Transport phenomena in neutral and ionized gases

by

Robert Edward Robson

A thesis submitted for the degree of Doctor of Philosophy at the Australian National University, Canberra.

acknowledged by appropriate references, this thesis contains the

results of original work carried out at the Australian National University.

I gratefully acknowledge the valuable guidance and the continual encouragement of my supervisor, Dr K. Kumar, throughout the course of this work. My thanks go to the members of the Ion Diffusion Unit, especially Drs R.W. Crompton and M.T. Elford, for their critical

evaluation of the work contained in the latter part of this thesis, and for many helpful discussions.

Transport processes in gases are examined from both the point of view of non-equilibrium thermodynamics and kinetic theory.

Thermodynamics is used to establish concise representations of the general properties of transport coefficients, and in particular, it is shown how the redundancy of Onsager relations in systems with

geometrical symmetries can be removed by using the theory of irreducible tensorial sets. The Boltzmann equation for fully and partially ionized gases is solved by a polynomial expansion of the distribution function, and expressions for transport coefficients are obtained in terms of the inverse of certain infinite-dimensional matrices. In the case of fully-ionized gases, it is shown that the approximations associated with the Fokker-Planck equation are accurate as long as the deviation from equilibrium is not substantial.

Partially-ionized gases are considered under significantly non­ equilibrium conditions brought about by application of an electric field. A new expression is obtained for ion mobility, and comparison is made with the earlier theories of Wannier and Kihara. For electrons, it is shown by way of direct numerical calculations that approximation of the distribution function by the first two terms of an expansion in Legendre polynomials is satisfactory even when the deviation from

Chapter 1.

I NTRODUCTION.

Chapter 2.

THE P H E N O M E N O L O G I C A L E Q U A T I O N S OF T H E R M O D Y N A M I C S :

A S E P A R A T I O N OF G E O M E T R I C A L A N D P H Y S I C A L P R O P E R T I E S

10

Section 1. Introduction 10

2. Irreducible tensorial sets. 12 3. Phenomenological transport equations 14

4. Symmetry effects. 20

5. Applications. 26

Chapter 3.

THE S I G N I F I C A N C E OF HIGH M O M E N T M U M - T R A N S F E R

C O L L I S I O N S IN IONIZED G A S E S

Section 1. Introduction. 41

2. The Lorentz plasma. 42

3. Matrix formalism. 47

4. The simple ionized gas. 52

Chapter 4.

M O B I L I T Y A ND D I F F U S I O N OF IONS:

T H E O R E T I C A L

D I S C U S S I O N

Section 1. Introduction. 61

2. Macroscopic theory. 65

2.1 Symmetry effects 65

2.2 Phenomenological approach 66 2.3 Comparison with solutions of the

Boltzmann equation 70

3. Transformation of the Boltzmann

equation. 74

4. Spatially homogeneous case. 86

4.1 Mobility 86

4.2 Special cases 81

5. Anisotropic diffusion. 88

5.1 General discussion 88

Section 1. Introduction. 106 2. Truncation problems and accuracy

considerations. 112

3. Realistic Interaction Potentials. 121

3.1 General discussion 121

3.2 Temperature-dependence of mobility 122 3.3 Field-dependence of mobility 125 3.4 Anisotropic diffusion of ions 139

4. Electrons 143

4.1 The two-term approximation 143

4.2 Anisotropic diffusion 149

Chapter 6.

C A L C U L A T I O N OF I N T E R A C T I O N I NT EG R A L S

Section 1. Introduction. 157

2. Cross-sections. 159

2.1 General potentials 159

2.2 Numerical quadrature 163

2.3 Tabulation of cross-sections 164 2.4 Special types of interaction 166

3. Interaction integrals. 175

I

3.1 General properties of 7 , 175

I

3.2 Analytic expressions for 7 , 180 3.3 Evaluation by quadrature 185 3.4 Interaction integrals for

Mason-Schamp potential 186

P U B L I C A T I O N S

1.

I N T R O D U C T I O N

Transport equations derived from kinetic theory must be consistent with the more general equations of non-equilibrium

thermodynamics, and in particular, the Onsager reciprocal relations [I] must be satisfied. It is therefore useful to initially proceed with the thermodynamic analysis, obtaining as much general physical information as possible, and to subsequently .carry out numerical calculation of transport coefficients via solution of an appropriate kinetic equation incorporating a particular molecular model. From this point of view, it is desirable to have thermodynamic properties, such as Onsager relations, represented in an unambiguous fashion. In the usual treatments [7] of systems with geometrical symmetries, especially when tensor phenomena (e.g., viscous flow) and large numbers of transport coefficients are involved, the consequences of microscopic reversibility (i.e., the Onsager reciprocal relations)

These results of thermodynamics have an importance of their own, even without reference to the kinetic discription of transport

phenomena, which is the subject of the remainder of this thesis. We consider some specific problems, which have been of continuing interest in kinetic theory, and where by use of a new computational technique we can carry out the discussion much further than has been possible

hitherto.

The oldest and perhaps best known kinetic equation is due to Boltzmann [3], and this has been used with varying degrees of success to describe a wide variety of transport phenomena [3, 4]. The

Boltzmann equation incorporates the assumptions of binary collisions and "molecular chaos" [3], and is therefore strictly speaking,

applicable only to rarefied gases in which intermolecular forces are of short range. It has also been applied to plasmas, but there, "screening effects" and long-range many-body interactions between charged particles are accounted for only in an ad hoc way [5].

