_{1}

^{*}

We analyze the line data from solar flares to present evidence for the emission spectrum of the recently discussed electron-proton pairs at high temperatures. We also point out that since the pairing phenomenon provides an additional source for these lines—the conventional source being the highly ionized high-Z atoms already existing in the solar atmosphere, we have a plausible explanation of the FIP effect.

Some time back, a finite-temperature Schrodinger equation was obtained to describe the pairing of an electron and a proton in a medium of such particles at finite temperature [^{6} K, the deepest of the states in the spectrum have binding energies in the keV range and can withstand the background thermal agitation of the medium. The transitions from the short-lived excited states to the deepest ones in this spectrum lead to spectral lines in the soft X-ray region. An application of the approach to the flaring regions of the Sun therefore leads to the prediction of such lines in the flare spectra. In the present note, we report on this matter: the calculated lines at a certain temperature from three Balmer-like series are essentially all seen in the flare data. We also point out that, since many of these lines are identifiable with the lines which in the conventional approach are presumed to originate from the low first ionization potential (FIP) elements, our approach seems to provide an explanation of the FIP effect which has been extensively discussed in the literature, e.g., in [

The equation for the bound state at temperature T, or the FTSE, is given by

where W is the binding energy,

with m_{a} and m_{b} as the electron and the proton mass, respectively. The function Q(W, p) is given by

where

and k is the Boltzmann constant. Using the approximation

we can write

where

Equation (7) can be solved in this approximation to give [

where l is the angular momentum quantum number, m the azimuthal quantum number and

the coefficients a_{k} in Equation (8) obey a difference equation given by

where we have put

The parameters c_{i} are given by

Equation (10) is a second order difference equation and will in general have two solutions; the dominant and the dominated. The eigenvalues of the equation are those values of _{a} (electron mass), m_{b} (proton mass) and α (the fine structure constant), we can use the program to solve for W at any given value of T. What emerges is a discrete spectrum for W: for each l, we get an increasing set of W values which can be labeled by a serial number n (which plays the role of the principal quantum number in the case of the hydrogen atom). The results are illustrated in

It is now important to make sure that the solutions obtained are indeed consistent with the approximation

given in Equation (5). This is done by verifying that the expectation values of _{n}_{,l} are quite small, say < 0.01. If not so, the solution must be discarded. It may be noted that since m_{a} is much smaller than m_{b}, it is sufficient to check that

small. To do this, we have to first calculate.

where

Note that M in the summations S_{1} and S_{2} denotes the highest k value for which a_{k} ≠ 0. _{k}. The latter are therefore yet to be determined. This can be done conveniently by using the backward Miller algorithm [_{k} and whence

