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THE EFFECTS OF LARGE FIELDS ON ISOTOPIC FERMION-BOSON MIXTURES
M. Miller
To cite this version:
M. Miller. THE EFFECTS OF LARGE FIELDS ON ISOTOPIC FERMION-BOSON MIXTURES.
Journal de Physique Colloques, 1980, 41 (C7), pp.C7-185-C7-188. �10.1051/jphyscol:1980730�. �jpa-
00220167�
THE EFFECTS OF LARGE FIELDS ON ISOTOPIC FERMION-BOSON MIXTURES
M.D. Miller
Laboratory for Low Temperature Physics, Department of Physics and Astronomy, University of Massachusetts, Amherts, Massachusetts 02003, USA.
Resume.- Nous examinons l ' e f f e t des f o r t s champs magnetiques s u r l e s proprietes thermodynamiques des melanges 3 ~ e - 4 ~ e en generalisant un modele du melange dQ Z Cohen e t van Leeuwen. Nous trouvons que l a pression osmotique d o i t c r o i t r e avec 1 ' i n t e n s i t e du champ aux basses temperatures ( e t pour des champs de l ' o r d r e de 50 T). Aux t r e s f o r t s champs, l e diagramme de phase du melange d o i t l u i - mOme changer, montrant une d6croissance de l a s t a b i l i t e maximum de le3He e t u n deplacement du point t r i c r i tique vers l e s temperatures plus elevees e t l e s concentrations d ' 3He plus f a i bl es.
.Abstract.- We examine the e f f e c t s of large magnetic f i e l d s on the thermodynamic properties of He 3
-
4He mixtures by examining a generalization of a model f o r the mixture due t o Cohen and van Leeuwen.
We find t h a t the osmotic pressure should increase with increasing f i e l d strength f o r low tempera- tures (and f i e l d s s50T). A t very large f i e l d s the mixture phase diagram i t s e l f will be changed with a decreased maximum 3He s o l u b i l i t y and the t r i - c r i t i c a l point moving t o higher temperature and lower (3He) concentration.
1. Introduction.- In recent years, important progress has been made i n understanding the oehavior of strongly interacting many-boson
systems i n t h e i r ground-state [ I ] . The theoretical tools t y p i c a l l y involve rewriting the Euler- Lagrange equations f o r coordinate space wave func- tions in terms of approximate d i f f e r e n t i a l ( o r in- tegral ) equations f o r the pair d i s t r i b u t i o n func- t i o n together
with
t r a c t a b l e , systematic methods f o r improving the r e s u l t s . The ground-state of the many-fermion system i s not nearly a s well understood.I f we crudely separate the contributions t o the f e r - mion ground-state energy ( i n the s p i r i t of Wu and Feenberg) i n t o those due t o the strong short-range correlations and those due t o s t a t i s t i c s then we may conclude from the aforementioned successful theories of boson systems t h a t t h e source of the problems with fermion systems l i e s i n not properly understanding the ( s t a t i s t i c a l ) "Fermi-correla- tions". Obviously one way t o d i r e c t l y probe these Fermi correlations i s t o study the thermodynamic properties of the system a s a function of magnetiza- tion. Such experiments f o r bulk 'He a r e very d i f f i - c u l t because of the combination of large "ambient"
energies (due t o t h e short-range c o r r e l a t i o n s ) ,
O(TF), and the small nuclear mon~ent, o ( ~ o - ' K / ~ ) , which require very l a r g e f i e l d s and low tempera- tures. In t h i s paper we shall consider the strong f i e l d e f f e c t s on an a1 t e r n a t i v e system, t h e He 3
-
4 ~ emixture, by examining a generalization of a model of the mixture f i r s t introduced by Cohen and van Leeuwen ( c v L ) [ ~ ] . In zero f i e l d , t h i s model was q u a l i t a t i v e l y successful i n describing the X-transi- t i o n , phase separation etc. i n the mixture. In Sec.
I1 we discuss the model and b r i e f l y c i t e i t s origin and limitations. In Sec. 111, the r e s u l t s and con- clusion, we discuss t h e zero-temperature properties, including the maximum He s o l u b i l i t y and spontane- 3
ous magnetic ordering. In addition, we present some preliminary r e s u l t s on the f i n i t e temperature be- havior.