Generally speaking, the non-linear, integral character of the Boltzmann equation allows it to be solved only approximately, and the method of polynomial expansions is often used for this purpose [6]. This

formulation undoubtedly facilitates algebraic manipulations in a problem which has for long suffered from inflexibility in this

direction [JO], perhaps the greatest improvement stems from the general expressions that can now be derived for matrix elements of the

Boltzmann collision operator [S]. These matrix elements are expressed in terms of Talmi coefficients [JJ] and interaction integrals [S] (the

n s )

latter play much the same role as the collision integrals ft 5 more familiar to kinetic theory), and are valid for any form of interaction potential.

The form of equations and transport coefficients obtained in the general thermodynamic discussion given in Chapter 2 fits in naturally with this method of solving the Boltzmann equation (see, e.g., [£], and also Sections 2 and 5 of Chapter 4). We adopt this matrix formulation both for this reason, and for reasons of greater computational efficiency.

It is well known that a Fokker-Planck equation can be derived from the Boltzmann equation by an appropriate expansion of the collision integral [72]. In Chapter 3, a generalized

(Boltzmann) Fokker-Planck equation is obtained in a form which is suitable for numerical computation of transport properties, and in which we can retain any number of terms in the momentum-transfer expansion. The calculations there show that the approximations associated with the usual form of the Fokker-Planck equation are accurate so long as the deviation from equilibrium is small; otherwise, high momentum-transfer collisions will be significant.

Another situation where large deviations from equilibrium are important is the case of a tenuous swarm of ions (or electrons) moving under the influence of an electric field in a gas of neutral molecules. In ion swarm experiments [73], it is observed that the mobility vs. field curve exhibits a maximum at intermediate field strengths, and it has been suggested [74] that this is due to a "balancing" of the effects of the attractive and repulsive parts of the ion-molecule interaction potential. A theoretical analysis of this problem can proceed via solution of the Boltzmann equation, incorporating realistic models of the potential; thus, by comparison of theoretically predicted and

experimentally measured mobilities, it is possible to obtain information concerning the actual forces operating between an ion and a molecule.

The main obstacle to such a theoretical investigation has apparently been due to mathematical and computational difficulties. The algebraical complexity of the Chapman-Enskog procedure [3], which deals with a comparatively simple problem, v i z . , calculation of

been mentioned. In the mobility problem, the deviation from equilibrium is substantial at even intermediate field strengths, and an extension of the Chapman-Enskog theory, as was performed by Kihara [

7 5

difficulties are removed, and an expression can be derived for mobility valid (at least formally) for all field strengths, and for any form of interaction potential (Chapter 4, equation (74)). Accurate numerical values of mobility can thus be calculated using realistic models of the potential over a wider range of field strengths than has previously been possible, and apart from certain small discrepancies [73], which the Boltzmann equation cannot account for, the experimental curves are well reproduced (see Chapter 5, Section 3.3).

Besides experiencing a nett drift motion due to application of an electric field, the ions also undergo an associated diffusion process if spatial inhomogeneities are present. This diffusion is anisotropic, ie.,diffusion coefficients parallel and perpendicular to the field direction usually differ. This phenomenon has attracted much attention recently, particularly in the case of electrons [

7 6

deduced (equation (102)), and numerical computation is carried out both for ions and electrons (Chapter 5, Sections 3.4 and 4.2).

The work in Chapters 4 and 5 is mainly devoted to calculation of ion transport coefficients by means of matrix solutions of the

diffusion tensor may thus be derived (Chapter 4, equation (28)). Generally speaking, electron transport phenomena may be treated quite adequately by methods other than those given here, since analytic

solutions to the Boltzmann equation are readily available if use is made of the smallness of the electron mass, and the so-called two-term

approximation [17]. The latter approximation amounts to retaining only the first two terms of an expansion of the electron distribution

function in Legendre polynomials, and its validity is based upon the assumed near spherically symmetric distribution of electron

velocities [

1 7

continues to be used without any prior justification by consistent numerical analysis. Once again, the present method of solving the

Boltzmann equation makes it possible to carry out such an investigation. In this case, the results of numerical calculations support the usual approximations and the associated physical arguments (Chapter 5,

Section 4.1).

It is well known that the evaluation of collision integrals

n s)

’ is one of the most difficult aspects of the Chapman-Enskog theory [IS]. Calculation of interaction integrals present similar problems in the present theory, and Chapter 6 is devoted entirely to discussions of some of the methods which may be employed for this purpose. For near-equilibrium situations, to which the procedures of Chapman and Enskog [

3

7 5

collision integrals) are required. To guarantee a suitable degree of computational efficiency and accuracy, it was necessary to develop the numerical procedures outlined in Chapter 6.

Various aspects of transport phenomena have been considered

separately in this thesis, and results of some value to each particular problem have been obtained. However, it may be useful to indicate how these results could form the basis for calculations corresponding to

more complex situations. For example, we have treated mutual interactions between charged particles, and collisions between charged particles and neutral atoms with ostensibly different objectives in mind. However, in actual plasmas occurring in nature, both types of interactions can be important, and the corresponding theoretical analysis could make use of both the method of solution of the Boltzmann equation discussed here , and the pertinent results of Chapters 3-5. If the even more complex problem of including internal degrees of freedom (e.g., angular

momentum) of particles were considered, then the procedure outlined in Chapter 2 (especially Section 5) could be used to provide at least the macroscopic equations; subsequent numerical calculation of transport coefficients would depend upon the availability of an appropriate kinetic equation [79].

