Temperature and stability properties of a radially constricted steady-state plasma between electrodes

TE1VIPERA'TURE il.ND STABILITY PHOPERTIES OF A RADIALLY CONSTRICTED STEADY -STATE PLASMA BETWEEN ELECTRODES

A Thesis submitted to The Australian National University for the Degree of

Doctor of Philosophy in the Department of Theoretical Physics

by

Patrick William Seymour

Research School of Physical Sciences, Institute of Advanced Studies, The Australian National University,

PREFACE

Each chapter of this thesis describes original work carried out by the candidate at The Australian

National University during an effective period of nine terms from April

1958

to July

1961

,

Vfuere it has been necessary to relate the candidate's work

v

to the work of others, proper acknowledgment has been made in the form of detailed, specific references.

Signed:

July

1961

iii ACKNOWLEDGNlENTS

In various ways many people have contributed to the completion of this thesis, and it is my pleasure to set down the follov'ling expressions of appreciation.

Within the Department of Theoretical Physics, I first sincerely thank my Supervisor, Professor K. J. Le Couteur, who at intervals suggested the problems considered in these pages, then constructively criti-cized my attempts at solution, and finally, when at certain stages of advancement my difficulties seemed insurmountable, provided timely advice and encourage-ment.

I run also very much indebted to Dr. F. C. Barker, Senior Fellow, who, in the earlier stages of my

researches, spent much time patiently providing me with some insight into the mathematical techniques of the theoretical physicist, and who assisted me, via medium of discussion, to clarify my views on the

topics discussed in Chapters 1 and 2 of this thesis. In connection with Chapter 2, I am also indebted to Dr. D. C. Peaslee, Reader, who 'perused an earlier form of Chapter 2, and offered helpful suggestions.

The presentation of results in convenient graphical form through the chapters involved me in the rather extensive use of an electric desk

calculator, and it is with gratitude that I acknow-ledge the many spot-check calculations carried out by Mrs. B. Nerdal, Departmental Assistant, in

iv

Within the Department of Particle Physics exists the Experimental Plasma Physics Group, led by Dr. A. H. Morton, Research Fellow. As I have produced for

publication manuscripts corresponding approximately to the four chapters of this thesis, Dr. Morton has kindly read them, and offered useful suggestions based on his knowledge and practical experience in the plasma

physics field. In voicing my appreciation to

Dr. Morton for this assistance, I should particularly like to stress my pleasure at his pronouncement, after study of my principal results, that the laboratory experiment finally proposed (pp. 128-131) is feasible. This is also a good opportunity for me to express my

gratitude to Mr. G. F. Cawsey, Principal Scientific Officer, Department of Supply, Victoria, who , during an earlier period of association with the Plasma Physics Group, taught me the meaning and use of

norder-of-magnitude" or "good!! physics, and provided me with much valuable food for thought in connection with the stability topic of Chapter

4.

Here my

indebtedness to the Plasma Physics Group would not be complete without a note of thanks to the Plasma

Physics Research Scholars, Messrs. I. S. Falconer and R. H. Hosking, with whom many pleasant and inform-ative discussions have been held.

The Australian National University possesses

a comprehensive Institute of Advanced Studies Library, associated with which is a growing specialised

particular I gratefully acknowledge the pleasant and efficient manner in which my numerous requests have been handled by Miss N. G. Cook, Assistant Librarian, and Mr. L. A. Delpratt, Library Assistant, at the former location; and by NIrs. E. Lennon, Library Assistant, at the latter location.

The four chapters of this thesis correspond approximately to the following papers by the writer:

(i) Drift of a Charged Particle in a Magnetic Field of Constant Gradient. Aust. J. Phys., Vol. 12, No.4, December 1959. (ii) Estimation of the Maximwn Temperature in a

Radially Constricted Gas Discharge Between Electrodes. Aust. J. Phys., Vol. 14, No.1, March 1961.

(iii) The Influence of Thermoelectric Effects on the Maximum Temperature in a Radially Constricted Gas Discharge Between Electrodes. Aust. J. Phys., Vol. 14, No.2, June 1961.

