Medical, Pharma, Engineering, Science, Technology and Business

**Oks E ^{*}**

Department of Physics, 206 Allison Labs, Auburn University, Auburn, AL 36849, USA

- *Corresponding Author:
- Oks E

Physics Department, 206 Allison Labs Auburn University

Auburn, AL 36849, USA

**Tel:**334-844-4362

**Fax:**334-844-4613 |

**E-mail:**goks@physics.auburn.edu

**Received Date:** April 25, 2017 **Accepted Date:** May 11, 2017 **Published Date:** May 15, 2017

**Citation: **Oks E (2017) New Source of the Red Shift of Highly-Excited Hydrogenic Spectral Lines in Astrophysical and Laboratory Plasmas. J Astrophys Aerospace Technol 5: 143. doi:10.4172/2329-6542.1000143

**Copyright:** © 2017 Oks E. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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High-n hydrogen spectral lines (SL), n=13–17, studied in astrophysical and laboratory observations at the electron density N_{e} ~ 10^{13} cm^{-3} by Bengtson and Chester exhibited red shifts by orders of magnitude greater than the theoretical shifts known up to now. Specifically, BC presented the shifts of these SL observed in the spectra from Sirius and in the spectra from a radiofrequency discharge plasma in the laboratory: both types of the observations yielded red shifts that exceeded the corresponding theoretical shifts by orders of magnitude. In the present paper we introduce an additional source of the shift of high-n hydrogenic SL. We show that for high-n hydrogen SL it makes the primary contribution to the total red shift. We demonstrate that for the conditions of the astrophysical and laboratory observations from paper by Bengtson and Chester, this additional red shift is by orders of magnitude greater than the theoretical shifts known up to now. Finally we show that the allowance for this additional red shift removes the existed huge discrepancy between the observed and theoretical shifts of those that of high-n hydrogen SL.

Astrophysical; Shifts of Hydrogen SL; Measurements; Red shift; Laboratory plasmas

Spectral lines (hereafter, SL) of hydrogenic atoms/ions in plasmas are typically red-shifted by electric microfields by Griem in 1997 and Oks in 2006 [1,2]. This Stark shift is important not only fundamentally, but also practically. In astrophysics, red shifts of SL are observed in various astrophysical objects: for deducing the relativistic (cosmological and gravitational) red shifts by Nussbaumer and Bieri [3] from observed red shifts it is necessary to take into account the Stark shift. In laboratory plasma diagnostics, measurements of the Stark shift can complement measurements of the Stark width for determining the electron density paper by Parigger et al. [4].

The best studied are shifts of hydrogen SL (especially of the H-alpha
line). For low-n hydrogen SL (n being the principal quantum number of
the upper level), studied experimentally mostly at the electron densities
Ne=10^{16}–10^{19} cm^{-3} and slightly higher, in the course of time there was
achieved an agreement between the experimental and theoretical shifts
– see, e.g., books by Griem [1] and Oks [2], as well as papers by Grabowski
and Halenka [5], Demura et al. [6], Demura et al. [7], Djurovic et al. [8],
Kielkopf and Allard [9] and references therein.

However, high-n hydrogen SL (n=13–17), studied in astrophysical
and laboratory observations at N_{e} ~ 10^{13} cmM^{-3} by Bengtson and Chester
[10], exhibited red shifts by orders of magnitude greater than the
theoretical shifts known at that time (in 1972) or at any later time up
to now. Specifically, in paper by Bengtson and Chester [10] there were
presented the shifts of these SL observed in the spectra from Sirius and
in the spectra from a radiofrequency discharge plasma in the laboratory:
both types of the observations yielded red shifts that exceeded the
corresponding theoretical shifts by orders of magnitude. In the same
year Baracza [11] presented observations of hydrogen SL of n=19-23 in
the spectrum from Sirius, i.e., higher-n SL than observed in the spectra
from Sirius by Bengtson and Chester [10]. In a later paper Baracza [12]
wrote “measurements in the spectrum of Sirius Baracza [11] did not
show any shift of Balmer lines lower than H21” and questioned the
shifts observed in the spectra from Sirius by Bengtson and Chester [10].
However, first Baracza [11] did not observe the same SL as Bengtson
and Chester [10]. Rather, Baracza [11] observed highern SL (n=19–23)
than SL of n=13–17 observed in the spectra from Sirius by Bengtson and Chester [10]. As the principal quantum number n increases, the SL
become much weaker in their absolute intensity, and also become much
broader and thus subjected to blending/merging with adjacent SL. (In
paper by Baracza [11] the author himself noted that the observed lines
H20 and H21 were subjected to blending.) For these reasons it is much
more difficult to reliably measure shifts of such higher-n SL observed
by Baracza [11]. In distinction, Bengtson and Chester [10] emphasized
that in their observations “no measurements of the profile center were
taken where there was strong blending in the wings”. Thus, in reality
observations by Baracza [11] in the spectrum of Sirius do not disprove
observations by Bengtson and Chester [10] in the spectra of Sirius.

