Journal of Modern Physics, 2011, 2, 564571 doi:10.4236/jmp.2011.26066 Published Online June 2011 (http://www.SciRP.org/journal/jmp) Copyright © 2011 SciRes. JMP Calculation of Complete Absorption and Intensity of Optical Radiation Spectrum of HeI ( = 5875 Å) with Fine Structure Reda A. ElKoramy1, Nazer A. Ashurbecov2 1Physics Department, Faculty of Science, Assiut University, Assiut, Egypt 2Department Physical Electronics, Daghestan State University, Daghestan, Russia Email: elkoramy@yahoo.com Received February 13, 2011; revised April 24, 2011; accepted May 13, 2011 Abstract Theoretical calculations which account for the complete absorption and intensity for the optical radiation He I ( = 5875 Å) spectral line with fine structure of the transition 23P2,1,0  33D3,2,1 during He nanosecond dis charge are presented. For different values of the absorption parameter ( 0ℓ), the absorption quantity A of the three components distorted as a result of reabsorption multiple process has been numerically obtained and graphically presented. The theoretical results for small values of 0ℓ (≤ 4) give a good agreement with the experimental data in literature. Keywords: Optical Radiation Spectrum, Complete Absorption, Intracavity Laser Spectroscopy, Simpson’s Formula, Computer Processing 1. Introduction While the classical theories gave a semiquantitative ex planation for the interaction between matter and radiation, no precise and general treatment was possible before the advent of quantum theory. In principle, the quantum the ory of matter allows one to calculate the various energy levels that any given atom or ion may possess. However, as long as classical electromagnetic theory is used, emit ted, absorbed or scattered radiation is still ambiguous. Much detailed work remains to be done to elucidate the nature of numerous approximate treatments and to assess the ranges of applicability and the degree of approxima tion that has been achieved. The variation of interatomic (intermolecular) forces as a function of interatomic (in termolecular) distance has been increasingly attracting the attention not only of theoretical and experimental physicists but also of those who are working on certain basic problems of chemistry, genetics and astrophysics. Plasmabroadened and shifted spectral line profiles have been used for a number of years as a basis of an important noninterfering plasma diagnostic method. The numerous theoretical and experimental efforts have been made to find solid and reliable basis for this application. This technique became, in some cases, the most sensitive and often the only possible plasma diagnostic tool. In the early 1960s a number of attempts [13] were made to improve and to check experimentally existing theories of spectral line broadening by plasmas. Most of these early works were concerned with the Stark broadening of hy drogen lines. Owing to the large, linear Stark effect in hydrogen, these studies were very useful for plasma di agnostic purposes. However, it is not always convenient to seed plasma with hydrogen, and sometimes this is not possible. Although the theory of Stark broadening was used on ly as a consistency check in the study of the influence of ion dynamics to the width and shift of visible helium lines in low electron density plasmas, an important con clusion was derived from the comparison with the results of semiclassical calculations [4,5]. This conclusion, which may be used for plasma diagnostic purposes with an av erage estimated accuracy of ±20% and ±30% for a neu tral atom and singly charged ion lines, respectively, led to the fact that the total width and shift is a sum of the electron and all ion impact widths. On the basis of com parison of selected data in [4] and calculations for the energy positions and widths of singlet and triplet (even and odd) resonances of heliumlike (Z = 2 – 10) systems lying between the n = 2 and n = 3 thresholds [6], an at
R. A. ELKORAMY ET AL. 565 tempt was made to extend these works and to identify lines of various elements which may be used with higher accuracy for plasma diagnostic purposes [7]. The study of emission radiation from atoms (or ions) in plasma constitutes an important area of plasma diag nostics. From the experimental point of view, plasma is sometime optically thick to its emission lines, then, the emitted photons may be absorbed within the plasma. Over the past, more than thirty years period of formation and development of spectroscopic techniques, selective intracavity laser spectroscopy (ICLS) provides the pos sibility of simultaneously recording the profiles of a number of spectral lines and bands in probing both brightly emitting and nonradiative objects. Also, of par ticular interest, ICLS combines high sensitivity in de tecting absorption centers [811] and, so, characterizes the integral absorption coefficient within the frequency limits of the absorption line [12,13]. Furthermore, transi tion probabilities and atomic lifetimes [4] and optical absorption and reflectance [14] have been reported. Pre dicted He I line intensities and line ratios in microwave generated plasmas and from a recently developed colli sionalradiative model were compared with experiment [15,16] It has been concluded that the intensity of triplet