Center-of-mass effects for hydrogen atoms in a nonuniform electric field: applications to magnetic fusion, radiofrequency discharges, and flare stars

We study whether or not the Center-of-Mass (CM) motion and the relative motion can be separated for hydrogen atoms in a nonuniform electric field. First, we show that in the general problem of two charges in a nonuniform electric field, the CM and relative motions, rigorously speaking, cannot be separated. Second, we use an approximate analytical method of the separation of rapid and slow subsystems to achieve a pseudoseparation of the CM and relative motions for hydrogenic atoms/ions in an arbitrary nonuniform electric field. Third, we further develop these results for the case of a hydrogen atom in the nonuniform electric field, where the field is due to the nearest (to the hydrogen atom) ion in a plasma. Fourth, we apply the results to the ion dynamical Stark broadening of hydrogen lines in plasmas. Fifth, we present specific examples of laboratory plasmas (e.g., magnetic fusion plasmas or radiofrequency discharges) and astrophysical plasmas (e.g., in atmospheres of flare stars) where the allowance for these CM effects leads to a significant increase of the width of hydrogen spectral lines.


Introduction
Center-of-Mass (CM) effects for hydrogenic atoms/ions in a uniform magnetic field are well-studied-see, e.g., papers [1][2][3] and references therein. The CM motion and the relative (internal) motion are coupled in a magnetic field and, rigorously speaking, cannot be separated. For hydrogen atoms it is possible to achieve a pseudoseparation leading to a Hamiltonian for the relative motion that still depends on a CM integral of the motion called pseudomomentum [3].
As for hydrogenic atoms/ions in a uniform electric field, it is well-known that the CM and relative motions can be separated rigorously (exactly)-see, e.g., [4]. As for hydrogenic atoms/ions in a nonuniform electric field, there seem to be nothing about the separation (or non-separation) of the CM and relative motions in the literature, to the best of our knowledge.
In the present paper we study this issue for hydrogenic atoms/ions in a nonuniform electric field. First, we show that in the general problem of two charges in a nonuniform electric field, the CM and relative motions, rigorously speaking, cannot be separated. Second, we use an approximate analytical method of the separation of rapid and slow subsystems to achieve a pseudoseparation of the CM and relative motions for hydrogenic atoms/ ions in an arbitrary nonuniform electric field. Third, we further develop these results for the case of a hydrogen atom in the nonuniform electric field, where the field is due to the nearest (to the hydrogen atom) ion in a plasma. Fourth, we apply the results to the ion dynamical Stark broadening of hydrogen lines in plasmas. Fifth, we present specific examples of laboratory and astrophysical plasmas where the allowance for these CM effects leads to a significant increase of the width of hydrogen spectral lines.
2. General case of the pseudoseparation of the center-of-mass and relative motions in a nonuniform electric field We consider a system of two charges e 1 and e 2 of masses m 1 and m 2 , respectively, in a nonuniform electric field. The Lagrangian of the system is r r r r r r  L  m d dt  m d dt  2 e e  e  e  ,  1   1  1  2  2  2  2  1 2  2  1  1  1  2 where r 1 and r 2 are radii-vectors of charges e 1 and e 2 , respectively and j is the potential of the nonuniform electric field. After the substitution so that R and r are the coordinates related to the CM motion and the relative motion, respectively, the Lagrangian takes the form is the reduced mass of the two particles, and is a nonuniform electric field (in the expansion of the electric potential we disregarded terms higher than the dipole one). In equation (6) and below, for any two vectors A and B, the notation AB stands for the scalar product (also known as the dot-product) of the two vectors.
The Hamiltonian, corresponding to the Langrangian from equation (3) is the Hamiltonian of the relative motion, p being the momentum of the relative motion. Thus, the above equations show that at the presence of a nonuniform electric field, the CM motion and the relative motion are coupled (by U(R, r) from equation (6)) and therefore, rigorously speaking, cannot be separated. However, in the case where m 1 =m 2 , the CM and relative motions can be separated by using the approximate analytical method of separating rapid and slow subsystems: in this case, the characteristic frequency of the relative motion is much greater than the characteristic frequency of the CM motion, so that the former and the latter are the rapid and slow subsystems, respectively. Below are the details of this method that can be found, e.g., in [5].
