Ind ian Journal of Pun: & :\pplicd Physics Vol. 41 . . \ pril2003, pp. 280-289

Statistical mechanics and thermodynamics for a mixture of bosons and fermions

K M Khan na 1 & Ayodo Y Kenncdl

Department of Physics, Moi Universit y, Box 11 25 Eldoret, Kenya

c-mai ls: I. profk mkhanna @hotmai l.com 2. yudah2000 @yahoo.com

Received 30 July 2002; accepted 8 November 2002

A stati stical distribution model is developed for a mi xture of bosons and fcrmions with different concentrat ions. The bosons and ferm ions arc assumed to be in teracting via a pair interaction. The most probable distribution in energy and the partition fun ction are derived . Thermodynami c qu ant ities such as entropy, internal energy and specilic heat ure obtained in terms of the partit ion functi on. The values of the internal energy, specific heat and en tropy increase as the concentrati on of hel ium-] in the mix ture decreases.

1 Keywords: Bosons . Fennions , Parti tion function, Di stribution, Entropy, Internal energy, Spccilic heat, I k lium-3 J

1 Introduction

Stati sti ca l mechanics as an apparatus is used to explain the mac roscopic properties o f a system, knowing the properties of the microscopic consti tuents of the sys tem. A ll the characte ri stics o f the ir constituents and the ir inte racti ons are assumed to be known. Thermodynami c qu antities, such as inte rna l energy, spec ific heat and entropy can be calculated from knowledge o f the stati sti ca l di stributi on and the partiti on functi on. So fa r, most of the studies deal e ither wi th a system of bosons or w ith a system of fe rmi ons . In nature, the re do ex ist sys te ms whic h are mix tures of bosons and fe rmi ons

such as : H , ~ H and ~ H, and the most interesting

mi xture is ~ He and ~ H e. The questi on to be

answered by the research workers is, as to how stati stica l mechanics could be used to dea l with a sys tem in whic h both bosons and fe rmi ons are present w ith diffe rent concentrati ons. S imultaneous ly, it should be c learl y understood that, in the mi xture, bosons obey Bose-Einste in stati sti cs and fermi ons obey Fermi-Dirac stati sti cs . What di stributi on law o r w hat will be the express ion for the most probable di stributi on in energy is the subject matte r of stuJy in the present pape r.

Hi storica lly, the first attempt to genera li se quantum Bose and Fermi stati sti cs for a mi xture o f boso ns and fermi ons was made by Gentile 1• He proposed stati stics in whi ch, up to N partic les

(N>> I) were a llowed to occupy a s ingle quantu m state, instead o f just one parti c le for Fermi case, due to the Pauli exclus ion princ iple, and inde finite ly many, for the Bose case. However, Gentiles approach was found to be too much of a genera li sati on and conta ined the vio lati o n of the conventiona ll y-accepted Pauli pr inc iple. Furthe rmore, hi s mode l did not eli ·tingui sh whic h parti c les were fe rmi ons and whi ch ones were bosons . Even the rati o of the two types of parti c les in the mi xture was not known. However, Gentil e's work la id the emphas is and the foundati on that, the stati sti ca l mechanics of a mi xture of bosons and fe rmions can be worked out.

The nex t serious attempt in thi s di rection was that of Mecl veclev2

• In hi s paper entitl cl , ' properties o f parti c les obey ing ambiguous stati stics', M ed veclev proposed a new c las <> of identi ca l partic les, whic h may ex hibit both Bose and Fermi

stati sti cs w ith respec tive probabili t ies, P,,' and P/. The mode l admits onl y primary Bose-E inste in and Fermi -Dirac stati sti cs, as ex ist ing. He assumed that, a partic le is ne ithe r a pure boson nor a pure fe rmi on. He let another parti c le, which inte rac ts wi th the first one, play the ro le o f an ex te rna l observer. During the inte rac ti on, it performs a measure ment at the first pa rtic le and identifies it as ei th r a boson or a

fe rmi on with respective probabiliti es, P~>' and P/ . According to the result o f thi s measurement , it inte racts with the first parti c le, as if the last is a

KHANNA & KENNEDY : BOSONS AND FERMIONS 28 1

fermion or a boson, respective ly . The first particle is the observer for the second particle and so the

process is symmetric. Note that, (P"'+P() is not necessarily equal to one, and , if not , it means that the second particle (observer) does not detect the

first partic le. The probability of thi s is ( 1-P"'-Pr'). The stati sti ca l uncertainty introduced here may be e ither the intrins ic property of a partic le itself or the experimental uncertainty of the measurement process.