Finally, a word about the notation used in this thesis. With minor exceptions, spherical tensor notation has been used throughout.

This has the disadvantage, of being unfamiliar, and perhaps unnecessary when only vectors are involved. However, it becomes essential in dealing with the tensors of high rank which frequently occur in

R E F E R E N C E S

[7] S.R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics". (North-Holland: Amsterdam, 1962).

[ 2 ] U. Fano and G. Racah, "Irreducible Tensorial Sets". (Academic Press: New York, 1959).

[ 3 ] S. Chapman and T.G. C o w l i n g , "The Mathematical Theory of

Non-Uniform Gases". (3rd Edition) (Cambridge Univ. Press, 1970).

[4] See, e.g., "Kinetic Processes in Gases and Plasmas". (Ed. A.R. Hochstim) (Academic Press: New York, 1969).

[ 5 ] R.S. Cohen, L. S p i t z e r , and P.Mc R. Routly, Phys. Rev. 8 0 , 230 (1950).

[ 6 ] K. Kumar, Annals of Physics 37, 113 (1966).

[7] H. Gr ad, Communs. Pure and Appl. Math. 2, 331 (1949).

IS] K. Kumar, Aust. J. Phys. 20, 205 (1967); 23, 505 (1970).

[9] S.C. Gupta, Ph.D. Thesis, Australian National University (1968); Phys. Lett.

28A,

[JO] See, e.g., ref. 3 Chapters 7-9.

[ J 7 ] K. Kumar, J. Math. Phys. 7 , 671 (1966).

[ 7 2 ] D.C. Montgomery and D.A. Ti dman, "Plasma Kinetic Theory". (McGraw-Hill: New York, 1964).

[ 7 4 ] G.H. Wannier, Bell Syst. Teah. J. 49, 343 (1970).

[ 1 5 ] T. Kihara, Revs. Mod. Rhys. 25, 844 (1953).

[16] J . H. Parker and J . J . Lowke, Rhys. Rev. 181, 290 (1969).

[17] W.P.AIIis, In "Handbuch der Physik" Vol. XXI (Ed. S. Flügge) (Springer-Verlag: Berlin, 1956).

[ I S ] H. O'Hara and F.J. Smith, J. Comp. Rhys. 5, 328 (1970).

2. THE P H E N O M E N O L O G I C A L E Q U A T I O N S OF T H E R M O D Y N A M I C S :

A S E P A R A T I O N OF G E O M E T R I C A L A ND P H Y S I C A L P R O P E R T I E S . +

1.

I n t r o d u c t i o n

Transport equations describing irreversible phenomena in fluids are usually written in terms of Cartesian tensors. While this

representation is satisfactory for equations coupling tensors of rank 1 (i.e., vectors), it becomes somewhat cumbersome when dealing with tensors of higher rank, particularly if the fluid medium is

anisotropic. For example, consider the phenomenological transport equation connecting the viscous pressure tensor with the gradient of fluid velocity (see De Groot and Mazur [/■])

i. .

l

k,1 = 1,2,3 Lijkl^lVk

(i, c = 1, 2, 3)

where 9^ = 9/ckc^ , F"l is the viscous pressure tensor, is

4

the fluid velocity, and is a fourth rank tensor whose 3

components are the viscosity coefficients.

Any spatial symmetries in the system are reflected in the

tensorial structure of the transport coefficients. The viscosity tensor above, for instance, is greatly simplified for a fluid medium with axial symmetry (see' [7], p. 312). Consider as a further example the

structure of the heat conductivity tensor, ., for a plasma subject

to an applied magnetic field,

H

f

^ ■

XT -XR 0

XR

X

0

0 0 A,

with A^C-H) = A^(H) , Al(-H) = Al(H) , and A^C-H) = -A^CH) . The

Onsager relations

V H> =

V'H)

are therefore identically satisfied.

In fact, this apparent redundancy of Onsager relations for systems with spatial symmetries is characteristic of the use of Cartesian

notation. One would expect that since Onsager reciprocity is a

consequence of microscopic reversibility, it should be independent of the specifically geometrical properties of the system. In the usual formulation of phenomenological transport equations (see, for example,

[7] and [2]), some "overlapping" inevitably occurs because, while the tensorial indices (£, j, k, ...) of the phenomenological coefficients determine the components of this tensor with respect to a particular coordinate frame, they are also involved in reciprocity relations.

It is proposed in this chapter to express transport equations as

f

relations between irreducible tensorial sets, and to exploit the advantages of using spherical tensor notation. While the idea of expressing tensorial "fluxes" and "forces" in terms of Cartesian

The authors of the standard reference for this subject, [4], have pointed out that the equations of physics take on their simplest form when expressed as relations between irreducible tensorial sets, and that such a representation is useful for the exploitation of any symmetry properties present in the system. The use of spherical notation has the advantages that (i) there are only two relevant

irreducible quantities is not new (e.g., see [/], p. 58 and [3]) it is hoped that the formalism developed here will show how the thermodynamical aspects of these relations can be separated from the geometrical effects.