The first step is to freeze the coordinates R of the slow subsystem and to solve for the motion of the rapid subsystem characterized by the truncated Hamiltonian where R is treated as a fixed parameter rather than as the dynamical variable. In the situation where the charges e 1 and e 2 are of the opposite sign (say, for definiteness e 1 <0 and e 2 >0), this becomes the Hamiltonian of a hydrogenic atom/ion in a 'uniform' electric field. By treating the last term in equation (12) in the first order of the perturbation theory, one obtains the following expression for the energy of the relative motion, i.e., the rapid subsystem (see, e.g., the textbook [ where there was used the well-known relation between the mean value 〈r〉 of the radius-vector and the Runge-Lenz vector A (see, e.g., [7,8] is the polar angle of the vector F(R) and q is the electric quantum number (q=n 1 −n 2 , where n 1 and n 2 are the parabolic quantum numbers). Physically, the quantum number q is intimately connected to the conservation of the Runge-Lenz vector A for the unperturbed hydrogen atom: the eigenvalue of the operator A is q/n-see, e.g., the textbook [6]. The second step of the analytical method of separating rapid and slow subsystems is to proceed to the slow subsystem (the CM motion), for which E(R) from equation (15) will play the role of an effective potential. The effective Hamiltonian H CM,eff (R, P) for the CM motion becomes (the first, R-independent term in E(R) has been omitted because it does not affect the CM motion) Thus, the application of this analytical method allowed the pseudoseparation of the CM motion and the relative motion for any two oppositely charged particles of significantly different masses in a nonuniform electric field.
It should be emphasized that in our paper, the CM coordinate R is considered as the dynamical variable (which generally depends on time) and that the Hamiltonian H CM,eff (R, P) from equation (16) can be used to solve for the CM motion. This is the primary distinction of our work from papers where the CM coordinate of a hydrogenic atom/ion in a nonuniform electric fields was considered to be fixed 1 . In section 3 we actually solve for the CM motion in the situation where the nonuniform electric field is due to the plasma ion nearest to the hydrogen atom, and apply the solution to the dynamical Stark broadening of hydrogen lines in plasmas. This would be impossible if the CM coordinate R would not have been treated as the dynamical variable.
We also note that higher order terms (quadrupole, octupole etc) in the expansion of the potential j(R) in equation (4) can be easily taken into account, if necessary, and this analytical method for the pseudoseparation of the CM motion and the relative motion, with R considered as the dynamical variable, would still work.
In the particular case of hydrogen atoms one has e e, e e, m m m m , 17 where e>0 is the electron charge, m e and m p are the electron and proton masses, respectively. Then equation (16) 3. Analytical solution for the center-of-mass motion of a hydrogen atom in the field of the nearest ion and its application to the dynamical Stark broadening of hydrogen lines in laboratory and astrophysical plasmas Now we consider the situation where the nonuniform electric field is due to the nearest (to the hydrogen atom) ion of the positive charge Ze and mass m i in a plasma located at the distance R from the hydrogen atom. Then the Hamiltonian from equation (18) can be rewritten as This Hamiltonian represents a particle of mass m in the dipole potential. Since this particle is relatively heavy (m?m e ), its motion can be described classically and the corresponding classical solution is well-known-see, e.g., paper [11]. For this physical system, the radial motion can be exactly separated from the angular motion resulting in the following radial equation: where E CM is the total energy of the particle. This equation allows the following exact general solution: It is well-known that in plasmas of relatively low electron densities N e , the Stark broadening of the most intense hydrogen lines, i.e., the lines corresponding to the radiative transitions between the levels of the low principal quantum numbers (such as, e.g., Ly-alpha, Ly-beta, H-alpha, etc), is dominated by the ion dynamical broadening-see, e.g., publications [12][13][14][15][16][17][18]. The corresponding validity condition is presented in appendix A. In the so-called 'conventional theory' of the dynamical Stark broadening (also known as the 'standard theory') [19][20][21][22], the relative motion within the pair 'radiator-perturber' was assumed to occur along a straight line-as for a free motion (in our case the radiator is a hydrogen atom and the perturber is the perturbing ion).
