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The Radio-Frequency Line Spectrum of Atomic Hydrogen and Its Applications in Astronomy

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The Radio-Frequency Line Spectrum of Atomic Hydrogen and Its Applications in Astronomy THE RADIO-FREQUENCY LINE SPECTRUM OF ATOMIC HYDROGEN AND ITS APPLICATIONS IN ASTRONOMY J. P. Wild Division of Radiophysics, Commonwealth Scientific and Industrial Research Organization, Australia Received September 8, 1951 ABSTRACT Formulae are obtained for the frequencies, transition probabilities, and natural widths of the dis- crete lines of atomic hydrogen that fall within the radio spectrum. Such lines are due to transitions within either the fine structure or the hyperfine structure of the energy levels. The conditions necessary for the formation of observable emission and absorption lines are examined. Thence an inquiry is made into which of the hydrogen lines are likely to be observable from astronomi- cal systems. It is found that the sun may give a detectable absorption line at about 10,000 Mc/sec, corre- 2 2 sponding to the 2 Si/2-2 P3/2 fine-structure transition, but that other solar fines are not likely to be observable. From the interstellar gas, the emission line already observed (i.e., the 1420 Mc/sec hyper- fine-structure line) is probably the only detectable hydrogen line. The importance of this fine in the study of the interstellar gas is discussed. Some general conclusions are drawn from the preliminary evidence regarding the motion and kinetic temperature of the regions of un-ionized hydrogen. The ratio data are used to obtain a measure of the product of “galactic thickness” and average hydrogen concentration. I. INTRODUCTION The first astronomical observation of a spectral line in the radio-frequency band has recently been announced by Ewen and Purcell1 and has had independent confirmation by two other groups of workers.2,3 The observed line is an emission line of frequency 1420 Mc/sec due to the transition between the hyperfine-s truc ture substates in the ground state of atomic hydrogen. It is observed from the general direction of the galaxy. The likelihood that this line is detectable from the galaxy was first pointed out by van de Hulst.4 In a subsequent discussion, Shklovsky5 concluded that no atoms other than hydrogen are likely to produce observable lines, although certain molecules could. The possibility of observing radio-frequency lines has also been discussed briefly by Reber and Greenstein6 and by Saha.7 The discovery of Ewen and Purcell now arouses new interest in the subject. The present paper has two objectives. The first is a detailed investigation of the radio- frequency line spectrum of atomic hydrogen. The second is an inquiry into which of these lines are likely to be observable from astronomical systems. Included in this part is a discussion of the 1420 Mc/sec galactic line and its importance to astronomy. The conclusions of the paper are summarized in the final section. II. THE RADIO-FREQUENCY LINE SPECTRUM OF ATOMIC HYDROGEN A radio-frequency spectral line originates in transitions between a pair of closely spaced energy states of an atom or molecule. In the case of atomic hydrogen sufficiently close spacing exists (1) between the main levels of high quantum number, (2) between 1 Nature, 168, 356, 1951. 2 C. A. Muller and J. H. Oort, Nature, 168, 357-358, 1951. 3 W. N. Christiansen and I. V. Hindman, reported by J. L. Pawsey, Nature, 168, 358, 1951. * Nederlandsch. Tijdschr. v. Natuurkunde, 11, 201, 1945. *AJ. U.S.S.R., 26, 10, 1949. 6 Observatory, 67, 15, 1947. 1 Nature, 158, 717, 1946. 206 © American Astronomical Society • Provided by the NASA Astrophysics Data System 6W O .2 RADIO-FREQUENCY SPECTRUM 207 .115. the fine-structure states of a particular main level, and (3) between the hyperfine-struc- ture substates of a particular fine-structure state. Transitions of the first of these kinds are of no particular interest in the study of discrete lines because they are so numerous 1952ApJ. that, without the presence of some selective excitation mechanism, they may be re- garded merely as contributing toward a continuous spectrum. We have therefore to consider only the fine-structure and hyperfine-structure lines. a) THE EINE-STRUCTURE LINES The splitting of each main level (characterized by the principal quantum number n) into a fine structure is ascribed to the combined effect of the spin and relativistic proper- ties of the electron. In describing the fine structure, we shall initially assume Dirac’s theory of the hydrogen atom. The effects of known departures