REGULARITIES OF THE DYNAMIC STARK EFFECT FOR AN ARGON ATOM IN A CIRCULARLY POLARIZED ELECTRIC FIELD
E. V. Koryukina,1 V.I. Koryukin2 UDC 539.184.5
In this work, the Stark effect for an argon atom acted upon by an alternating circularly polarized electric field is studied theoretically. The calculations are carried out by the method of energy matrix diagonalization. The algorithm of the method is implemented using a special software package written in FORTRAN, which is used for computer simulation of shifts and splitting of the argon-atom states in the electric field. Based on the calculation results, regularities in the behavior of shifts and splitting of the energy levels of the argon atom depending on the electronic structure of the levels as well as on the changes in the strength and frequency of the electric field are derived.
Keywords: calculation method for the dynamic Stark effect, circularly polarized electric field, regularities of the Stark effect for an argon atom.
INTRODUCTION
Investigation into the influence of the electric field on the atomic and ionic energy spectra is a topical problem in plasma spectroscopy and atomic spectroscopy. The external electric field excites emission spectra in active media where the Stark effect is observed, and an internal electric field of plasma also affects the emission spectra. The Stark effect is widely used for theoretical study of the processes taking place in plasma, and for plasma diagnostics, in particular, for determining the energy distribution function, concentration, and temperature of electrons and for estimating the electric-field strength inside the discharge [1]. Argon is a rare gas often used in plasma physics and spectroscopy, therefore, examination of the energy spectrum of this atom in the electric field is of theoretical and practical interest. An investigation into the energy spectrum of an Ar atom is a topical problem for the case of an alternating circularly polarized electric field which is realized in an inductive high-frequency discharge and under laser excitation. In addition, the solution to the Schrödinger equation for circular polarization of the electric field is a part of the problem on collisions of atomic hydrogen with charged particles at ultralow energies [2].
In the case of circular polarization of the electric field, the non-stationary Schrödinger equation can be reduced to the stationary one within the rotating-wave approximation [3]. Undoubtedly, the solution to the stationary Schrödinger equation is much more straightforward than that to the non-stationary one, however, in this case, no general method for calculating shifts and splitting of the spectral lines in the electric field of an arbitrary strength and frequency has been developed so far [4, 5]. Application of perturbation theory is restricted by the requirements of low electric-field strength and smallness of splitting of energy levels in the field, moreover, resonant and non-resonant perturbations by the electric field must be considered within different approximations. In the framework of perturbation theory, a number of formulas was obtained for energy-level shifts in the one- and two-level approximations [6], and
1National Research Tomsk State University, Tomsk, Russia; 2Siberian State Medical University, Tomsk, Russia, e-mail: [email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 88–94, January, 2012. Original article submitted June 1, 2011.
also for the case of an isolated atomic level (the requirement is satisfied at the electric field frequency ~1015 Hz) under non-resonant perturbation by the electric field [5, 7]. However, all formulas derived in [2, 4, 5, 7] are valid only for Rydberg atoms.
A theoretical approach suitable for calculating atomic spectra in an alternating circularly polarized electric field with the strength and frequency changing in a wide range was put forward and developed in [8–10]. In the present work, this method free from limitations inherent in perturbation theory and allowing the calculations within the many-level approximation to be made is applied to calculating the dynamic Stark effect in an Ar atom. It should be noted that the shifts and splitting for a number of spectral lines of an argon atom in the electric field were calculated earlier [11, 12], however, due to computation complexity, no consistent study of the dependence of the Stark effect on the electric field frequency and strength was possible so far. It is writing additional programming modules to the software package permitting to carry out computer simulation of the Stark effect that allows us to perform successfully a systematic study of the energy spectrum of an argon atom in a wide range of the electric-field frequencies and strengths. Based on the obtained results, general regularities inherent in the behavior of the energy levels of an argon atom depending on the electronic structure of these levels and electric-field parameters were revealed.