As for further observations of shifts of high-n hydrogen SL in laboratory plasmas in the same range of n as in paper by Bengtson and Chester [10], such an experiment was reported by Himmel [13] who observed hydrogen SL of n=12–19. The comparison with the shifts of hydrogen SL of n=13–17 observed in the laboratory plasma by Bengtson and Chester [10] had shown the following. The shifts of SL H13, H14, and H16 agreed in these two experiments within the error margins, but the shift of SL H15 and H17 observed by Himmel [13] were significantly smaller than the corresponding shifts observed by Bengtson and Chester [d that the observed lines H20 and H21 wer] and Himmel [13] suggested that the larger shifts measured by Bengtson and Chester [d that the observed lines H20 and H21 wer] “are caused by some kind of systematic error … as far as the investigation of the laboratory plasma is concerned”. However, first Himmel [13] emphasized that “direct comparison of observations is not possible because different plasma parameters were used”. Second, it seems inconsistent to assume that in the laboratory measurements by Bengtson and Chester [d that the observed lines H20 and H21 wer], two SL (H15 and H17) were subjected to a some kind of systematic error, while three other SL (H13, H14, and H16) were in agreement with the shift measurements by Himmel [13] and thus were not subjected to the same systematic error. Third, measurements of the widths of the high-n hydrogen SL by Himmel [13] versus n showed that in the range of n=12–17 the width increased with the growing n, but in the range of n=17–19 the width decreased with growing n accordfing by Himmel [13]. The decrease of the width with the growing n in the range of n=17–19 contradicts to any modern theory of the Stark broadening of hydrogen SL and thus could indicate some systematic error in the experiment by Himmel [13].

Let us summarize the above situation–First for the astrophysical observations and then for the laboratory observations:

1. In the spectra of Sirius, the relatively large, theoretically unexplained shifts of the high-n hydrogen SL in the range of n=13– 17, observed by Bengtson and Chester [10] actually have not been disproved by Baracza [11,12] who observed significantly higher-n hydrogen SL (n =19–23) that are much weaker, broader and thus subject to blending (making them less reliable) compared to the SL of n=13–17 observed by Bengtson and Chester [10]. In other words Baracza [11,12] compared “apples with oranges” instead of comparing “apples with apples”.

2. The two laboratory experiments on the shifts of high-n hydrogen SL - by Bengtson and Chester [10] and by Himmel [13]–agreed with each other (within the error margins) with respect to 3 out 5 SL measured by Bengtson and Chester [10]and disagreed with respect to 2 out 5 SL measured by Bengtson and Chester [10]. However, plasma parameters in the two experiments differ from each other and [13] emphasized that the direct comparison was not possible. Besides, the dependence of widths of the high-n hydrogen SL measured by Himmel [13] on n contradicts to the modern Stark broadening theories and could be symptomatic of a systematic error.

3. Himmel wrote that “it seems desirable to determine which theoretical model if any qualifies for explaining detectable line shifts” of these high-n hydrogens SL as studied by Himmel [13]. Up to now there was no theoretical explanation of the relatively large, detectable line shifts of high-n hydrogen SL observed in the spectra of Sirius and in the spectra from the corresponding laboratory plasma. So, there is still the need for such explanation.

In the present paper we introduce an additional source of the shift of high-n hydrogenic SL. We show that for high-n hydrogen SL it makes the primary contribution to the total red shift. We demonstrate that for the conditions of the astrophysical and laboratory observations from paper by Bengtson and Chester [10], this additional red shift is by orders of magnitude greater than the theoretical shifts known up to now (below we refer to the latter as the “standard shifts”). Further we show that the allowance for this additional red shift leads to the agreement with the astrophysical red shifts for all four high-n hydrogen SL observed in paper by Bengtson and Chester [10] and to the agreement with the laboratory red shifts, for four out of five high-n hydrogen SL observed in the same paper. The last but not least–the theory developed in the present paper has the fundamental importance in its own right: it can be applied also to the high-n SL of hydrogen like ions and thus motivate further observations of the shifts of not only hydrogen SL, but also of SL of hydrogen like ions.

For the electron densities N_{e} ~ 10^{13} cm^{-3}, there are the following two
major “standard” contributions to the shift of high-n hydrogen SL–in
order of diminishing magnitude. The largest standard contribution
is due to quenching (non-zero Δn) Griem [14] and elastic (zero
Δn) Boercker and Iglesias [15] collisions with plasma electrons–
hereafter, the electronic shift, see also paper by Griem [16]. For
high-n lines, the primary component of the electronic shift comes
from the quenching collisions: it scales as ~ n^{4}, while the secondary
component (originating from the elastic collisions) scales as ~ n^{2}.