(iv) A Stability Criterion for a Radially Constricted Gas Discharge Between Electrodes. Aust. J. Phys., Vol. 14, No.3, September 1961.

This is a convenient opportunity for me to express my sincere appreciation to the Board of

Standards and the Advisory Co®nittee of the Australian Journal of Physics for the extremely cooperative

manner in which these papers were accepted for publication.

Solution of the problems considered in this

vi thesis was made possible by the award to the writer of a Research Scholarship, and I am much indebted to the Council of The Australian National University for making this opportunity available to me. In this connection, I am also indebted to lVIr. J. Vi. J. Byrne, Assi stant Secretary, Department of Supply, N.S.W., who kindly released me from my duties as Supervising Engineer, Telecommunications Branch, N.S.Vv., to permit me to

undertake my doctoral studies; and to the Commonwealth public Service Inspector, N.S.W., who approved leave of absence for this purpose under the provisions of

Section 71(1) (b) (1) of the Public Service Act.

Pursuit of this course of study was considerably aided by my wife , Gladys Joyce Seymour, who carried out the full-time duties of Secretary, Department of History, Institute of Advanced Studies, during my

studies here, typed the manuscripts of my four papers on an evening basis, and then, very recently, retired from official life - to type the stencils for the entire thesis! All who read this appreciation will understand how very much I am indebted to Gladys Joyce.

There are others who have helped me in general ways in connection with completion of this work, and here I place on record my appreciation of their

efforts.

To conclude these acknowledgments, the pleasant period from April 1958 to July 1961 at The Australian National University is epitomised for me in the

Edward FitzGerald's first version of the Rubaiyat of

Omar Khayyaro:

Myself when young did eagerly frequent Doctor and Saint, and heard great Argument About it and about; and in the end

Came out a wiser man than in I went.

Canberra, A.C.T. July 1961.

viii

TEMPERATURE AND STABILI'rY PROPERTIES OF A RADIALLY CONSTRICTED STEADY-STATE PLA9~ . BETV'lEEN ELECTRODES

SUMMARY

The above subject is discussed in four chapters through this thesis. The first chapter bears on charged particle orbit theory and a simple case of

plasma instability, while the second and third chapters are concerned with the flows of heat and electricity in a centrally-constricted plasma geometry under assumed steady-state conditions. In the fourth chapter a sufficient stability criterion is derived for this particular plasma configuration.

A table of contents, giving the section titles within each chapter and the corresponding page numbers, is included for ease of reference. To facilitate

cross referencing, combined chapter and section numbers are included at the top left-hand side of each page. Specific references given in the text correspond to alphabetical reference lists provided at the close of each chapter. The fourth chapter is followed by a review of topics covered in the thesis, and some

remarks on possible future experimental and theoretical studies. Additional references, relevant to the

research undertaken, appear at the end of the thesis. A detailed swnmary of the work covered now follows.

In Chapter 1 charged particle motions are considered. Using the Lorentz transformation of special relativity to relate the electric field, E'

-.1. ' and the magnetic field, ordinate system, to and H _-.1.

co-ordinate system (where the subscript ~ on a field vector signifies that it is perpendicular to the constant relative velocity, ~, between the two co-ordinate systems), the familia.r E x H drift velocity expression for a charged particle in crossed electric and magnetic fields is obtained by a method not

ix

usually employed in the textbooks. This result, which is independent of the sign of the charge, is then

generalised to obtain the usual sign-dependent drift velocity result for charged particle motion in crossed force a.nd magnetic fields.

To obtain physical insight into the mechanism of plasma instability in simple cases, the Rosenbluth-Lon~aire particle orbit approach to the hydromagnetic analogy of the classical Rayleigh-Taylor instability of fluid dynamiCS is reviewed non-mathematically, using as principal factors the sign-independent and sign-dependent drifts mentioned above. Since in this case initiation of instability is found to arise from a sign-dependent g x H type of drift (g the gravitation acceleration vector), mention is also made of the Alfven

sign-dependent drift velocity for a charged particle moving in an inhomogeneous magnetic field as a possible source of instability.