1. THEORETICAL METHOD
In a circularly polarized electric field, the non-stationary Schrödinger equation is written as
0( , )
( ( ) ( cos sin )) ( , )nn
ti H eF x t y t t
t∂ψ
= − ω ± ω ψ∂
rr r , (1)
where nψ is the wave function of n-th state of the system, 0 ( )H r is the unperturbed Hamiltonian, and the operator ( cos sin )eF x t y t− ω ± ω describes perturbation induced by the interaction of an atom with a circularly polarized
electric field of frequency ω and strength F. The “+” and “–“ signs correspond to the right and left polarization of the field, respectively. To go to the stationary Schrödinger equation, use is made of the rotating-wave approximation [3].
In the framework of this approximation, the wave function in the coordinate system rotating around the Z-axis with the frequency ω is as follows:
( , ) exp( ) ( , )zt i tJ tϕ = ω ψr r , (2)
where zJ is the z-component of the total angular momentum operator. On substituting the wave function (2) in Eq. (1), we have
( , ) ( , ),ti Q tt
∂ϕ= ϕ
∂r r 0( )z xQ H J FD= −ω ± . (3)
As seen from Eq. (3), the operator Q is time-independent. Hence, in the rotating-wave approximation, it is possible to go from the non-stationary Schrödinger equation (1) to the stationary one, and we get
( ) ( )Qϕ = εϕr r , ( , ) exp( ) ( )t i tϕ = − ε ϕr r . (4)
The operator Q is the operator of energy of an atom in the electric field, and ε and ( , )tϕ r are the energy and wave function of the atom in the electric field in the rotating coordinate system. Instead of solving the Schrödinger equation (4) within perturbation theory, it is much more convenient to solve this equation by the method of energy matrix diagonalization developed in [8, 9]. It was shown that the wave functions and energies of an atom, being solutions to the
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Schrödinger equation (4), are found from diagonalization of the energy matrix of an atom in the field with the following elements:
(0) (0) (0) (0) (0)( ) ( )mn n mn m z n m x nQ E J F D= δ −ω< ϕ ϕ > ± < ϕ ϕ >r r , (5)
where (0)nϕ and (0)
nE are the wave function and energy of the n-th atomic state in the absence of external electric field, F and ω are the strength and frequency of external electric field, and xD is the x-component of the dipole transition operator. Diagonalization of the energy matrix with elements (5) results in a set of wave functions and the energy spectrum for the n atomic states in the electric field. Upon diagonalization of the Q matrix, we get the energies nε and
wave functions nϕ as
(0)( , ) ( )ni tn nk k
kt e C r− εϕ = ϕ∑r (6)
for the n atomic states in the electric field in the rotating coordinate system. The coefficients Cnk in the wave function (6) depend on the electric-field frequency and strength. To find the average atomic energies in the initial coordinate system, it is necessary to perform averaging over the oscillation period. Upon averaging, the average energy of the system in the electric field in the initial coordinate system is written in the following form:
( , ) ( , ) ( , ) ( ) ( )n n n n n z nE t H t t J=< ψ ψ >= ε ±ω < ϕ ϕ >r r r r r . (7)
Here the sign “±” corresponds to the right and left polarization of the field, respectively. It is seen from Eq. (7) that nE does not depend on time and the atomic spectrum in the electric field is the same for the left and right field polarizations, because
n m n mE E− = ε − ε +ω (8)
for both polarization types. This result is confirmed by conclusions drawn within perturbation theory in [5]. The matrix elements of the Dx operator in Eq. (5) are determined as
where the reduced matrix elements J D J′ ′< γ γ > are calculated depending on a coupling scheme. For an Ar atom,
the ground state 3p6 1S0 is calculated in the LS coupling scheme, whereas the excited 3p5nl2S+1[K]J states are computed in the Jl coupling scheme. The reduced matrix elements J D J′ ′< γ γ > are calculated using the following formulas:
