The generalized Holstein‐Biberman transport theory developed in Part I is used to calculate the effect of reflecting walls in increasing the decay time of Doppler broadened resonance radiation in cylinders and slabs. The results are compared with the corresponding results obtained from the Cayless diffusion theory for optical thicknesses between about 20 and 3000. It is shown that the Cayless theory grossly underestimates the effect over the entire range, giving values for the fractional increase in decay time which are too small by about a factor of 3 at the smaller optical thicknesses, and too small by about a factor of 30 at the larger thicknesses. The discrepancy is shown to arise partly from the inaccurate boundary condition used in the Cayless theory and partly from the inapplicability of the mean free path concept to processes involving the transport of resonance radiation. The consequences for the behavior of low‐pressure mercury rare‐gas discharges are discussed, and it is estimated that wall reflectance must begin to have a significant effect on the characteristics of such discharges at reflectances as low as 20% rather than at the 50% reflectance predicted by the Cayless theory.
REFERENCES
1.
2.
3.
4.
5.
6.
The attempt to calculate a mean free path for the resonance quanta leads to expressions which depend on the size of the container and which diverge as the size of the container becomes infinite. Similar difficulties arise when one tries to use the Fokker‐Planck approximation to derive a diffusion equation for the transport process.
7.
M. A. Cayless, Proc. 4th Int. Conf. Ionization Phenomena in Gases, Uppsala, 1959, (North‐Holland, Amsterdam, 1960), Vol. 2, p. 271;
8.
M. A. Cayless, Proc. 5th Int. Conf. Ionization Phenomena in Gases, Munich, 1961 (North‐Holland, Amsterdam, 1962), Vol. 1, p. 262;
Brit. J. Appl. Phys.
14
, 863
(1963
). The theory developed in these papers represents the most successful of a long series of attempts by several different authors to overcome a host of uncertainties concerning the basic processes which take place in these complex discharges.9.
10.
11.
The imprisonment time in the active discharge is assumed to be roughly the same as the decay time calculated by Holstein in Ref. 2; the diffusion coefficient is then estimated by assuming the distribution of atoms in the resonance state to be parabolic, i.e., roughly the same as in the decay problem. Serious disagreement with the experimental results is found only for the dependence of the characteristics on wall temperature (see also Ref. 10); a decrease of about a factor of 2 in the diffusion coefficient is required to give agreement with experiment.
12.
M. A. Cayless, Proc. 6th Int. Conf. Ionization Phenomena in Gases, Paris, 1963 (North‐Holland, Amsterdam, 1964), Vol. 2, p. 151.
13.
P. J.
Walsh
, Phys. Rev.
107
, 338
(1957
). The connection between the decay time and the imprisonment time in an active discharge is discussed in this reference.14.
The dependence of and on R arises from the dependence of on R.
15.
In I, Eq. (66), the exponent in the first factor of Eq. (10) was inadvertently omitted. The values given in Table I of I, however, were obtained from the correct expression.
16.
Reference 1, Eq. (4.18).
17.
J. H.
Ingold
, J. Appl. Phys.
39
, 5834
(1968
). Ingold’s numerical values for are between about 10% and 18% smaller than those given by the Holstein approximation [Eq. (7)]. Since this error is partly compensated by the fact that we have used approximations similar to those of Holstein in evaluating we have preferred to use Holstein’s values in obtaining Eq. (10).18.
Phelps and McCoubrey (Ref. 5), using somewhat better approximations than those of Holstein, found values of about 20% smaller than those given by Eq. (12). For the reasons mentioned in Ref. 17, however, we have preferred to use Holstein’s values in obtaining
19.
The expression for γ, which was not given explicitly by Holstein, is obtained in Appendix B.
20.
