JOURNAL OR THE OPTICAL SOCIETY OF AMERICA AUGUST, 1956 Total Absorption and Overlapping Effect Due to Two Spectral Lines of Equal Width and Strength FiuTz REICHE Physics Department, New York University, University Heights, New York (Received November 29, 1955) The total absorption due to two narrow, identical spectral lines, of equal strength and width, is computed as a function of two parameters, -y and a-, where y characterizes the frequency distance of the two lines, while ar is proportional to the product of the number of absorbing particles (per unit volume) and the thickness of the absorbing layer. This represents the so-called "curves of growth," for the case in question. Of special interest, besides the two limiting cases, y -) 0 and -y - o, is the influence of the overlapping effect for relatively small y. 1. INTRODUCTION ABOUT forty years ago, R. Ladenburg and the author published a paper' on selective absorption, in which, among other problems, the total absorption (Gesamt-A bsorption) of a primary radiation by a single narrow spectral line was computed. The primary radi- ation was considered to have constant intensity over a broad spectral region, outside of which the absorption due to the spectral line could be neglected, in first approximation. The intensity distribution of the spectral line, as determined by the coefficient of extinction, was assumed to be given by the expression derived from the classical theory of dispersion. It is well known that this intensity distribution, for narrow spectral lines, is identical with the line broadening due to radiation damping and to collisions, as shown by H. A. Lorentz. It may be mentioned that Ladenburg and the author later discussed the total absorption also in the case where the intensity distribution of the absorbing spectral line is given by the Doppler effect of the random motion of the absorbing particles. 2 Also the superposition of collision broadening, radiation damping, and Doppler effect, and their combined effect on the total absorption have been discussed by several authors. 3 The original problem solved by Ladenburg and the author has later been amplified and generalized in different ways. Matossi, Mayer, and Rauscher 4 ex- tended the discussion of the total absorption to the case of n overlapping spectral lines of similar shape (dis- persion theory), equal widths, but different strengths. Elsasser 5 considered the case of an infinite number of spectral lines of equal intensity, equal width, and equal mutual distance, important for the theory of radiative transfer in far infrared atmospheric bands. l R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913). 2 R. Ladenburg and F. Reiche, Compt. rend. 158, 1788 (1914); R. Ladenburg and S. Levy, Z. Physik 65, 189 (1930). 3 W. Schiitz, Z. Physik 64, 683 (1930); M. W. Zemansky, Phys. Rev. 36, 219 (1930); E. F. M. van der Held, Z. Physik 70, 508 (1931); H. J. Hiibner, Ann. Physik 17, 781 (1933). Computations by Reiche, Fleisher, and Wilke, partly published in Tech. Rept. No. 10, Duke University, Department of Physics, August 15, 1954 [E. Hinnov and H. Koln]. 4 Matossi, Mayer, and Rauscher, Phys. Rev. 76, 760 (1949). I W. Elsasser, Phys. Rev. 54, 126 (1938). See also, W. Elsasser and J. I. King, "Stratospheric radiation," Tech. Rept. No. 6 Department of Physics, University of Utah, September 30, 1952. The following paper takes up again a special, simple case of the work of Matossi and co-workers, namely, the discussion of the overlapping effect of two exactly identical spectra lines; it uses however another method of approach than those authors. The essential integral P determining the total ab- sorption depends on two parameters: a, that is propor- tional to the product of the strength of the lines and the thickness of the absorbing layer, and y, which is proportional to the square of the frequency distance of the two lines. P can easily be expanded into a power series of o-. This leads, in a rigorous way, to the limiting cases y -+ 0, of completely coinciding lines, and y -oo, of completely separated lines. Another approach is considered for values of y< 3 where the integral P is expanded in a power series of y, while for > a numerical evaluation of P is preferred. The more general case of two lines of equal width but unequal strength is being worked out, at present. 2. FORMULATION OF THE PROBLEM Writing the absorption coefficient of a single, narrow spectral line in the well-known form 4 : a K(co) = - (c.)-cw) 2 +b 2 (1) (as given by the theory of dispersion), the total ab- sorption, in a broad frequency range 2, of two identical spectral lines will be: 2A + [ az 2A6= d 1exp_ J~-3 L c-l)2+b 1] (2) w, and W2 are the circular frequencies of the centers of the two lines, z is the thickness of the absorbing layer, b is half the width of the two lines, O=2(col+W 2 ) is the 590 VOLUME 46. NUMBER
Transcript
JOURNAL OR THE OPTICAL SOCIETY OF AMERICA AUGUST, 1956
Total Absorption and Overlapping Effect Due to Two Spectral Linesof Equal Width and Strength
FiuTz REICHEPhysics Department, New York University, University Heights, New York
(Received November 29, 1955)
The total absorption due to two narrow, identical spectral lines, of equal strength and width, is computedas a function of two parameters, -y and a-, where y characterizes the frequency distance of the two lines,while ar is proportional to the product of the number of absorbing particles (per unit volume) and thethickness of the absorbing layer. This represents the so-called "curves of growth," for the case in question.Of special interest, besides the two limiting cases, y -) 0 and -y - o, is the influence of the overlappingeffect for relatively small y.