**Table 1** presents the electronic shift Se of the hydrogen SL H_{13}–
H_{17}, calculated by formulas from papers by Griem [14,16], and their
comparison with the shifts from paper by Bengtson and Chester [10]
observed in astrophysical and laboratory plasmas. It is seen that the
electronic shift is by orders of magnitude smaller than both the shift of
the SL H_{14}, H_{15}, H_{17} observed in the spectrum of Sirius and the shift of
the SL H_{13}, H_{15}, and H_{17} observed in the laboratory plasma.

n | λn (A) | Se (A) | S_{Sirius} (A) |
S_{exp} (A) |
---|---|---|---|---|

13 | 3734 | 0.0017 | -- | 0.03 ± 0.03 |

14 | 3722 | 0.0021 | 0.03 ± 0.05 | 0.00 ± 0.04 |

15 | 3712 | 0.0026 | 0.09 ± 0.07 | 0.15 ± 0.05 |

16 | 3704 | 0.0032 | –0.007 ± 0.05 | 0.00 ± 0.05 |

17 | 3697 | 0.0038 | 0.21 ± 0.08 | 0.30 ± 0.08 |

**Table 1:** Electronic shift Se of the hydrogen spectral lines H13 – H17, calculated
by formulas from papers by Griem [14,16] and their comparison with the shifts from
paper by Bengtson and Chester [10].

This comparison already shows the inability to explain the observed shifts from paper by Bengtson and Chester [10] by the “standard” sources of the shift. This is because the second largest “standard” shift is by one or even two orders of magnitude smaller than the first “standard” shift (the electronic shift). For this reason it is sufficient to estimate the second largest standard shift just by the order of magnitude, as we do below.

Specifically, by evaluating the second largest standard shift we mean the standard approach to calculate the contribution to the shift from plasma ions–hereafter, the standard ionic shift.

For the parameters relevant to the laboratory experiment from
paper by Bengtson and Chester [10] (n=13–17, N_{e}=1.2 × 10^{13} cm^{-3},
T=2000 K) and the similar parameters for the astrophysical observations
from the same paper, the ions can be considered quasistatic. In the
standard approach the first step in calculating the contribution of the
ionic shift is to use the multipole expansion with respect to the ratio
r_{rms}/R (in the binary description of the ion microfield) or with respect
to the analogous parameter r_{rms}F^{1/2} (in the multi-particle description
of the ion microfield F), where r_{rms} is the root-mean-square value of
the radius-vector of the atomic electron (r_{rms} ~ n^{2}/Z_{1}, where Z_{1} is the
nuclear charge), and R is the separation between the nucleus of the
radiating atom/ion and the nearest perturbing ion. Here and below we
use the atomic units.

It was quite obvious that the dipole term of the expansion (~ 1/R^{2} or
~ F) does not lead to any shift of a hydrogenic SL. This is because each
pair of the Stark components, characterized by the electric quantum
numbers q and –q, is symmetric with respect to the unperturbed
frequency ω_{0} of the hydrogenic line–in terms of both the displacement
from ω_{0} and the intensity. (Here q=n_{1}–n_{2}, where n_{1}and n_{2} are the
first two of the three parabolic quantum numbers (n_{1}n_{2} m).) As for
the quadrupole term of the expansion (~ 1/R^{3} or ~ F^{3/2}), it does not
shift the center of gravity of hydrogenic lines, as rigorously proven analytically in paper Oks [17]. Namely, after taking into account the
quadrupole corrections not only to the energies/frequencies, but also
to the intensities, and then summing up over all Stark components of
a hydrogenic SL, the center of gravity shift vanishes at any fixed value
of R or F^{1} [18]. So, within the approach of the multipole expansion, the first
non-vanishing ionic contribution to the shift of hydrogenic SL should
come from the next term of the multipole expansion: from the term
~ 1/R^{4} or ~ F^{2}. While considering this term, some authors limited
themselves by the quadratic Stark (QS) effect (such as, e.g., in papers
by Griem HR [16, 18,19]):

(1)

Where Z_{2} is the charge of perturbing ions; the superscript (4) at
ΔE_{QS} indicates that this term is of the 4^{th} order with respect to the small
parameter r_{rms}/R. Here and below we use the atomic units ħ=e=m_{e}=1,
unless specified to the contrary.

However, first, it is inconsistent to take into account the quadratic
Stark corrections to energies, but not to the intensities of Stark
components Könies, Günter [19,20]: the corrections to the energies are
of the same order as the corrections to the intensities, as noted in paper
by Demura et al. [6]. Second, the following deficiency of papers by
Griem [16], by Könies and Günter [19,20] is even more important with
regard to the energy correction of the order ~ 1/R^{4}. The above Equation
(1) originated from the dipole term (of the multipole expansion)
treated in the 2nd order of the perturbation theory. However, the
quadrupole term, treated in the 2^{nd} order of the perturbation theory,
and the octupole term, treated in the 1^{st} order of the perturbation
theory, actually also yield energy corrections ~ 1/R^{4}, as it was shown as
yearly as in 1969 by Sholin [21]. The rigorous energy correction of the order ~
1/R^{4} has the form (as in
1969 by Sholin [21] and presented also in book Komarov et al. [22]):

(2)

Obviously, it is inconsistent to allow for one term and to neglect two other terms of the same order of magnitude.