Another attempt in thi s direction is the so-called 'stati stical independence mode l' of Landau & Lifshitz' in which two weakly-interacting subsystems (bosons and fermions) are together regarded as one compos ite system (a mixture), and the subsyste ms are assumed to be quas i-c losed . The stati stical di stribution or count for such a mixture is the product of the indi vidual probabilities for two subsystems, one corresponding to bosons and the other corresponding to fermions. With these assumptions, the stati stical independence mode l will hold only for an ideal gas assembly of bosons and fermions . In reality , such an assembly does not exist and hence the stati stical independence model cannot be used for real mi xtures of bosons and fermions , or for ' He and 4He mixtures .

Chan4 et al. studied the 'effect of di sorder on super-fluid ·'He-4He mixtures' and the ' thermodynamics' of 3He-4He mixtures in aerogel'. But, the studi es re lated to more of an ideal system rather than a real syste m.

Therefore, the properties of a mixture of bosons and fermions, assuming, there ex ists a pair interaction between the bosons and fermions; and that the concentrations of the bosons and fermions are different from each other have been studied. Since the concentrat ions of the bosons and fermions will not be the same, and they are considering only pair interacti on, in a g iven state of equilibrium, some bosons or fermions will be left un-paired depending upon whether boson concentration is more or fermion concentration is more. Whatever be the relative concentration of bosons and fermi ons, the value of the occupation number of fermions in a given state will not exceed, will be rather much less, than the degeneracy of that state so that, Pauli exclus ion princ iple is not violated. With these bas ic assumptions , the authors have derived expressi ons for the followin g:

Statistical count for an assembly that is a mixture of the bosons and fermions , assuming a pair interaction between the bosons and fermions ; the most probable di stribution in energy for a mixture of the bosons and the fermions in the assembly; the partition function for such an assembly; the internal energy , the specific heat and the entropy .

The above express ions are used to study the thermodynamic properti es of a mixture of liquid helium-3 (fermions) and liquid he lium-4 (bosons) with different concentrations. The theoreti cal resu lts obtained are compared with the ex perimental observations on the properties of a mixture of liquid he lium-3 and liquid helium-4 .

2 Theoretical Derivations

To study the properties of a mixture of bosons and fermion s with different concentrati ons, consider an assembly of N partic les in which there are N" bosons and N, fermions such that:

. .. ( I )

Let £,, £2, £, ... Ei ... be the energy states of the assembly, and in the stati stical equilibrium the number of particles ass igned to these energy levels be n, , n2, n3 • . • ni ... , respectively , such that, the numbers ni must sati sfy the conditi ons requiring the conservation of partic les, N, and conservat ion of energy,£, i. e .,

~

I,n . =N j =l 1

and

~

... (2)

In.i£ . =E ... (3) j = l 1

such that:

n J = n;" + n .if . . . ( 4)

where

n Jh =number of bosons in the energy level j . . . (5)

and

nif =number of fermions in the energy leve l j ... (6)

Let l4 be the number of states in the j-th level, i.e, l4 is the degeneracy of the j-th leve l. The n, the number of ways Pjl, in which nj, bosons can be ass igned to 01 states in the/' level is g iven by:

282 INDIAN 1 PURE & APPL PHYS, VOL 41 , APRIL 2003

. . . (7)

Similarly, the number of ways, P11, in which nif

fermions can be ass igned to the £0 states is given by:

wj ! P .. = -----"--

.ll (W ; -nil) ! .. . (8)

To sati sfy Pauli exc lusion principle, it is necessary to assume that, £0 >> nJJ··

Once the particles are placed in the £0 sub-levels in the j -th level, it shall be further assumed that, bosons and fermions may interact in pairs. However, not all the bosons may form pairs with fermions, since the bosons and the fermions will not be in equal proportions in the mixture . Thi s statement implies that, nj,> njJ. But, the derivations will be independent of the fact whether njl, > njr or nif

> nj/•· Fig. 1 is a pictorial illustration of the di stributi on of the two types of particles.