In Section 2 some aspects of the algebra of irreducible tensorial sets are discussed, mainly to fix the notation, and in Section 3 we apply this to develop a formalism which yields phenomenological transport equations in the form of relations between irreducible tensorial sets. Onsager relations amongst the new phenomenological coefficients are here expressed in a form independent of the geometry of the system. Section 4 deals with the effects of spatial symmetries, and several applications are considered in Section 5.

2. Ir re duc ibl e Tensorial Sets

The standard reference for the matter contained in this section is that of Fano and Racah [4]. In what follows, their notation will be used exclusively. Doubly-underlined symbols correspond to the German characters of [4].

(1)

A standard irreducible set of order 2 l + 1 , .a , is a set of (

i

2 l + 1 quantities, a^ ( m = - l ... + 1 ) , which transform under

rotations of the coordinate frame according to the Z-th standard unitary representation of the three-dimensional rotation group. Such a transformation may be written in matrix notation as

(la)

= I

DllUlV

(It is understood that a summation over an index is over all allowed

(

(

values of that index.) The rotation matrix, D , = D , (ip, 0, <j>) is

mm mm r

a function of the Euler angles (ip, Q, (p) o f the rotation. Note that it is unitary, i.e.,

(2)

A contrastandard set of order 2 1 + 1 , jiL , is defined to be one which transforms according to

^

where

- I a r l = D ( Z ) <

(l)

The full expression for Dp as a function of (ip, 0, (J)) is given by [4]. However, this is not required here, and we note only for future reference that

D a ] (0, 0, 4)) = e t m ^s , ,

mm m m (4a)

D U \ 0 , IT , 0) = (- ,

mm m 9-m (4b)

r ( q ) (z2) i ( Z )

a, » h.

m i m 2

[ l 1 m 1 l 2 m 2

m i m 2

(5)

where the transformation coefficients are the Clebsch-Gordon or Wigner coefficients

1 4

6 2

1'2m2 I

~

0

m =

and l + 1 2 - Z > |Z^-Z2 | •

They have the following symmetry and orthogonality properties

Z,+Z -Z

( Z ^ i Z

2^2

|

(-)

(Z

2^2

| Zm)

Z +Z -Z

= (-)

& ± m± Z

2

-

?2

I Z-m)

l -vm — -—

= (-)

J W T I

z2™2 I

(6)

£ ^ i m l ^ 2 m ~ m l I Zm) (Z1^ 1 I ^ ,?77} = <5zI z

£ (Z^m^ Z2^1

I

Zm'jfZ^m^

|

Zw)

m ?77

We may therefore invert equation (5) to obtain

= l H

i im i l 2 m 2

(Z2)l(Z)

* = J

3. Phenomenological Tra n sp ort Equations

For a fluid system, the thermodynamic fluxes and forces

Cartesian notation. Let a be an index specifying the physical nature of the flux and its- conjugate force, and let n be the

tensorial rank. Then the entropy production per unit volume and time,

ö , is given by an expression of the form [7]

Tö = I J0. . .

a .. .t 1 n 1 n

I n a a

a

where . and

...t

I n I n

a a

£ -| - 1 s2,3; £ - 1,2,3;...; £ - 1,2,3

1 2 n

a

are conjugate fluxes and forces respectively, and T is the temperature. For simplicity, we abbreviate the set of indices £ , ..., £ by i ,

a a

and therefore write

r* = I

1

?

•

a, i

n a a

a

(

)

Proceeding as usual, we would postulate the existence of phenomenological equations of the form

1

I

aa}

i J J a' ,j n n , n ,

n , a a a

a

(9)

,

raa

_ raa

where L. - L . .

"'n ^ t'l‘ ’ ^1" ‘ ,

a a a a

are the phenomenological

coefficients of rank na ^ na ' ’ The 0>nsager Relations [7] may be

written as

, ( H ) = t t

,La

y

l 3 a a J 1

n ft ,

a a ft a' , fta

where T is the parity of cf? under the operation of time-reversal, n

a and

H

However, we choose to express the entropy production in terms of irreducible spherical tensors. Following Fano and Racah [4] and Kumar [73, this can be achieved by making the following transformations:

4

n Im ^ a

t ' L ("

i

n Im a

aJ m Lall

(

)

The transformation matrices are unitary

Im

f ' * f >

Im i n a Im Jn aJ

= 6. . = 6 ...6 . . ,

1n Jn 'l ±') ± %n °n

a a a a

( _{* f} >

Im

n 7 V | i 1 n aJ

=

67,76

(

)

It is not necessary at this stage to know the actual values of these coefficients (see Appendix) and we note here only that

Im I i

aJ

- 0 unless 7 5 n . We may invert equations (11) a J X (at) Lall \lm i l n a J (13)

f Im i

l n

a J

Tö

_I

m m (14)

New phenomenological relations are now postulated:

I K(alm Ia'I'm' )XL°i' V ]

a'l'm' W

or (15)

e7^a ) = £ x ( a a T

a ’

)x

where the set of indices (a, Z, m) has been abbreviated by a .