n | Binding energy | ||
---|---|---|---|

1 | 5003.429 | ||

2 | 929.218 | 934.085 | |

3 | 360.299 | 360.398 | 360.421 |

4 | 197.600 | 197.607 | 197.609 |

5 | 125.586 | 125.587 | 125.587 |

6 | 86.994 | 86.994 | 96.994 |

7 | 63.845 | 63.845 | 63.845 |

8 | 48.856 | 48.856 | 48.856 |

9 | 38.591 | 38.591 | 38.591 |

10 | 31.254 | 31.254 | 31.254 |

11 | 25.827 | 25.827 | 25.827 |

12 | 21.700 | 21.700 | 21.700 |

13 | 18.489 | 18.489 | 18.489 |

14 | 15.942 | 15.942 | 15.942 |

15 | 13.887 | 13.887 | 13.887 |

16 | 12.205 | 12.205 | 12.205 |

17 | 10.811 | 10.811 | 10.811 |

18 | 9.643 | 9.643 | 9.643 |

19 | 8.655 | 8.655 | 8.655 |

20 | 7.811 | 7.811 | 7.811 |

21 | 7.085 | 7.085 | 7.085 |

22 | 6.455 | 6.455 | 6.455 |

23 | 5.906 | 5.906 | 5.906 |

24 | 5.424 | 5.424 | 5.424 |

25 | 4.999 | 4.999 | 4.999 |

26 | 4.622 | 4.622 | 4.622 |

27 | 4.285 | 4.285 | 4.285 |

28 | 3.985 | 3.985 | 3.985 |

29 | 3.715 | 3.715 | 3.715 |

30 | 3.471 | 3.471 | 3.471 |

∞ | 0.000 | 0.000 | 0.000 |

for any given W_{n}_{,l} only if the temperature T is above a certain value T_{n}, and furthermore, T_{1} > T_{2} > T_{3}…etc. The relevant results are displayed in _{1,0} is acceptable only for T > T_{1} ≈ 10^{7} K, that corresponding to W_{20} or W_{21} only for_{n}_{,l} for n > 2 are now acceptable if T > T_{2}. We must also keep in mind that a bound state with binding energy W will survive in a medium at temperature T only if ^{6} K, the deepest possible states that can form in our spectrum are at the level n = 2, i.e., W_{2,0} and W_{2,1}. We note that for these levels

The wavelength λ of the spectral line corresponding to any transition involving the energy difference

Substituting for any allowed

Let us now apply our approach to Solar flares. The latter are appropriate for such an application as they occur in a medium with low particle densities (≈10^{12}/cm^{3}) at temperatures around a few million degrees Kelvin and are known to emit soft X-ray lines. Furthermore, since the flaring phenomenon is a prolonged affair, we may expect that between the initial (growing) phase and the final (decaying) phase, there should be a period over which the flare burns at a reasonably constant temperature (with fluctuations of, say, not more than^{6} K and 10^{7} K, to see if there is any temperature in this range at which our calculated lines are reproduced in the flare data. As will be seen in the following, we find that there indeed is such a temperature given by T = 4.26 × 10^{6} K. The binding energy spectrum

T(W_{n}_{,l}): Temperature for energy level ^{6} K) | |||||||||
---|---|---|---|---|---|---|---|---|---|

T (W_{1,0}) | T (W_{2,1}) | T (W_{2,0}) | T (W_{3,1}) | T (W_{3,0}) | T (W_{4,1}) | T (W_{4,0}) | T (W_{5,1}) | T (W_{5,0}) | |

≈0.20 | 4.0 | 1.05 | 0.875 | 0.436 | 0.38 | 0.229 | 0.209 | 0.141 | 0.126 |

≈0.10 | 5.25 | 1.38 | 1.22 | 0.575 | 0.501 | 0.302 | 0.275 | 0.186 | 0.174 |

≈0.01 | 10.9 | 2.63 | 2.63 | 1.20 | 1.20 | 0.650 | 0.650 | 0.398 | 0.398 |

≈0.0 | 20 | 5 | 5.0 | 2.0 | 2.0 | 1.0 | 1.0 | 0.630 | 0.630 |

S. No. | Transitions (Series B) | Calculated | Observed | Convention-al source | Transitions (Series A & C) | Calculated | Observed | Conventional source |
---|---|---|---|---|---|---|---|---|

1 | 3.1 → 2.0 | 21.797 | 21.798+ | O VII | 3.0 → 2.1 3.2 → 2.1 | 21.608 21.613 | 21.602+ 21.602+ | O VII |

2 | 4.1 → 2.0 | 16.947 | 16.956+ | ? | 4.0 → 2.1 | 16.835 | 16.821 | ? |

3 | 5.1 → 2.0 | 15.428 | 15.428 | ? | 5.0 → 2.1 | 15.335 | - | - |

4 | 6.1 → 2.0 | 14.721 | 14.737 | Fe XIX | 6.0 → 2.1 | 14.636 | - | - |

5 | 7.1 → 2.0 | 14.327 | 14.311 | ? | 7.0 → 2.1 | 14.247 | 14.258 | Fe XVIII |

6 | 8.1 → 2.0 | 14.083 | 14.076 | Ni XIX | 8.0 → 2.1 | 14.006 | 14.017 | ? |

7 | 9.1 → 2.0 | 13.921 | 13.934 | ? | 9.0 → 2.1 | 13.845 | 13.842 | ? |

8 | 10.1 → 2.0 | 13.807 | 13.796 | Fe XIX | 10.0 → 2.1 | 13.733 | 13.738 | Fe XIX |