1. INTRODUCTION
ABOUT forty years ago, R. Ladenburg and theauthor published a paper' on selective absorption,
in which, among other problems, the total absorption(Gesamt-A bsorption) of a primary radiation by a singlenarrow spectral line was computed. The primary radi-ation was considered to have constant intensity over abroad spectral region, outside of which the absorptiondue to the spectral line could be neglected, in firstapproximation. The intensity distribution of the spectralline, as determined by the coefficient of extinction, wasassumed to be given by the expression derived from theclassical theory of dispersion. It is well known that thisintensity distribution, for narrow spectral lines, isidentical with the line broadening due to radiationdamping and to collisions, as shown by H. A. Lorentz.
It may be mentioned that Ladenburg and the authorlater discussed the total absorption also in the casewhere the intensity distribution of the absorbing spectralline is given by the Doppler effect of the random motionof the absorbing particles.2 Also the superposition ofcollision broadening, radiation damping, and Dopplereffect, and their combined effect on the total absorptionhave been discussed by several authors.3
The original problem solved by Ladenburg and theauthor has later been amplified and generalized indifferent ways. Matossi, Mayer, and Rauscher4 ex-tended the discussion of the total absorption to the caseof n overlapping spectral lines of similar shape (dis-persion theory), equal widths, but different strengths.Elsasser5 considered the case of an infinite number ofspectral lines of equal intensity, equal width, and equalmutual distance, important for the theory of radiativetransfer in far infrared atmospheric bands.
l R. Ladenburg and F. Reiche, Ann. Physik 42, 181 (1913).2 R. Ladenburg and F. Reiche, Compt. rend. 158, 1788 (1914);
R. Ladenburg and S. Levy, Z. Physik 65, 189 (1930).3 W. Schiitz, Z. Physik 64, 683 (1930); M. W. Zemansky, Phys.
Rev. 36, 219 (1930); E. F. M. van der Held, Z. Physik 70, 508(1931); H. J. Hiibner, Ann. Physik 17, 781 (1933). Computationsby Reiche, Fleisher, and Wilke, partly published in Tech. Rept.No. 10, Duke University, Department of Physics, August 15, 1954[E. Hinnov and H. Koln].
4 Matossi, Mayer, and Rauscher, Phys. Rev. 76, 760 (1949).I W. Elsasser, Phys. Rev. 54, 126 (1938). See also, W. Elsasser
and J. I. King, "Stratospheric radiation," Tech. Rept. No. 6Department of Physics, University of Utah, September 30, 1952.
The following paper takes up again a special, simplecase of the work of Matossi and co-workers, namely,the discussion of the overlapping effect of two exactlyidentical spectra lines; it uses however another methodof approach than those authors.
The essential integral P determining the total ab-sorption depends on two parameters: a, that is propor-tional to the product of the strength of the lines andthe thickness of the absorbing layer, and y, which isproportional to the square of the frequency distanceof the two lines. P can easily be expanded into apower series of o-. This leads, in a rigorous way, to thelimiting cases y -+ 0, of completely coinciding lines,and y -oo, of completely separated lines.
Another approach is considered for values of y< 3where the integral P is expanded in a power series of y,while for > a numerical evaluation of P is preferred.
The more general case of two lines of equal width butunequal strength is being worked out, at present.