For our purpose, it is sufficient to evaluate the standard ionic shift contribution by calculating the multipole corrections only to the energies–for three reasons. First, the ionic shift contribution caused by the corresponding multipole corrections to the intensities of the Stark components of a hydrogen line are of the same order of magnitude as ionic shift contribution caused by the multipole corrections to the energies. Second, frequently there is a partial cancellation of these two sources of the standard ionic shift: multipole corrections to the intensities of the Stark components frequently result in the shift to the opposite direction compared to the shift contribution caused by the multipole corrections to the energies, as noted in paper by Demura et al. [6]. Third, the calculations will show that the total standard ionic shift is several times smaller than the electronic shift, so that evaluating the former by the order of magnitude would be adequate.

While calculating the ionic multipole corrections only to the
energies, we took into account not only the rigorous expression for the
term ΔE^{(4)} = ~1/R^{4} given by Equation (2), but also the rigorous analytical
expressions for the terms ΔE^{(5)} ~ 1/R^{5} and ΔE^{(6)} ~ 1/R^{6} presented in
book by Komarov et al. [22] in Equation (4.59). Specifically for the
parameters corresponding to the observations from paper by Bengtson
and Chester [10] (N_{e}=1.2 × 10^{13} cmM^{-3}, Z_{1}=Z_{2}=1), the results are shown
in **Table 2** in the column S_{i, standard}.

n | 13 | 14 | 15 | 16 | 17 |
---|---|---|---|---|---|

Se (A) | 0.0017 | 0.0021 | 0.0026 | 0.0032 | 0.0038 |

Si,standard (A) | 0.00035 | 0.00055 | 0.00083 | 0.0012 | 0.0017 |

**Table 2:** Comparison of the electronic shift Se with the estimated standard ionic
shift for hydrogen spectral lines H13 – H17.

It is seen that the total standard ionic shift S_{i, standard} is indeed several
times smaller than the electronic shift Se. Thus, regardless of whether
or not the calculations of S_{i, standard} would include also corrections to the
intensities of the Stark components, S_{i, standard} would remain a relatively
small addition to Se and the huge discrepancy with the observed shifts
from paper Bengtson and Chester [10] would remain unexplained.
So, while introducing below a new source of the red shift, which is by
orders of magnitude greater than Se, we then compare the observed
shifts from paper Bengtson and Chester [10] just with the sum of
the new shift and Se, while S_{i, standard} will be included in the theoretical
estimate of the error margins of the final results. We also note that the
so-called “plasma polarization shift”, which plays an important role in
explaining the observed shifts of the high-n SL of hydrogenic ions, was
found in paper by Theimer and Kepple [23] to be negligibly small for
the high-n SL of hydrogen atoms.

The standard approaches to calculating the ionic contribution
to the shift of hydrogenic SL, discussed in the previous section, used
the multipole expansion in terms of the parameter r_{rms}/R that was
considered small. All terms of the multipole expansion, starting
from the quadrupole term, at the averaging over the distribution
of the separation R between the nucleus of the radiating atom/ion
and the nearest perturbing ion, led to integrals diverging at small R.
These diverging integrals were evaluated one way or another, e.g., by
introducing cutoffs. However, the mere fact that the integrals were
diverging, was an indication that the standard approach did not
provide a consistent complete description of the ionic contribution
to the shift. The fact is that the standard approaches disregarded
configurations where r_{rms}/R > 1, i.e., where the nearest perturbing ion
is within the radiating atom/ion (below we call them “penetrating
configurations”). For low-n hydrogenic SL, the statistical weight of
penetrating configurations is relatively small, but it rapidly increases
with n approximately ~ n^{6}.

For penetrating configurations, it is appropriate to use the expansion
in terms of the parameter R/r_{rms} < 1 in the basis of the spherical wave
functions of the so-called “united atom”, which is a hydrogenic ion of
the nuclear charge Z_{1}+Z_{2}. The energy expansion has the form e.g., book
by Komarov et al. [22] Equation (5.10 - 5.12)):

(3)

Therefore, the first non-vanishing contribution to the shift of the energy level is

(4)

Since s(n) decreases as n increases, it might seem that for the radiative
transition from the upper level n to the lower level of the principal
quantum number n_{0} < n, the shift of the SL would be dominated by the
shift of the lower level. However, in reality, for any level of the principal
quantum number n_{1} (n_{1} is either n or n_{0}) the shift would the product
of two factors: s(n_{1}) from Equation (4) and the statistical weight I(n_{1})
of the corresponding penetrating configuration. It will be shown below
(Equation (12)), that I(n_{1}) increases with growing n_{1} much more rapidly
than ~ n_{1}^{2} (e.g., for relatively low density plasmas, I(n_{1}) scales as n_{1} 6, so
that the shift S_{i, penetr} due to penetrating configurations scales as n_{1} 4).