Alternatively, equations (11) and (13) may be used in equations (9) to obtain equations (15) once more, with the additional result that

a'I'm' ) - \ \Zm | i

[z'm'

1 jn ,

i I - K a}t na'}

n n , a a

rCLCL

J •

a a

This may be inverted by using equations (12)

.aa

L. . - J \lm 1

1 j - 7 , , \ 1 n .

n n , Zml m K aJ

a a

I ' m '

n ,

a

K(alm Ia'l'm' ) . (16)

Onsager Relations in the new notation are

*(a I

a,)H

= Va'x(a'

I

a ) -H •

(17)

By a transformation similar to equation (5), the indices Z and Z1 in

K(alm Ia'l'm') may be coupled to give a new tensor defined by:

L(al, a'l')yL) = I (Zm Z'M-m \ LM)KXalm |a'Z'M-m) . (18)

Because of the Wigner coefficient on the right hand side, we know that L is restricted to values such that

l + ll > L > Il-l' I .

Equations (18) may be inverted (see (7)) to give

K{a1m I a'I'm') - £ {1m I'm' \ Lm+m' )L{al, a' l' )^^, • (19)

From equations (15) and (19) we have, therefore,

J (aZ) =

y

(Zw Z'M-m I LM)L(al, a'Z' )f(,L

^ ] .

rr, L

1

’

M

M-m

a ' V I M

(

)

[Z] (Z)

The contragredient set a^ is related to the set a. as follows [4]

[Z]

( VL-mAl)

{I)*

a = (-) a = a

/77 - m m

(

)

Using the symmetry properties of the Wigner coefficients, i.e., equations (5), and equations (21), equations (20) become

j{clI )

m l ■

a'l'LM * ‘ 2L+1

2Z+1 {l'm-M IM 1m)L{al, a' V

A L ) {a'V )

m-M . (2 2)

This may be written symbolically as

J{at)

a'l'L

(23)

Aal) {a'V) T (aL) , . , . .

where J. , _X , and _L denote sets whose elements are

Aal) {a'V) , {clL) . . .

J , X , and L.. respectively, and the index ot is

m m M * J 5

(aZ,

a'V)

LCal, a'V

a a

l+l'-L

Ua'l' , al: -H )CL) (24)

where the symmetry properties of the Wigner coefficients, equations (6), have also been used. In set notation, this is

L(al, a ' V ; H)(£) = T T ,(-)Z+Z’

Lh(a'V

Because of the irreducibility of the sets involved, we know that

Onsager reciprocity, as exhibited by equations (24) and (25), is expressed in the form of a maximum number of separate independent relationships. Furthermore, in this notation we need not refer to any coordinate system in order to state the reciprocity, as we must when using Cartesian notation. This stems from the fact that there are no reciprocity conditions on the index M of the phenomenological tensors, and this is the index which determines the components of a tensor with respect to a particular coordinate system. Thus we have separated those effects arising from time-reversal symmetry (i.e., Onsager relations) from those pertaining to geometry alone.

The Cartesian phenomenological coefficients are obtained from their spherical counterparts by using equations (15) and (19)

Laa i j

n n , a a

Il'mLM

1 \ l m

I 1.

I)

( T )

Finally, note that the L(al, a 1l')j'J~J^ are, in general, complex, and

obey conditions of the type (21).

4.

Symmetry Effects

If a physical system is symmetrical under a given rotation of the

coordinate f r a m e , then we postulate that the transport equations

describing irreversible phenomena in this system must be form 'invariant

under the rotation. This implies that the set of transport

coefficients remains unchanged, and from equations (1) and (22) it follows

that

, (cxL)

M

n (£),.,, n X M (OIL)

(27)

where (i|j, 0, <}>) are the Euler angles of rotation, and the index,

a , has the same meaning as in equation (23). Certain conditions on the

coefficients ^ may be inferred from equation (27), depending on the

type of symmetry involved. Some special cases of geometrical

symmetries are considered below.

Spherical Symmetry

Systems having this property are also called isotropic. In this

case, equation (27) holds for arbitrary (41, 0, 4>) , so that we must have

I (olL) M

I ( 0^0 ) r r

L

SL0&M0

i.e., the phenomenological coefficients are just scalars. (Note that

by (21), i_(0L^>) ps real.) Using the conditions (28) in equation (22),

i.e., for an isotropic system only fluxes and forces of the same tensorial character are coupled.

J{.at) 1

✓21+1 a'

I Hal,a'l){0)X (-a 'l) (29)

Since the direction of the magnetic field is immaterial in this isotropic system, it follows that

L(a°)(_H) =

LCa°)(H) t

Using conditions (28) and (30), the Onsager Relations, (24), become

L(al, a'l)(0) = t t ,L(a'l, al)

(

)

a a (31)

Cylindrical Symmetry

Suppose that a system is anisotropic solely because of the

presence of a field, f . (In the absence of such a field, the system is isotropic. ) Then the system is cylindrically symmetric about an axis defined by the direction of the field, T . If we are making measurements with respect to the set of axes ( x, y, 2) , relative to which f has polar coordinates (/, 0^., cjy) , then, by rotating the

coordinate frame through the Euler angles (o, 0^., <Jy) , we obtain a

new set of axes (#, y, z) , relative to which f is directed along the 2-axis. In this new coordinate frame, the system is cylindrically symmetric about the 2-axis.

j(al)

= D (J) . j(al)

t al) - R a ) ■ ^ a l )

— ( a l ) A L ) _{. r(al)}

L = _D • _L

where

(32)

Da ) - D U ) (o, 0

*/)

Since is unitary, the inverted form of the last of equations (32) is:

L(aL) = S(L)* • l (aL)

Transport equations connecting quantities measured relative to the rotated frame are of the same form as equations (22)

j(al)

m l

a ' V m

[2L+l

/ 21+1 Cl'm-MM lm)L(al, a' V )

(L)j(al)

M m-M (34)

The form of the Onsager relations (24), is unchanged by a rotation of axes

L(al, a'V ; H)(L) = T T

a a,(-)U V T l a ' r , al. (35)

Because of the cylindrical symmetry, the system must be invariant for arbitrary rotations of the coordinate frame about the 2-axis. Using (4a) in equation (27), we find

t-(oL) t(olL)x o._ .