--ce---<_1)-----

0 Boson I Fermion C.Boson-Fenni·m pair

Fig. I - Distribut ion of two types of particles

In general, njl;>nif, and it is the number of fe rmions nif that will determine the number of boson-fermion pairs. Hence, the number of boson­fermion pairs will be njJ and the number of un­paired bosons will be (nj1,-njJ) . Since the permutations among the particles and the permutations among the pairs in the same energy level do not give a new complexion in the statistical distribution model proposed here, the following permutations must be excluded from the number of ways in which n1, bosons, njJ fermions and njJ

boson-fermion pairs are distributed in the /" level:

(i) Permutations among identical pairs should be excluded by di viding by (njJ) !

(ii ) Permutations among identica l un-paired bosons should be excluded by di viding by (njl,-nj11) !

Hence, the total number of ways, P;,1 in which nil, bosons, nif fermions and nif pair of bosons and fermions can be di stributed among the £0 sub-levels in the/' level is given by:

(w )"1" (w .)! p - J J

jh/ - ( ) I ( ) I ( ) I nil . w i -n iJ . n ih - n .if . ... (9)

The stati stical count, Chi, for such a di stributi on among all the levels (i= I ,2,3 ... ) avai I able to the assembly is the product of such express ions as given in Eq . (9) since every arrangement in a given energy level can be considered independently of the other energy ·levels. Thus we can write:

c "f = I1 Pjbf - J J ~ IJ~ { (w )"i" (w .) ! }

i = l i=l (n 11 )!(w ; - n 11 )!(n ;1, -n;1 )!

... ( I 0)

Eq. (I 0) wi II now be used to calculate the most probable distribution in energy for a mi xture of bosons and fermions in which bosons and fermi ons interact in pairs.

3 Most Probable Distribution in Energy for a Mixture of Bosons and Fermions

The objective is to calcul ate, for what values of njl, and njJ·· the stati stical count C hr is the max imum, under the conditions of N and E bei ng fi xed. The distribution numbers and the corresponding energies must satisfy the following re lations:

~

I.ni" = N, j=l

~

L fl;bEj = £, j = l .

~

I,n ~E = E i = l 1 J J

... ( I I )

. . . ( 12)

... ( 13)

. .. ( 14)

where Nh and Nr are the total number of bosons and fermions in the mixture such that, the total number of particles N is given by:

KHANNA & KENNEDY: BOSONS AND FERMIONS 283

. .. ( I 5)

Similarly, Eh and Er are the total internal energies of bosons and fermions such that, the total energy of the mixture E is given by:

... ( 16)

Now, to find the values of ni, and nif for which chr is maximum, the procedure is to allow the variation of Chr with respect to np, and nif, and put the result equal to zero.

For Chr to be maxi mum, it follows that :

a ln(C,1 ) =

I {-a-ln(C,1 )dn;,} + .I{-a-ln(C,1 )dn 11 } = 0 1=1 an 1, J= l an JJ

... ( 17)

The variations dnjh and dn il are not independent since the n ih, s and nil's must continue to satisfy the restri cti ons given in Eqs 11-14. Since N and E are fixed, the variations in nih and nir must satisfy the following equations:

~ ~

"Ldn1, + "Ldn11 = 0 1=1 1=1

. . . ( 18)

and

~ ~

'IE dn 1 +'IE dn if = 0 j= l 1 1) j = l .I 1

. .. ( 19)

Hence, along with Eq. ( 17), Eqs ( 18) and ( 19) must also be sati sfied. These equations can be combined by the method of Lagrange' s un­determined multipliers which are denoted by a and ~ - Thus, multiplying the first and the second terms in Eq. ( 18) by (-a") and (-a,), respectively, and Eq. ( 19) by(-~) and adding to Eq. (17), we get :

i{-a-ln(C"1 ) -(a,+ /3E)}dn .i, + J=l an1"

.. . (20)

Now, Eq. 20 demands that, all the terms should be separately equal to zero, and the terms for which dni" and dn ir are not equal to zero, then the coefficients of dn," and dnir should be, respectively, equal to zero. If we assume that one of the dnjh's and

dn/ s is non-zero , then the corresponding coefficients will be zero. Thus we can write:

a -ln(C,1 )-(a" + /3E) =O ani,

... (2 1)

a -ln(C,1 )-(a1 + /3E.i) =0 anjj

... (22)

