This paper describes a new type of filter radiometer-the Selective Interferometer Filter (SIF)-which has extremely high selectivity to the radiation from a given gas.
Filter radiometer:
the selective interferometer
J. E. Harries and John Chamberlain
This paper describes a new type of filter radiometer-the
Selective Interferometer Filter (SIF)-which has
extremely high selectivity to the radiation from a given gas. The basic device is a double output Michelson interferometer, with mirrors fixed at zero geometrical path difference. It is shown that if a cell containing a gas is placed in each of the arms, the ensemble can be made to operate as a highly selective filter, passing radiation only very near the center frequencies corresponding to the absorption lines of this gas. This is a result of direct absorption effects and of the variation of refractive index with frequency and consequent The device may, theremodification of the optical path difference of the two arms of the interferometer. fore, after calibration be operated as a highly selective ir radiometer; and a feature of importance in atmo-
spheric and space applications is that it requires no internal moving parts to achieve this selectivity. Details of the device are presented, and a thorough examination of the practical realization of it is given.
1.
Introduction
Over the past decade a number of novel radiometers and spectrometers have been proposed and developed for use in atmospheric research, based on the general principle of identifying the presence or concentration of a gas in an atmosphere by what may be
generally classified as correlation methods. Sometimes the observed spectrum is correlated with a recorded spectrum (or interferogram) or sometimes with the spectrum of a sample of the gas itself carried in the instrument. Of the former type, the Barringer Research Corporation' has developed the correlation spectrometer in which a mask is placed in the focal plane of a disper-
sive spectrometer. Slits are cut in the mask at positions corresponding to the wavelengths of the spectral lines of the gas to be investigated. These workers and others 2 have also developed a correlation interferometer which compares a small portion of the
observed interferogram with a recorded interferogram of the required gas. The second idea, that of using a sample of the gas
within the spectrometer itself, has been extensively pursued by Houghton et al. at Oxford and Smith et al. at Heriot-Watt Universities.3 - 5 The selective chopper and pressure-modulator radiometers developed by these groups operate by either chopping between an empty and a full cell or by periodically
National Physical Laboratory, Divisions of Quantum Metrology and Electrical Science, Teddington, Middlesex TW1l OLW, En-
gland. Received 20 November 1974; revised 20 October 1975.
modulating the pressure of a small amount of gas in a
single cell: the signal detected at the modulation frequency gives information on the radiation transmitted in a narrow band of frequencies at and near the line centers and, therefore, has a very high degree of selectivity. Goody independently suggested a variant on the Houghton and Smith type of device using two cells of gas, 6 in which the signals from a source transmitted
by the two cells are compared. Of the various methods, the use of a gas cell offers the highest selectivity to a specific gas, since only radiation very near the line centers is detected. Effective linewidths at a few millibars pressure are only 10-3 cm-1 , so a device using this principle has a very high spectral resolution. In addition, Houghton et al. have shown5 that by integrating over a band of individual lines the signal from a gas in the atmosphere can be enhanced so a good SNR can be achieved. A general disadvantage is, of course, that a sample of gas has to be
contained in the instrument.
For certain gases that
may be corrosive (e.g., nitric acid) or somewhat unstable
(e.g., ozone), this can present difficulties. The pressure modulator radiometer is, even so, an extremely elegant and valuable device. A drawback in its use is the need to modulate the pressure: this requires a fairly sophisticated and somewhat sensitive mechanical system, which utilizes moving parts and which should be avoided if possible, in particular when thinking of spaceborne applications. It was to find some way around the need to modulate the pressure that the present study began. It was inspired primarily by the previously mentioned paper by Goody,6 which concluded with a brief description of an interferometer that had a gas cell in November 1976 / Vol. 15, No. 11 / APPLIEDOPTICS
2667
gasx pressure p cell lengt9 12
0E0 path difference
SourceS
>
()
C
4X
M
1U
/
F
,r
p-
jf
I
*
gas
pressurep
lines, as we shall now show analytically.
The case we
shall treat will be more general than Goody's, having a cell in each arm containing gas at different pressures.
cell length
I
BS
Ill. aV path difference
Fig. 1. A schematic diagram of the Selective Interferometer Filter (SIF). Radiation from a source S is modulated by a simple
chopper C, passed through an interference filter F to isolate a chosen spectral band, and then into a. Michelson interferometer. Cells of length 1l and 12 are placed in the arms containing pressures p land P2 of gas x, respectively. The interferograms that would be
obtained if one of the reflectors were scanned away from zero optical path difference are shown against outputs 1 and 2.