M 0 MO (36)

axis perpendicular to the 2-axis (say the y-axis), provided that

f

f

2-axis, then the quantities of interest (viz., the transport

coefficients) may depend on

f

f E f

2

reversal of the field is therefore equivalent to the operation

f

-f

f

by

-f

r ^ h - / 6 ) = f 8 ) .

With the condition (36) this gives

L^aL)(-f

<.-)LL<

oaL) if) .

On the other hand, if

f

f

L(0a £ > ( / ) = <.-. (38)

Hence, tensors having odd L-values vanish.

If we now put the condition (36) into equation (33), we find

,

(a

M

XL)*

M

(o, e

_{v )T}

0

XL)

0

M

rV

4 7 T _ y U ) * ( ? )

Appendix for their relationship to the usual spherical harmonics), it

follows that

T h u s , for a system in which there is an anisotropy-producing

field,

f

respect to any set of coordinate axes may be represented by a product

of a spherical harmonic, whose arguments are the polar angles of

f

in this coordinate system, and a coefficient which depends only upon

f

The transport equations, (22), become in this case, using

equation (39),

Equation (39) should be compared with equation (7) of Williams [S],

who invokes a "representation theorem" in order to find the functional

dependence -of phenomenological transport coefficients on the magnetic

field H . In dealing with fluid systems with spin, some authors [3]

make expansions in terms of irreducible Cartesian tensors formed from

the spin density,

o .

= O

proportional to

Y ^ \ o ) .

(.39)

a'l'L

-j{al)

m , L M k a 'm L0 1 l m ) T ( a l ’ C41)

a l L

and so only fluxes and forces with the same index, m , are coupled.

Suppose now that anisotropy is caused by applying a magnetic

field, H , to the system, i.e., f E H . As explained before, the

coefficients can depend only upon the magnitude, H , of the

field, so that from equations (35) and (35), we have

Ual,

a’l'i

TaTa,(-)l+l'

Equations (37) are in this case

= (-)£r(°i ) (ff) . ₍₄₃₎

From equations (42) and (43), the new form for the Onsager relations is

t

Hal, a'l')(nL) = t T Lia'V , al)(rL)

0 a a 0 (44)

If the anisotropy producing field is an intrinsic property of the medium, and anisotropy effects due to H may be neglected, then equations (38) and (35) lead to the result

U a l, a'l'i H )4^ = T x , (-)U V L(a' V , al-, -H)'£)

0 a a 0 (45)

Both members of Eq. (44) are, of course, functionally

Symmetry under Reflections

Let p be the parity of and 0f under

ra V J m m

reflections of the coordinate axes. Then if the system is invariant under a reflection of coordinates, this leads to the familiar result that fluxes and forces of different parities are not coupled, i.e.,

where S is the total entropy flux per unit area and time, E is the electric field, y the chemical potential of the mobile charged

particles (usually electrons), z is their charge to mass ratio, and

J is the electric (conduction) current. Here, all fluxes and forces

are vectors, so that their spherical counterparts all have degree l = 1 .

The transformations corresponding to (11) are

(46)

5. Appl icat ions

Thermomagnetic and Galvanomagnetic Effects

The expression for entropy production is (see [7])

To = -TS * V ln T + J • ( E- Vy /s )

E .

Table 1

Fluxes, Forces, and Their Time-Reversal Parities for Thermomagnetic and Galvanomagnetic Effects

a = S a = E

m m ll (£,(1)- 9 ( 1 V / 2[ m m i—1

<

0

x ial)

m - 9 (1) ln m T 6 ll J am \ ,ll

T

a - +

The entropy production written in terms of spherical tensors is

Td =

I

“

m m

\

m

m

m

m=0,±1 K J

We choose any arbitrary set of coordinates axes

(x,

y, z)

H

H

i

iLL(Sl,

i—

1

Co _[-3(1)

In T, y(l)( H ) ] U )

Am < — i o II ,2

+ p T o s i , E D ^ p (1)_{, I}(L)(H)] (1)

m

l i L L ( E l

9

( H ) ] (1)

1—

1

o

II

,2 1/1

+ £’1)qL) [j(1), I(L)<fi>](l)

m . (48)

L(

51

,

S

1

)

q

L) = -L(

51

,

51

)

q

L)

(5

=

0

,

1

,

2

)

i.e. , three relations amongst twelve coefficients.