The absorption coefficient at the center of the line is where c is the velocity of light, is the frequency at the center of the line and is the most probable thermal speed of the gas atoms, i.e., where is the gas temperature, M is the mass of a gas atom, and k is Boltzmann’s constant. The reduced frequency is From I, Eq. (2), so that where is the wavelength at the center of the line, and are the statistical weights of the excited and ground states, and is the number of atoms per unit volume in the ground state.
21.
22.
For a cylinder, the diffusion coefficient used by Cayless (Ref. 8) is smaller than that given in Eq. (30) by the factor [see Eq. (36)]. The use of the smaller diffusion coefficient in the present problem would increase the discrepancy between the transport and diffusion theory results.
23.
For large optical thicknesses Eq. (33) can be simplified by taking advantage of the fact that is a very small quantity. Neglecting compared with unity, we have, from Eq. (29), where is the first root of φ(ζ). Eq. (33) then reduces to the simple expression Since the first root of cot ζ is and the first root of is We then have and These expressions are accurate to within about 10% [compared with Eqs. (35) and (37) ] when the value of or is about 3, and improve in accuracy with increasing optical thickness.
24.
From Ingold’s numerical results (Ref. 17), increases fairly slowly with R at small R, so that Eq. (20) remains approximately valid for reflectances as high as about 20%. The use of more accurate values of would increase the values of in Table II. It should also be noted that the percent increase in with R according to Eqs. (27), (29), and (31) parallels that of the numerical results quite closely up to reflectances of about 50%. Thus, the ratios of the values of for the transport and diffusion theories given in Table I are valid up to much higher reflectances than the values themselves.
25.
And fairly small up to very high reflectances. The predicted reduction in the 2537 Å production efficiency, for example, is only about 10% at a reflectance of 90%; large effects occur only at extremely high reflectances. The changes in the other discharge parameters (voltage gradient, electron temperature, etc.) are determined essentially by the change in the 2537 Å efficiency, and are also quite small up to extremely high reflectances.
26.
The possibility of small but not negligible effects at reflectances as low as 20%–30% is of great importance in understanding the behavior of the fluorescent lamp discharge. It now appears possible, for example, that a small increase in the 2537 Å production efficiency may contribute to the rather mysterious increase in lumen efficiency which occurs when the particle size distribution of the phosphor coating is shifted towards larger sizes [see, for example,
K. H.
Butler
and H. H.
Homer
, Illum. Eng.
55
, 396
(1960
)]. Since, in practice, the main effect of this shift appears to be to decrease the uv reflectance of the coating from about 30% to about 10% while keeping the total uv absorption constant, this possibility was completely ruled out by the Cayless theory.27.
While it is not possible to calculate at large R from Eq. (20), a very rough estimate on the basis of Ingold’s numerical results for the plane‐parallel case (Ref. 17) suggests that for the 2537 Å line, corresponds to values of R in the neighborhood of 80%–85%. The reduction in the 2537 Å production efficiency must therefore be expected to be less than 10% for wall reflectances smaller than about 80%.
28.
Assuming the actual wall reflectance to be the same for both lines.
29.
This, again, would contribute to the increase in lumen efficiency referred to in Ref. 26, a possibility completely ruled out by the Cayless theory.
30.
While it seems clear that the predictions of the Cayless theory must be modified rather severely in the directions indicated, the quantitative details presented in the above discussion are not very reliable because the transport and diffusion theories have been compared under the conditions prevailing in a decaying plasma rather than an active discharge [see, for example, Ref. 13 and I, Eq. (76)]. More accurate results could be obtained by modifying the Cayless formulation of the discharge equations (Ref. 8) to include the generalized resonance radiation transport theory from the outset.
31.
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw‐Hill Book Co., New York, 1953), Vol 2.
32.
Reference 31, p. 97.
33.
G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, Eng., 1962), p. 405.
34.
Reference 2, Eqs. (5.13), (5.26)–(5.31).
This content is only available via PDF.
© 1970 The American Institute of Physics.
1970
The American Institute of Physics
You do not currently have access to this content.