from this theory will be discussed later. An outcome of Dirac’s theory is that pairs of states having the same n and 7 (but dif- ferent l) quantum numbers coincide. The energy levels are given with sufficient accuracy by Rh c Rh ca2 /I 3 Wnj= ¿, c (i) n n (tV¿) where h is Planck’s constant, c the velocity of light, R Rydberg’s constant, and a the fine-structure constant. The quantum numbers l and7 can assume the following values: 1=0, 1, 2, . , n — 1 ; j = l±h- The selection rules for electric dipole radiation allow transitions to take place when the l and7 of the initial and final states differ by AZ = ± 1, A7 = 0, ± 1. It follows that, apart from degenerate transitions of the type in, l,j) — (n,l — 1,7), allowed transitions between fine-structure states of the same n are of the type (n, l, j) - (n, l — 1, 7 - 1) . The frequency, v, of the transitions of this type is given, from equation (1), by 7... _w w _Rhco2 ( \ 1 ^ hv TUny Wnj-i — c ( ,* _ 1 .-11)’ n 2 J ~r 2/ where7 may assume the values f, . , ^ Numerically^ 1.7 5 10 X 10nn~d{ (7-è)-1- (i+è)"1 sec -1 (2) The frequencies of the fine-structure lines associated with main levels up to n = 4, calcu- lated by equation (2), are given in Table 1. We shall now calculate the transition probabilities of this series of lines. The Einstein transition probability of spontaneous emission, ^421, for a transition between an upper state 2 and a lower state 1, is given by óIttV A 21 c 0) where g2 = 27 + 1, the statistical weight of state 2, and S21 is the “strength” of the line. The strengths of the fine-structure transitions may be calculated by combining four equations given by Condon and Shortley,9 namely, their equations numbered 745, 1138, 8 The values of physical constants used throughout this paper are those given by R. T. Birge, Rev. Mod. Rhys., 13, 233, 1941. 9 The Theory of Atomic Spectra (Cambridge: At the University Press, 1935). © American Astronomical Society • Provided by the NASA Astrophysics Data System 6W O .2 208 J. P. WILD .115. 663, and 662. For transitions of the type (n, l,j) — (n, l — l,j — 1) we obtain 2 2 1952ApJ. S21 = (4 f - 1) (w - P) a¡e , (4a) where ao is the radius of the first Bohr orbit and e is the electronic charge. Also, for the degenerate transitions of the type {n, l,j) — (n, l, — l,y)> we obtain 9n2 2 j \ 521 = i6ÿ T+T(n -l ) a°e ' Since 521 is expressed in terms of ale2, the most convenient numerical form of equation (3) is 1 ,421= 7.521 X 10-38 —sec“2 . (5) g2 Va 0eV Transition probabilities of the fine-structure lines associated with levels up to w = 4, calculated from equations (5) and (4¿z), are included in Table 1. TABLE 1 Fine-Structure Lines for n = 2,3, and 4, According to Dirac’s Theory Transition Natural Designation Frequency Probability Half-width (Me/Sec) (Sec-1) (Mc/Sec) 22P3/2- 10,944 8.9X10-7 100 Tt" 2 2 7 <=/ , p. 3 P /2- -3 S1/2. 3243 1.4X10“ 30 2 3 2 8 3 D /2- -3 P1/2. 3243 8.7X10" 40 2 3 2 -9 3 D5/2- -3 P3/2. 1081 3.9X10 40 42P3/2- -42SÍ/2. 1368 3.5X10-8 13.6 2 8 4 D3/2- 1368 2.8X10" 17.2 2 2 9 4 D5/2- -4 P3/2. 456 1.2X10- 17.2 ? •< 2 -42D /2. 456 7.2X10-10 6.6 42F5/2- 2 3 11 4 F 7/2 - ■ 4 D5/ 2. 228 9.6X10- 6.6 Finally, we calculate the natural widths of these lines. Even in the absence of all extraneous causes of line broadening (e.g., Doppler effect and collisions), a spectral line has a finite “natural” width due to the finite lifetime of the excited state. In optical spectroscopy this natural width is rarely of practical importance because instrumental limitations and other causes of fine broadening tend to make it insignificant. However, the natural widths of the radio-frequency fine-structure lines are relatively very large and may be the chief factor in determining the widths of actual lines. The natural shape of a line is given by (l') = Tôî I ITTT(v-vo)T&P \ ’ (6> where/(f) dp denotes the probability that a transition of the type (1,2) occurs within the frequency interval (v, v + dv), f0 is the center frequency of the line, and ôvis the half- width of the line (defined as the total width between half-maximum points). The half- width is related to the lifetimes ti and T2 of the lower and upper levels by © American Astronomical Society • Provided by the NASA Astrophysics Data System 6W O .2 RADIO-FREQUENCY SPECTRUM 209 .115. This equation and values of lifetimes tabulated by Condon and Shortley9 have been used to calculate the half-widths of the fine-structure lines given in Table 1. 1952ApJ. b) THE HYPEREINE-STRUCTURE LINES In the above discussion we have ignored the effects of the nuclear magnetic moment upon the electron.
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