Delta }}}_{2}} = {\text{|}}{{n}_{2}} - {{{\bar {n}}}_{2}} - 1{\text{|}}, \\ \Delta = n - \bar {n}, \\ \end{gathered} $$

$${{N}_{1}} = \min ({{n}_{1}},{{\bar {n}}_{1}}),\quad {{N}_{2}} = \min ({{n}_{2}},{{\bar {n}}_{2}}),$$

$${{\bar {N}}_{1}} = \min ({{n}_{1}} + 1,{{\bar {n}}_{1}}),\quad {{\bar {N}}_{2}} = \min ({{n}_{2}} + 1,{{\bar {n}}_{2}}).$$

In this expression, the following relations were taken into account:

$$\begin{gathered} {{\cot }^{2}}\frac{\varphi }{2} = \frac{{4n\bar {n}}}{{{{\Delta }^{2}}}},\quad \sin \frac{\varphi }{2} = \frac{\Delta }{{n + \bar {n}}}, \\ \cos \varphi = \frac{{4n\bar {n} - {{\Delta }^{2}}}}{{{{{(n + \bar {n})}}^{2}}}}. \\ \end{gathered} $$

(57)

Now, it is easy to show for the nα series that in the case of n ≪ Δn. expression (55) coincides with the classical Born result. We should use the asymptotic forms for Jacobi polynomials.

$$\begin{gathered} P_{k}^{{(\alpha ,\beta )}}(\cos \varphi ) \simeq \frac{{\Gamma (k + \alpha + 1)}}{{k!{{U}^{\alpha }}}} \\ \times \sqrt {\frac{\varphi }{{\sin \varphi }}} \frac{{{{J}_{\alpha }}(U\varphi )}}{{{{{\left( {\sin \frac{\varphi }{2}} \right)}}^{\alpha }}{{{\left( {\cos \frac{\varphi }{2}} \right)}}^{\beta }}}},\quad \varphi \ll 1, \\ \end{gathered} $$

(58)

where U² = $A_{0}^{2}$ – $\left( {\frac{{{{\beta }^{2}}}}{4}} \right.$ + $\frac{{{{\alpha }^{2}}}}{{12}}$ – $\left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}\frac{1}{{12}}} \right)$ and A₀ = k + $\frac{{\alpha + \beta + 1}}{2}$.

Then we have to simplify U in the case of n ≫ 1:

$$\begin{gathered} {{U}_{1}} = \sqrt {{{{\left( {{{N}_{1}} + \frac{{{{\Delta }_{1}}}}{2}} \right)}}^{2}} - \frac{{{{{(m - 1)}}^{2}}}}{2} + \frac{{\Delta _{1}^{2}}}{{12}} - \frac{1}{{12}}} \\ = \sqrt {\left( {{{N}_{1}} + \frac{{m + {{\Delta }_{1}}}}{2} + \frac{{m - 1}}{2}} \right)\left( {{{N}_{1}} + \frac{{m + {{\Delta }_{1}}}}{2} - \frac{{m - 1}}{2}} \right) + \frac{{\Delta _{1}^{2} - 1}}{{12}}} \\ \end{gathered} $$

If we take into account the fact that Δ₁ ~ 1, we get

$${{U}_{1}} \simeq \sqrt {{{N}_{1}}({{N}_{1}} + m)} $$

Similarly, it can be done for U₁ = U₃, U₂ = U₄.

After the substitution of expression (57) into (55) and simple mathematical transformations, we can obtain the classical Born formula for the X-matrix element (16).

APPENDIX B

1.1 Selection Rules for σ Component for H _nα Series

Let us consider system (32) with +1 in the second equation:

$$\left\{ \begin{gathered} ({{i}_{2}} - {{i}_{1}}) - ({{{\bar {i}}}_{2}} - {{{\bar {i}}}_{1}}) = 0 \hfill \\ {\text{|}}{{i}_{2}} + {{i}_{1}}{\text{|}} = {\text{|}}{{{\bar {i}}}_{2}} + {{{\bar {i}}}_{1}}{\text{|}} + 1. \hfill \\ \end{gathered} \right.$$

(a) i₂ + i₁ and ${{\bar {i}}_{2}}$ + ${{\bar {i}}_{1}}$ ≥ 0:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} + \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} + \frac{1}{2}. \hfill \\ \end{gathered} \right.$$

(b) i₂ + i₁ ≤ 0, ${{\bar {i}}_{2}}$ + ${{\bar {i}}_{1}}$ ≥ 0. These transitions are forbidden by the selection rule for m.

(c) i₂ + i₁ ≥ 0, ${{\bar {i}}_{2}}$ + ${{\bar {i}}_{1}}$ ≤ 0. There is only one transition that satisfies the selection rule for m: 0 → –1. It may be included in the next item (d).

(d) i₂ + i₁ and ${{\bar {i}}_{2}}$ + ${{\bar {i}}_{1}}$ ≤ 0:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} - \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} - \frac{1}{2}. \hfill \\ \end{gathered} \right.$$

Doing the same manipulations, we can obtain the result for system (32) with –1:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} - \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} - \frac{1}{2}, \hfill \\ \end{gathered} \right.$$

when i₂ + i₁ and ${{\bar {i}}_{2}}$ + ${{\bar {i}}_{1}}$ ≥ 0, and

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} + \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} + \frac{1}{2}, \hfill \\ \end{gathered} \right.$$

when i₂ + i₁ and ${{\bar {i}}_{2}}$ + ${{\bar {i}}_{1}}$ ≤ 0.

We can combine different cases in the following way:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} \pm \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} \pm \frac{1}{2}. \hfill \\ \end{gathered} \right.$$

(59)