Theory
Figure 1 illustrates the two-cell system proposed. Radiation from a source, say the sky, is modulated by a simple mechanical chopper C, passed through a filter F, which isolates one spectral band (we have used the 15-gm CO2 band as an example in our calculations), and then into a Michelson interferometer set at zero geometrical path difference. Cells are placed in each of the arms as shown, of length 11and 12, initially evacuated, but later to contain CO2 at pressures P1 and P2, respectively.
The interference functions observed at outputs 1 and 2 may be generally described as a function of optical path difference x as follows
one arm and that could be used as a bandlimiting fil-
Interference at output 1:
ter. In this paper we shall take up Goody's rather brief comments and develop a complete description of such a system in some detail. We shall then indicate how this may be utilized and improved to produce a highly selective ir filter with no internal moving parts. We have called this device the Selective Interferometer Filter (SIF).
I,(x) = I + F(x) =f
power reflectance
= 0.5), output 2 shows zero signal
at zero path difference (the effects of unequal reflection and transmission will be considered later). Introducing a gas cell into one of the arms of the interferometer causes a change in optical pathlength in that arm near the centers of the absorption lines of the gas due to the effects of dispersion. Thus, in the vicinity of the lines, as the refractive index changes value, there will, in general, be alternate dark and light fringes observed at output 2. In addition, the introduction
of the gas cell causes an imbalance be-
tween the intensity of the beams in the two arms of the interferometer, by a pure absorption effect. The device acts by a combination of these two effects to pass radiation at the frequencies of the absorption 2668
APPLIEDOPTICS/ Vol. 15, No. 11 / November 1976
2|2It2B(cos2rTx
da. (1)
Interference at output 2: 12(x) = - F(x)
11. Basic Principles
Goody suggested that a dynamic range problem he had encountered with his own double cell spectrometer (see Ref. 6.) might be solved by limiting the bandpass of his instrument; he suggested this might be done by including a Michelson interferometer in one of the two channels, with a cell of the gas under investigation in one arm. Figure 1 illustrates a Michelson interferometer. We will consider output 2. Beside each output is an inset schematically showing the interferogram of a broad band source that would be obtained if one of the mirrors were scanned away from zero optical path difference. Assuming that the beam splitter divides the radiation perfectly (i.e., power transmittance =
21I 12tI2Bodu +f
(114 + ItJ4B d-
f2rJ2It2Bo
cos27rxdr. (2)
P and are the amplitude reflectance and transmittance of the beam splitter; and B is the incident spectral power, all functions of wavenumber a. The terms 11,2 and F(x) are the background and interference functions of the interferogram.7 At zero optical path difference (x = 0) for a perfect instrument, the intensity at output 2, I2(0), is zero since 12(0)
-J =-
0 if
oi
21;121112 +
4-
rI2
=
(3)
1(
JtjJ2
Inserting gas cells as shown in Fig. 1 may be repre-
sented mathematically by complex insertion losses of the form L = S exp(-io)
exp[-27rior(n -
1)1]
(4)
loss due to gas in loss due to cell of length cell windows
In this expression, S and iP are the amplitude and phase shift describing the loss at the windows of the cells (assumed the same for the two cells), and is the complex refractive index of the gas. Inserting this loss expression into Eq. (2) and setting x = 0, gives
12(0) =j
(Ir|4|L212 + ItI4 LiI2)Bodo,
I L2IBOcos27Ta[(n2 -
-J'21rj2ItI2IL
-
=fs2B 0
1)1 4 ]d;
-
(5)
I ,B4 {4,2 exp(-a212) + exp(-al1 )
24) exp
-
(n1
1)2
[-t2
Ill
+
212)] cos2rul(n -
(n,
-
2
-
1)12
1)l1} do. (6)
In Eq. (6) al and a2 are absorption coefficients (functions of a), and 0 = I 2/jjt2 is a measure of the optical efficiency of the beam splitter ( = 1 for the perfect instrument). Clearly, from a consideration of Eqs. (5) and (6), 0 can also be used to discuss other imperfections, such as Si, (4)]; so calculations for 4i1
S 2 and , 1 5" 12 [see Eq. 1 may be taken to repre-
sent a number of possible real optical imperfections within the device. To investigate the performance of the SIF as a function of wavenumber, we may define an effective normalized transmittance X2 (0) for a single absorp-
center wavenumber UO = 667.00 cm-', = 0.2 cm-l atm-1, line half width Yo line strength So = 2.0 cm- 2 atm-', and. background refractive index no
= 1.0000.