2. FORMULATION OF THE PROBLEM
Writing the absorption coefficient of a single, narrowspectral line in the well-known form4:
aK(co) = -
(c.)-cw) 2+b2(1)
(as given by the theory of dispersion), the total ab-sorption, in a broad frequency range 2, of two identicalspectral lines will be:
2A + [ az2A6= d 1exp_
J~-3 L c-l)2+b
1](2)
w, and W2 are the circular frequencies of the centers ofthe two lines, z is the thickness of the absorbing layer,b is half the width of the two lines, O=2(col+W 2) is the
590
VOLUME 46. NUMBER
August1956 TOTAL ABSORPTION DUE TO TWO SPECTRAL LINES
frequency halfway between the lines, and
2irNe2ba=mo
innoc(3)
is the "strength" of the lines. In Eq. (3), N is thenumber of absorbing particles per unit volume, (includ-ing the "oscillator-strength factors" f), e and m theircharge and mass, respectively, and no the index ofrefraction, (as it would be without the absorption inthe frequency range considered).'
Introduce (see Matossi-Mayer-Rauscher 4 )
3. EXPANSION OF P' IN POWERS OF r. THELIMITING CASES y- O AND yT-0
Expanding P in powers of a, we obtain
where
1 n
(1 + COSP) n1
r,=fd (1 +, cos2^) n
(13)
The computation of the Sn's is straightforward, startingfrom Si= Er/ (I+y)] and yields generally
X=-coo.
r(n- 1)! mcn-i (2m) !(- 1)m
(1+,y)l =O (!)z(n-m-1)!22.(1+y)
(5)Then
+8 r f az az2A3= d LX 1-exp -_ C2
fa^ x2+blx+c x2 -blx+ e2
where
(6)
e2= b2 (1+ 12) (7)
Going over to lim 6 -> oo,* one obtains:
+.0r 2az (02+C2) lim(2A)= f dx 1-exp - + j. (8)W0 u t e i (x 2+C2)2b212X2s
We introduce the new variable 4, by the substitution
This is the integral which must be discussed, as functionof the two parameters o* and 7y.
* See the correction for finite 5 in Sec. 6. Actually, the lowerlimit of x cannot sink below (-coo), which is here replaced by(-cc).
wherer*= (I-1)/2, for odd I
r*=l/2, for even Ijif r*-<m,
r* =m, otherwise.
Written down explicitly, the first terms of the powerseries for P' are
a 2+'y 02 4+3 y+y 2
P'=7rr 1---+_4 1+y 16 (1+7)2
a3 40+367+2172+573
384 (1+7)3
a4 112+1127+8772+3873+7y44-
3072 (1+7)4
Oa 672+7207y+670,y2+40573+140y4+217y6
61 440 (1+7)0
(16)
CW2-W
I=b
and
where
(14)
(15)
591
FRITZ REICFHE
(A) For y -- 0, Eq. (16) goes over into the followingpower series in -
Pt Do = Pso
cy a2 5 7 7= 7r 4--+-- -3+ 124 0 + .
2 4 48 192 640(17)
or, generally, from Eq. (15), where r has to be equalto m
with
P,-o = r (- 1)l M l1a0 1+1 I
_11/2, for event 1i (18)(I-1)/2, for odd 1
E . Im=O 22-(m 1)2 (- 2m) !
It can be proved rigorously, by mathematical inductionthat
(21)!
21(1 !)3
for even and odd values of 1, and that
.0 al (21)!Z=°( 1+1 21- (I )3=eCoi-J~e (20)
where Jo and J are the Bessel functions of order zeroand one, respectively, with the argument i. Hence,
(21)
Eq. (25) can be rewritten as follows:.
lim(2A6) =RL.R.
= rbae-01[( ) ( - )(2 ~2/
(27)
Comparison of Eqs. (27) and (22) shows, that a in Eq.(27) has to be replaced by 2o- in order to obtain Eq. (22).This means, that the total absorption for two equalcoinciding lines is obtained from the total absorptionby a single line, by simply replacing the parametercy= az/b 2 of the single line by the double value, anobious result.
(B) Fory -a oo, Eq. (16) goes over into the followingpower series in a-:
/ 7 a 2 51
-- +-- -c 3
4 16 384
7 7 1+074- 2 +.
3072 20480
or generally, from Eq. (15):
00 (- 1)1( /cyIP' 0+W = 70er E- - Tz-o 1+1 2
withI (-1) 21(2m) !
m=0 22- (M )3 (I-In)
(28)
(29)
(19)
Pso = re-:Jo (i) - iJ1 (in) ]
and from Eq. (12)
lim(2A5),o= 2brae,-Jo(i)-iJi(i)].