Partly as an exercise in itself, and partly because of its possible aggravating influence on instability, an extension of Alfven's first-order perturbation theory is included in the form of an

exact solution, in terms of complete elliptic integrals of the first and second kind, for the motion of a

x having constant gradient perpendicular to the direction

of the main field. This exact solution yields as

approximations Alfven's result and the case of circular orbit, and includes the somewhat novel case in which the magnetic field vanishes at points in the particle's orbit, for which perturbation methods are inappropriate. In this latter case larger drift velocities, of the

order of the particle velocities, are expected in the regions of vanishing magnetic field, and hence larger charge separation effects than in the Alfven region may be encoID1tered. One likely interpretation of this result is that plasma instabilities may be increased in growth rate.

In Chapter 2 a steady-state deuterium discharge between two electrodes is considered and the free boundary surface of the plasma is assumed thermally

insulated when pinched away from the walls of the discharge tube. Cooling is therefore by heat conduction to the electrodes , compared to which

bremsstrahlung loss is shown to be negligible if the dischargo is not too long. The main question exruained is how much the plasma maximum temperature T

ill can be raised by constricting the cross section of the dis-charge near the centre.

The analysis is confined to substantially-ionized deuterium, and curves based on the Saha equation are provided to show the minimum gas temperatures required for various particle densities. With neglect of

thermoelectric effects, there exists a median plane, normal to the longitudinal axis of the discharge,

xi

are symmetrical. Defining we as the electron gyro-frequency and ~e as the electron self-collision time, the analysis is carried through both by specialising the current density and heat flux vectors, j and q,

to cater for an isotropic plasma (We~e«1), and for

a plasma made anisotropic by a strong, external magnetic field H (w~»1) o

e e Prior to detailed mathematical analyses, however, a simple continuity

argument yields the important relationship,

q + V j = o ,

where V is the electric potential, provided that everywhere within the discharge q is parallel to j.

This type of flow is for convenience termed lon gi-tudinal flow.

The detailed axi-symmetric analyses for slightly and greatly constricted discharges show that

where

a

_{o and Ko} are related to Spitzer's formulae

for the electrical and thermal conductivities of a highly-ionized gas, and T

m is the temperature on the median plane, where V

=

0. For strictly longitudinal flow, with j and q parallel to H

at every point within the discharge, this result is

valid for all values of we~e ' and hence for all

values of the radial compression ratio

v

from unity upward, where, in t erms of the maximum and central radii of the discharge, P₁ and Po respectively,

xii

are held at zero temperature, we obtain for the

constricted discharge the useful numerical relationship

where V* _e

*

* - 1

T ~ 5899 V volt deg. ,

ill e

is the semi-voltage across the discharge, and the star superscript is used when necessary to indicate that the discharge is no longer Itlinear".

It is emphasized that this result is independent of v. For these detailed analyses, initial use of an electric stream function 'f and the electric potential V as orthogonal curvilinear coordinates simplifies the plasma energy equation. An analytic solution of this equation if the temperature T is a function of ~ only shows that the heat flow can be everywhere perpendicular to the flow of electricity only when the streamlines are straight, and parallel to the longitudinal axis of symmetry. Analytic solutions if T is a function of

V, representing the longitudinal flow, are given for (1) straight strerualines parallel to the longitudinal axis of symmetry, (2) hyperbolic streamlines to

represent a discharge constricted at the median plane. A curve giving the variation of T with axial distance

along the linear discharge is included.

expressions for tne central temperature

Upper-limit

T a r e m

obtained in terms of the total current carried by the longitudinally stabilized discharge and its character-istic dimensions, and a curve gives the dependence on constriction of T and of the resistance R between

m

semi-xiii

length. If large radial constriction at the median plane is achieved by use of a strong guiding magnetic field which makes the conductivities anisotropic, a tensorial analysis is required, but leads to the same results for all values of we're when the flow is

strictly longitudinal. Vfilere the thermal insulation and neglect of bremsstrahlung approximations apply, the direction of heat flow is not expected to depart

significantly anywhere from that of the flow of electricity when we_{'t e}« 1, and so the above curve should provide a useful guide to the increase of Tm

and R due to constriction. For an area constriction of 400:1, Tm and R are increased by a factor of

about

4

.

field

Characteristics relating T and the magnetic H when W 't

=

1

e e are given for various values of the total particle density. In terms of the known central temperature and total central confining

magnetic field ~, the total concentration can be

estimated from the pressure balance as

14 2

*

-3

n

=

2.88 x 10 H

IT

particles Cill. m m

and so the characteristics given can be used to obtain an order of magnitude estimate of we'te' and hence of

the state of the plasma.

