Line Shapes in a Magnetic Field: Trajectory Modifictions II: Full Collision-Time Statistics
Hellenic Army Academy, Varis—Koropiou Avenue Vari, P.O. 16673 Vari, Greece
Atoms 2019, 7(4), 94; https://doi.org/10.3390/atoms7040094
Received: 22 August 2019 / Revised: 17 September 2019 / Accepted: 18 September 2019 / Published: 5 October 2019
(This article belongs to the Special Issue Plasma Spectroscopy in the Presence of Magnetic Fields)
In a previous paper, a variation of the Collision-time Statistics method was applied to identify the relevant perturbers for line broadening under the action of a constant magnetic field. As discussed, that version was simplified and inadequate for low magnetic field and/or large perturber mass (ions). The purpose of the present work is to augment the previous work, so that such cases can be handed efficiently. The results may also be used to construct analytic, i.e., impact/unified models under the usual assumptions in these models.
In a previous paper , a variation of the Collision-time Statistics method was applied to identify the relevant perturbers for line broadening under the action of a constant magnetic field. As discussed, that version was simplified and inefficient for low magnetic field and/or or large perturber mass (ions), because an unnecessarily large number of perturbers, the vast majority of which only contribute negligibly to broadening, were generated. The purpose of the present work is to augment the previous work, so that such cases can be handled efficiently. The main problem with the previous treatment was the case of large Larmor radii and corresponding slow cyclotron frequencies, as in the case of large perturber mass and/or low magnetic field. In such cases the motion of the perturbers in the plane perpendicular to the magnetic field may be quite far from completing a full cycle. As a result in the simplified treatment of  a large number of ineffective perturbers would be included in that case. It is believed  that “as long as the gyro-radii of the electrons are much larger than the Debye sphere” the particle trajectories, or, equivalently, the dielectric function would be unaffected by the magnetic field, since the perpendicular motion involves presumably much smaller length scales than the parallel motion, although this belief remains to be quantified. It is shown here that this criterion is not quite correct and that the key parameter determining the adequacy of neglecting spiralling is the ratio of the cyclotron frequency to the width of the line in question.
2. Theoretical Formulation
As in  we assume that:
- the distribution functions, e.g., the Maxwellian velocity distribution is affected by the B-field.
- the shielding is also not affected, e.g., Debye screening
We also consider a neutral emitter in this work.
2.1. Collision-Time Statistics
To include and only the relevant perturbers, we use a modification of the collision-time statistics method of Hegerfeld and Kesting  with Seidel’s improvement - see Ref.  for details, as discussed in .
As in , perturbers move in a helical path characterized by the parallel constant velocity , where the magnetic field direction defines the z-axis (passing through the emitter), the perpendicular velocity with magnitude and impact parameter , which is the distance of the center of the spiral to the z-axis, i.e., the perpendicular motion in the x-y plane is a circular motion with the Larmor radius around the center , with the cyclotron frequency and Q the perturber charge. For the impact parameteri.e., the impact parameter lies in a disk or annulus depending on whether the range of the interaction, discussed below, is larger or smaller than .
The relevant quantities for the helical trajectory R(t) are as follows: The z-coordinate of the trajectory iswithwith the times of closest approach representing the time the perturber trajectory intersects the x-y plane and being uniformly distributed. thus represents how far from the x-y plane the perturber is at t = 0.
Hencewhere describes the position of the impact parameter vector in the x-y plane and is uniformly distributed in (). is an angle describing where on the circular trajectory projection the perturber finds itself at and is also uniformly distributed in () and ultimately related to the time the B-field was turned on. Each perturber is thus characterized by the vector , or equivalently instead of .
As in  we consider as “relevant " perturbers those that, at any time during the interval of interest (0,), come closer to the emitter than a distance , defined so that the interaction is negligible for distances larger than . For a Debye interaction, we usually take , where denotes the shielding(Debye) length. This is because the interation becomes negligible ( for larger distances).