The absolute values chosen are merely reasonable assumptions, and their magnitudes do not affect the substance of what follows. Lorentzian line shapes have been used for a(a) and n(cr): () =
-zr = _ (C
( n(o) = no + 4,2 47 (c
2 -. )Y.P + (P)
-
-a)
2
(8)
x
)2 ++ (P yp) 2]
(9)
2
The exact line shape used has no bearing on the principles of the device at all-a change to, say, the Zhevakin-Naumov or the Gross line shape would affect only the detailed properties of the parameter X 2(o-) to a small degree.
(1) The first calculation was carried out to inves-
tion line, when viewed at output 2:
tigate how the properties of the single cell (i.e., one cell empty of gas) and double cell modes of the device
X2 (uf) _2= s22(0) s2 0fi 4
differed.
-
4)2
+
exp(-a212) + exp(-u 1tl) - 2 212)] cos 2ffc[(n
- 1)2 -
exp
A single cell calculation (A) was carried out for 11 = 1 cm and P' = 1 mb, with 0 = 1 (perfect optical com-
(!
ponents). The pressure P2 in cell 2 was put equal to (n -
1)11]-
(7)
This expression gives the normalized transmittance for a single spectral line; the transmittance for a whole band of lines will be given at each wavenumber
by the product of the X 2(U)s for the individual lines in the band. It should be noted that S, Bo, , , ' all a2,
l, and n2 are all functions
of wavenumber,
though this has not been written explicitly. In the following study of the device any a dependence of S, Bon P, and t will not be included since this would not
alter the principles of our argument, though in the real device, of course, such effects will be important.
It is not necessarily required to know such dependences explicitly in practice, however, since they may be allowed for by empirical calibration of the device before use. IV.
Instrument Performance
Study-the
Ideal
zero, and 12 = 1 cm.
Then in a second calculation
(B) P2 was made equal to 5 mb, with the other parameters remaining the same. The results of these calculations
are shown in Fig. 2; curve A represents
the single cell case, and B the double cell. Immediately we see one important
property of the device,
which is that by using it in either the one- or two-cell modes, we can produce either a transmission profile with a maximum at the ao s of the lines, or (with two cells) a profile that maximizes in the wings of the
lines and falls to zero at the aos. Thus, in a simple way, we can choose to investigate either the centers or the wings of an incoming band of emission lines
from the atmosphere. This facility, as is well known from past remote sensing work,8 9 is important if it is
*80
Instrument
To begin with, an instrument for which
60h A
and
b 40
1,
12,
was investigated. The transmission, X2 (0f),of the instrument about a single absorption line at o-o observed at output 2 has been calculated with Ipl 2 a Ijj 2 = '/2, for various values of 1 = 11 a 12, and various
pressures pi and P2. X2(o-)has been evaluated at up to ±0.025 cm-1 on either side of line center, in steps
of 0.0005 cm-'. For our example calculations we have taken a typical line from the 15-gm CO2 band
and have used
20
-0 02
-...--
-0 01 Wavnumbers
a.
0 01
. .. . . .. . . .
-
.
0 02
from line centres (cm-)
Fig. 2. The effective normalized transmittance for a single absorption line X 2 (a) seen at output 2 (Fig. 1) for a single cell (A) and double cell (B) SIF. Case A: 11 = 12= 1.0 cm, pi = 1 mb, P2 =0;caseB: 11 = 12= 1.0cm,p=1mb,P2=5mb. November 1976 / Vol. 15, No. 11 / APPLIEDOPTICS
2669
800. 700600*500t
400 *._..
b
/
-
300 200 *100
-0
02
-0 01
01
0 01
0-02
Wcvenum hers from line centres (cm-l)
Fig. 3. The effective normalized transmittance X 2 (a) for different pressures in the two cells. Case B: l = 12 = 1.0 cm, p = 1 mb, P2 = 5 mb; case C: l = 12 = 1.0 cm, p = 2 mb, P2 = 12 mb; caseD: 11= 12=1.0 cm,p= 4mb,P 2 =40mb.
