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Generalized Thermoelectric Thomson RelationsL.I.Anatychuk, 0.J .Lus te

Institute of ThermoelectricityGeneral Post-Office, Box 86, Chernivtsi, 58002, Ukrainee-maiI:anatychuk@ite.cv.ua, fax: (380-3722)-41917, tel.: (380-3722)-41917

Abs t rac tThermoelectric Thomson relations interrelating theSeebeck, Peltier and Thomson coefficients were generalizedfor anisotropic thermoelectric media in magnetic field withregard for possible Umkehr effect. From the restrictionsimposed by space and time symmetry on the properties ofanisotropic media in m agnetic field it was shown that the firstThomson relation relating the generalized Peltier coefficientn to the generalized Seebeck coefficient a. is of the form

differing from the Thomson formula for isotropic media. Thesecond Thomson relation was also generalized for anisotropicthermoelectric materials i n magnetic field. It was shown that

as pi_ = - d i v a n T +-at TFrom ( 5 ) it can be seen that in this case the specific changein entropy is determined only by two nonthermoelectricmediu m parameters, i.e. thermal conductivity and electricresistivity. Theref ore, if we arbitrarily conside r onlythermoelectric phenomena, being abstracted from thenonequilibrium processes of electric conductivity and thermalconductivity they are accompanied by, then the second law ofthermodynamics for solely thermoelectric components ofprocesses under study can be used as a strict equality

ds = -divq/T, valid for equilibrium processes, but not theodd with respect to magnetic field Thomson effect can inequality &, -d iv q/ T ofnonequilibrium thermodynamics.originate in thermoelement material even if thermoelectriccoeff icient of the material is temperatu re independent. Thus, a deeper study of the Thomson relations based onthermodynamics o f irreversible processe s did not result in theIntroduct ion

The Thomson relations interrelate fundamentally theSeebeck, Peltier and Thomson effects. These relations werefirst established by Thomson for isotropic media in his bookOn Dynamic Theory of Heat [ l ] based on the use ofclassical thermodynamics o f equilibriu m reversible processesin the formn = l Z T , (1 )

aaaTr = T - - ,

ll , a , are the Peltier, Seeb eck and Thomso n coefficients.As is known, the use of classical thermodynamics ofreversible equilibrium processes is not quite correct forthermoelectric phenomena, so it would be natural to study theThomson relations on the basis of thermodynamics ofirreversible processes after its fundamentals were developedby Onsager 121. This work was performed by Domenicali 131and DeGroot [4]. Nevertheless, the use of theory ofirreversible processes did not change the expressions ( I ) , (2 )for isotropic media. The reason for this is that thermoelectricSeebeck, Peltier and Thomson effects, though alwaysaccompanied by the nonequilibrium electric conductivity andthermal conductivity processes, in the isotropic medium understationary conditions these phenomena in themselves can beconsidered as quasi-e quilibrium and quasi-reversible inthe sense that their contribution to local change in the entropyof volume unit under these conditions is zero.as-ar T

Really, in ( 3 ) heat flow vectorq = -KvT +nj ,

and, hence,

(3 )

(4 )

specification or establishment of new regularities for isotropicmedia. However, for anisotropic media in the presence ofmagnetic field within thermodynamics of irreversibleprocesses the Thomson relations take on the form [5]A(B) = T;(-B), ( 6 )

(7)where tilde sign means tensor transposition.

As can be seen from (6) , (7), the left and right-handtenso rs in these relations differ in the transposition and sign ofvector B. Therefore, in terms of physics, ( 6 ) , 7) are formalassertions on the equality of values belonging to twoabsolutely different physical states of medium. The first state(left-hand side of equalities) is characterized by certainorientation of external influence vectors j , q (or V 7J and Bimposed on the medium. A distinguishing feature of thesecond state (the right-hand side o f equalities) is that vector jhere has the orientation o f the first-state vector q, vector qhere has the orientation of the first-state vector j, and magneticfield B has reversed its direction. Thus, directly from ( 6 ) , (7)the relation between the Seebeck, Peltier and Thomsoncoefficients for one certain physical state of the medium is notestablished, in contrast to the original Thomson definitions( I ) , ( 2 ) .Our purpose in this paper is to generalize the Thomsonrelations for gyrotropic and anisotropic media i n the presenceof magnetic field in terms of one physical state ofthermoelectric m edium.General izat ion of th e Thomson re la t ionsLet us use the generalized resistivity, thermal conductivity,Seebeck, Peltier and Thomson scalar coefficients for a certainfixed in crystal orientation of temperature gradient, electriccurrent density and magnetic field vectors [6]

