Neglecting the Doppler width, the author has calculated the shift of maxima caused by mutual overlapping of intensity patterns of very close spectral lines for a number of spectroscopic instruments. A table, which may be used to correct for shrinkage effect has been given for Fabry Perot ctalon. The value of the resolving power of Fabry Perot etalon according to Abbe’s criterion has also boon refined taking into account the shrinkage effect

Sodha, Mahendra Singh

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SHRINKAGE EFFECT OF CLOSE COMPONENTS IN SPECTROSCOPIC INSTRUMENTSB y  MAHENDRA SINGH SODHA*Department of Physics, Allahabad University, Allahabad*( R e c e i v e d  f o r  p u b l i c ^ i o n ^  A p r i l  9, 1955)A B S T R A C T . Neglecting the Doppler width, the author has oalrulated tho shift of m axim a caused by mutual overlapping of intensity patterns of very close spectral liros for a  number of spectroscopic instrumonts. A  table, which imiy be used to correct for shrinkage effect has been given for Fabry P^rot otalon, 1 ’h(^  value of tlie resolving power o f F ab ry  Perot etalon according to Abbe’s criterion lias also boon n»fincd taking into account the shrinkage effect.I N T R O D U C T I O NIn every spectroscopic* instrument, the mutual overlapping of intensity pat­terns of two very close speetral lines results in their spurious closing together. Very little work has been clone on this important aspect, which introdiuies consi­derable errors if disregarded, particularly when tlie instruments are used to the limit of their capacities.The analysis of the problem becomes extremely difficult when both the Dopyiler and the instrumental widths are taken into ac^couiit. Oldenburg (1922) has dis­cussed the ease in wfiich thcTe is partial overlapping of the Doppler distribution of two lines. In his treatment he implicitly assumed that thc^  Doppler width is much larger than the instrumental width. Hen<;e this treatment is not applicable to a number of ceases in which the natural line width is reduce ]^ to he less than the instrumental width, as has hcen pointed out by Tolansky (1947). A small gap Fabry Perot etalon is generally employed in the examination of a widespread spectrum and the observable line width is mainly duc^  to the instrument and not the line. Similarly with the prism and the grating and even with high resolving power instruments, when modern hvperfme-structure sourc^ es like the atomic beam are used, the condition essential to Oldenburg’s treatment that the instrumental width be less than Doppler width breaks down.In this communication the author has calculated the shrinkage effect for a number of instruments taking only the instrumental width into account.P R I S M  A N D  G R A T I N GThe intensity pattern of a spectral line after diffraction by a grating is given hjr48* Now at Defence Science Laboratory, New Delhi-12.461(1)462 Mahendra Singh SoAkawhere N  is the number of lines in the grating and 2/f=27rve(sin sin 0) is the the phase difference between the rays diffracted by two adjacent elements of grating, where the symbols have usual meanings.The maximum intensity say / q is given byh  -  b n ^and hence Eqn (1) may be expressed asr  sin  ^Nfi sin*-* x(2)where x and <  1 .Eqn (2 ) also represents the intensity pattern in case of a prism if ,t? =3 nlv sin 0, the symbols having usual meanings.The quantity Ax is proportional to the angle between two close spectral lines and therefore we shall use Ax instead of M  to re}»reseiit the latter in this investi­gation. The intensity distribution of another spectral line separated by an angle Ax =  a is given by/" sin* (x—a)h  ~  (*-«)* (3)The resultant intensity pattern of two such lines with intensities I q and IJb  (intensity ratio 6 : 1 ) is///„= . ( r + n / / o  =sin*-* X I sin*^  (x—a)■+ 6 (x ~ a fFor very small values of x we haver , ,  / . ^Ilh=\  1 -  3 4- 45 ) +2** \ 1 sin* (x~a)3 45 6 (*-«)*The maximum near a; =  0 , is given by d Ijdx — 0 i.e.