1 2 1 2( 1) ( , )J D J A Q T T J U U J l r lϕ′ ′ ′ ′ ′ ′< γ γ >= − < > ,
3 3
2( 1) max( , )l l
nl n ll r l l l R r R′+ +
′ ′′ ′< >= − < > , (10)
2
0( )nl n l nl n lR r R R r rR r dr
∞
′ ′ ′ ′< >= ∫ , (11)
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where
, , , ,c c c c c cS S L L J JA ′ ′ ′ ′γ γ= δ δ δ δ , 12 cl J J′ ′ϕ = + − + ,
1
( , ) [ ][ ][ ][ ] 211
cc c
lKJKJQ J KJ J K J K K J J
K lJ K
⎧ ⎫⎧ ⎫⎪ ⎪′ ′ ′ ′ ′= ⎨ ⎬⎨ ⎬′ ′⎩ ⎭⎪ ⎪′ ′⎩ ⎭
(12)
for the transitions between the excited states, and
, , ,c c c cS S L LA ′ ′ ′γ γ= δ δ δ ,
cl L S Jϕ = + − + ,
1
( , ) [ ][ ][ ][ ][ ][ ]
1 / 2
c
c c c
c
SL K
Q SLJ J K J L S J K J J S L J l
J l J
′⎡ ⎤⎢ ⎥⎢ ⎥′ ′ ′ ′ ′ ′ ′ ′= ⎢ ⎥⎢ ⎥⎢ ⎥′⎣ ⎦
(13)
for the transitions between the ground and excited states. In Eqs. (12) and (13), the following designations are used: , ,c c cS L J are the quantum numbers of the atomic core, ... are 6j-coefficients, [ ]... are 12j-coefficients of the second
kind, and [X] = 2X+1. The radial integral nl n lR r R ′ ′< > in Eq. (11) is calculated using a semiempirical formula, which
is an improved modification of the Bates-Damgaard formula [12]. The specific form of our semiempirical formula and details of its derivation are reported in [10].
It follows from the above reasoning that the proposed theoretical approach is free from limitations inherent in perturbation theory and may be used for calculating the dynamic Stark effect in a circularly polarized electric field with the strength and frequency changing in a wide range. In addition, in the framework of the given method, the calculations are performed within the many-level approximation, which allows us to follow the influence of energy-state interactions on the behavior of these levels in the electric field. Finally, this method makes it possible to calculate resonant and non-resonant excitations within a single approach rather than using two different approximations as is the case with perturbation theory.
2. RESULTS AND DISCUSSION
The algorithm of the developed theoretical approach is implemented in a special software package written in FORTRAN. Special programming modules permitting graphical data processing to be performed make it possible to investigate atomic energy-level shifts and splitting in the electric field of any strength and frequency. In this work, the electric-field strength was considered in the range up to 10 kV/cm with different frequencies generated by real excitation sources: ω = 100 MHz (electrodeless high-frequency lamps (HFD) [13]), ω = 151.91·103 MHz (an NH3 laser), and ω = 243.52·104 MHz (an HCN laser) [14]. In calculations of the Q matrix, ns-, np-, nd-, and nf-states with n ≤ 10 were taken into account. The behavior of all examined nl[Jc,K]JM states was studied as a function of the electric- field frequency and strength. The dependence of the behavior of the energy states on the degree of their mixing in the electric field was also studied. Let us consider the regularities obtained from simulation of the Stark effect in the electric field of different strength and frequency.
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State mixing in the electric field (ω and F are fixed). In the first place it is obvious, that the higher the excited level (that is, the bigger the principal quantum number n of an outer shell electron) the more mixed the level is with other energy levels. Further, an analysis of the wave functions has shown that the degree of the Stark state interactions increases with the orbital quantum number l of the outer shell electron. The ns states are practically pure, and the maximal mixing of the states is typical for the nf states. The interaction of Stark states in the electric field increases with the quantum number J .