For small values of a, Eq. (22) yields
lim(2A5),-o j!-2b7ro-a smallj
while, for large values of cy,
lim(2A5),-o 2_2b(27ro-)ka large
(22)
(23)
(24)
We compare these results with the expressions whichR. Ladenburg and the author obtained for the totalabsorption by one narrow line
lim(2A5)=RL.R.=7rrL.Re.rL.R.
2
r /trL.R. /i irL .R. , 1 25
Xl~~ot-)-iJ~t-) 1 (25
(where the subscript L.R. refers to the notation in thequoted paper by Ladenburg-Reichel). Since
PL. =2b and rL =Oy,
It can be proved rigorously, by mathematical induction,that
T=M=(211!) 3
Hence, from Eq. (20):
Pf Yo = 7re-2 [Jo(ioy/2) - f (i/2)]
and from Eq. (12):
lim(2A),y.OO = 2b7rue-o4 2[Jo (ir/2)-i J(i,/2)].
For small values of
lim(2A6)- )0: l2bro,e small
while for large values of
lim (2A3) ) -4b (X '2.
e large
Equation (34) is a good approximation for
1<<<<Y.
(30)
(31)
(32)
(33)
(34)
(35)
Comparison of Eqs. (32) and (27) shows that thetotal absorption for two equal, completely separatedspectral lines is twice the total absorption due to asingle line, also an obvious result.
592 Vol. 46
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August1956 TOTAL ABSORPTION DUE
4. DIFFERENT APPROACH TO THEDISCUSSION OF P
For the discussion of the essential integral P, one canuse a different way of approach, based on the idea ofintroducing combinations of Bessel functions at anearly stage. This can easily be done by the followingmethod. Starting from Eq. (12) and integrating byparts, one obtains
P=aJ dtg(4P) exp{--f(P)},
where
g()(1-cos) (1-2y cos4'-) cos%
(1+,y COS14)2
1+cos,6
1+y coS21p6
*(36)
The function f (4) shows a different behavior for y < 13and y> 4. For y l , it is steadily decreasing, from itsmaximum value 2/ (1+y) (for = 0) to its minimumvalue zero (for 1' = 7r).
For y> 4, however, f(4i) increases first, from thevalue 2/ (1+,y) (for t'= 0) to its maximum
I ((I +1/,Y) + 1)
at -=4,1 , with cosi/q= (1+1/-y)1-1, and decreasesafterwards to zero (for fP = 7r). We therefore treat thesetwo cases separately.
(A) r-<1/ 3
A new variable x is introduced by putting
1 +cos,6 1+cosxf (4)= (37)
1+Y cos,4, 1 +'
x=0 corresponds to 4'=0, while x=7r corresponds to1=7r.
After a lengthy calculation the following result isobtained, which is still rigorous:
_1 exp 1 +cosx I
P' = a'/ ( 4+z)fJ sinx ( l+cosx) 2 exp I-a1 1 +'
X [2y+y cosx-cosx+ ( . ldx
with
(... )I=1+2,y(1-2 cosx-2 cos2x)
+y (5+4 cosx)] .I
We now expand the square root of the integrand inpowers of y.
First, we obtain
(- * *, *)_+(1 -2 cosX -2 cos~x)+,y2 (2+4 cosx-4 cos 3 X-2 cos 4x)
Introducing, instead of the powers of cosx, the cos ofthe multiples of x, we obtain for the brace in theintegrand of Eq. (40) the following expression, up to
593TO TWO SPECTRAL LINES
FRITZ REICHE Vol. 46
the fourth power of y:
(I -*2 }-(-) cosx - 1(7- -y) cos QX)-(3y2 5-y3) cos(3x)
Inserting these expansions in Eq. (45), one obtains
451 3 15P'_(1+y) (2,ry) 1
8 y 1 28y2 1024y
9 45 105 -y/2 1-+ +-~
8y 128y2 1024y3
17 35 5985+3y2/8 1--- +
8y 12 8y9 1024y3
I 25 243 14553-5y 3 /16 1--- + -
8y 128y2 1024y3
I 33 579 227371+35'y 4/128 1--- + .