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P' = , ( b o ), (9 )IC* =t, (%T,), (10)a* t o( 2 ~ ~ ) (12)r*= jo (? to ) , (13)n' = t o(hio), (14)where to, j, are unit vectors of temperature gradient andelectric current density. These scalars yield the value ofapplicable coefficients for an arbitrary symmetry of tensors

$ , i , ? ,d ,h and generalize for the case of nonsymmetricaltensors the kn ow , Nye coefficients for the symmetricalsecond-rank tensors [7].

Le t us find a new form o f the first Thomson relation.Compare coefficient n * to the product a T with regard

fo r (6):n'r=rr.(&jo)=n'6, (15)

where

is Umkehr effect parameter [SI. This effect occurs in somecrystals and is manifested in that the reversal of magnetic fielddirection on retention of the absolute value of field intensityleads to a marked change in the Seebeck coefficients. Thedimensionless parameter 6 can be represented as

(17)whence it is evident that this parameter in the absence ofmagnetic field

6(j0,70,0)= 1. (18)In the presence of magnetic field 6 can be the valuedifferent from unity only in the case whenIn [9] it was s h o w that parameter 6 should satisfy the

-1C65l (20)From (15) with account of thermodynamic boundaries for

d ( B ) # B(-B). (19)condition

6 20) it is seen thatn* ta*T . (21)

Thus, in the general case in the presence of magnetic fieldin low-symmetry crystals a new form of relation (21), incontrast to ( I ) , is an inequality. Only for the isotropic mediumthe relation (21) has the form of an absolute Thomso n equality(1). The relation (21) is a new form of the first Thomsonrelation.

Le t us apply the above approach to the second Thomsonrelation as well. Resolving the tensors of the Seebeck andThomson coefficients into symmetric and asymmetric, as wellas even and odd with respect to magnetic field components

&=& ' + d " = ~ I t + & - = B " + + i r y + + C i S - + h Y - , , ( 2 2 )f i = f i s + h a = f i + + f i - h ' + + f i " + + h ' - + n " - , ( 2 3 )

4

we have

wherea*+ t o &+ j, ) , a*- t, ( d - j,) , . (25)

The relation (25) is a new form of the second ThomsonIn contrast to (2), it comprises two odd with respect to

magnetic field components -T--Za' - . In so doing, thepresence of -2a'- compo nent is responsible for possibleexistence of odd Thomson effect in the medium wherethermoEMF is temperature-independent.Conclusions

Thus, a new form of the Thomson relations yields twoimportant results - the inequality n*za*T and odd withrespect to magnetic field Thomson effect.

Account of the generalized Thomson relations in thetheory of thermoelements is one of important aspects ofcalculating current and potential distribution, technicalparameters and characteristics, in particular when using inthermoelements the anisotropic and gyrotropic media andeddy thermoelectric currents, where the orientation oftemperature gradient and current density vectors is reversed inthe thermoelement operating medium.References1 , W. Thomson, Mathematical and Physical Papers,Cambridge.-1882.-V.1,558p.,V.2,306p.2. D .Nye. Physical properties of crystals and theirdescription by means o f tensors and matrixes. - Moscow :

Mir, 1967, 385 p.3. Domenicali C . Irreversible thermodynamics ofthermoelectricity. Rev. Mod. Phys..- 1954.- 26, N 2. - P.237-275.4. de Groot, S.R. Thermodynamics of irreversible processes.North-Holland, Amsterdam, I 95 1.5. R. Fieschi, S.R.DeGroot, P.Mazur, J . Viegler.

Thermodynamical theory of galvanomagnetic andthermomagnetic phenomena 11. Phisica, 1954.- V.20, N4.- P. 245-258.

6. O.S.Gryaznov, B.Ya.Moyzhes, V.A.Nemchinsky.Generalized thermoelectric figure of merit. Zhum altekhnicheskoi fiziki, 1978.-Vo1.48.-P.8.0.J.Luste. Thermodynamic limits of the thermoelectricfigure of merit Journal o f Thermo electricity.-I 993.- V.1,N 1.- P. 21-34.

relation.da'-dT

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