2. Theory.- Cohen-van Leeuwen Model. We consider a system with NB bosons and NF fermions i n a volume,
V , a t temperature, T, and i n a uniform magnetic
f i e l d , H. Then t h e model of CVL i s defined by the Helmholtz f r e e energy
N ,",T) .=
N;
EBB(2-tZ)'
2NFNB 1 E B F F(N&"+. F-Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980730
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where
= 41/ma
+
l/mg)'/mas 2 (3
1
and = B,F. Also, i n ' ~ ~ . ( l ) , )!F is t h e Helm- holtz f r e e energy f o r an ideal Bose or Femi gas and the magnetization i s given by
M = vn(NF+
- NF-1
U, 5 NF, (4
where
r
i s the spin-order parameter, pn i s . t h e nuclear moment and NF, a r e the number of fermions a1 igned para1 l e l o r anti-para1 l e l t o t h e f i e l d . The parameter, 5, i n Eq.(l) i s t h e Bose-Einstein condensate f r a c t i o n and i s defined bywhere p = N/V is the number density, x = NF/N i s the fermion concentration, g is a Bose integral
3/ 2 and
f A2
@, = B , F,x - Z a - - ma k ~ T (6)
i s the thermal de Broglie wave length. Equation (5) i s valid f o r T < TI where T, will be i d e n t i f i e d with the superfluid t r a n s i t i o n . For T > TI we have
where z f ) i s the ideal gas a c t i v i t y . Thus, f o r T < T, we have z k ) = 1 and f o r T > TI, 5 = 0.
4nalogously, f o r t h e fermion system we have
where f i s a Fermi integral.
3/ 2
If the mixture phase-separates, the two phases will be a t the same pressure and f i e l d thus we a r e led t o consider the Gibbs f r e e energy,
where the pressure, f i e l d and chemical p o t e n t i a l s
a r e given, respective1 y , by
Thus ateachconcentration, x, and temperature, T , Eqs. (11 ) and (12) allow us t o associate with each density and magnetization a pressure and f i e l d . Then Eqs. (13) and (10]determines the Gibbs f r e e
energy which contains a l l t h e thermodynamic informa- t i o n concerning the mixture.
In t h i s system t h e r e a r e t h r e e possible con- current phase t r a n s i t i o n s : Bose-Einstein condensa- tion (BEC) i n the Boson constituent, magnetic order- ing i n the fermion constituent and phase separation.
The BEC and magnetic ordering are second order tran- s i t i o n s i n t h e Ehrenfest sense ( f o r t h i s model) and t h e phase separation i s f i r s t order.
The CVL model represented by the model f r e e energy, Eq. (I
1,
together w i t h Eqs. ( 5 ) , (7) and (8) is = t h e lowest order term i n some expansion of F i n powers of a , but i s a reasonable form which con- t a i n s enough physics t o mimic, q u a l i t a t i v e l y , real 3 ~ e-
4 ~ e while s t i l l being simple enough t o be t r a c t a b l e . For a discussion of t h i s model see Kin- caid and Cohen [3].111. Results.- J u s t as t h i s model e x h i b i t s a d i s t o r - ted pressure response r e l a t i v e t o the real mixture by having too large a compressibility [ 2 ] i t a l s o e x h i b i t s a d i s t o r t e d magnetic f i e l d response w i t h too large a s u s c e p t i b i l i t y . A t T = OK, f o r densi- t i e s greater than p i C (zpFca3 = n/24) the fermions undergo ferromagnetic ordering. Therefore t h e H = 0 phase diagram although extremely i n t e r e s t i n g i s not a representation of He 3
-
4 ~ e . The behavior of t h i s model in high f i e l d s may, however, provide some clues as t o t h e behavior of t h e real mixture.F i g . 1
A t T = OK, those mixture q u a n t i t i t e s of i n - t e r e s t include t h e maximum fermion s o l u b i l i t y and the osmotic pressure. In Fig. 1 we show the reduced maximum fermion s o l u b i l i t y
x i
xa(H)/X,(o) a t P = 5bar f o r H = 10
-
50 T. I t i s c l e a r t h a t increasing the f i e l d reduces the maximum s o l u b i l i t y . This i s simply a r e s u l t of the f i e l d driving down the He 3chemical p o t e n t i a l , i .e., a l l ~ = -(M/N) i s ( m ) p . T . x = i
manifestly negative d e f i n i t e .
he-decrease
in u F i s l a r g e s t in the region x a 1 and becomes negligible as x -t o. Thus from a Maxwell construction point of view [ i F ( l ) = p (x ) I the value of xR must decreaseF a as H i s increased.
