Singh, B N

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8 THERMODYNAMIC FUNCTIONS FOR TWO DIMENSIONAL QUANTUM STATISTICS* By B. N. SINGH (University of Delhi) (Rcrfi1'l'd for puvlication, July 5, 19.10) In recent years the Fermi-Dirac statistics applied to the case of electron gas has found numerous applications. The case in which the electron gas is restricted to two dimensions is also of interest in certain special cases. In th~ first part of the present paper the various thermodynamical fUlIctions for a v-dimensional space for a gaseous assembly obeying Fermi-Dirac statistics or Bose-Einstein statistics have been evaluated in the usual familiar way, 1 and the values of these functions for a two dimensional space havc been deduced as a special case. Au application of the results has heel! made ill a following paper2 in deducing the Illagnetic susceptibility of a free electron gas whell the electrons are confined to a plaue. 1. We first determine the distribution laws for a v-dimensional space. The wave fUlJction for all allowed state of a free particle ill a space of' volume' L I' IS 117i"~:rl 1,1 = C sin 5111 L 12"X~ L ......... Sill 0.=-1' . IllITx lI SIll -, I. =c II o=} where The wavelength ,\ associated with a normal 11lode of vibration is given by and, therefore, where p is the momentum. * Communicated by tht' Indian Phy~icaJ Sodety 74 B. N. Singh Now, taking account of relativistic mechanics, since we have h 2 f:O =_._., .. 8111L2 where The number of states C H for which the energy is less than e is equal l' , 1 to 2 -v times the volume of a sphere of radius {2:(l,J2}~ and is given by v 'J -1' C)e)= 11'(2 r ~+I 2 v )1,'/ ";;,, +')f' 1'herefore av(e) d€, the number of states of energy lying between 6 and 6+ d.:, is given by ( rr L2 )~; I .) 2. " ~ - I = 2g h2'(~·2 J~(:) (" + III C- )(.: 'j 2111 C -.:) -, (3) g being the weight factor. In the completely non-relativistic case (lIi~i~ > > I). this reduces to v g 27TmI2 ,'- I ( )'2 v "1'(6) de =;i":) ~"- .:~ £Ie, I I 1 ·,· (mc 2 < < ) 1 and in the comp ete y re atIvlstlc case,-,;- I, we lave kl v 2g (d 2 )2 v - I a)e)de = rG ) h,l~2': df'. Equations (4) and (5) for the two limiting cases can be written as a (6)de=D€,'-lde, v (6) Thclmodynamic Functions for Tuo Dimentlional Quantum etc. 75 where, in the non-relativistic case, s=; , D= I~sl-21r1;;f t and, in the relativistic case, S=t', J)= 2~ (71"1. 2 )s/2. J~(~ ) ,il-Z ~~ (Ii) It is to be noted that for the three dimensional ('a5<.' s takes the vall1c~ 312 and 3 ill the non-relativistic and relativistic clses respectively. For the two dimensional case its values arc I in the non-relativistic case and 2 ill the relativistic casco The distribution function i("') =-- 1_, is not affected by the number I C .:/ kT + {3 A of dimensions. (3= + 1 for Fermi-Dirac statistics and f~ = - I for Rose-Einstein statistics. The numbl?r of particles, therefore, possessing energy in the range 6 to ~: + d€ is given by De"-'1 N(f:)de=, de. I ej k.T -c +/3 A (9) 2. We shall first consider the case of non-degeneracy. In this case A < < I and the distribution function is easily expanded as a series. The total number N of the particles in the assembly is given by S<X> SOO u"- I N = N(.:)de= D(kT)" " u du L + (J o 01\ (10) 1 (3A (32 A 2 f = D(kT)" Al'(s) 1-' + -- - ... . 2 ~ 3'~ (11) And, therefore, 'l'he total energy of the assembly E is given by where '16 B. N. Singh From (12) and (14), we easily obtain, noting that E=sPV. the general formulae for the non·degenerate case I<~_. = G_-:-p~= log Al -1 + afjA} -Ab{-J2 A}2+ .