Chapter J has for its topic a generalisation of the cases examined in Chapter 2. The same a pproxi-mations are made, but the analysis is extended by

including in the expressions for j and q vector

xiv

effects of Seebeck and Peltier. This results in the temperature and voltage distributions no longer having symmetry about a median plane. For simplicity the analysis is limited to strictly longitudinal flow, so that a vector method is sufficient.

Again a simple continuity argument shows that

q + V j = o ,

where V is the electric potential, but now detailed analyses reveal that the equipotential surface on

which q

=

V

=

°

is displaced from midway between the

electrodes nearly to the cathode, and that the maximum temperature is displaced somewhat from the midway

position towards the anode by amounts which decrease as central constriction increases. The important influence of inclusion of thermoelectric effects is

•

that the maximum temperature is increased by approxi-mately

14%

for about the same total applied voltage

producing a given current in a particular discharge geometry.

The characteristic relating the maximum

temperature and resistance ratio e and the radial compression ratio v obtained in Chapter 2 is not changed by thermoelectric effects. Comparison of voltage and also temperature versus distance

characteristics for linear and constricted discharges without and with thermoelectric effects is given by means of graphs. For the temperature characteristics

simple physical interpretations are provided.

In Chapter

4

the stability of the

xv

electrodes is considered. Ini t ially, the noriIlal lllode

and energy principles for the examinat ion of pl asma

stability are reviewed and the results contrasted, and

it is recalled that the diffusion equat ion,

1 2

:::: _\I H

a

t

4 7C (5 f..l

suggests that the magnet ic field l eaks through the plasma with a characteristic time of decay

't:d

=

4 1t f..l (J L 2

where f..l is the permeability, (J the electrical

conductivity, and L a l ength comparable with the

dimensions of the system. Teller's power ful stability

criterion is next obtained in a useful integral form by

means of a thermodynamic dnalysis of interchange

instability. This general geometrical result is then applied to the centrally-constrict ed plasma between

electrodes, and a sufficient stability criterion

d.erived. For this type of system, initially

stabilized by a strong, external guiding magnet i c

field, it is found that the onset of instability occurs for a discharge current

where ¢E froln the external sol enoid is the total

flux through any discharge equipotential surface, is the discharge semi-length, and practical simpli

-z _e

fying asswnptions of high radial compression ratio v and maximum discharge radius P

xvi In terms of the temperature, the decay time can be written as

-13 2 3/2 -2 -3/2 'L" d ~ 2 x 1 0 L T cm . de g . sec 0 ,

a result which confirms, for practical values of L and T, that the plasma !?steady state1l is in the region of milliseconds.

To conclude this detailed sUflhuary, it is thought that the procedure outlined in Section

4

and discussed in Section 5 of Chapter 4 could form the basis of an interesting laboratory experiment for the observation of transition from stability to instability in a

xvii

CON'rENTS

PREFACE.

ACKNOWLEDGI\lIENTS , SUMMARY.

Page ii

Section 1.

Chapter 1

CHARGED PARTICLE MOTIONS

Introduction.

iii viii

1

2. Charged Particle Drift in Crossed

3.

4.

Electric and Magnetic Fields. Charged Particle Drift in Crossed

Force and Magnetic Fields. Physical Mechanism of Plasma Instability. Charged Particle Drift in an

Inhomogeneous Magnetic Field;

Introductory Remarks.

2

5

8

5.

Exact Solution for Charged Particle

6.