Therefore, for high-n hydrogenic SL, for which n >> n_{0}, the shift of
the lower level can be disregarded compared to the shift of the upper level. Then the contribution of penetrating configurations to the shift of
hydrogenic SL can be estimated as follows:

(5)

Here we consider the ions of charge Z_{2} as the dominating ion
species, while atoms/ions of the nuclear charge Z_{1} are considered as a
small minority (in case where Z_{1} differs from Z_{2}). In Equation (5),

(6)

Here <…> means the average over those sublevels of the same
principal quantum number n that are involved in the radiative
transition to the level n_{0} << n, i.e., over values of l=0, 1, …, n_{0} (according
to the selection rules).

The distribution P_{w}(w)=P_{w}(R/R_{0}) of the interionic distances in
Equation (5) can be obtained from the binary distribution P_{u}(u)=P_{u}(F/
F_{0}) of the ion microfield (where F=Z_{2}/R_{2} and F_{0}=Z_{2}/R_{0}^{2}, so that u=1/w)
presented in papers by Held B [24,25] where these authors took into
account ion-ion correlations (i.e., the ion-ion interaction) and the
screening by plasma electrons– (Appendix A) Since,

(7)

Then for P_{w}(w) we get,

(8)

Using the results from papers by Held and Held et al. [24,25] for
the case of Z_{1}=Z_{2} ≡ Z, the ion microfield distribution can be normalized
analytically and brought to the form

(9)

Where Meijer G […] is the Meijer G-function and,

(10)

The latter being the Debye radius. A practical formula for the quantity ‘v’ is:

(11)

Then according to Equation (8), for the distribution P_{w}(w) entering
Equation (5) we get

(12)

We note that the quantity k in Equation 12 scales with the electron
density Ne as Ne^{1/3}. Therefore, for relatively low electron densities one
has k << 1 and Equation 12 can be approximated as P_{w}(w) = 3w^{2} exp(–
w^{3}). Then the integral in Equation 5 can be calculated analytically to
yield S_{i,penetr} = s(n) [r_{rms}(n)/R_{0}]^{3}. Since rrms scales as n^{2}, then S_{i,penetr} scales
as n4. As the electron density increases, the shift S_{i,penetr} scales with n
slower than n4, but still very rapidly increases with growing n. The
results of calculating this contribution to the shift the hydrogen SL H_{13} – H_{17} by formulas (5), (6), (12) for the parameters corresponding to the
observations from paper by Bengtson and Chester 1972 (N_{e} = 1.2 × 10^{13} cmM^{-3}, Z_{1} = Z_{2} = 1), are shown in **Table 2** in the column S_{i,penetr}. The sum
S_{i,penetr} + Se is shown in the column Stot. The theoretical error margin is
shown only for the latter and it is primarily due to the approximate way
of estimating S_{i,penetr}.

The following can be seen from **Table 3**:

n | Λn (A) | Se (A) | Si, _{penetr} (A) |
Stot (A) | S_{sirius} (A) |
S_{exp} (A) |
---|---|---|---|---|---|---|

13 | 3734 | 0.0017 | 0.032 | 0.034 ± 0.010 | -- | 0.03 ± 0.03 |

14 | 3722 | 0.0021 | 0.043 | 0.045 ± 0.014 | 0.03 ± 0.05 | 0.00 ± 0.04 |

15 | 3712 | 0.0026 | 0.057 | 0.060 ± 0.018 | 0.09 ± 0.07 | 0.15 ± 0.05 |

16 | 3704 | 0.0032 | 0.073 | 0.076 ± 0.023 | –0.007 ± 0.05 | 0.00 ± 0.05 |

17 | 3697 | 0.0038 | 0.093 | 0.10 ± 0.03 | 0.21 ± 0.08 | 0.30 ± 0.08 |

**Table 3:** Shift Si, penetrate due to penetrating ions and its sum Stot with the
electronic shift Se for the hydrogen spectral lines H13 – H17, and the comparison
with the shifts from paper by Bengtson and Chester [10] observed in astrophysical
(S_{Sirius}) and laboratory (S_{exp}) plasmas. All shifts are in Angstrom.

1. For the SL H_{13}, there is an excellent agreement between the total
theoretical shift Stot and the experimental shift S_{exp}. No data for the
shift of this SL from Sirius.

2. For the SL H_{14}, there is a good agreement of the total theoretical
shift with the shift of this SL observed from Sirius and a satisfactory
agreement (within the error margins) with the experimental shift of
this SL.

3. For the SL H_{15}, there is a good agreement of the total theoretical
shift with the shift of this SL observed from Sirius and a satisfactory
agreement (almost within the error margins) with the experimental
shift of this SL.

4. For the SL H_{16}, there is a satisfactory agreement (within the
error margins) of the total theoretical shift with both the shift of this SL
observed from Sirius and the experimental shift of this SL.

5. For the SL H_{17}, there is a satisfactory agreement (within the error
margins) of the total theoretical shift with the shift of this SL observed
from Sirius, but a disagreement with the experimental shift of this
SL; however, the latter disagreement is not anymore by two orders of
magnitude, as it was the case before the present paper, but rather just
by a factor of two (after allowing for the error margins).