Suppose that we arrange the experiment so that H is now along the 2-axis. Then equations of the type (41) hold

* (1) m

TS^ = -A + tt J^

m m m m m

a"( 1 ) y

= -n 3 ( 1 V + /? 7 (1 )

m m m m m

m - 0 , ±1 , (50)

where A , tt , n » 5 are the thermal conductivity coefficients, the

m m m m

Peltier coefficients, the differential thermoelectric power coefficients, and the resistivity coefficients, respectively, and are defined by:

A

=

VT(

51

,

* L=0;1,2 m °

(

)

TT = I 0)£T ( 5 1 , 5 1 )

* L=0;i,2 w °

(5)

n =

VT(

51

,

51

)

^

5

=

0

,

1,2

*

°

(

)

i? = y co L ( 5 1 , 5 1 ), 5 = 0 , 1 , 2

(51)

with

5 /2 5 + 1 , I ^

co = / —-— (lm 50 lm) /?? ■ ✓ 3

Using equation (49) and the definitions (51), we have

as an alternative expression for the Onsager reciprocity.

Note the particularly simple form of equations (50) when compared with th.e corresponding expressions in Cartesian notation (see [7], Chap.

XIII, equations (96)-(99)).

Equations (50) may be expressed in terms of spherical harmonics (see Appendix)

t|s|y(1)(§) = -A IVT\Y (1)($f) + tt |J|Y(1)(J)

1 1=777 m ' ' = m m 1 ' = m

IEIxl1}(t) - |Vu/2|y(1)( Wi/2)

I I \ = rn

=

-n

77?1 1 m' 1 = m

(m = 0, ±1) , (52)

where the use of "barred" notation to signify measurements in a coordinate frame in which

H

experimental determination of the coefficients A , tt , q , and R

r m m m m

Further, we note that by using equation (26), we can express, for example, the Cartesian thermal conductivity tensor A^ . , in terms of

the quantities L(S1, Sl)^^ (L = 0, 1, 2) . Using equations (36) and

(A5) (see Appendix) in equation (26), we have

A.. = y (1/7? I Y)*(l-m I j)*(lm 1-m I Z/0)L(5'1, 5 1 ) ^ ^ , mL

=

I (L0

1(5i, 5i)

/3

(

)

0 1 0 + —

US1,

/2 0

1 0 0

_0 0 1 _0 0 0_

1.(51, 51)

(

)

0 -2

(53)

From equation (43), we may deduce that 1(51, 51)^°^ and L(51, 5 1 ) ^ ^

are even functions of

H

H

conductivity tensor (see Introduction).

Fluid Systems with Spin

The thermodynamics of irreversible processes for fluids whose

molecules possess intrinsic angular momentum has been studied by

several authors (see, for example, [3] and [9]). For simplicity, we

consider a one-component fluid only. In such a system, there exists

at each point of space an intrinsic angular momentum density,

O

which results in the system being anisotropic. Just as a density

gradient results in a diffusion process, a gradient in

o

a "diffusion" of intrinsic spin. We therefore introduce the spin

diffusion tensor,

I

Besides considering the usual balance equations for mass,

momentum, and energy, we must also consider the conservation of

intrinsic angular momentum in calculating the entropy production

do

it -

I + Ypö x H + e : fl (54)

Here p is the density, Y the gyromagnetic ratio, the viscous

(ei23 = 1 = -£213

et°-)-The Cartesian form for the entropy production is [3]

Ti> = -q • V In i - n : [e • r 1 Ca-yrH)+Vv] - I : [v r ^Ca-yTH)] > 0-1

where q is the heat flux,

T

Fluxes

q^r1)

I Om

i)qi to

i

, U )

l (to

ij

I

a)

l {Im

ij )I .

.

id

to

a = 0, 1, 2)

-i,

i)

Forces

9^ - J (1m ^)*^ .T (m = 0, ±1)

m

1

0^ ] = r '1(Zm I ’

=

1

T

1

C

1

to \

m'

Cl{m Im>) E X. 1 i H l m 1 j k y = " ^ h l &m'm ’

^

J

S ^ C v ) = £ (Zm I

ij)*i.v. ,

m

ij

3 1

r ' V n Ca-YrH) =

I

_1(

r

Z

m

ij)*3 (a.-yra }

I

U

= ° ’ ls 2)

m

ij

3 1

1

-l, ..., I) .

Table 2

Fluxes, Forces, and Their Parities for Fluid Systems with Spin

a - I a - q

ln T 6

The fluxes and forces are classified according to Table 2. Note the Cartesian expansion of a second rank tensor, . , in terms of

its irreducible parts is given by

T . . = t& . . + . + TS. . •Z'J V

where t - trace !Z\ . is the scalar part, !Z^ . is the antisymmetric

part, and . is the traceless symmetric part. The spherical tensor,

(l )

T , is thus related to T . . by

T^l) = I (lm

m “ .