lines is strongly affected by the local metastable state (21S and 23S) populations and the initial metstable frac tion plays an important role in determining line intensi ties. In spite of there has been a lot of work done within the past decade, both theoretical and experimental, were given [1723]; and, recently, resonance parameters and autoionization widths of the doubly excited states of the helium isoelectronic sequence [24] have been studied, however, there are some important aspects of the subject which have not been clearly discussed. Presently due to progresses of computer processing, the need of another review is quite obvious. The present work is merely to aid in this process of focusing thought upon crucial is sues. The present work is intended to derive analytic ex pressions for calculating dependencies of complete ab sorption and intensity of radiation spectrum on optical depth along the line sight in a nanosecond gas discharge. We report results of complete absorption for the optical radiation of HeI ( = 5875 Å) spectral line with fine structure of the transition 23P2,1,0  33D3,2,1 through He nanosecond discharge. 2. Theoretical Analysis 2.1. Advanced Spectroscopic Treatment Spectroscopy is a powerful tool to diagnose plasmas and has played and will continue to play a major role in plasma physics research. Interest in line broadening has been renewed in the last few years because of the large amount of recent plasma work. The line shapes obtained by spectroscopic examination of the emission of plasma can yield information about plasma density. 2.2. Doppler Producing Broadening and Shift of Spectral Lines The line profiles emitted from the plasma are governed by Doppler and Stark broadenings. Doppler effects give rise to further broadening rather than ones. So, other broadening mechanisms can be neglected for our plasma condition, which represent a typical emission spectrum from laboratory plasma. It consists of continuous brems strahlung and recombination radiation superposed by line radiation. We now visualize the plasma to be homoge nous and to increase steadily its optical depth in the di rection of observation. The spectral radiance will in crease until it reaches the Planck value of the blackbody emission according to the existing temperature. We con sider an emitting atom in motion to infinity and reduce temperature to the point where, classically at least, no translational motion exists. Every influence exerted on a radiating or absorbing atom (or molecule) modifies its spectral lines in one way or another. The effect of its own thermal motion, which broadens the lines statisti cally, is known as the Doppler Effect. In the case of emission or resonance absorption, the process of radia tion damping, or a finite width of each of the two energy levels associated with spectral lines, is responsible for the natural width. If the atom is in a gas of the same kind, the spectral lines are broadened and sometimes are shifted with asymmetry of the line shape. The essential cause of line broadening lies in the finite difference of interaction energies of the radiating atom, in the initial and final states involved in the radiation process, with a colliding atom. When the interacting atoms, one of which is excited, are identical, a resonance case will be established. This resonance introduces twofold degener acy (symmetric and antisymmetric). As it is known that the intensity of emission (absorp tion) is defined as the radiant energy emitted (absorbed), which penetrates a unit area in a plasma volume in unit time per unit solid angle perpendicular on the area, and, per unit of frequency . The sodefined intensity encom passes radiation of all frequencies and hence, should be called “complete intensity”. If one measures the spectral intensity of one line and its vicinity, the frequency (or wavelength) gives information that a certain element is present in a certain stage of ionization. The absorption and emission of a spectral line are determined by the population of the lower and the upper quantum level re Copyright © 2011 SciRes. JMP
566 R. A. ELKORAMY ET AL. spectively, and, by Einstein’s probabilities. For the cal culation of the line intensities emitted from plasma, and, vice versa, for spectroscopic plasma diagnostics, the knowledge of the transition probabilities of all investi gated lines is indispensable. We suppose, our emitting atom has a velocity compo nent in the line of sight of observer. Now, if the atom is emitting a spectral line radiation of wavelength 0, and a frequency 0, Doppler effect causes our observer to re ceive radiation of wavelength ', such that 01c , (1) assuming c (light velocity). Velocity distribution in the line of sight is given by the Maxwellian distribution factor as exp [−βξ2], where 2 kT , (2) [Mthe atomic mass, k –Boltzmann’s constant and T is the absolute temperature]. From (1), it can be written 2 22 0 . (3) In the last relation 0 may now be considered as the wavelength corresponding to the maximum intensity of the line (line center), and as the frequency whose displacement corresponds to the line of the sight velocity , or in other words, the line shift. Although there are some kinds of broadenings (such as Stark, van der Waals, ...etc), Doppler effects