Justification of the quasistatic description of the penetrating configurations in hydrogen plasmas can be found in Appendix B.

The present paper was motivated by the fact that high-n hydrogen
SL (n=13–17), studied in in the astrophysical and laboratory
observations at N_{e} ~ 1013 cmM^{-3} by Bengtson and Chester [10], exhibited
red shifts by orders of magnitude greater than the theoretical shifts
known up to now. We introduced an additional source of the shift
of high-n hydrogenic SL arising from the configurations where the
nearest perturbing ion is within the radiating atom/ion (“penetrating
configurations”). We demonstrated that for high-n hydrogen SL it
makes the primary contribution to the total red shift. We showed that
for the conditions of the astrophysical and laboratory observations from paper by Bengtson and Chester [10], this additional red shift is
by orders of magnitude greater than the theoretical shifts known up to
now. The comparison with the red shifts observed in paper by Bengtson
and Chester [10] demonstrated that the allowance for this additional
red shift removes the existed huge discrepancy–the discrepancy by
orders of magnitude–between the observed and theoretical shifts.

We emphasize that the primary focus of the present paper was to
bring to the attention of the research community a new source of shifts
of hydrogenic lines and to show that it is the dominant source of shifts
of spectral lines corresponding to the radiative transitions from a level
n to a level n_{0 }<< n. This is an important fundamental result in its own
right. As for the application of this fundamental result to the laboratory
and astrophysical observations by Bengtson and Chester [10], we note
the following. While the allowance for this shift brought the theory
by orders of magnitude closer to the observations by Bengtson and
Chester [10], it cannot be interpreted as the ultimate explanation of the
observations by Bengtson and Chester [10] (e.g., for lines H_{14} and H_{16},
while there is an agreement within the combined error margins, there
is still no explanation why the most probable value of the observed shift
for these two lines was zero).

Another potential application of this new source of shifts might
have been to Radio Recombination Lines (RRL), i.e. to hydrogen lines
corresponding to radiative transitions from a level n >>1 to one of the
neighboring levels n_{0}=n–p, where p << 1. However, it turns out that for
RRL, this shift is about 5 orders of magnitude smaller than the width,
thus making practically impossible to detect such shift (Appendix C).
There are two reasons for this result. First, this shift is proportional to
the electron density Ne and for H II regions emitting RRL, Ne is by 9 or
10 orders of magnitude smaller than for laboratory and astrophysical
plasmas studied by Bengtson and Chester [10]. Second, for RRL one
has n and n_{0} very close to each other (both being in the range between
100 and about 200). In this situation, the contribution from level n to
the shift (which would lead to the red shift of a particular PRL) and the
contribution from level n0 to the shift (which would lead to the blue
shift of the same PRL) almost cancel each other.

More rigorously, the resulting shift of PRL increases with growing
n much slower than for radiative transitions from level n to level n_{0} << n (Appendix C). Therefore, the fact that for PRL the values of n
are by about one order of magnitude greater than for observations by
Bengtson and Chester [10] cannot override the decrease of the electron
density by 9 or 10 orders of magnitude.

Finally we emphasize that in the present paper we calculated this new red shift approximately–just the get the message across. We hope that our results would motivate further observational and theoretical studies of the shifts of high-n hydrogenic spectral lines in astrophysical and laboratory plasmas.

Held et al. [24] derived the ion microfield distribution at a
charged point (P_{u}(u) in our notation, u=F/F_{0}), as well as the related
distribution of interionic distances (P_{w}(w) in our notation, w=R/R_{0}),
taking into account ion-ion correlations (i.e., ion-ion interaction)
and the screening by plasma electrons. For relatively small interionic
distances, relevant to our study of the shift by penetrating ions, the
unnormalized distribution given by Equation (67) from Held et al. [24]
can be represented in our notations as follows:

(A.1)

where the factor g(w) incorporates ion-ion correlations and the screening by plasma electrons. If one would disregard ion-ion correlations and the screening by plasma electrons, so that it would be g(w)=1, then the normalized distribution would simplify to:

(A.2)

The allowance for ion-ion correlations and the screening by plasma
electrons adds additional exponential factor, which in our notation is
exp(– k/w) where k is given in Equation (10). As a result, the normalized
distribution P_{w}(w) takes the form given by Equation (12).

From the theory of the Stark broadening of hydrogen lines
in plasmas it is well-known that the quasistatic description of the
interaction of the perturbing ion with the radiating hydrogen atom is
valid as long as the internuclear separation is much smaller than the ion
Weisskopf radius ρ* _{Wi}* as studied by Lisitsa [26], where,

(B.1)

Here <1/V_{i}> is the inverse ion velocity averaged over the Maxwell
distribution, M_{reduced} is the reduced mass of the pair perturber-radiator.
(In this Appendix we do not use the atomic units.) For hydrogen
plasmas, M_{reduced}=M_{p}/2, where M_{p} is the proton mass, so that

(B.2)

The largest internuclear separation, involved in calculating the shift of hydrogen lines due to penetrating configurations, is

(B.3)

where a_{0} is the Bohr radius (according to Equation (6) with Z_{1}=1).
Therefore, for the ratio r_{rms}/ρW_{i} we obtain:

(B.4)

Thus, the quasistatic description of penetrating configurations in hydrogen plasmas is valid as long as the temperature is:

(B.5)

Obviously this condition is fulfilled in the laboratory and astrophysical plasmas studied by Bengtson and Chester [10], as well as in many other laboratory and astrophysical plasmas.