Osmotic equilibrium i s a powerful probe of mix- t u r e properties, especially in He 3
-
. 4 ~ e mixtureswhere one has superleaks available t o play the r o l e of the semi-permeable membrane [4]. If we imagine two vessels (L and R) connected by a superleak then osmotic equilibrium i s attained whenever
In the t r a d i t i o n a l , but by no means only, experiment, we s e t TL = T and HL = HR and ask i f xL f xR then
R
what pressure difference (PL
-
PR = a , t h e osmotic pressure) i s necessary t o s a t i s f y Eq.(14). In Fig. 2 we plot t h e change in osmotic pressure as a function of concentration a t T = OK and P = 5 bar as the f i e l d i s increased from zero t o 50 T or 100 T. The change in n i s a 0 (0.1 Torr) which i s e a s i l y observable.Fig. 2
The differences show t h e i n t e r e s t i n g feature of saturating w i t h increasing x. In f i r s t order one would expect the difference i n a t o be dominated by a term l i k e S B MH (where f o r 3 ~ e
-
4 ~ e , p B = 2.26~10 3Torr/K) which i s monotonic in x since the susceptibi- l i t y a t fixed pressure
x
x1I2. I t i s possibleP
t h a t a competing e f f e c t , the change in density a s a function of f i e l d i s making i t s e l f known, however, t h i s point has not y e t been examined in d e t a i l . The differences in n a t f i n i t e temperature ( f o r T<0.25K) show the same s o r t of behavior as i n Fig. 2 with magnitudes again %O(O.l Torr)
.
[These f i n i t e tem- perature numbers await t o be confirmed in a more accurate calculation and then will be published e l s e - where.] For T > O K we find t h a t the maximum He 3s o l u b i l i t y decreases w i t h increasing f i e l d as i t did a t T = OK. The x-line i s basically unaffected by magnetic f i e l d s since the coupling between density and f i e l d i s so weak. The t r i - c r i t i c a l point, on the other hand, "moves up" the A-line in response t o the decreased He maximum s o l u b i l i t y 151. 3 A t P = 5b, as H i s increased from 0 to 100 T, xt decreases by
% 0.05 and Tt increases by %0.05 Tx. These numbers are only approximate and need t o be confirmed in a more accurate subsequent calculation.
Thus, in conclusion, we find t h a t although the
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coupling of He 3
-
4 ~ e t o external f i e l d s i s weak, present i n He 3-
4 ~ e i t would present importantt h i s model predicts i n t e r e s t i n g responses i n the challenges t o the microscopic, f i r s t - p r i n c i p l e s osmotic pressure f o r H
*
50 T, field-dependent boun- theories.d a r i e s f o r the phase-separation curve and position of This research p a r t i a l l y supported by ARO grant t h e t r i - c r i t i c a l point. I f t h i s behavior i s indeed DAAG29-78-GO163 and NSF g r a n t DMR76-14447.
References
1. See, f o r example C.E. Campbell, in Progress in 4. C. Ebner and D.O. Edwards, Physics Reports 2, Liquid Physics, edited by C.A. Croxton (John Wiley 77-154 (1970).
& Sons, New York, 1978), Chap. 6. 5. I t i s i n t e r e s t i n g t o note t h a t i n t h i s system t h e 2. J.M.J'. van Leeuwen and E.G.D. Cohen, Phys.Rev. 3 ~ e
-
rich boundary of t h e two phase region l i e s out- 176: 385-397 (1 968).-
s i d e t h e A - l i n e unlike the 5 = 0, H = 0 system3. J.M. Kincaid and E.G.D. Cohen, Phys.Reports
22,
treated in Ref. 2, where the A-linewas
the57-143 (1 975). boundary.