It should be noted that the behavior of the nl[Jc, K]J levels depends on the core type. At Jc = 1/2, the energy states conventionally designated as nl′[K]JM practically do not mix with any neighboring states under the influence of the electric field even at sufficiently large n. At Jc = 3/2, the Stark states conventionally designated as nl[K]JM, strongly mix in the electric field, and the degree of their mixing increases fast with the principal quantum number n. Strong interaction of the argon-atom states leads to occurrence of forbidden lines. In particular, owing to the fact that the nd[K]JM states have a significant admixture of the nf[K]JM states, and the np[K]JM states have a high admixture of the nd[K]JM states, lines forbidden by the selection rules for the quantum numbers J and M may appear in the atomic emission spectrum. In the case of very strong interactions between the states, there arises a problem of identification of these states within the Jl coupling scheme.
Further it will be shown that the degree of mixing of the Stark states has a determining influence on the behavior of these states under changes in the strength and frequency of the electric field. Knowing the electronic structure of the level under consideration, one can estimate the degree of its mixture with other levels and predict the behavior of an atom in this state in the electric field of the given strength and frequency.
Let us consider the regularities revealed in the behavior of the energy states as functions of the electric-field strength and frequency.
The F-dependence of the states (ω is fixed). In the case of weak interaction of the Stark states, the following regularities were derived in the behavior of shifts and splitting of the levels under changes of the electric-field strength:
1) The dependence of the Stark-state shifts on the electric-field strength is quadratic, that is, ΔE ~ F 2, where ΔE is the shift of the state under consideration with respect to the position of the state in the absence of electric field. The quadratic dependence is known from perturbation theory [4, 5, 7], thereby, our result confirms the validity of the developed theoretical approach and reliability of the software package. As an illustration, Figure 1 shows the shifts of the np[1/2]0 and nd[1/2]0 energy states of an Ar atom in the electric field of frequency ω = 100 MHz.
2) The Stark-state shift in the field increases monotonically with the principal quantum number n of an outer shell electron (see Fig. 1). Splitting of the levels in the electric field also increases with n.
3) All examined ns-, ns′-, and np′-states are shifted to the IR region with an increase in the electric field strength, whereas the nf-, and nf ′-states are shifted to the UV region (see Fig. 1 and Table 1).
Strong interaction between the energy states leads to the breakdown of at least one of these regularities. As an illustration, Fig. 2 shows the breakdown of the quadratic dependence of the shift ΔE on F and monotony of the
0 2 4 6 8 10-0.4
-0.2
0.0
0.2
0.4
np[1/2] 0
6p
7p
8p
7d 6d
nd[1/2] 0
Δ E,
cm
-1
F, kV/cm
8d
Fig. 1. The dependence of the np[1/2]0- and nd[1/2]0-states of an Ar atom on the electric-field strength, ωHFD = 100 MHz (a weak interaction of states).
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dependence ΔE on the principal quantum number n for the nd[1/2]0 states of an Ar atom for the electric-field frequency ω = 243.52·104 MHz. As for the direction of the shift of the nd states, which strongly mix in the electric field, they can shift both to the IR and UV regions with an increase in the electric-field strength (see Table 1).
The ω-dependence of the states (F is fixed). To study the dependence of the Stark effect on the change in the electric-field frequency, it was necessary to find frequency dependences of shifts and splitting of argon-atom levels at a fixed electric-field strength. These dependences for a number of levels at the electric-field strength F = 10 kV/cm are listed in Table 2 and 3. In the tables, ΔE is the energy-level shift with respect to its position in the absence of the electric field, and ΔESplit is a maximal splitting of the energy level calculated as a difference between the states with maximal and minimal deviations from the unperturbed position.
