Chowdhuri, A G

Singh, B N

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IACS Institutional Repository

4 0A  NOTE ON THE ENERGY AND W AVELENGTH MAXIMA IN FERMI-DIRAC AND BOSE-EINSTEIN DISTRIBUTIONS*By  a . G. CHOW DHURlUniversity of DelhiANDB. N. SINGH{R eceived  fo r  publication, Ju ly  4, ig.^o)During recent years there have been a number of jihysieal and astrophysical applications of quantum statistics. In the present note we shall consider the properties of the Fernii-Dirac and Bose-]i)instein distribution laws wdiich corres­pond to the W ien’s displacement law' in the case of black-body radiation. It is useful sometimes to know' the value of the energy corresponding to the maximum in the energy-distribution curve of the particles in the assembly or the value of the de Broglie wavelength corresponding to the maximum in the distribution w'ith respect to the wavelength associated w'ith the particles. These are given in the present paper for the various cases of degeneracy and non-degeneracy.The number of photons of black-body radiation lying within the energy range e to e + de at temperature T  is given by Planck's lawv{e)d<-' =SO that K(e)dp =Sn’c'h-de,d>-,(r ' -  i)where E(e)dK is the energy per unit volume of the photons lying in the energy range e to e +  de.E(fi) is maximum for e = e„, w'here e„, is given byde■ 0,which gives 3^T• Comniunicated by the Indian Physical Society.368 A. G. Chowdhuri and B. N. Singhor =2-822./:TIf uc rousickr tlie pliolon distribution from the standpoint of wavelength instead of energy, we lla^ 'e for the energy of photons in the wavelength'intcivalA to A-i JASttcJiKfA)(iA=- a Aand D(A) is maximum for A = A,„, where \m is given bydli(A)_ ^ d\which gives - e/t/A„./cT CjtA..//1'orchIWc shall now discuss the case of Kermi-Dirac statistics. The energy*distribution \pw in the completely non-relativistic case [ » i  | isTm c ‘ru  ^ j 2TT^ (^2m)- • jh I €  k - T , ^e +1Awhere is the energy per unit volume of the particles lying in the energyrange c toDifferentiating the above with respect to e and equaling to zero we gel for g„,, the value of e corresponding to the niaxiinuni of the distribution curve,J k T3 kT— I A . . .  ( i )or‘V - ' rwhere •However, if we consider the energy distribution in terms of wavelength, we haveI____  .E(A)cU-~ rC "r TAdX.PuttingJEWJAw e ob ta in .. (2)where ^2— 2^ mrnkTNote on the Energy and Wavelength Maxima, etc. 389The Fermi-Dirac distribution law for the relativistic case ( <<Ci 1 is\ kTT e l k l \C ' -f 1Aand this will have a niaxiiimni at where is given by-  ( 3 )a being equal to t„J kT,As before, if we consider it from the wavelength standpoint, we getK(\)dA.- r^, IC 1 IAdAwhich will have a niaxinniin at A = A„, where A,„ is given bydXI . C . ,where' ' i s ' : - ] ' ' '_ Am/^ TJic '(4)Tile equations (i), (2), (3) and (4) have been solved gia].hically and positions or the maxima due to the variation in A arc shown in figures j and 2,In fiiiinc (i) are shown curves for the non-degenerate case (o < A < i) and the details are as follows—cui-ve (a) shows A for non-relativistic case,curve (6) shows 2= against A for relativistic case,hecurve ic) shows a against A for non-relativistic case,curve id) show’s — against A for relativistic caseIn figuie (2) are show’u curves for the degenerate case fi<-^A<'^). curve (a) shows y against A for the non-relativistic case,8390 A. G. Chowdhuri and B. N. SinghThe uppermost scale x refers to curves c and d.The middle scale 3' refers to curve a. The lowermost scale r refers to curve h.F ig ure  2The upper scale for 3/, r refers to curves a and 6.The lower scale for x refers to curves c and d.A /I’Tcurve {b) shows z'— - - against A for the relativistic case,hecurve (c) shows A:=c,„/fcT against A for the non-relativistic case, curve (d) shows A — Cm/fcT against A for the relativistic case.The ordinate in figure 1 represents A and that in figure (2) log A. The abscissa represents a', y or 2 as the case may be.Table i shows the limits of the values of a , y and as A varies from o to i in the non-degenerate case and from i to ^  in the degenerate case.T able INon-relativislic case| -^” ,^ > > iRange of, A„ yafcTRange of —Degenerate case ,A varies from 1 to oc 1756 Ijo oc 1 0*5652 to 0Noii-degemTute case,A varies from 0 to iji ’5 to 1756 j  o'5774 to 0-5652Note on the Energy and Wavelength M axima, etc. 391Rdativistic case i -Jv 1}}}C*Range of e,JkT Range- of A,„/, T//trDegenerate case,A varies from i to ocNon-clegenerate laise,A varies from o to i3-13 to 3*0 to 3-13o-ic)g4 to o O'l,’ to (V1U().|Thanks arc due to Dr. D. S. Kolliari for his interest in the work.

A Note on the Energy and Wavelength Maxima in Fermi-Dirac and Bose-Einstein Distributions

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/1248/1/A%20note%20on%20the%20energy%20and%20wavelength%20maxima_Chowdhuri%2c%20A%20G.pdf

A Note on the Energy and Wavelength Maxima in Fermi-Dirac and Bose-Einstein Distributions

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