Drift in a Magnetic Field of

Constant Gradient. 9

(a) Derivation of Basic Equations. 9 (b) Comparison with Alfven's Result. 12 (c) Exact Solution in Terms of

Elliptic Integrals.

Case 1. Electron does not enter Region of Reversed

13

Magnetic Field. 13 Case 2. Electron enters Region of

Reversed Magnetic Field. 15 (d) Discussion of Results. 18

Section

1.

2.

Chapter 2

ESrrDIATIOH OF 'rHE IJlA.XIlviUM TEMPERATURE IN A RADIALLY CONSTRICTED

P:r.A'SMA-BE'rWEEN ELEC'EHODES Introduction.

Application of the Saha Equation.

25

25

3.

Preliminary Discussion of the Current

Density and Heat Flux Vectors. 26

4.

j and q for Conditions of Isotropy

and Extreme Anisotropy (Hall and

Righi-Leduc effects only included) .

30

(a) j and q for Isotropic

Conditions (we're < < 1 ) .

₃₅

(b) j and q for Conditions of

Extreme Anisotropy (we'r e > > 1).

36

5.

Establishment of the Plasma

Energy Equation. 38

6.

Solutions of the Plasma

Energy Equation. 42

6

.1.

Curved Stream Tube of a

Constricted Discharge. 43 6.2. Use of the Stream Function '¥

and the Electric Potential V as Orthogonal Curvilinear

Coordinates (we're «1).

46

6

.3.

Solution of the Energy Equation

when the Heat Flow is every-where Perpendicular to the

Flow of Electricity (W_{e 're«1). 50}

6

.4

.

Solution of the Energy Equation

for Longitudinal Flow (w_e're«1).

51

Case 1. Streamlines Parallel to

the Axis of Symmetry. 52

xix Section

Case 2. Streamlines Curved. 57

6.5.

Solution of the Energy Equation

for Longitudinal Flow (w_e't"e»1). 62 7 . Compari son of Maximum Temperatures

8.

1.

and Discussion of Results. -Keferences.

Chapter

3

THE INFLUENCE OF THERMOELECTRIC EFFECTS ON THE IVLAXINiUM TEMPb;H.ATURE IN A RADIALLY

CONSTRICrrED PLASMA BE'rvVEEN ELECTRODES Introduction.

2. Isotropic Forms of j and q (Seebeck

71

and Peltier effects only included). 72

3.

Solution of the Plasma Energy Equation

1.

for Strictly Longitudinal Flow. 3.1. Curved Stream 'rube of a

ConstricteQ Discharge.

3.2. Solution of a SL~plified Form of the Plasma Energy Equation using Complex Variables.

3.3.

Specialisation of Results for a

Linear Discharge.

3.4. Specialisation of Results for a Discharge having Hyperbolic Strear1l1ines.

Discussion of Results. References.

Chapter 4

A STABILITY CRITERION FOR A RADIALLY CONSTRICTED PLASNIA l3EThEEN ELECTRODES Introduction.

75

76

78

85

88

96

99

xx

Se ction .fage

2. Stability Analyses of Cylindrical

Gaseous Cqnductors: TellerYs

Stabilit~ Criterion, 102

(a) Examples of Normal Mode Analysis,

( b) EX8J.Jlples of Energy Principle

1

Analysis.

102

106

3. Thermodynamic Derivation of Teller's Stability Criterion.

(a) Calcula~ion of OW.

(b) Calculation of OU,

(c) The Flute Instability and

Teller's Criterion.

I I I

I I I

112

114 4. .Application of Teller's Criterion to

5.

6.

a Radially Constricted Gas Discharge. 118

Discussion of Results.

References.

REVIEW OF TOPICS COVERED ~~D POSSIBLE FUTURE EXPERIMENTAL Al\JD THEORETICAL

123 125

STUDIES. 128

Chapter 1

CHARGED PARTICLE MOTIONS

1. Introduction.

The motion of a charged particle under various

magnetic and electric field conditions has been discussed by authors such as Alfven (1950), Spitzer

(1952, 1956), Allis (1956), Post (1956), Simon (1959), Chandrasekhar and Trehan (1960), Delcroix (1960), and

Linhart (1960). The subject is important because,

although the macroscopic equations describing the

behaviour of an ionized gas can, in various

approxi-mations, facilitate certain plasma calculations (see,

for example, Chapman and Cowling 1953; Brueckner and

Watson 1956; Chew, Goldberger and Low 1956;

Chandrasekhar, Kaufman and Watson 1957; Green 1959;

and several Geneva 1958 papers), it becomes increasingly obvious that the subtleties of topics such as plasma

instability can, in simple cases, be better understood physically via the microscopic or particle orbit theory.