Eqs (21) and (22) are true for all values of j. Substituting for C"r from Eq. ( I 0) in Eq. (2 1) gives:

[ w i ]= exp(a "+ /3E;) ... (23)

(nih - nil)

Similarly, substituting for C"r from Eq. ( 10) 111

Eq. (22) gives:

[(OJ - n 1 )(n , - n JJ ) l 1 1

.J = exp(a + /3E ) ,, 1 } • jj"

... (24)

Eqs (23) and (24) can be solved for ni" and n ,r to get:

( J-L, - E i l n ., =OJ . exp

1 .J kT

2[ r- f-L1 - /1,, + 2Ej ]l]-l + OJ OJ . + ex p _ _!__....::..._ _ ___.!._

1 1 kT ... (25)

. .. (26)

where a = _.!!:E__ a . = _!!.1_ and f3 1

" kT ' .t kT = kT . .. (27)

Eq. (25) gives the most probable di stributi on in energy for bosons in the mixture, and Eq . (26) gives the most probable distribution in energy for fermjons in the mixture. In Eq . (27) J..lh is the chemical potential for bosons and J..lr is the chemical potential for fermion s. It should be noted that, nir is contained in the expression for nih·

4 Partition Function for a Mixture of Bosons and Fermions

The general express ion for the partition function Q for a mixture of bosons and fermions can be written as:

284 INDIAN J PURE & APPL PHYS, VOL 41, APRIL 2003

Q N [-Jlr -p,] = exp ---'-·--

kT

[ ~ ( )1l [ - J1 r - Jl" ]

= ~~~ n;, + nif Jexp kT . . . (28)

Substituting for nih from Eq. (25) and n i1 from Eq. (26) in Eq. (28), we get after simplifications:

+W exp 1 i (

- f.l -E l I kT

.. . (29)

Eq. (29) is the expression for the partrtron function for a mixture of bosons and fermions in which pair interaction between the two types of particles is considered. It must be understood that, the partition function Q has two significant terms .

The first term of the summation has the sum (/-lh+/-l1) of two chemical potential s for bosons and fe rmions, and a factor containing 2-times Ei. These are as a result of the pair interaction between the bosons and fermions. The second term in Q contains only 1-lr and this term emphasizes the conservation of the number of fermions in the mixture . Thi s actually shows that, the di stribution of fermions is not affected by the distribution of bosons, because in whichever way the bosons are distributed, there should be only one fermion in each energy sub-level.

Thus, in this mixture, even though the distribution of fermions is not affected by how the bosons are di stributed or arranged or how many they are, the distribution of bosons on the other hand is directly influenced by how many fermions are there in the mixture for the following two reasons:

The number of fermions is a determining fac tor of the number of boson-fermion pairs that shall be formed and the number of fermions sha ll also determine how many free or unattached bosons shall remain to be distributed among the available energy states.

5 Thermodynamic Relations for a Mixture of Bosons and Fermions

As a model calculation, the authors have derived express ions for the entropy S, the internal energy E and the specific heat Cat constant volume in terms

of the temperature T and the fermion

Nf concentration7J = - , 111 addit ion to other

N parameters in vo I ved .

The general expression for the en tropy is :

S = kNf [~~l Q7J ]+T ~lnQ] 1J l N r ()T

.. . (30)

Substituting for N1and Q, we get the expression for S for the mixture of bosons and fermions as:

k ~ S=-I

1] j = l

[ { ( l ( ~- 1 a ? f.1 . + p, 2£ .