due to the relative motion of radiating systems give rise to further broadening ra ther than ones. If the velocities of radiating system have thermal distribution, collisions are negligible; otherwise, and furthermore, if one arbitrarily equates the line center intensity to unity, the Doppler effect alone produces a distribution of intensities over the spectral line which may be represented by the following form, using (3) 2 2 0 exp 2 M II kT . (4) Doppler broadening results in a Gaussian line shape, whose halfwidth D is defined as the frequency width of the spectral line at an intensity I equivalent to one  half the maximum intensity (I0/2). Accordingly, using (4), we can write 12 0 2ln2 2 D RT c , (5) where R represents the gas constant. It may be remarked that the emitted spectrum of any homogenous source becomes more and more continuous (smooth) as the depth of the source increases. This is due to the fact that the strongest lines tend to be reabsorbed, then the weaker ones and eventually also parts of the continuum, until the spectrum resembles a blackbody continuum. 2.3. Spectroscopic Treatment of Radiation Absorption An observer collects radiation along the line of the sight and, in general, absorption of radiation will occur, and this has to be taken into account. This leads, neglecting scattering, to the following treatment. We assume an absorption line at the frequency with a dispersion pro file and frequency—dependent absorption coefficient ( ), such that its line width is much smaller than the generation line width. The generation spectrum corre sponding to the entire pulse exhibits a valley which also has the dispersion profile described by the Lambert Beers relation 0 0 ,exp d It t (6) Here: I( ) represents the spectral radiance intensity of generation in the modes with narrowband selective ab sorption (in the absence of saturation) (sometimes re ferred to as flowout) recording from the absorbent 2 , as shown in the schematic diagram of Figure 1; I0( ,t) = I0f(t)  the spectral radiance intensity of generation in the absence of additional selective absorption, f(t) is a func tion describing the shape of the radiationpulse envelope in the spectral region (flowin) at the absorbent surface 2 comes out of the radiant source 1; and = (ctx/L) is the effective thickness of the absorbing layer at the moment of time t, x is the length of the absorbent, and L is the cavity base. It must pointed out here that, in the particu lar case where a laser pulse can be approximated by a Figure 1. Schematic diagram of the illustrated positions of the 1—radiant source and 2—absorbent surface of a speci men. Copyright © 2011 SciRes. JMP
R. A. ELKORAMY ET AL. 567 function which describes a rectangle, so, 1 0 0 . t ft t Let us consider basic dependences and relationships which rather adequately describe intracavity absorption in pulsed lasers which are useful for measuring the mag nitude of the frequencydependent absorption introduced into the cavity. Under the conditions where the apparatus width of the spectral instrument used for recording the generation spectra of a laser with a selectively absorbing medium in the cavity is commensurable with the actual width of the absorption line or exceeds it, it is advisable to apply the method of complete absorption [13]. How ever, the basic aspects and special features of the meas urements of the spectrumintegral absorption in the ab sorption line were given previously [11,5]. The complete line absorption and intensity depend es sentially on ( ) and . The quantity of the complete absorption A of radiation is defined as: 0 1,. IIt (7) 2.4. Limits of the Presented Treatment For simplicity, we restrict our attention to the following conditions concerning the spectrum generation: 1) We consider that the radiant source 1 has a rectan gular form within its interval , such that outside this in terval no radiation may be emitted or absorbed, i.e.; 0 0 0 ,0 . It It t 2 . 2) On the other hand, since the absorption spectrum is different over the profile of the absorption line, the inte gration given in (6) extends over a limited frequency range of the absorbed line, such as 01 0 2 ,0 . I It Here ( 2  1) represents the frequency optical depth along the line of the sight, at which Doppler effect ex hibits a constant value. Therefore, Equation (6) becomes: 2 1 0exp dII (8) If the radiated layer 1 and the absorbed one 2 (Figure 1) are identical in all its parameters [temperature, pres sure, thickness and energylevel distribution function as well as the same identical atoms (or molecules) …etc], so, for small values of 0 and using Equation (8), Eq uation (7) takes the form [26] 2 1 2 1 0 0 d, or 1exp d, II AI A (9) taking into consideration that ( ) has the following ex pression [5] 2 0 0exp 2.7726, D (10) where 0 represents the absorption coefficient in the spectral line center and D is, from its definition, taken at ordinate ( ) = 1/2 0. 