In this chapter we first dwell briefly upon the

derivation of two important charged particle "drift

velocity"results. Application of these results to a

simple case of plasma instability, following Rosenbluth

and Longmire, is then quickly reviewed, and, after some

remarks on Alfven's approximate drift velocity solution

for a charged particle moving in an inhomogeneous

magnetic field, an exact solution is obtained in terms

of elliptic integrals for the motion of a charged particle in a magnetic field having constant gradient perpendicular to the field direction.

Ch .1-2 2

In this work we adopt the symbols E and H . respectively for the space-time averaged Lorentz

electric and magnetic field vectors (Rosenfeld 1951) , and, by considering the current and charge densities explicitly in the averaged form of Maxwell's field equations for a vacuwn, draw no distinction between either E and the electric displacement, D, or Hand the magnetic induction, B (See also Spitzer 1956, p.23). The space-time averages are evaluated for regions

which, although small on the macroscopic scale, are nevertheless large compared to regions in which the microscopic electric and magnetic fields can be

expected to vary significantly; that is, for regions which must be large enough to contain many charged particles.

2. Charged Particle Drift in Crossed Electric and Magnetic Fields.

lf ~ is the constant relative velocity between chosen moving and fixed frames of r eference, and the subscript 1 on a field vector indicates that it is perpendicular to

y,

then, from the special theory of

relativity (well discussed by Stratton 1941, McRea 1954) it is possible by means of the Lorentz

trans-,

formation to relate the electric field, Eo , _{- ..1-} and

,

the magnetic field, _{H 1.'} in the moving frame of reference, to _E.l and _H.l in the fixed frame of reference, as follows:

E +

_Y

X g.l

,

-.1

EJ. :: (2.1)

J

2

Ch.1-2 3

and

1

=

H1. -

c

__

V x E c2 _{- -1.}

(2.2)

J

1 - (V/C)2

where c is the light velocity, and electromagnetic

*

units are employed .

The components of the total electric and magnetic field parallel to v are unchanged by the Lorentz transformation.

As pointed out by Alfven (p.6) and Post (p.348), when the el ectrical conductivity of an ionized gas is high, electrostatic fields are not readily supported. Hence if we can assume that the el ectric field is

small compared to the magnetic fi eld, and that v < < c, equations (2.1) and (2.2) reduce to the simpl e forms

E' = E + v x H

(2.))

and

H'

=

H

where the components of the field vectors parallel to v have been included in these results.

Alfven remarks that equations (2.3) and (2.4),

which indicate the independence of H and the

dependence of E on the coordinate syste~, are omitted in most treatises on electromagnetism.

The relativity of the el ectric field can l ead to a familiar drift velocity result as will now be shown.

Using the superscripts 1. and II to denote respectively the electric field components perpendicular and

parallel to the total magnetic field,

g,

it is

*Electromagnetic units will be employed throughout this work in conformity with those of Spitzer

Ch.1-2 4

evident from (2.3) that the term v x H cannot cancel

E~

but that it can cancel

E~,

so that

E'~

can be

made to vanish. From E' 1

=

E.l + V x H == 0 we obtain by operation with H x the equation

H x E.l + H x (,y x H) =:: 0, w.hich l eaves the component of v parallel to H undefined. Expansion of the

vector triple product gives

v

.