(4)(4^)I 2x 8+3 ’ 46 ) +  I F  (X) =  0orwhere F ( x )2 sin*(a- x) sin 2 (o—x)(5)(a--x)« (o-x)*Eq|i. (0) may be solved by the method of successive approximations. If  x'„ is the solution to approximation' >1+1 — 15 * n +  2b (6a)Similar equation for the shift of maximum of the other line is obtained by putting 1/6 for 6 . Thus (6a) becomes—4 36=  2 W )  ... WPutting Xq =  — 0 wc get the first approximation as^ 1 = 1  F(0) == 3(sin aja^ ){{Qm a/a)—cos a) ^  f(a) ... (7)The higher approximation values can fee calculated by the use of equations (6 ) and (6 )The calculations are particularly valuable when the two lines are close togethe* and when their separation is small compared with the fringe width. Tn this case first approximation is sufficient and the displacements of two lines from their true positions are given by tShrinkage Effect in Spectroscopic Instrnmenta, etc., 46SAv a 6KVhf(a)... (8a) ... (86)where f l a ) =ri® I a )A V is the separation of the two lines and and Sv'' are the shifts of stronger and weaker components (6 >  1 )Table I gives the values of rfv/Av for values of a — tt, tt/1 .1  16 and tt/1 .28 which correspond to the separations required by the Rayleigh, Abbe and Sparrow cri­teria respectively.TABLE IValues of Svj/^v1/2 1 2 3 4 5•IT 0.0000 0.0000 0.0000 0.0000 0.0000 0.00007T/1.115 0.0906 0.0453 0.0227 0.0151 0.0113 0.0091ir/1.26 — 0.1214 0.0607 0.0405 0.0304 0.0243For all values of 6, ^v/Av =  0 when a  = n  , There will be no maxfor h =  1/2 or less when a  =ff/1.26 and for b =  1/3 or less when a  =  jr/1.118,464 M ahendra  S ingh  SodhaIt is interesting to note that f(a) can also assume negative values thereby giving rise to what may be called an expansion effect of close eomponents eg when a is slightly greater than n.The treatment and results are also applicable to the reflection echelon, which has an intensity pattern similar to the grating.F A B B Y  P E R O T  E T A L O NThe intensity of a spectral line in the order n„ being an integer andn a small quantity is given by/'//fl — 1/{1+ jF' sin* #(to0+ w)} =  1 /(1 +X*) nearly, where X  = nF^n, F  being the coefficient of fineness and / q the intensity of the incident beam.The intensity pattern of another line of an intensity 1 jb times that of the first separated by an order A»i isFITg= ljh{l-^(X —a f]  where a — nF^An.Hence the resultant intensity pattern is given by=  =  ... (1 0 )The maxima are given by~ ( l/2 /,)(d//dX) {X/(l+X*)*}+[(Z-a.)/6{l-f (^ -a )* P 3 =  0 ... (1 1 )Eqn. (1 1 ) is a fifth degree equation and its solution is difficult. Hence we will follow the method of successive approximations. If  X'„ is the solution to approximation«±i_ I ^  »+i ~  ®{ l + X \ r  ^  &{l+'(Jt'„-a)*}* -or =  a(l +X';i)*/[(l+X'=S)*+6{l +(X'„-a)*}*]The shift of maximum for the other line **„+! is given byX"n,i =  a(HTaking X„' — X^" =  0 we get to the first approximation( 12)(13)X ' = and X" =■{l+6(l+a*)2} “  ■ *The displacements Sv' and of the two lines from their true positions are therefore to a first approximation given byS v '_x ' __ 1Av aiv"{l+ft(l+«*)*)I(14)Av a {l+(l+a*)*/6}where Av is the real wave number difference between the two lines,(16)%Table TT skives values of shifts for a raiiffe of values of a and b. The values given in the table are such that they differ from the next approximation by less than .005, The table is graphically illustrated in figure 1, The corrections for shift of maxima may be applied with the help of the table, and the graphs byShrinkage Effect in Spectroscopic Instruments 465/;= iFig. Ithe method o f  suceossive approximationn, i.c*. Moling Iho co rrectio n s for the measured values of a, adding them to a and again determining tlic corrections.TABLE 11Values of shift of maxima (a:) in F. P. etaloa.. V  -1/3 1/2 1 2 3 4 51.4 0.2286 0.0900 0.0572 0.0420 0.03121.5 0.1815 0.0786 0.0504 0.0370 0.02791.6 0.1550 0.0680 0.0442 0.0328 0.02491.7 0.1342 0.0560 0.0391 0.0294 0.02221.8 0.1170 0.0531 0.0347 0.0268 0.01981.9 0.3000 0.1020 0.0471 0.0308 0.0229 0.01772.0 0.2398 0.0896 0.0418 0.0263 0.0198 0.01562.1 0.2030 0.0778 0.0373 0.0236 0.0178 0.01432.2 0.3060 0.1610 0.0693 0.0317 0.0213 0.0160 0.01282.3 0.2620 0.1399 0.0619 0.0287 0.0192 0.0144 0.01162.4 0.3194 0.2150 0.1232 0.0555 0,0269 0.0174 0,0131 0.01052.5 0.2752 0.1830 0.1080 0.0500 0.0236 0.0158 0.0118 0.0095

Shrinkage Effect of Close Components in Spectroscopic Instruments

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Shrinkage Effect of Close Components in Spectroscopic Instruments

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