3. Results and Discussion The choice of a transition involving only levels with well understood kinetics for this study is essential. The helium (He) discharge is one of the most thoroughly studied discharge systems [27]. We give now certain analysis needed for the calculation of the complete absorption of the fine structure HeI ( = 5875 Å) spectral line, ob tained in a laboratory work [28] during a nanosecond He discharge, due to the optical transition 23P2,1,0  33D3,2,1, schematic representation diagram of which is shown in Figure 2. This transition is ideal because of its conven ient wavelength ( = 5875 Å), and most of the kinetics of the 33D level [29], the 23P level [30,31], and other nearby levels [32] are known. The energy levels 3D3,2,1 of the state 3D practically are coincident (splitting of that state consists of hundreds and thousands submultiplet fractions of percents) whereas the state 2 3P is reversal. By the rule of brightness, the intensities of 2 3P components Figure 2. Schematic representation diagram of: a) the en ergy levels of the spectral line transition 2 3P2,1,0  3 3D3,2,1 of HeI (λ = 5785 Å), illustrating its fine structure and; b) the intensity ratios (1:3:5) of its components with respect to their statistic a l weights. Copyright © 2011 SciRes. JMP
568 R. A. ELKORAMY ET AL. are in the ratios of 1:3:5, respectively, with their statisti cal weights of the levels 23P0, 23P1 and 23P2. On the other hand, these ratios are the same as that observed between the absorption coefficients in the line center of each individual component. So, the relationship between 0 and I0( ) of the stimulated emission at the frequency of the absorption band edge of the transition depends on the relative population of the energy levels (Nm/Nn). Since the two energy levels m and n have a fine structure, for individual components i j, it may have the following relation [33] 2 3 00 2 j i i ji ij N g cI gN h ij , (11) where (0) and I(0) represent the absorption coefficient and intensity at the line center. It is more easily established a resonance distribution for the nearest superposition sublevels rather than that for faraway remaining ones. Since, the frequencies of the three components, of which are close to each other, it can be used for the following substitutions: , ji mi mn gg NNN gg n N and, therefore, Equation (11) may be rewritten as 2 3 00 2 nm ii mn ij gN cI gN h j (12) If m n N N is constant, the absorption coefficients of the spectral line components are proportional to its bright ness. This partial fulfillment of Boltzmann’s law also allows one to keep the processing results of measure ments to be carried out for lines with a fine structure since Nm/Nn is constant during the experimental condi tions. Neglecting splitting of the upperlevel, so, for the summation of the transition probability Anm and the tran sition probability of the individual components, it may be written in the following form nj nm j a AA a , (13) where An1:An2:An3 = a1:a2:a3 = 1:3:5 ; and, according to the rule of brightness, then aj = gj, where gj represents the statistical weights of the sublevels on which the lev el m splits. In the experimental conditions, one mirror has been used for the radiant source 1. So, if we suppose that I' represents the intensity of the flowin radiation, then, taking into account the reabsorption and loss on the front window, rI' represents the reflected flowin inten sity, where r is called the reflection coefficient of the mirror. We suppose that the defined relationships of ini tial intensities of the three components and their absorp tion coefficients in the line center, respectively, are as follows 01 0203123 :: ::; II aaa 0102031 2 3 :: ::bbb; i.e., generally speaking, 00 , jj j CaC b where C and C' are constants in the experimental condi tions, therefore, the intensity distribution function of the jth component takes the following form [33] 2 00 2 0 00 1exp exp j jj j jj I a C rCbc Cb . (14) Accordingly, the intensity distribution function of the sum flowin radiation from all components can be writ ten as 2 00 2 0 00 1exp exp jj j jjj I a C rCbc Cb (15) On the other hand, during the passing of the absorbent source 2, a part of the radiant flowin would be absorbed, so, ( ), defined in Equation (10), must be equated with the sum of the absorption coefficients of all individual components, i.e., j j , and, for the radiant flowout, exp j j II . (16) As it is known, an integrated intensity equals the area S( ) under the curve of the corresponding intensity dis tribution function. Figure 3 shows, experimentally, the reabsorption contours of the two components of inves tigated HeI ( = 5875Å). Then, assuming that I'( ) cor responding to the sum intensities for the three compo nents distorted as a result of reabsorption during one multiple passage is proportional to its corresponding area S'( ), and also S''( ) corresponding to I'' ( ), the defini tion of the complete absorption coefficient [Equation (9)], may be given as Copyright © 2011 SciRes. JMP