H El x H

v

-

H

-

=

(2.5)

2 2

H H

Without loss of generality we can now impose the condition v ' H =:: 0, to obtain the simple form

v

E.l X H

= : : -2

H

E x H

=

- - -

₂

H

If E and H are

asslli~ed constant in space and time, this result may be

displayed as

v =

-e

E x H

Since, for this velocity, there is no electric

field perpendicular to H in the moving frame of

reference, it follows that a charged particle will

(2

.6)

perform either circular orbital motion about a magnetic line of force, in a plane perpendicular to the direction of H'

-'

or helical motion of constant or increasing pitch about a line of force , depending on whether there is a uniform or accelerating component of the motion parallel to H

--

'

i . e., depending on whether Ell is zero or finite, Here we neglect the effect of

collisions on the motion parallel to H.

From the fixed frrune of reference, plasma in the moving frame of reference will appear to be moving with

Ch.1-3 5

Wi th E,II _{pr esent, the resultant charged particle}

motion in the moving frame as viewed from the fixed

frame will be a sideways-drifting helical motion of

increasing pitch. From these r emarks it follows that

ve must bo the electric drift velocity particularly

well discussed by Spitzer

(1

956

,

pp.3-5), and also

treat ed by Alfv~n (pp.

47-48),

Spitzer

(1

9

52,

pp. 30

1-307),

Allis (pp.

3

8

4-3

8

5),

Post (pp.

353

-

354)

,

Simon

(

p

.26),

Chandrasekhar and Trehan

(

pp

.

1

8

-1

9)

,

Delcroix

(

p

.

45)

and Linhart (pp.

21-24).

result

(

2

.

6

)

for particles of the

Consideration of the

having charge q.

1 and mass n1i ,

·th

1 species ,

l eads to the

important conclusion that ve(i) is independent of

q.

1m

..

1 1

3.

Charged Particle Drift in Crossed Force and Magnetic Fields. Physi cal-Me"chaIiTsm of .Plasma Ins ta bili ty .

The electric drift veloci ty ,Y_e can be

generalized by replacing E with _,F _-i

Iq

_{i '} where _{- 1} F·

is a constant external force field. 'rhen, wi th H

again constant,

v

=

-d

F. x H

-1

dependent on the sign of qio

( 3 .1)

Various drift velocities (polarization, curvature,

magnetic gradient and those given by

(

2

.6)

and (J.l)

above) are discussed and tabulated by Chandrasekhar

and Trehan (pp.

14-36,

6

5-

96

).

Here we note particularly that the qi have the

direction of motion E x H in constant, crossed

Ch.1-3

whereas for constant, crossed ext ernal force and

magnetic fields, the q.

l of opposite sign proceed in opposite directions.

Hence, for a plasma having characteristic

dimensions much in excess of the Debye length (Post ,

pp.

349

-

350;

Spitzer

1

956,

pp.

16

-

17;

Francis

1960)

,

these results indicate that where an uninhibited drift

of charged particles can form, the E x H drift will

produce, under steady-state conditions, zero el ectric

current, while the F. x H drift can produce an - l '

-electric current of finite value.

These sign-independent and sign-dependent drifts

can lead to a clear appreciation of the detailed

physical mechanism of plasma instability where the

geometry is simple, as has been emphasised by

Rosenbluth and Longnire

(1

957).

To illustrate this 6

point we sketch the argwuent given by the above authors

for the hydromagnetic anology of the classical Rayleigh

-'Taylor instability of non-conducting fluid dynamics.

This type of instability was first analysed hydro

-magnetically by Kruskal and Schwarzscnild

(1954),

who

used a macroscopic approacn; the hydrodynamic case is

treated by Lamb

(1

945)

.

A horizon tal plasma (heavy fluid) is Lllagined

supported against gravity by a uniform horizontal

magnetic field (weightless fluid), the resulting

in-terface being plane horizontal, as in Fig. l(a).

The uniform magnetic field that penetrates the

plaSIlla is asslJIlled attenuated with J~espect to the

oxternal field. The combined effect of the change

Ch.l ₇

j

!

J

k

· .

.

.

. . . .

.

0.!=L j

j

,k

unit vectors

·

_·

_.

.

.

.

.

.

.

. . . .

.

.

.