TABLE 1. The Direction of Shifts ΔE (cm–1) for the Stark States nl[K]JM Depending on the Frequency and Strength of the Electric Field (M = –J...J)
State ω nl K; J HFD NH3 HCN ns′ ½; 0, 1 IR IR IR
np′ ½; 0, 1
IR IR IR 3/2; 1, 2
nd′ 3/2; 1, 2
IR (n ≤ 7), UV (n > 7) IR (n ≤ 7), UV (n > 7) IR (n ≤ 6), UV (n > 6)
5/2; 2 IR (n ≤ 7), UV (n > 7) nf′ 5/2; 2 UV UV UV ns 3/2; 1, 2 IR IR IR
np
1/2; 0
IR IR
IR (n ≤ 6), UV (n > 6) 1/2; 1 IR (n ≤ 8), IR – UV (n > 8)
3/2; 1, 2 IR (n ≤ 8), IR – UV (n > 8) 5/2; 2 IR
nd
1/2; 0 UV UV UV (n ≤ 7), IR – UV (n = 8) 1/2; 1 UV (n ≠ 7), IR – UV (n = 7) UV (n ≠ 7), IR – UV (n = 7) IR – UV (n = 6.7), IR (n = 8) 3/2; 1
IR IR IR – UV (n = 8), UV (n = 7.9)
3/2; 2 IR (n < 7), IR – UV (n = 8) IR (n = 7), UV (n = 8) 5/2; 2 IR IR (n = 7), UV (n = 8)
nf 3/2; 1, 2
UV UV UV 5/2; 2
0 2 4 6 8 10-0.03
0.00
0.03
0.06
0.09
0.12
4d
7d
6d
5d
nd[1/2] 0
Δ E,
cm
-1
F, kV/cm
Fig. 2. The dependence of the nd[1/2]0- states of an Ar atom on the electric-field strength, ωHCN = 243.52·104 MHz (a strong interaction of states).
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The simulation results have allowed us to reveal the following regularities in the behavior of shifts and splitting of argon-atom levels under changes in the electric-field frequency:
1) In the absence of Stark-state interactions in the electric field, the splitting and shift of the energy level do not depend on the electric-field frequency. As an illustration of this regularity, the calculation results are listed for the 6s′[1/2]0, 5d[1/2]0 (see Table 2), and 6s[3/2]1 levels (see Table 3).
2) Under weak interaction between the Stark states, an increase in the electric-field frequency leads to a decrease in shifts and splitting of these states. As an illustration, the calculation results for the shifts and splitting of the nl′[1/2]0, and nl′[K]J levels are listed in Tables 2 and 3 (except 8p′[3/2]2). This conclusion does not contradict perturbation theory, because according to this theory, an increase in the electric-field frequency leads to the fact that the atomic level in the electric field can be considered as isolated in the optical limit (at ω~1015 Hz), that is, one can neglect the interaction between this and the neighbouring levels [5]. At a strong interaction between the states, this regularity is broken (see Tables 2 and 3 for the calculation results for shifts and splitting of the nl[1/2]0, 8p′[3/2]2, and nl[K]J levels except for 6s[3/2]1 and 5d[1/2]0 levels).
3) Under weak interaction between the Stark states, an increase in the electric-field frequency leaves the directions of shifts of the levels and states in the electric field unaffected, whereas strong interaction does change the shift direction, and it is a characteristic phenomenon (see the calculation results for the np and nd states in Tables 2 and 3, and the results in Table 1).
As for the states with J ≥ 3, they were also computed within the developed approach. However, these states interact with other states so much that their classification within the Jl coupling scheme is impossible, and the behavior of such states in the electric fields of different strength and frequency requires special investigation.
TABLE 2. The Shift ΔE (in cm–1) of the Stark States nl[K]0 (M = 0) as a Function of Electric-Field Frequency (F = 10 kV/cm)
The results obtained are of interest both from a theoretical point of view and for solving practical tasks of spectroscopy. The simulation data have allowed us to reveal the regularities inherent in the behavior of the Stark states of an Ar atom under the changes in the frequency and strength of the electric field. In doing so, we showed that the interaction between the Stark states, which is specified by the electronic structure of the level, plays the key role in the behavior of the energy-level shifts and splitting in the electric field. The results of modeling can be used for accounting for the processes taking place in plasma, for plasma diagnostics, and for solution of spectroscopic tasks.
It should be noted that the quadratic dependence of the Stark-state shifts is well known from perturbation theory, and this regularity is confirmed within the developed theoretical approach. The rest regularities found in this work have been derived for the first time.
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