T-ln2wi- W;exp .I + xp--' ] CJT kT kT

(-f.l -E Jll + W; exp ~T i ~

{ 2[ (- f.1 f - f.l , + 2£ j l~ - l1 1 1 kT

w . w . + exp J

. . . (3 1)

The internal energy E is g iven as :

... (32)

Subst ituting for Nand Q we get:

E=kT I - In 2 ~ { a i =l ()T

KHANNA & KENNEDY: BOSONS AND FERMIONS 285

{ 2[ (-Jl j - Jl" + 2t:j J~ - l 2w w . + exp + I I kT

( {L , - E j ]~ w

1. exp · kT

... (33)

On including fe rmi on concentrati on 11 in Eq . (33), we get:

kT 2 ~ E=- 2::

'1J joo l

... (34)

The expression for the specific heat Cv at constant volume is g iven as:

Cv = 2 NkT( a ln(Q)) + NkT 2(a2

In \Q) l ... (35) aT , aT-

Substituting for N and Q and express ing the result in te rms of fe rmi on concentrati on 11 , we get:

kT = C=- 2::

1J i= l

[ { )

- 1

T ~In 2w 2 w exp( Jl r + Jl , ]+ exp( ?!J_ ] aT 2 1 1 kT kT

... (36)

The thermodynamic quanti ties whose expressions are derived in thi s section are the ones that will be calculated and the theoretical results obtained will be compared with the known experimental results.

6 Results and Discussion

The molar thermodynamic quantities are the ones of interes t. The experiments done by Chan et al.45

, mainly focu sed on the molar quantities of liquid he lium-3 and liquid he lium-4. Wilks & Bett6

g ive the mol ar vo lume o f liquid he lium-3 as 40.0 em\ and its mo lar dens ity as 0 .07gc m·' ; and the molar volume of liquid he lium -4 as 28.0cm' and its molar density as 0.14 gem·'. The refo re, thi s means that, the molar mass of liquid he lium-3 is 2 .80g and that of liquid helium-4 is 3.92g. Although the chemical potential should have temperature dependence, at low temperatures, it assumes a nearly constant value g iven by the expression6

:

... (36)

where m is the molar mass, V the molar volume and N is the number of partic les in one mo le, and this is Avogadro number = 6 . 025x l01 .~ parti c les mol-1

•

Substitution of the empirical data into Eq. (36) g ives the chemica l potentialll1, for fe rmi ons as :

p1 = 3.184 x 10-27 e V . . . (37)

and for bosons as:

.. _(38 )

Since the transiti on temperature'' o f liqui d he lium-3 is far much lo we r than the transiti on temperature of liquid he lium-4, it is inte res ting to

286 INDIAN J PURE & APPL PHYS, VOL 41, APRIL 2003

know what happens in the vicinity of the transition temperature of liquid helium-4. Furthermore, the temperature ranges used in experiments45 7 on such mixtures are so much higher than the transition temperature of liquid he lium-3, which is lower than 2.5 mK. However, given that, both liquid he lium-3 and liquid he lium -4 are at the same temperature, they are considered to be in thermal equi librium.

:'I 20 0 ~_ .......... +--+-+--+". c• /.,... 0 / . ,..; • ~ 15 0 I

CJ I " "' I t:: u I 0 0

0 / .... () +' .... () +' J t:: ...

til 50 2 0.. /I t:: .--.--0 ·-.... +' .... +' ... til 2.0 2 .I 2.2 2.3 2.4 ~

Temp erature, T (K)

Fig. 2- Variation of partition function with temperature

6.1 Calculation of partition function Q

Using the essential parameters for liquid helium-3 and liquid he lium-4 given in Table I, the partition function Q was calculated for the mixture in the temperature range 2.00-2.40 K, in steps of 0.02 K. Table 2 gives the values of Q for the mixture at different temperatures. Fig. 2 depicts the variation of Q with temperature T Fig. 2 shows that, there is an exponential ri se in the value of the partition function Q from temperatures of 2.00 K up to about 2. 18 K. Above 2.18 K and as temperature approaches 2.40 K, Q assumes a nearly constant value of about 2.00x I 024 particles . The reason for this kind of behaviour is that , at very low temperatures 1, there are very few energy states~ that can be occupied by the particles in the assembly. But, at higher temperatures, which can be taken as above the critical temperature, the number of energy levels available for partic le occupation cou ld be large, and these could be called the excited states of the assembly of particles . Leaving the zero­momentum-state (ZMS), the occupation number of the states just above the ZMS will not be large.