necessary to obtain height or range resolution of the distribution of a gas. (2) To demonstrate
how the effective width of
the filter profile could be altered, further double cell calculations were carried out for
i1
p = 2 mb,
= 1 cm,
2 = 12 mb
(C)
and
and though multiplied by an attenuating factor exp[-'A(all + a212)], this term can cause X 2(^) to increase above one. It is fairly easy to show that X 2(^) as defined in Eq. (7) can never increase above 4.0 for any conditions. We should also note that the examples E-I demonstrate how a very wide transmission profile may be achieved easily by choice of suitably large pressures and pathlengths. The range of altitudes over which the SIF could be used is, therefore, very large, ranging from pressures of 1 atm for use near sea level, to 10-3_10-4 atm in the stratosphere, where the effective transmission profile width is limited by Doppler broadening rather than by pressure broadening. V. Instrument Performance Study-the Instrument
To investigate how the behavior of the real SIF might depart from the above results calculated for the ideal case, we have considered a number of possible imperfections and how to model them mathematically. There are three areas that have been considered. (1)
4 =
Unequal cell lengths [11# 12in Eq. (7)].
(2) Variations of background refractive index across the spectral band being investigated (no not constant and not equal to unity). (3)
11=
Real
I4p2AjI2
The effects of
1 (i.e., imperfect
#
beam splitter, unbalanced optical beams).
1 cm, p = 4 mb, 2 = 40 mb (D).
The profiles of X2 (^) calculated for these cases and
Point (1)
that, as expected, the effective width of the filter pro-
term in Eq. (7) is small, i.e., for low pressure X path
for case B are shown in Fig. 3, where it can be seen .file increases with gas pressure.
Furthermore,
a sec-
ond effect is illustrated by this diagram, that the magnitude of X2 (^) at the maximum points increases with the ratio P2/P 1. (3) In the low pressure and small cell length regime we have considered so far, the dominant contributions to X2 (^) have come from the absorption terms in Eq. (7) because at pressures below several hundred millibars the argument of the cosine term is small, so cos 2ro-[(n2
-
1)12 -
(n,
-
1)11]
It is clear that when the argument of the cosine conditions, then cos[ ] 1, and variations in the pathlengths of the cells (1 5` 12) will not be important. Indeed, the effect of unequal cell lengths will only affect X 2 (^) significantly once the cosine term (the effect of interference) becomes significant, i.e., for higher pressures X pathlengths. Even for such conditions, the difference 1 - 12must be of the order of the cell length itself to have a significant effect;
1. To il-
lustrate how the cosine term (i.e., interference effect rather than absorption effect) becomes important at higher pressures and cell lengths, we have calculated X 2 (^) for the single cell mode withp, = 1000 mb,P 2 = 0, and for 1 = 12 = 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm.
The results are shown in Fig. 4, where the calculations are labeled E, F, G, H, and I, respectively. As the cell length increases, the effect of the first dark fringe-as the cosine term changes from positive through zero to negative-may be seen. Going back to Eq. (7) we can see that, if q0= 1, then at a =
6
1-2
-
b
the
-
0(0)
[where 0(0) means that the term is tending towards zero].
As we move away from the line center, the argument of cos[ ] grows, so this term becomes negative 2670
APPLIEDOPTICS/ Vol. 15, No. 11 / November 1976
'I
-
.E
__---2 5
-2 0
- 5
.*
-l 0 W.venumber
1
\
I'
*80
40
maximum value of X 2 (a) that can possibly occur is one, since the three terms in X 2() are, respectively, X 2 (aO-) = 1 + 0(0)
r
2
-05
0
from line centre
N
\
-...