~C{:PA13 ......... , N kT N k T -. ,,/- (16) aud Nk + 3S--1 ./);1,\ :; () "1 ......... , 3 where C, F and S are respectively til ... TheflllouynullIic l'otcutial. Frce Encrgy and Entropy of the assembly. As a special case we writc down the values of the above fUllctions ill thc nOll· relativistic two· dimensional case. We have and l' s= -=1, 2 _ N _ lIh 2 A I - nid - g2 1TmkT-' where n is the surface density. Also a=\, b = ~ilO and [= o . ..!:~ = I + (JJ\l + 1 #2 ~.q2 + ......... . NkT 4 36 C A /3 2A 2 =log Al +$_1 + _I + ......... . NkT 2 24 F (-3 A (3 21\ 2 - - = log Al - 1 -I- ---- J + 1 + ......... , Ni/!' 4 72 and S /121\'2 ,- =2-log AI + 1 + ........ .. Nk 72 We shall now takc up the degenerate case. (18) In degeneracy the Fermi·Dirac and Bose·Einstein cases are considered separately. In Bosc-Einstein statistics A = 1. f3 = - I and equation (13). therefore, easily gives G=o. F= -~ = -J'(s)D(kT)'+ l1'(s+ 1), S " Thermodynamic Functions Jor Two Dimensional Quantum etc. 77 s= S+I·TE =(s+I)I'(s)D(kT)'k~(S+I), s c .. =(-~~ X = (S+l)~' wherc Ct. is the specific heat when the size of the SIJace (" volume ") occupied by the assembly remained fixed. In the two-dimensional non-relativistic case these reduce to E= c==o. F=-Jo:, 2E s= 'I' ' of C =.~~ (1' T' ",hew C denotes specific beat at constant area. (T The degencrate case of the Fermi-Dirac statistics is characterised by A > > I and /3= + I. The integrations are carried out by using Sommerfeld's formula according to which, for large A, where Uo = log A. Equation (10) yields N = D(kT)' uA 11 + ___ ~:r~ ____ I)S_ + 7;r4 S 6 uo2 3bo And, defining A 1 by relation (12), we have uo=log A= , 1/sl ;r2(S-I) ;r.," .. (s-I)(4s2-30s+54) {l (s+ I)Ad l--~-··-···· .... -- --~-. . . . -6{r(s+ I)A 1 }2/S 720 {l'(s+ I)A 1}4/S ... f '" -......... J ... Similarly, integrating equation (13) for large A, we have S , }l/S [ ",2(s+ I) E= - NkT{I (s+ 1)A 1 . 1+ .. -,---c.-----. -s+ I 6{I'(s+ 1)A1}2/S from which we easily obtain r; ={I'(S+I)Al}1Isj1- -"'6-=- __ ('-S_----'I). __ _ NItT {I'(s+ 1)A}}2/S ......... I' 78 B. N. Singh 11'2 {r;(~~ I'}A~}I7s' , Nb2 C = ..:~.-.-.----.---' . 3 {r(s+I)A1}I/S In the two dimensional case s= I and the values obtained for the various thermodynamic functions are entered in Table I, where for the sake of comparison results for the three dimensional case are also shown. My thanks are due to Dr. D. S. Kothari for his help and interest in this work. TARI.E 1 Two dimensional 'l'hree dimensional -----r---··--·------.--.-... ------. - .'-._-(31T~)~ j~ log A \ (AI + 1)* 1T~ 2~ ... \ I - AI 1 3A Al~ I 4 • ( I } (31T!)~AI~ j 1+ 4-.2 ... ~ E ~ j I + --~-:y ... ~ 511'!'I 23 NkT 7. A 4-2 3A, 33 J:J I (311'~ Y ~~ 11'~ 2~ f G i A1 +l i _ .. AJ 1-NkT I 4 ••• j 4 3* A1lJ .A21_jl - 3~2J2'" f ( ) 'j ~'f F I ~ 311'1. . a _ .511" 23 NkT I Al I 1- ~ ••• 5 4 33 A J • S 1 11'2 CIT 1T2mkT (.4!TR )1 = ~.v_ Nk ... --= 3 AI Nk 1,2 3n Nk * In the case of two dimensions the equation can be int~gratcd exactly and log A comes out equal to A1+x, It may be noted that the series in Sommerfeld's integration formula for the degenerate case: are conect to an order of x/A. , . REFERENCES t Fowler & Guggenheim. II Statistical Thermodynamics," p. 239. I II Magnetic Susceptibility of two dimensional electron gas." 

Thermodynamic Functions for Two Dimensional Quantum Statistics

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Thermodynamic Functions for Two Dimensional Quantum Statistics

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