In the well-known paper by Bell et al. [27], the authors measured widths of Radio Recombination Lines (RRL) from several H II regions, including Orion A. Their primary finding was that the width of RRL of the principal quantum number n up 180 increased with the growing n, but for RRL of n > 180 the widths decreased with the growing n. Recently Alexander and Gulyaev [28] presented the newest observations of RRL from Orion nebula. They also found that the width of RRL of the principal quantum number n up 180 increased with the growing n.

As for the RRL of n ~ 200 and higher observed by Bell et al., [27] and Alexander and Gulyaev [28] point out the following. Bell et al. [27] applied the frequency switching method to the same spectrum six times successively, which made their method increasingly insensitive to line broadening as the line width increased and exceeded the frequency switching offset parameter.

Alexander and Gulyaev [28] demonstrated that the narrowing of RRL reported by Bell et al. [27] is apparent: their method effectively filtered out Stark broadening for n ~ 200 and higher. For this range of n, most of the width measurements by Bell et al. [27] were below the Doppler width and increasingly below 3σ in signal-to-noise ratio, which is a manifestation of limitations in the use of the frequency switching method multiple times in succession.

Therefore, here we estimate the shift by penetrating ions for RRL up
to n=180 and compare it with the measured width. For the extremely
low electron densities characteristic for H II regions (Ne ~ 4000 cm-3 or
less according to Bell et al. [27], Ne ~ 5000 cm-3 according to Alexander
and Gulyaev [28], the distribution P(w) of the relative separation
w=R/R0, given by Equation (12), can be simplified, so that after the
integration in Equation (5) the result for the shift by penetration ions
simplifies to (for Z_{1}=Z_{2}=1):

(C.1)

The relative shift δS_{i, penetr}, defined as ratio of this shift to the
unperturbed frequency of the particular RRL is:

(C.2)

Since RRL are characterized by n_{0}=n–p, where p << n, then
Equation (C.2) can be simplified to:

(C.3)

After substituting 1/R_{0}^{3}=4πN_{e}/3 and returning from the atomic
units of N_{e} to the CGS units, we obtain:

(C.4)

(a_{0} is the Bohr radius).

For n=180 and N_{e} = (4000–5000) cmM^{-3}, Equation (C.4) yields δS_{i,
penetr} = (1-2) × 10^{-6}. The relative width δW (i.e., the ratio of the width
to the unperturbed frequency of the PRL), as measured by Bell et al.
and Alexander J, Gulyaev S [27] was δW ~ 0.1 for n ~ 180. In other
words, the measured width exceeds the shift (caused by penetrating
ions) by 5 orders of magnitude, so that it was impossible to detect it.

To conclude this Appendix, let us briefly discuss the experimental
results by LaSalle, Nee and Griem [29]. These authors presented
profiles of hydrogen lines measured in a laboratory plasma–the lines
corresponding to the radiative transitions between level n and n-1
(so called n-alpha lines) for n=12, 13 and 14. While their focus was
on the experimental width of these lines, they also mentioned some
experimental red shift, such as, e.g., (0.2 ± 0.2) μm for Ne between 3.4 ×
1014 cmM^{-3} and 5.8 × 1014 cmM^{-3}. For example for the line originating from
n=12, it corresponds to the relative shift δS_{ exper} = (0.003 ± 0.003). If one
would formally apply Equation (C.4) to this case, one would get the
theoretical relative shift δS_{i, penetr} at least two times higher.

However, in the experiment by LaSalle, Nee and Griem [29], the
electron density Ne was by 11 orders of magnitude higher than in
observations by Bell et al. [27] and by Alexander and Gulyaev [28] (and
by 30-50 times higher than in the experiment by Bengtson and Chester
[10] Therefore, the low density approximation of the distribution
P(w) of the relative separation w=R/R0, employed in the derivation of
Equation (B.4), is not justified. The more accurate calculation, based
on Equations (5) and (12), brings δS_{i, penetr} to practically an agreement with δS_{ exper} = (0.003 ± 0.003) within the combined error margins of δS_{i,
penetr} and δS_{ exper}.