'l'~blt: 1 - ~ ~~ s l· ntial Parameters for lil1uiJ hcl ium-3 and hclium-4

Parameter lillllid I il l lid hclium-3 hcl ium-4

volume (em ') 40.00 23.00 Jc:n siry (gem· 1) 007 0 14 mass (g) 2.80 3 92 chem ical potcntia! (cV) 3.184X1 0-17 6.215x 1 u-~"

6.2 Calculation of Internal Energy

Eq. (34) is used to calcu late, how the internal energy E varies with temperature and he lium-3 concentration, by using the parameters listed in Table I . The helium-3 concentration . TJ , values that are used in the calculation are, 0.35, 0.50, 0.76, 0.84 and 0.96. Table 2 gives the value of E in the temperature range 2.00-2.40 K in steps of 0.02 K. To convert the value of E from electron volt (eV) to Joules (J), E values are multiplied by a factor of 1.6x JO·'Y. Table 3 also contains the va lues of E for different TJ values. The graph of internal energy variation against temperature for different values of TJ is given in Fig. 3.

1200 . s 1000 / '\ ~ I . ;>: - •-E l OJ) 800

/;¥''\ \ ... --+- E2 ., ~ 600 -o-E3

03 lh ,, \\ ·-•-E4 E 400 ~"'"'~ \\ --E5 2 ..s , \\ 200 , .. ,

~~·~ 0

2.0 2.1 2.2 2.3 2.4

Temperature, T (K)

Fig. 3-Vari ation of internal energy with temperature for given heliurn-3 concentrations

The graph depicting the variation of E against 7 shows that, as the temperature is lowered, starting from 7=2.40 K, the value of E increases and reaches a maximum value at 7=2. 14 K, a temperature close to the transition temperature of liquid he lium-4 (7,=2.166 K for liquid 4He). The value of E decreases below 7=2.14 K and becomes qu ite small for 7::::2.00 K, and this is consistent with the nature

KHANNA & KENNEDY: BOSONS AND FERMIONS 2R7

of the super-fluid state. The value of E increases as ll decreases, and its value is the maximum around the transition temperature. However, the internal energy decreases gradually, below the transition temperature, because the particles have to lower their energy due to the existence of pair interaction .

Table 2- Valu e:< of the partition function again:< t temperature

T (K) Q (1)X1 022 particb

2.00 4.05

202 7. 16

2.04 12.40

2.06 20.80

2.08 33.50

2.10 51.40

2.12 74.10

2.14 99.50

2.16 125 00

2.18 146.00

2.20 164.00

2.22 176.00

2.24 184.00

2.26 190.00

2.28 193.00

2.30 196.00

2.32 197.00

2.34 198.00

2.36 199.00

2.38 199.00

2.40 200.00

6.3 Calculation of Specific Heat c.

Eq. (36) is used to calculate, how the specific heat at constant volume for a mixture of liquid 'He and liquid 4He varies with temperature T and liquid 3He concentration TJ . Since the molar quantities of the constituents of the mixture are the ones that were of interest, the values of the specific heat were multiplied by a factor of 1.6x iO· '~ and divided by 297 .619 to convert the units of specific heat from eVkg·'K' to Jmoi ·'K'. This is because, from Table I, 6.72 g is the mass of 2 moles of the mixture and thus I kg of the mi xture is roughly equal to 297 .619 moles. Table 4 gives the values of C at different temperatures and for different values of TJ , and Fig. 4 shows the variation of C with t and for different values of TJ .

60

Q' 50 i '\ l 40 /;\\

-•- C1

--c2

C- -o- C3 u 30

~ ·- \\ - "-C4 .... ~ y·-'l, \\ --C5 til

20 u ~ \ . ~

~·" 1.::: 10 ·;; J ~·, u b.o..o c.. 0 '

{/)

2.0 2.1 2.2 2. 3 2.4

Temperature, T (K)

Fig. 4- Variat ion of specili c heat capacity at constant volume with temperature for given he lium-3 concentrati ons

Tabk 3- Values of internal energy at Jiffcrent temperatures for given helium - 3 concentrations

Internal cnergv (J)

T (K) 11 1=11.3> 11 2=11. '"1 11 1=11.7(, 11 ~ = 1J.Xf ll .;=u.w.