Therefore for a perturber to be relevant the condition must hold for at least one time t in , where is the time of interest, i.e., a time large enough that the Fourier transform of the line profile has decayed to negligible levels, or an asymptotic form is identifiable. is a linear combination of products of time evolution operators (U-matrices) of the upper and lower levels. These time evolution operators -needed for times - are determined by solving the Schroedinger equation in the Debye-shielded field . Therefore a particle will only be relevant if for at least one time in the interval [0,] it comes closer than to the emitter(if not, then the perturbation produced by that particle is negligible due to Debye screening), which means that for at least time t in [0,]:
Thus we generate and as before, but also draw , uniformly distributed in ( as illustrated in in Figure 1 and effectively only accept perturbers if, for at least one time in (0,) Equation (6) is satisfied. The value of the LHS occurs for . This in general imposes (i.e., not all contribute for a given ) on the values of and that a perturber can have and still contribute effectively to broadening, specifically:
Note that for Case b, the argument of the inverse cosine is absolutely ≤1. Specifically:
- If , thenThe left of the inequality follows because alone. The right part also follows sinceHence
Hence the angle difference must be in the shaded area shown in Figure 2. So for at least one time t in (), the following must hold for the perturber with parameters to contribute:i.e.,or
The net result is that for each , only a a range of ,contributes (this means that a fraction contributes compared to the simplified case discussed in the previous work, i.e., the collision volume is smaller by ). If this is we effectively have rectilinear trajectories for the time of interest. If this turns out to be larger than , we have a full revolution and we can use the simplified formulas discussed in . As mentioned, we are mainly interested in the situation where and , as this is the case of large , but slow , otherwise the relation between and is always satisfied for at least one t in .
We can use the variable with and write the argument of the inverse cosine as
Note that for low B (large ) this tends to −1, hence the inverse cosine is close to . This means that in this limit , e.g., we get a (but note that in that limit we had divergencies in the relevant functions when computing the collision volume in ).
This situation is depicted in Figure 3, which shows the typical situation for the phase space of the quantity that contributes. This is also illustrated in Figure 4, which shows, for the same , 4 different angles , which determine the centers of the spirals and the parts of the circular projections of these spirals that are effective. For instance if the center of the spiral, i.e., the vector of the impact parameter is the the right (), then (the leftmost of the circular trajectory projection) for . Similarly, if the impact parameter vector is to the south (), then the relevant is to the north of the circular trajectory projection, e.g., ).
In the limit (or infinite perturber mass), and we get from the and integrations a term .
2.2. Collision Volume
The collision-time statistics method first computes the number of relevant particles, i.e., the density times the relevant volume, i.e., the above cylinder. This volume is as before , except that we also account for the polar angle , describing the orientation of the impact parameter with respect to the x-axis and the angle describing the position of the particle on the perpendicular x-y plane at time , i.e., we have the extra integrations . The integration simply returns a factor of 1 if , but it does so even under the weaker condition , or
Otherwise it gives a factor of with defined in Equation (15).
The nonnegative root of Equation (17) isi.e., the results of  are also valid for , which in turn requires that(else ), which also guarantees the reality of , i.e.,
The angular integration simply returns 1 in either case. As a result, the results of  need for . Otherwise, the collision volume calculation runs as follows:
For , i.e., , . However, as already discussed, this is valid (e.g., no restriction on is required ) also for , hence for .
For small and is the maximum of the two. The collision volume reads:with and denoting a one and two-dimensional Maxwellian velocity distributions respectively and with redefined as in Table 1:
and are:with The integrals are given explicitly below. However, we first define the dimensionless quantities:andand(essentialy the averaged inverse s ).andwhilewithwith and being of course functions of x.and
Note that the difference from the previous work  is the factor for . Also note that in , the corresponding integrations to infinity, e.g., the equivalents of and diverged as . This divergence has been eliminated here due to the factor. This is shown in Appendix A, which evaluates the and integrals.
The remaining contributions vanished in  as and clearly continue to do so here.
As already mentioned in , the number of particles that are in this volume, and hence need to be simulated, is simply the volume multiplied by the perturber density.
2.3. Generating Perturbers
To generate perturbers we proceed as in , but also generate for each perturber an angle , uniformly distributed in (0,). Once we have generated and , we also generate uniformly distributed in .
In more detail, we first draw a random number uniformly distributed in (0,1). If this is smaller than , then we generate from the distribution by generating independently a with the probability distribution , a with the probability distribution and a with the probability density in .
Otherwise we generate from the distribution . The generation of impact parameters was done by a rejection method, as straightforward inversion is not possible.
Once and have been generated, is selected as a uniformly distributed time in . and are also generated as discussed above.