9 | 11.1 → 2.0 | 13.724 | 13.719 | Ne VIII | 11.0 → 2.1 | 13.651 | 13.645 | Ne VIII |

10 | 12.1 → 2.0 | 13.662 | 13.669 | Fe XIX | 12.0 → 2.1 | 13.589 | - | - |

11 | 13.1 → 2.0 | 13.614 | 13.630 | ? | 13.0 → 2.1 | 13.541 | 13.551 | Ne IX |

12 | 14.1 → 2.0 | 13.576 | - | - | 14.0 → 2.1 | 13.504 | 13.504 | Fe XIX |

13 | 15.1 → 2.0 | 13.545 | 13.551 | Ne IX | 15.0 → 2.1 | 13.474 | 13.463 | Fe XIX |

14 | 16.1 → 2.0 | 13.520 | 13.520 | Fe XIX | 16.0 → 2.1 | 13.449 | 13.446 | Ne IX |

15 | 17.1 → 2.0 | 13.500 | 13.504 | Fe XIX | 17.0 → 2.1 | 13.429 | 13.426 | Fe XIX (?) |

16 | 18.1 → 2.0 | 13.483 | 13.463 | Fe XIX | 18.0 → 2.1 | 13.412 | 13.402 | Fe XIX (?) |

17 | 19.1 → 2.0 | 13.468 | 13.463 | Fe XIX | 19.0 → 2.1 | 13.397 | 13.4-2 | Fe XIX (?) |

18 | 20.1 → 2.0 | 13.456 | 13.446 | Ne IX | 20.0 → 2.1 | 13.385 | 13.375 | Fe XVIII |

19 | 21.1 → 2.0 | 13.445 | 13.446 | Ne IX | 21.0 → 2.1 | 13.375 | 13.375 | Fe XVIII |

20 | 22.1 → 2.0 | 13.436 | 13.446 | Ne IX | 22.0 → 2.1 | 13.366 | 13.354 | Fe XVIII |

21 | 23.1 → 2.0 | 13.428 | 13.426 | Fe XIX (?) | 23.0 → 2.1 | 13.358 | 13.354 | Fe XVIII |

22 | 24.1 → 2.0 | 13.421 | 13.426 | Fe XIX (?) | 24.0 → 2.1 | 13.351 | 13.354 | Fe XVIII |

23 | 25.1 → 2.0 | 13.415 | 13.402 | Fe XIX (?) | 25.0 → 2.1 | 13.345 | 13.354 | Fe XVIII |

24 | 26.1 → 2.0 | 13.410 | 13.402 | Fe XIX (?) | 26.0 → 2.1 | 13.339 | 13.322 | Fe XVIII |

25 | 27.1 → 2.0 | 13.405 | 13.402 | Fe XIX (?) | 27.0 → 2.1 | 13.335 | 13.322 | Fe XVIII |

26 | 28.1 → 2.0 | 13.400 | 13.402 | Fe XIX (?) | 28.0 → 2.1 | 13.330 | 13.322 | Fe XVIII |

27 | 29.1 → 2.0 | 13.396 | 13.402 | Fe XIX (?) | 29.0 → 2.1 | 13.326 | 13.322 | Fe XVIII |

28 | 30.1 → 2.0 | 13.393 | 13.402 | Fe XIX (?) | 30.0 → 2.1 | 13.323 | 13.322 | Fe XVIII |

- | - | - | - | - | - | - | - | - |

29 | ∞ → 2.0 | 13.343 | 13.354 | Fe XVIII | ∞ → 2.1 | 13.273 | 13.279 | Fe XIX |

We now turn to the relevant experimental information. The data between 5 and 20 Ȧ were obtained by Phillips et al. [^{6} K. Any line from the data which is within