,,.;..---_
0-5
La
Is
2 0
25
(cm-I)
Fig. 4. The effective normalized transmittance X 2(^) at high pressures and pathlengths. Case E: 11= 12= 1.0 cm, Pi = 1 atm, P2 = 0; case F: Il = 12= 2.0 cm, p = 1 atm, P2 = 0; case G: 1l = 19=
3.0 cm, pi = 1 atm, P 2 = 0; case H: 1 = 12 = 4.0 cm, pi = 1 atm,P 2 = 0;caseI: 11= 12= 5.0 cm,p1 = 1 atm,P2 = 0-
To study the problem systematically we have repeated calculation A (Fig. 2) for
a
= 1.0, 0.9, 0.7 and
0.5; these have been labelled cases A, J, K, and L, respectively. The results for X 2(a-) are shown in Fig. 5 on a logarithmic scale. The effect of f 5 1 may be seen as a background leakage transmission whose
C Io I
00
-002
CCI °°
-CI
002
Wovenumber from line centre (cm-)
Fig. 5. The effect of X •d 1. The effective normalized transmittance X2 (G). Case A: 1 = 12= 1.0 cm, pj = 1 mb, P2 = 0, 0 = 1.0; caseJ: 1 = 12= 1.0 cm,pi = lmb,P 2 = 0,40 = 0.9;caseK: 1 = 12=1.Ocm,pi 1mb,P2 =0,0=0.7;caseL: 11=12= 1.0cm,pi
= 1mb,P2=0,0=0.5.
and thus for cell lengths of the order of 1 cm, it is
concluded that small differences of length are unlikely to have any effect whatsoever X 2 (a-).
on the function
Point (2)
In our calculations so far we have assumed that the
background level of the refractive index is equal to unity and, moreover, is independent of wavelength. Neither of these assumptions is valid in the real world, where the background no will not be equal to 1 due to the effects of lower frequency dispersion and also where no will differ at either end of the absorption band in question, again due to the additive effects of the refractive index with increasing frequency within the band. We have used experimental data on the ir dispersion in gaseous CO2 (Refs. 10 and 11) to model the
variations of no more realistically. The magnitude of (no
-
1) is always much less than An (the peak-peak
range in n at a dispersion feature) at least in the gaseous state.
Thus, (n
-
1) effects will always be
much smaller than the dispersion effects themselves, which as we have already seen are generally smaller
than, or of the same order as, the direct absorption effects. Therefore, the effect of (no - 1) variations on X2 (a-) is very small. Point (3)
When we come to study the effect of 0 # 1 [see Eq. (7)], we can (as we have discussed in Sec. III) consid-
er this to represent not only beam splitter imbalance (IPIi I) but also any other optical or geometrical effects that might cause an intensity imbalance between the two interfering beams of the interferometer, other than that caused by the presence of the gas itself. Such imbalances can arise from asymmetric geometrical attenuation of the two beams (shad-
magnitude depends on the degree of imbalance, (0 1). In this example the leakage transmission is independent of wavelength, simply because 0 has been assumed independent of wavelength. More generally, the leakage will follow any wavelength dependence that 0 (and therefore P and t) may have. A consideration of the level of the leakage transmission will reveal that this problem is the most important so far investigated.
If
X
= 0.9, there is a 0.01
leakage transmission over the whole band; if 0 = 0.5, this factor becomes 0.25. The narrower the filter profile is, the more important is the leakage because it represents a larger fraction of the total integrated signal observed at output 2 (Fig. 1). Clearly it is important to maintain 0 as close to unity as possible, and in any case to maintain X > 0.9 at least. The reflectance IPI and transmittance
I il of a practical beam
splitter will certainly not be identically equal, and it is likely that values of 1.0 > 0 > 0.8 will be found in practice. Thus it may prove necessary to improve the balance between the two beams by introducing a gray attenuator into the more intense beam, or using a mechanical stop to limit its intensity. Either of these approaches may be adopted without difficulty and, once the 0 value of the beam splitter is known, a suitable remedy may easily be devised. VI.
Aspects of instrument Design
The SIF we are currently building for use in an extensive testing program will operate in the 15-gm region for studies of the intense CO2 band at this wavelength. The instrument will use surface-reflecting optics based on an aluminium-alloy main structure. The beam splitter will in the first instance be stretched Mylar, but other beam splitters, including substrate-coated types, will also be investigated. The cells will be 2.0 cm in length, with an aperture of 2.0 cm; and the windows of the cells will be germanium, bloomed for 15 ,im.
The transmission of the device will be measured using a separate high resolution spectrometer (probably an interferometer itself), though it will not be possible to measure the transmission function X 2 (-) explicitly for widths less than about 0.05 cm-l. The stability of the device and the problem of quantitative calibration for atmospheric measurement will also be studied. A flight model will be developed out of these considerations, which will be flown on either aircraft or balloon platforms in due course. VII.