R. A. ELKORAMY ET AL. 569 Figure 3. The absor ption and reabsorption co ntours of two components of fine structure of the investigated HeI (λ = 5875 Å). SS II ASI . (17) It should be pointedout here that we have fitted a superposition of three Voigt profiles to the measured line profiles. The frequency and relative intensities of the three fine—structure components are used for the Voigt profiles. Each of the Voigt profiles consists of the appropriate Doppler and apparatus profile. Substituting Equations (15) and (16) for Equation (17), and taking into account Equation (9), we finally, have 2 1 2 1 ee 1 1e d jj jj j j j j A d . (18) The numerical calculations of the dependencies of A, using Equation (18), on 0ℓ have been made for the in vestigated HeI spectral line with or without its fine structure, with the aid of Simpson’s formula [34] and computer processing. In the calculated program, the fol lowing relations and values have been considered: 1) 2 2.7726 0e, 1 j Dj jj j ,2,3; 010 020 030 13 ; ; 99 5 9 2) The wavelengths of the components of the fine structure and its corresponding frequencies and Doppler halfwidths calculated by using Equation (5) are as fol lows 1 = 5875.669 Å, 1 = 5.1057995·1014 s −1, D1 = 3.1659745·10 9 s−1; 2 = 5875.648 Å, 2 = 5.1058138·1014 s −1, D2 = 3.1659844·109 s−1; 3 = 5875.618 Å, 3 = 5.1058458·1014 s −1, D3 = 3.1660042·109 s−1; 3) For the natural spectral line, we have 0 = 5875.000 Å, 0 = 5.1063809·1014 s −1, D0 = 3.1663351·109 s−1. The obtained results of the relation between A and 0ℓ are represented in Figure 4. From the graphs of that fig ure, one can see that, at small values of 0ℓ, there is a great loss (~ twice) in the absorption process for the in vestigated spectral line with its fine structure. Further more, with the increasing layer thickness, the fine struc ture behaves at much smaller values in A rather than that in the natural spectral line. This result should be taken into account, since the population is spread from one line to three with close but well resolved frequencies. The results of comparison with semiclassical calcula tions for neutral atom lines [4] and Monte Carlo simula tion for radiation reabsorption [35] are very encouraging. For a number of He I, C I, N I, O I and F I lines the agreement is well within ± 20% and in several cases, He I and C I, better than ± 15%. In comparison with neutrals, for singly ionized atom lines the results of the test ex periment versus semiclassical calculations [4] are rather meager. The results for multiply ionized atom lines are a pleasant surprise. The average deviation between semi classical electron impact calculations [36] does not ex ceed 20% and in several cases is smaller than 10%. If one takes into account the unsettled situation for Ca II, Si II and Xe II, it seems that a lot of theoretical and experi mental effort has to be involved to improve the present situation. 4. Conclusions The main conclusion to be drawn from these results is that the absorption of fine structure of the investigated spectral line occurs in a certain frequency range around 0 owing to various processes in the thermal motion of Figure 4. Dependence of the calculated complete absorption A of the investigated HeI (λ = 5875 Å) on the optical pa rameter 0ℓ, with and without fine structure. Copyright © 2011 SciRes. JMP
570 R. A. ELKORAMY ET AL. the gas atoms. Apart from its intrinsic interest in the study of Doppler broadening, the phenomenon described essentially in this review is important in connection with the limitation of the resolution close spectral lines. This limitation arises in a twofold manner; the absorption is reduced, making the lines more difficult to observe, but this reduction is much less at the edge of a line than in the center. Another specific conclusion is that multiplets are very useful in the assessment of the optical depth. The inten sity ratio of their components usually is well known for the free atoms and ions, and any deviation will indicate the magnitude of the selfabsorption, the strongest com ponent being affected. Within limits the optical depths of the individual components can even be derived. The results of the present work predict that, if plasma is seeded with atomic species having high absorption coefficients for ground states atoms or ions, the respec tive resonance transitions easily become optically thick and their spectral radiance allows a temperature meas urement. With respect to applications, these plasmas can be used thus as highpower sources radiating at the blackbody radiance in narrow spectral regions [37]. Furthermore, the results of this work support the as sumption made in its theoretical analysis that the line broadening due to Doppler effects varies directly the absorbed layer thickness. So, it is more convenient in an experimental work of plasma laboratory to make the ab sorbent layer thickness as small as possible to avoid the greater loss in the emitted radiation. On the other hand, it is pointed out here that Doppler broadening calculations have made possible a different and very convenient ap proach for determining the electron densities in dense plasmas with an accuracy comparable to other spectro scopic methods [38]. Finally, the theoretical results of the calculations offer a satisfactory agreement with the experimental data in literature [20]. To extend the study to other ions of the sequence, plasmas with known density and temperature especially at higher values and/or spectrometers with higher resolution are necessary. 5. Acknowledgements The authors wish to express his gratitude to Prof. Dr. Salama A. A, Professor of Numerical Analysis at Assiut University, Assiut, Egypt for his guidance and valuable assistance in the statistical calculations. 6. References [1] H. R. Griem, M. Baranger, A. C. Kolb and G. 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