2.00 87.680 61 .370 40.380 36.'i30 31.970

202 152.500 106.800 70.230 61.540 55.600

2.04 256.300 179.400 II ROO 106.800 93.450

2.06 410.700 287.500 189.200 171 .100 149.800

2.08 615.700 431.000 283 .500 256.500 224.500

210 843.000 590.100 388.200 351.300 307.300

212 1030000 720.800 474.200 429.100 375400

2.14 11 04.000 772800 509.400 460.000 402.500

216 1037.000 726.000 477600 432. 200 378.100

2.18 866.300 606.400 399 000 361.000 315.800

2.20 659.500 461.600 301.700 274.800 240.400

2.22 469 .900 328.900 216.400 195.ROO 171.300

2.24 320.700 224.500 147.700 133600 11 6 900

2.26 213.400 149.400 98.370 88.910 77800

2.28 140.000 98.020 64.490 58.350 51 050

230 91 .350 63.950 42.070 38.060 33.300

2.32 59.530 41.670 27.420 24 .810 21.710

234 38.880 27.220 17.910 16.200 14.180

2.36 25.490 17.840 11.740 10.620 9.294

2.38 16.800 11.760 7.735 6 998 6.124

2.40 11.130 7.791 5.125 4.637 4058

The shape of the spec ific heat cu rve is more or less the same as that of the interna l energy curve, but the peaks in the specific heat values occur at a temperatme of about 2.167 K, which is exactly the

28~ INDIAN J PURE & APPL PHYS , VOL 4 1, APRIL 2003

A.-t ransiti on te mperature for liquid ~He. Like internal energy, the value of C increases, as we lower the temperature, starting from 2.40 K, reaches a maximum at T=2. 167 K, and then dec reases gradually , as the te mperature is lowered to arou nd 2.00 K. If, however, the mi xture is heated, starting from T =2 .00 K, the value of C first inc reases, reaching a maximum at T=2. 167 K, and then decreases as the temperature is increased to around 2.40 K.

Table 4- Values of specific heat at Jiffcn:nt temperatures anJ hdium -3 concentrations

Tempe-~pecific heat C (!mol-l 1(-1)

r:tlu re 111 == 0.35 11 2 == 0.50 fl I:= 0.76 11-1 := 0.84 11 ; == 0.%

(K)

2.00 0.'1765 0.1236 0.08 13 0.07355 0.06436

202 0.5320 0.372--1 0.2450 0.22170 0.194000

2.04 1.5150 1.0600 0.6975 0.63110 0.5:i220

2.06 3.'1920 2.7950 1.8390 1.66300 145600

2.08 9.--1700 6.6290 4.36 10 3.94600 3.45300

2. 10 19.521l0 13.6600 8 9880 8.13200 7. 11500

2.12 33.7200 23.6000 15.5300 14.05000 12.29000

2. 1-1 --17.6800 33.3700 21.9600 19.87000 17. 38000

2.16 55.0400 38.5300 25.3500 22.93000 20.07000

2. 18 53.0300 37. 1200 24.4200 22.10000 19.33000

220 --l--1251l0 30.9800 20.3800 18.--1--1000 16.13000

2.22 33.3100 23.3 100 15 .3400 13.88000 12.14000

2.24 23.4000 16.3800 10.7800 9.75100 8.53200

2.26 15.7600 ll.o300 7. 2570 6.56600 5.74500

2.28 '10 3500 7.2480 4.7690 4.31400 3 77500

2.30 6.7180 4 7030 3.0940 2.79900 2.44900

2.32 -1.3360 3.0350 1.9970 1.80700 1.58100

2.34 2.7970 1.9580 1.2880 1.16500 1.02000

2.36 1.8080 1.2660 0.8328 0.75350 0.65930

2.38 1.17-10 0.8218 0.5406 0.48910 0.42800

2.-10 0.7660 0.5362 113527 0.319 10 0.27930

The va lue of C is hi gher for lower values of 11 or C increases as 11 decreases and these results are consis tent with ex pe rimental observations45

.

However, experimental observations by Chan et aU showed shifts in the transition temperature at which peaks in the va lue of the specific heat occurred. Thi s can be accounted for, due to the fact that, experimenta ll y, a hi ghly porous materi al ca lled aeroge l was used to contro l the flow of li quid ' He into liquid ~He and changes in the thermodynamic quantities of the mi xtu re were observed for di fferent

liquid 'He concentrations. Whereas our theoretical model assumes a bulk mix ture, meaning without aerogel, of the two liquids. Furthermore, calcul ations of the authors do not inc lude the flow properties of the two liquids, for i stance, super­fluidit y in liquid ~He is supposed to disappear above a certain c ritical ve loci ty. The normal super-fluid phase transiti on in pure liquid ~He is a second-order phase trans iti on, whereas the phase changes in the mi xture of liquid ' He into liquid ~He . This may be due to the fact that, in the case of the authors, the atoms of the two liquid s have not been cons ide red to be entire ly independent but , exchange energy through pair interaction.