8y 128y2 1024y3
(45)
.1I(49)
Remembering that y=o/ (1+y), and that
1-y/ 2 +3y 2/8-5 3/16+35-y4/128- . (l.+y)-,
594
,V2 VP4 V,6 ,8
August1956 TOTAL ABSORPTION DUE TO TWO SPECTRAL LINES
one can write Eq. (49) in the following form:
(l+z)3/ 9 51 125P'-_(27a)1 8a[ 1- _Y+__z2 _ ___z3
8a 2 8 16
1155 l 3 (I+) 5 /2 r 15 35+_ _ 4 _1 +--y+--,2
128 128 f2 2 8
405 6755 l 15(1+y) 7 /2F 7___Y 3+__Y Y4 _1 +--Y
16 128 1024 a3 L 2
1197 4851 53053___ y2 +____y3 - _____y4..
8 16 128or, if -y is small,
1 -31Y 3(1+5,y)2P_ (2ra) 1 _
8 128o-2 where
For y= , the two limits of the second integral becomeequal, (= (3/2)1). Even for special values of y, it doesnot seem to be possible to reduce P to known functionsof a, except for very large values of a. For these largevalues of a, only the immediate neighborhood of u=0gives an appreciable contribution to the integrand.Hence, introducing w=olu, assuming also yu2 to besmall compared to 1, and expanding the first squareroot [the second integral gives no appreciable con-tribution, since in it u(y+1) _2], one obtains fromEq. (54)
f exp{ -w2)
3,y- 1 (5+ 1)2X 1 1+ _2_ - _ _W4. ... dw,
4 32a0 -(55)
15- (1+7Y - 1332 - 203y'3)
1024a 3 *(B) > / 3
Corresponding to the behavior of f(V/), we subdiP into two parts:
P=aff g(1J') exp(-af(4,))d44-o
+aj g(,t) exp{-af (4))dP=Pi+P2.A=*,
In order to compute P1 and P2 , we put
1+cos46f (+) = -r = U2.
I +' Costs/
a a / +-y i iWl= » >>1.
(51)
vide
Hence, using the properties of the error function
3-y-1 3(5y+1)2 1P'=(27ra) 1+ - *-- . ,
I 8a 128a2(56)
a result which is in formal agreement with Eq. (51).This asymptotic expansion is valid for small values of
y/a, where a>>1. It is usable also for large values of y,provided that 1<<y<<a. It fails, however, for values of-y, which are comparable with or much larger than a,
(52) for instance in the case: 1<<a<<y.For very large values of y, the upper limits of both
integrals in Eq. (54) can be taken approximately as 1,while the lower limit of the second integral is approxi-mately zero. Hence,
The values of P'/2- were computed, mainly fromEqs. (54) and (60), for various values of y and a.tFrom these, lima., (2A3) is found by multiplicationwith
8riVze2
4ba =
[see Eqs. (11) and (12)] that is, essentially with a.Table I and the curves of Fig. 1 show the corre-
sponding values of
1-im (2A5)=1P05- 2e
for various values of y and a. We put
e (- ; o-) = l (2A)-lim (2A6) ];b--.0 3 .0
(61)
(62)
in agreement with Eq. (32).If y and a are of the same order of magnitude, and
both very large,
[Pt [ 1 ( (2f 1 2 ]Ilyv largel~ I~ 1--4 7
I~-J argej \~ aI 2 \VyI/
+ ' (1-6-101/) I2a-
lhere
2 et,D(x)=-f exp(-V2)dtis the error function.
, r 0
-(59)
For medium values of , where y (> ) is of the order- 1, it seems best, to compute the two integrals in Eq.(54) numerically, as functions of a-.
It may be emphasized, that also in the case (A),where y-<1, formula (38) can be written in a similarform as (54). As a matter of fact, by introducing
1±cosx
1+,
one obtains
u-U2
P'/2a = f uU du exp -aiur}undo
X [(ey-)2+ 1+ (+4,yi(l _ ))-iji. (60)
Expansion of the square root in powers of y leads, ofcourse, back to the results of part (A). For values ofy which are not too small (but still a4), the numericalevaluation of the integral seems again to be the bestway of computation.
and define [1-e(-y; a)] as the "overlapping coef-ficient." e(-y; a-) represents the ratio of the total absorp-tion due to two equal lines, to the total absorptiondue to the same two lines, if they would be atinfinite distance from each other. Numerical values of[1- eY; a)] are given in Table II. There is clearly nooverlapping effect for y= o, due to the definition ofE(y; a). The largest overlapping effect occurs for y=0,where, from Eqs. (22) and (32)
e(0; a-) = a Jo(ia) - iJiio-)Jo (ioa2)-iJi(i,/2)
a a,2For small a: e(O; u)_1--+-...-
4 8(63)
1 1 13For large a: e(0; a-)_- 1+-+ 1 .
v2\ 8a 128a2
The maximum value of the overlapping coefficient, for-y=0 and a -- , o, is [1-1/V2}]=0.2928.