The author is grateful to Dr. A.V. Demura for his comments

^{1}/We note in passing paper by Caby-Eyraud et al. [18] focused at the theoretical
study of the hydrogen SL Ly-alpha at much higher electron densities than in the
observations by Bengtson & Chester [10] and Himmel [13]. After noting that for
the “unshifted” (more rigorously, central) Stark components of hydrogen SL, the
quadrupole shift is to the red (which was actually well known already 6 years earlier
from paper by Shoiln [21], Caby-Eyraud et al. [18] wrote: “This would explain, at
least partially, the red shift observed … in the Balmer lines arising from odd upper
levels” with reference to Bengtson & Chester [10]. However, first, while for the
Ly-alpha line (which is the only one SL studied by Caby-Eyraud et al. [18]. the
contribution of the quadrupole shift of the central Stark component to the total shift
of this SL could have been significant (because for the Ly-alpha line the central
component contains 2/3 of the total intensity of this line), the contribution of the
quadrupole shift of the central Stark component of hydrogen SL H13, H15, H17
(studied by Bengtson & Chester [10]) to the total shift of these SL would have been
miniscule (because for these SL the central component contains only between
4% and 5% of the total intensity of these lines). Second, but most importantly: for
any hydrogen (or hydrogenlike) SL, after summing up the quadrupole shift over
all Stark components with the allowance for the quadrupole corrections to both
the energies/frequencies and the intensities, the quadrupole shift of the SL as the
whole vanishes, as rigorously shown analytically byOks [17].

- Griem HR (1997) Principles of plasma spectroscopy. Cambridge University Press, UK, Sect. 4.10.
- Oks E (2006) Stark broadening of hydrogen and hydrogen like spectral lines in plasmas: The physical insight (Oxford: Alpha Science International), Sects. 2.6, 6.
- Nussbaumer N, Bieri L (2009) Discovering the expanding universe. Cambridge University Press, UK.
- Parigger CG, Plemmons DH, Oks E (2003) Balmer series Hβ measurements in laser-induced hydrogen plasma. Applied Optics 42: 59-92.
- Grabowski B, Halenka J (1975) On red shifts and asymmetries of hydrogen spectral lines. Astron Astrophys 45:1 159.
- Demura AV, Helbig V, Nikolic D (2002) Spectral line shapes (16th edn) ICSLS AIP Publishing, CA. Proceedings of the 12th International conference on spectral line shapes. Los Angeles, CA. 645: 318.
- Demura AV, Demchenko, GV, Nikolic D (2008) Multi-parametric dependence of hydrogen stark profiles asymmetry. Eur Phys J D 46: 1-111.
- Djurovic S, Cirisan M, Demura AV, Demchenko GV, Nikolic J, et al. (2009) Measurements of Hβ Stark central asymmetry and its analysis through standard theory and computer simulations. Phys Rev E 79: 401-402.
- Kielkopf JF, Allard NF (2014) Shift and width of the Balmer series Hα line at high electron density in a laser-produced plasma. J Phys B: At Mol Opt Phys 47: 155-701.
- Bengtson RD, Chester GR (1972) Observations of shifts of hydrogen lines. Ap J 178: 565.
- Barcza S (1972) Highly excited hydrogen lines in stellar spectra II Astrophys Space Sci 16: 372.
- Barcza S (1979) Energy levels in Debye field. Astronomy and Astrophys 72: 26.
- Himmel G (1976) Plasma effects in the spectrum of high Balmer lines. J Quant Spectrosc Rad Transfer 16: 52-59
- Griem HR (1983) Shifts of hydrogen lines from electron collisions in dense plasmas. Phys Rev A 28: 1596.
- Boercker DB, Iglesias CA (1984) Static and dynamic shifts of spectral lines. Phys Rev A 30: 2771.
- Griem HR (1988) Shifts of hydrogen and ionized-helium lines from Δn=0 interactions with electrons in dense plasmas. Phys Rev A 38: 29-43.
- Oks E (1997) New type of shift of hydrogen and hydrogen like spectral lines. J Quant Spectrosc Rad Transfer 58: 821.
- Caby-Eyraud M, Coulaud G, Hoe N (1975) Contribution of strong interactions to the wings of hydrogen lines broadened by the charged particles in plasmas. J Quant Specrosc Rad Transfer 15: 593.
- Könies A, Günter S (1994) Asymmetry and shifts of the Lα-and the Lβ-line of hydrogen. J Quant Spectrosc Rad Transfer 52: 825.
- Günter S, Könies A (1977) Rapid communications: General methods of statistical physics. Phys Rev E 55: 907.
- Sholin GV (1969) Physics of highly excited atoms and ions. Opt Spectosc 26: 275.
- Komarov IV, Ponomarev LI, Slavyanov SY (1976) Spheroidal and coulomb spheroidal Functions. (Moscow: Nauka) in Russian.
- Theimer O, Kepple P (1970) Statistical mechanics of partially ionized hydrogen plasma. Phys Rev A 1: 957.
- Held B (1984) Electric micro-field distribution in multicomponent plasmas. J Physique 45: 1731.
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- Bell MB, Avery LW, Seaquist ER, Valee JP (2000) A new technique for measuring impact-broadened radio recombination lines in HII regions: Confrontation with theory at high principal quantum numbers. PASP 112: 1236.
- Alexander J, Gulyaev S (2016) Stark broadening of high-order radio recombination lines toward the Orion nebula. Ap J 828: 40.
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