However, from the preceding discussion it follows that in the more advanced approach, the relative motion within the pair 'radiator-perturber' should be treated as the motion in the dipole potential-(D/R 2 )cos θ, as seen from equation (19). The relevant setup of the problem is to choose the instant t=0 as the instant of the smallest distance (the closest approach) within the pair 'radiator-perturber'. Then v 0 =(dR/dt) t=0 =0, so that equation (21) By considering the motion within the pair 'radiator-perturber' in the reference frame where the perturbing ion is at rest, so that P 0 =mV 0 , where V 0 is the relative velocity within the pair 'radiator-perturber' at t=0, the energy E CM can be rewritten as  Now we consider a radiative transition between hydrogen energy levels a and b. In the general case, the ion dynamical broadening operator Φ ab is defined as follows (by analogy with the electron dynamical broadening operator defined, e.g., in paper [19]): Here 〈K〉θ o denotes the averaging over the angle θ 0 , and the operator σ(V 0 , θ 0 , t) has the form: Here N i is the ion density, f(V 0 ) is the distribution of the velocities (usually assumed to be Maxwellian), ρ is the impact parameter of the perturbing ion, U a and U b are the corresponding time-evolution operators, the symbols * and [K] ang.av stand for the complex conjugation and the angular average, respectively. If the time t would be considered as a parameter, then the diagonal elements of the operator σ(V 0 , t) would have the physical meaning of cross-sections of so-called optical collisions, i.e., the cross-sections of collisions leading to virtual transitions inside level a between its sublevels and to virtual transitions inside level b between its sublevels, resulting in the broadening of Stark components of the hydrogen spectral line. By using the trajectories from equation (25) and averaging over the polar angle θ 0 , one can obtain the evolution operators and then the ion dynamical broadening operator with the allowance for the effect of the CM motion. However, in this general case, the results cannot be obtained analytically.
Therefore, for obtaining the final results analytically (which should help getting the message across in the simple form), we now employ the so-called impact approximation and substitute the evolution operators by the corresponding scattering matrices (see, e.g., papers [20,21] or books [18,22]): In the case where non-diagonal matrix elements of the Φ ab are relatively small, the lineshape is a sum of Lorentzians, whose width γ αβ and shift Δ αβ are equal (apart from the sign) to the real and imaginary parts of diagonal matrix elements 〈α|〈β|Φ ab |β〉|α〉, respectively: Here α and β correspond to upper and lower sublevels of the levels a and b, respectively. Here and below, for any operator G, for brevity we denote its matrix elements 〈α|〈β|G|β〉|α〉 as αβ G βα . As we calculate the scattering matrices by the standard time-dependent perturbation theory, we obtain the following expression for the operator σ R , V , dR 2 R K Q R , V , R . 33 where a B is the Bohr radius, K interf represents the so-called interference term. In the conventional theory [20][21][22], in equation (34) instead of V eff (R 0 , θ 0 ), it would be V .