Conclusion
beams, by different reflection and absorption proper-
This paper summarizes the findings of a study of the new Selective Interferometer Filter (SIF). This
ties of the cell windows, as well as by the beam splitter effect, PI id il -
exploits the dispersion and absorption due to the
owing), by unequal reflectivities of mirrors in the two
device is based on a Michelson interferometer
November 1976 / Vol. 15, No. 11 / APPLIEDOPTICS
and
2671
spectral lines of a gas to create an extremely high resolution spectral filter, specific to that gas. It has been shown that the spectral transmission characteristics of the SIF are highly suited to remote sensing applications with the single cell mode being applicable to studies of line centers and the double cell mode being useful for studies of line wings. The effective resolution of the filter profile may be readily adjusted by varying the gas pressure and cell length. The highest resolution achievable will be limited by Doppler broadening at the temperature of the instrument, i.e., to about 10-3 cm-' at 15 ,umor 10-4 cm- 1 at 150 gim. In all cases of application to the stratosphere or troposphere, however, it is possible to match the filter width, or effective resolution, to the width of the atmospheric emission lines being observed. Though not considered explicitly in this paper, foreign gas broadening as well as self-broaden-
ing may be employed when choosing the effective resolution of the instrument. The probable sources of mechanical and optical imperfection within the proposed SIF are unlikely to be severely restrictive in the operation of the instrument. Indeed, with careful preliminary tests on the beam splitter and gas cells, it should be possible to assemble and operate the device quickly and easily. with no further adjustment. Since there are no moving parts required to achieve the very high spectral resolution, the whole assembly can be designed to stay in alignment for considerable periods. It is thus an extremely attractive instrument for use in remote sensing from space. Because of its promise of simplicity and reliability, it is possible to consider the application of the SIF to a number of fields of remote sensing, including the monitoring of stratospheric composition, the measurement of atmospheric temperatures, and the detection of local or regional pollution in the surface layers of the atmosphere. The effective spectral resolution of the SIF may, as we have shown, be varied at will over a large range to accommodate the widely different spectral linewidths encountered in these different applications. A program of development of this device is under way in our laboratory, which includes an experimental study of the optical principles involved and of the specification of the SIF for various applications. Also, further theoretical consideration is being given to the definition of the transmission properties of the SIF and to its use in quantitative radiometric measurements.
References 1. A. R. Barringer and-J. H. Davies, Investigations in Correlation Spectroscopy and Interferometry in Proceedings of the Joint Conference on Environmental Pollutants Paper 71-1105(Am. Inst. Aeronautics and Astronautics, New York, 1971). 2. A. W. Goldstein, M. H. Bortner, R. N. Grenda, A. M. Karger,
and P. J. LeBel, Paper 73-515 in Proceedings of the Joint AIAA/AMS Conference June 1973, Denver (Am. Inst. Aeronautics and Astronautics, New York, 1973). 3. S. D. Smith and C. R. Pigeon, Mem. Soc. R. Sci. Liege 9, 336 (1964). 4. P. G. Abell, P. Ellis, J. T. Houghton, G. Peckham, C. D. Rodgers, S. D. Smith, and E. J. Williamson, Proc. R. Soc. A320, 35 (1970).
5. F. W. Taylor, J. T. Houghton, G. Peskett, C. D. Rodgers, and E. J. Williamson, Appl. Opt. 11, 135 (1972).
We are grateful to N. W. B. Stone for reading this
manuscript and suggesting several improvements to it. The device described in this paper has been registered in the U.K. under patent number 38485/74. Patent applications have been filed for the U.S.A., Canada, France, Germany, and Switzerland. 2672
APPLIEDOPTICS/ Vol. 15, No. 11 / November 1976
6. R. M. Goody, J. Opt. Soc. Am. 58, 900 (1968). 7. M. Francon, Optical Interferometry (Academic, New York, 1966). 8. L. Kaplan, J. Opt. Soc. Am. 49, 1004 (1959).
9. J. T. Houghton and F. W. Taylor, Rep. Prog. Phys. 36, 827 (1973). 10. International Critical Tables (McGraw Hill, New York, 1930), Vol. 7, p. 9. 11. C. Statescu, Philos. Mag. 30, 737 (1915).