T ab le 5- Values of entropy at Jifferent temperatures for g-i ven hdium-3 concentrations

l·:n rropy (! / K)

T (K) 111 ==0.35 11 2==0.5 11 ,== 0.74 11-, == 1).86 11;==094

2.00 43.570 30.690 20.330 18431) 16.160

2.02 75.020 52850 35.030 31.7.)1) 27.840

204 124.800 87 960 58.310 52.851) 46.360

2.06 198.000 139.600 92.570 83.920 73.620

2.08 293 800 207.200 137 500 1247 ()() 109.400

2.10 398.000 281.000 186.700 169.301) '148_(,()()

2.12 --180.800 3--10.000 226.--1 00 205.-100 180.--100

2.14 509.200 361.100 241.200 219.001) 192 500

2.16 471.900 336.100 225 600 205.' ()() 180.600

2.18 387.700 278.200 188.300 171.'>00 151.--100

2.20 288.900 209.800 143.900 131.500 116.500

222 200.000 148.200 103.800 95.29l• 84.980

2.24 130.900 100.200 72.570 67.10(1 60.380

2.26 81.780 66.090 50.320 47.0()1• -12.850

2.28 48.540 42.990 35.250 33.400 30.'180

2.30 26.680 27.800 25.340 24.--1.0 23. 170

2.32 12.530 17.960 18.920 18.650 18.110

2.34 3.41 7 11 .630 1--1.790 14.9:;0 14.860

2.36 -2.--138 7. 561 12.130 12.520 12.770

2.38 -6.208 4.'140 10420 10.'180 II --121J

2.40 -8.6~6 3.246 9.317 9.982 10.550

6.4 Calculation of entropy S

Eq . (3 1) is used to ca lcul ate the variation of S

with T for diffe rent values of 11 - The va lues are given in Table 5 and the graphs show ing the

variation of S with T and for different values of 11 are plotted in Fig. 5. The graphs have the same shape, as the ones for the va ri a ti on of E and C with

KHANNA & KENN EDY: BOSONS AND FERMIONS 289

T. The Tables show maximum values for E,C and S at T =2 . 16 K, whereas the graph for S shows peak values at T =2. 14 K, the same temperature as for E. Graphically , the peak values of C occur at 7=2.167 K. The value of entropy increases as the

concentrati on 11 decreases, the same trend as for E and C.

600

500

g 400

1-. 300 . .....,

- •- S1

-t-S2

-a-S3 r.n ;::... 200

- ><- S4 c.. 0 - - S5 .p 100 :::: ~

0

·100 2.0 2.1 2.2 2.3 2.4

Temperature, T (K)

Fig. 5- Variat ion of entropy wi th temperature for given he1ium-3 concentrations

In the temperature ranges considered here, liquid 3He cannot become super-fluid , since its transition temperature to super-fluid state is lower than 2.5 mK, whereas the transition temperature for liquid ~He to the super-fluid state is 2. 166 K, which lies in the temperature range considered by the authors . The value of entropy, like other thermodynamic quantities, increases as the value of

11 decreases .

7 Conclusions

Different authorsu5 studied the stati sti ca l thermodynamics for a mi xture of bosons and fermions by putting forth, different mode ls. In these models , partic les were cons idered to be independent or weakly interac ting. In the mode l of the authors , the bosons and fermi ons are supposed to be interac ting via a pair interacti on, and the whole assembly is supposed to be in thermal equilibrium. From the calculations presented in Tables 1-4; and the graphs presented tn Figs 1-4, it can be concluded as follows:

The transition temperature of the mi xture is around 2.16 K, a lthough the graphs for E and S seem to show that, the transiti oP 0ccurs at T =2 . 14 K. The values of E, S and C" are maximum at the transition temperature; the magnitudes of the thermodynamic quantities increase as the value ofT) decreases; the phase transition is smeared.

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