= £

ij

id) ts . .+1^ .+TS. . . VC VC T'O,

From the properties of the coefficients (lm | ij ) given in the Appendix, it follows that

T ( 0 ) = I ( 0 0 I vj)tS . . ,

id ^

T (1) = l (lm I ij)Ta. . , (55)

m k. vC

T (2) = £ (2m I id)??.. .

m id V

The entropy production when expressed in terms of spherical tensors is

Tb = - I q (1)3 [1] ln T - I n (^ }

u m b; m

m lm

- I j ^ W ^ t a - y r H ) > o . (56)

~ m m

lm

Since, in general, both O and

H

anisotropy of the system, there is no cylindrical symmetry unless they happen to coincide. However, when the magnetic field is very strong, the anisotropy effects of the spin field may be neglected, and in this case,

H

f

H

parity, p , do not couple, equation (46) holds. Hence the transport S CZ,](V) - /2 6 7 r 1

m ll

a^-yl T / ^

r4TT

h i h i q l , 91)^£)[3(1) ln T , Y (£)(fl)]^ }

''L

+

l

dTOjl,JZ')5,Z')[r'1S (Z')(a-yrH), Y(£)(fl)] (1)1 ,

(57)

LV

= - / I j , ^

^ \ v ) - / 2 6 7 r F 1 ( a ( 1 ) - y r ^ ( 1 ) ) , ^ ( L ) ( H ) 1(l)

(58)

—1/??

7-(£) _ / 47T / V ;L~Tf-ri N(L)X r ( D m w(^)/U\l(^) Jra - - / r n i \ L\, £ L(JZ’ 0 6Z.' 1 Ls ln X (H)Jra

+ I

h i l l , IV

)^£)rr'1S a '

)(a-yrH), Y(£)(fl)l (i)\ .

(59)

LV

0

-

m )

Equations (57 )-(59 ) should be compared with equations (37)-(43) of [3], bearing in mind that in our case, the anisotropy effects of intrinsic

spin have been neglected.

If the direction of the magnetic field is taken to be the s-axis, then from equations (41)

- (1)

<7™ - I /

I ' L V

2L+1

„ « )

m

2L+1

V L

( I ' mLO | l m ) [ i ( q l , q ± ) ^ \

+

r-1r((7i,

) (Q

( I ' mLO | l m ) T ( M ,

nv ) (qL)

x f t 1 ' } ( v ) -

SI

( I ' mLO | Im

|l(n, ?

)<L).

+ r_1I

a t , i v ) (Q

The transport coefficients are classified according to Table 3.

By equations (44),

rc<?i,

n)(

0L)

=

Knz, n

=

UIl, iv

)gL) =

the Onsager relations

(-)lun, q

d

(0

l)

(-)l+l' l ( n v , m ) (0Ll

(-)l+l'l(iv, i i)qL)

are now

(7 relations), (63)

(5 relations), (64)

(5 relations). (65)

Table 3

Classification of the Transport Coefficients

Thermal conductivity L(q1, q l ) ^ ^ 1 = 1 = 1 ' , L = 0, 1, 2

(3 coefficients) Viscosity L (flZ-, W V )q^

o

II _{0 0 1 1 1 2 2 2}

o

II

r-'

i _{1 2 0 1 2 0 1 2}

O

II _{1 2 1 0, 1, 2 1, 2, 3 2 1, 2, 3 0, 1, 2,} (19 coefficients)

Spin diffusion L(Il, I V )q^^ Z-, Z' and L as for viscosity.

(19 coefficients)

Cross-effects between heat conduction and spin diffusion I V ) (0L)

l = 1 1

i

0 1

L = 1 0, 1, 2 1

o

ii 1“

1

■' = 1 2

II _{0, 1, 2} i—

1

1 2 2, 3

(7 coefficients) 1(1 l,ql)(QL) 2

1 2, 3

Bloch Equation. The balance equation for intrinsic angular

momentum, equation (54), when written in terms of spherical tensors is

do

(

)

dt = - y/2 py[a(1), - y/2 n Cl) (66

)

U )

where it has been assumed for simplicity that I - 0 . It is

advantageous to work in the coordinate frame with H along the

(1 ) 77(1)

2-axis , i .e . , H =

5 9 m 0 6 , for then equation (66). becomes

vr(l)

p -U2— = - /2 pY [ö(1), Ä (1>] (1) - ff(1)

dt L Am m

= - r/2 py(lm 10 I 1 - /2 n (1)

1 1 m 0 m (67)

From equation (61) it follows that if ) = 0 (equivalently,

curl V = 0 ), then

n (1) = 3 r 1 a (1)- y r ^ 1}6

m m \ m 0 mO

(

)

where

~ = ^ l /■2L^ - Urn L0 I Ji??)L(ni, f l l ) ^ (69)

From (67) and (68), it follows that

* (1)

m

dt = - r/2 {(1( m 10 I1 l^)a(1)^^1)m 0 + - j rpf 1( a ^ - y l V 1 ^m 0 m0 (70)

fr

dt

(71)

do

dt

2

T-pT

uni, ni)

(DV-CD

o

J°±l

where

t

:1

e

2

(um , ni)(0) -

um, ui)^2)l ,

(

)

L /ä pr ^

Q

t

: 1 5 —2— (uni, ni)(0> + -i. um, nih2)} .

(74)

T

/ 3 Pr i

0 '

Note that T r and T are both real, since

L„ ^

L I U U

are real, by equation (21). They may be interpreted as longitudinal and

transverse relaxation times respectively. We can also show that they are positive from the condition

W

Equations (71) and (72) thus describe the relaxation of a nonequilibrium spin state , and may be compared with the more usual form of the Bloch equation (see, e.g., [70]). The coefficient

L(fll, ni)^1) is related to a chemical shift.

A P P E N D I X

The fundamental transformation coefficients (lm |

k)

are :

k

(10 1

k) =

(1±1 I

k)

1/^2,

-i)