The formula and Table II show that this limitingvalue 1-1/V21 for [1-e(O; a)] is approximatelyreached for a-100, and that 1-e(O;oc)] does notchange appreciably with a-, for high values of a-. More-over, Table II shows that, for large values of a, [1- e]does not change appreciably with y, as long as is stillsmall compared with a-. For example, for a= 1000, thevalue of [1-el for y=25 is only -2% lower thanthe value for y= 0.
6. CORRECTION FOR FINITE
In Sec. 2, the total absorption had been defined byEq. (2) or Eq. (6). The "absorptivity" A,'that is the
t These numerical calculations and the drawing of the curveshave been carefully performed by Mr. Richard Budd, New YorkUniversity, to whom I am most gratefully indebted.
596 Vol. 46
w
August 1956 TOTAL ABSORPTION DUE TO TWO SPECTRAL LINES
ratio of the absorbed energy to the incident energy, isgiven by
+A -- +IA = If 1, (x) dx f lo (x)
X lexp 2az(x2+c
2) 112
) dx
(,2+C2)2_ -b212X2 J(64)
where Io(x) is the intensity of the primary radiation,assumed to be constant over the whole range of inte-gration so that actually Io cancels and the denominatorbecomes 28. With a approaching infinity, the numeratorof A goes to a finite limit, so that A approaches zero, forall values of the parameters, while lima_,,0 (2A3) remainsfinite and represents the limiting value of the totalabsorption, which is discussed in the previous sections.On the other hand, the absorptivity A itself obviouslymust be expected to approach the value 1, for a -and remaining finite. Therefore, the case of finite 6is considered in this section.
Using the same transformation as in Sec. 2, one finds
2A= 2b(1+y)1 (P-Q), (65)
where P is given by Eq. (12), while
T df r I 1+cos' 11
Ifv* iLpl 1+,ycos24] (66)with I
A*= 2 tan'1[(5/b) (1+y)-1].
In order to simplify the calculation, we assume
3>>b(+-y)1 = [b2+ (21 )2 1 )
4,*_-=r-(2 b16)(1 +,) .Then
a-
co.
WM
it
135 E _ --
5
5 _ - I _____
as~~~~~~~~~- -l
0.1 025 0.5 1 4 9 25 100
FIG. 1. Relative total absorption 2P'=lima . (2A )/4b as functionof y, for various values of a.
(67)
TABLE II. Values of [1-e(; )]. Overlapping coefficient.
Introducing a = ir- VI as a new variable, which has onlysmall values in the range of integration, and performingan integration by parts one obtains
8
b
+2(1± ) 1I(b/b)(2a,1 exp(- 2)dS.\ 1+eo
(68)
Hence, formula (12) is replaced by the "corrected"equation
2A6 = 2bP'+26[1-exp{-2b 20/62}]
-48b(2a.)if (b/I) (2a)i
fo(69)
provided that 8 fulfills the condition (67).For very large a, more exactly, if
b (1+y)i<<6<<b(2')
implying that >>y, 2A8 becomes
2A682bP'i arge+26-2b(2bron),~a>>yl
or, using Eq. (56)
2A8--2b(27ro-) +28-2b(27ro)I~ 28.
(70)
(71)
Hence, lim A -4 1, as expected. From Eq. (69) it
la> Y
follows that thefirst term on the right-hand side, whichhas been thoroughly discussed in the previous sections,can be regarded as a good approximation to the valueof the total absorption 2A8, as long as the sum of thetwo additional "correction-terms" remains small com-pared with the first term.
If, for instance, >>b(2a)1 , the two additional termsreduce to (-4b2u/6). Dividing Eq. (69) by 4b andputting 8= bX, where X>>1, one obtains
2A6 0f-lp -,
4b X(72)
so that aIX must be small compared with 2P', thevalues of which are given in Table I, for various valuesof a and -y. For fixed X, this imposes an upper restrictionon .