The present work extends the simplified theory for spiralling motion in a constant magnetic field presented in  which was typically efficient for electron perturbers to more cases of practical interest, i.e., ion perturbers and/or weak magnetic fields. The results of  are seen to hold for and are here extended to , i.e., a regime typical for ions or weak magnetic field, thus validating the common wisdom that ion trajectories are usually unaffected by spiralling. In addition, this work identifies relevant parameters (e.g., q, ) and criteria for using a straight line and also allows the efficient treatment of spiraling ion trajectories if needed. The results of this work are also a useful basis for approximate standard treatments, i.e., impact/unified theories, if (strong) collisions are isolated/disentangled and further, to perturbative impact/unified treatments if these collisions may be handled in perturbation theory.
With regard to the notion , discussed in the introduction, that the relevant quantity for neglecting spiralling is that , the present paper shows that although the idea is qualitatively correct, i.e., one may indeed neglect spiralling for small B and/or large perturber mass, leading to a large Larmor radius, the actual situation is more complex and described by Table 1. Specifically, to neglect spiralling, it is necessary that or equivalently , i.e., the ratio of the cyclotron frequency to the width(HWHM) of the line is important. This is because if this ratio is small, then the perturber motion does not cover a full revolution and if very small, the motion is essentially unaffected by the magnetic field. However, if the line is very narrow ( is large) there may be enough time to complete at least a sizeable portion of a revolution, even if the Larmor radius is larger than the Debye length.
This research received no external funding.
Conflicts of Interest
The author declares no conflict of interest.
Appendix A. Calculation of the I3 and J3 Double Integrals
Note that the integrands are the same except that involves and extra factor of .
Note that these integrals are the contributions for .
Appendix A.1. The I3 Integration
For and , the non-trivial integrals reduce to ()
For both and , the upper limit if , for whichand
Hence the middle term vanishes, while the arctangent is
If (as in ) the lower limit is , the argument of the square root is also 0, and
Note that unlike the simplified version , does diverge because the cancells the .
The J3 Integration
We break the integral aswhere using , we have:which does diverge for andwith
Using this becomes:withand
If , and , hence the argument of the inverse cosine is very nearly −1, and thus we have which gives a vanishing contribution for . However, for this is multiplied by so the result is not immediately clear. Since Taylor-expanding the inverse cosine around -1 does not work due to the infinite derivative, we can use Frobenius’s method or writeand write
Next, use L’Hopital’s rule to evaluate aswith the final result that
Therefore we again have no divergence for small q (large s); furthermore the above large s asymptotic result is useful numerically due to possible underflows of .
- Alexiou, S. Line Shapes in a Magnetic Field: Trajectory Modifications I: Electrons. Atoms 2019, 7, 52. [Google Scholar] [CrossRef]
- Günter, S.; Köenies, A. Diagnostics of dense plasmas from the profile of hydrogen spectral lines in the presence of a magnetic field. JQSRT 1999, 62, 425–431. [Google Scholar] [CrossRef]
- Hegerfeld, C.G.; Kesting, V. Collision-time simulation technique for pressure-broadened spectral lines with applications to Ly-α. Phys. Rev. A 1988, 37, 1488–1496. [Google Scholar] [CrossRef] [PubMed]
- Seidel, J.; Verhandl, D.P.G. Spectral Line Shapes 6; Frommhold, L., Keto, J.W., Eds.; AIP: New York, NY, USA, 1990; Volume 216. [Google Scholar]
- Alexiou, S. Implementation of the Frequency Separation Technique in general lineshape codes. High Energy Density Phys. 2013, 9, 375–384. [Google Scholar] [CrossRef]
Figure 1. x-y trajectory projections for . Shown is the annular region between concentric circles with the origin (the emitter position) as center and radii and , respectively. For an impact parameter at a distance from the center in the annular region, a circle with radius (dashed) represents the projection of the perturber path in the x-y plane. Hence a point on that circle is , with the angle on the dashed circle. This must be no more than away from the center, else this perturber does not contribute.
Figure 2. The shaded area shows the difference that satisfies Equation (6).
Figure 3. The shaded area shows the difference that satisfies Equation (6).
Figure 4. Illustration of the relation between and for the case . Impact parameters lie between the dashed and dash-dotted circles with radii and . The part of the circular trajectory projections that are within (bold circle) of the emitter (i.e., the center) are in the opposite direction of the impact parameter vector, e.g., to the north for the southern circular trajectory projection.
Table 1. vs. relevant parameters.
© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).