The experimental results in

A look at

It may be noted that 29 distinct lines from the Solar flare data have been used up in the above as evidence for our generalized Balmer series. The data of course contain hundreds of lines over a wide range in the X-ray region. Our object is not to suggest that they all originate from our pairing mechanism. In fact, following the pioneering works of Grotrian and Edlen, we take it for granted that these lines follow from the so-called forbidden transitions in highly ionized high-Z atoms, the presence of which in stellar plasmas is a natural consequence of their having been formed in the interior of stars. The sources of a great many of these lines have accordingly been identified and the effort in that direction continues. Nevertheless, we may mention that, of the 29 lines from the data which we matched with our calculated lines, eight remain unidentified (marked [?] in

The conventional approach is thus not all-encompassing. In fact, the lack of proper identification in this approach of as many as eight (possibly 10) lines out of 29 does suggest that these lines may have a different origin. But what evidence could there be for such an additional origin for the other 19 lines which have been identified in the conventional approach? Interestingly enough, some evidence is indeed there. As already noted, it comes from what is called the FIP effect [^{6} K than in the photosphere (T ≈ 6000 K). How does one infer this? One compares the relative intensity pattern of emission lines for the same elements from a) the high temperature sources like the flares and the coronal active regions and b) the photosphere. One finds that the intensities of lines corresponding to the low FIP elements from the former are anomalously enhanced. This may be interpreted to mean that these low FIP elements are relatively more abundant in the high temperature sources, which is in disagreement with the classical stellar atmospheres theory. Let us now look at the situation from the point of view of this study. We observe that of the 19 lines predicted by us, which are also identifiable as lines from known elements, almost all can be traced to a single low FIP element Fe (see

It may also be pointed out that the detailed theoretical calculations [

In the context of the present study, we note that the generally accepted and empirically corroborated picture of the uniformity of elements in the upper reaches of the Sun, e.g., the photosphere and the Solar flares and coronal regions, would imply that the relative intensities of spectral lines from various elements would not show any significant variation from one region to another. It turns out that this is not so. The reason is: while 19 of the 29 lines in the data analyzed here and attributed to our pairing mechanism at T = 4.26 × 10^{6} K can also be identified with those from a single low FIP elements Fe XIX, their intensities in the flares region are found to be anomalously enhanced as compared with the intensities of lines from the photosphere. It then makes sense to conclude that this enhancement is due to the existence of another mechanism operative in the region of flares, but not in the photosphere. As has been argued above, our pairing mechanism takes place at temperatures exceeding about a million K and not at temperatures around 5000 K that characterize the photosphere. We further note that 8 (possibly 10) of the 29 lines in the data analyzed above are not identifiable (or have questionable identification) on the basis of transitions from Fe XIX or any other elements. This lends support to the view that they may well be due to the additional pairing mechanism presented here.

Finally, we should note that while a recent review [

The author is grateful to Dr. Bernhard Haisch for pointing out the relevance of the FIP effect to the present work. He is also grateful to Dr. J.J. Drake and Dr. Anand Bhatia for providing some valuable information. Finally, it is a matter of great pleasure to thank Dr. G.P. Malik without whose encouragement the present investigation would not have been completed.

L. K.Pande,11, (2016) Electron-Proton Pairing at High Temperatures, Solar Flares, and the FIP Effect. Journal of Modern Physics,07,25-35. doi: 10.4236/jmp.2016.71003

We evaluate here the expectation value

where, in view of Equations (8) in the text,

and

Upon using

we obtain, after going through some elementary algebra,

With the help of the standard result

and the recurrence relation

we then get

where

Proceeding in a similar manner, we also obtain

where

We thus obtain [

Note that M is the summations in S_{1} and S_{2 }denotes the highest k value for which a_{k} ≠ 0, and

determinant method, which also determines the value of M. The method does not require an explicit knowledge of coefficients a_{k}, which can be determined by using the backward Miller algorithm [

whence Equation (10) can be written as

where

We thus have

These equations have to be supplemented by

Equation (A14a) then determines_{l} (which can be taken as unity because the ratio S_{2}/S_{1} above will be independent of this choice),

Thus knowing W_{n}_{,l} corresponding to any value of T as calculated from Equation (10), and the corresponding values of l and M, we can calculate the coefficients a_{k} and, finally, using Equation (A10), the value of