2
The next step is the averaging of 1/V eff (R 0 , θ 0 ) 2 in equation (34) over the angle θ 0 : so that the quantity Q(R 0 , θ 0 ) after the averaging over θ 0 becomes The way the quantity D (entering equation (37) was defined in equation (19) as D=[3n|q|ħ 2 /(2μ)] Z is valid only for the Lyman lines. For all other hydrogen lines one should use the arithmetic average of the values of D for the upper and lower Stark sublevels-as suggested in the similar case in paper [24] and used in paper [25]. Therefore, in the present paper for all other hydrogen lines we use the following value of D D 3 n q n q Ze a 4, 39 where the quantum numbers with the prime symbol and without it relate to the lower and upper levels, respectively. The next step is the averaging over R 0 . The integral over R 0 in equation (33) has a weak, logarithmic divergence at both small and large impact parameters-just like in the conventional theory [19][20][21][22]. Therefore, as in the conventional theory, we subdivide collisions into 'weak' (R 0 >R min ) and 'strong' (R 0 <R min ), and introduce also the upper cutoff R max (just as in the conventional theory) discussed later. Then the diagonal elements of the cross-section of optical collisions can be represented in the form  (though more rigorously, it should have been R max =min(R Debye , V 0 /Δω), where Δω is the detuning from the center of the spectral line; physically, the requirements R max <V 0 /Δω being the allowance for incomplete collisions).
By integrating analytically over R 0 in equation (40) and substituting into the result the expression for the strong collision constant C from equation (41) we obtain: The boundary R min between the weak and strong collisions in equation (44) is the solution of equation (41) with respect to R min : The next step is the averaging of several quantities from the above equations over Stark sublevels of the upper and lower levels, so that each of these quantities will have the unique value for the particular hydrogen spectral line. First, the square root of the averaged matrix element (〈α|〈β|K 2 |β〉|α〉) is asserted to be The averaging over Stark sublevels (since (q 2 ) av =(m 2 ) av ) results in the following leading term in the quantity [ αβ (K 2 ) βα ] av We mention that the same result (49) can be obtained after the corresponding averaging in the spherical quantization.
We denote This quantity has the meaning of the so-called Weisskopf radius: it is defined here more accurately than in the conventional theory by Griem [22] (which is why here and below the superscript 'A' stands for 'accurate')-see appendix B. The next quantity to be averaged over Stark sublevels of the upper and lower levels, so that it will have the unique value for the particular hydrogen spectral line, is the quantity D from equation (39) (44), and also introduce dimensionless parameters 2 On page 43 of book [22], Griem explicitly chose 3/2 for the quantity |1-S a (R 0 , V 0 , θ 0 ) S b * (R 0 , V 0 , θ 0 )| that we denoted as C. To avoid any confusion we note that what Griem called 'strong collision term' was C/2. The extra factor 1/2 arises from the following integral for the strong collision term:

Conclusions
We studied the general problem whether the CM motion and the relative motion can be separated for hydrogenic atoms/ions in a nonuniform electric field. We demonstrated that, strictly speaking, they cannot be separated. Then we used the approximate analytical method of the separation of rapid and slow subsystems to achieve the pseudoseparation of the CM and relative motions for hydrogenic atoms/ions in an arbitrary nonuniform electric field. This is a fundamental result in its own right. Next we further developed these results for the case of a hydrogen atom in the nonuniform electric field, where the field is due to the nearest (to the hydrogen atom) ion in a plasma. We showed that the effect of the CM motion can be formally taken into account via the substitution of the initial relative velocity V 0 in the pair 'atom -ion' by an effective velocity V eff that depends on the quantum numbers of the hydrogen atom, as well as on the initial separation R 0 in the pair 'atom-ion' and on the ion charge Z.
Then we applied the results to the ion dynamical Stark broadening of hydrogen lines in plasmas. We obtained analytical results for the cross-sections of the optical collisions that control the corresponding Stark width. We presented specific examples of laboratory plasmas (such as magnetic fusion plasmas or plasmas of radiofrequency discharges) and astrophysical plasmas (such as in atmospheres of flare stars) where the allowance for these CM effects leads to a significant increase of the width of hydrogen spectral lines-by up to (15)(16)(17)(18)(19)(20)%.
Thus, in addition to the fundamental importance, the results of the present paper seem to have also practical importance for spectroscopic diagnostics of laboratory and astrophysical plasmas.