/

H

urw~rcLs, il1 x.-d..irec.t,·oh.

eeL)

PLa..ne.,

hori:z.ont"-a..L in"terfo-ce- be.tween.

t

La..

sma- a.. n cL m

a...j

net i c... fie

L

~}

H·

z

o

J

.

.

Pos·,rive ·,on. bouncLa.r

J /

E Lec.tron. bouncLa.r

J

H

ufwa.rols, in )(- oLif'€.c.ti 0 I'\....

Fi

_J

.

i

I

LLustra..tiovv

of+he.

~Jd..roW\.o...Jn~tic­

A

no...Lo~LJ

ot the..

R-CL

C11.1-4 8

pressure gradient in the plasma initially balances the

gravitational force. It is then assumed that the

in-t erface is perturbed as shown by the solid line boundary in Fi g. 1 ( b) . Here the q. (ions and el ectrons) in the

l

electrically neutral plasma are under the influence of crossed F. _{- l} = m. _{l _} g and H fields (g the gravitational acceleration vector), and so proceed in opposite

directions by equation (3.1). Since charged particle drifts are soon inhibited in the perturbed boundary region of this plasma configuration, charge separation results within a surface layer, and consequently charge appears on the undul ating boundary surface shown in Fig. 1 (b) . Attendantly, el ectric field vectors ' E

~ Y and k Ez are creat ed, and, from equation (2.6) , mass motions in the direct ions j Ey x

li

and _{k Ez x H}

take place. It is at once seen that the

1

Ey x H drift is in the z-direction, and, as the detailed

mathematical treatment of Rosenbluth and Longmire shows, the corresponding mass motion t ends to encourage growth of the perturbation at A and B, Fig. l(b), and the

syst em is unstable , with a rate of growth identical

with that found in the conventional Raylei gh-'raylor case,

An interesting practical identification of R ayleigh-Taylor instability in a stabilized linear pinched plasma between electrodes is described by Albares et al (1961). However, it is not appropriate to take up a more

general study of plasma stabil ity her e, and so this is conveniently deferred until Chapter

4.

4.

Charged Particle Drift in an Inhomogeneous Magnetic Field; Introductory Remarks.

A sDnple expr ession for the sign-dependent drift

Ch.1-5 9

magnetic field has been obtained by Alfv~n

(1

9

4

0;

1

950

,

pp. 13-23), who, in his first-order theory, considered

the inhomogeneity as a small perturbation of a uniform illagnetic field. Spit zer

(1

956

,

pp.

6

-7)

di scusses

Alfven's drift result, and emphasises that, in contrast

to the cases of E X Hand g X ~ drifts , the magnetic gradient drift vel ocity, of direction H X V H for a

positively-charged particle, can now only be found by

means of an approximat e theory. While this r emark is probably true for rather general magnetic fiel ds, it i s neverthel ess possi ble to obtain an exact solution

for the drift vel ocity of a charged particle in a magnetic field having constant gradient perpendicular

to the direction of the main fi eld (Seymour

1

9

5

9

).

This exact solution, which is developed in Section 5,

yields as approximations Alfv~n's result and the case

of circular or bit, and includes the case where the

magnetic field vanishes at points in the particle's

orbit , for which perturbation illethods are inappropriate.

5. Exact Solution for Charged Particle Drift ~~~agnetic Field of Constant Gradient.

(a) Derivation of Basic Equations.

Consider f irst the motion of an electron, and suppose the i.nagnetic field is in the z-dir ection and

is represented by Hz

=

"'A x. Then, from the non

-relativistic equation of motion we have, in the absence of el ectric field,

e

ill ₌ vXH ( 5 _.l)

d t c

Ch

.1-

5

10

is - e/c (Spitzer

1956,

p.l).

The Z-coillponent of v is constant, and need not

be considered explicitly, The other components vary

as

V2 ₌₌ v2 +

X

d Vx e )

ill ₌₌ A x Vy )

d t c )

)

(

5

.2)

)

d Vy e )

ill _ _

== + A x Vx )

dt c )

)

is constant, and in the notation of

Fig. 2, Vx == v cos W, v y == v sin W •

these relations in

(5

.

2)

gives

Substitution of

d W e A x

=

( 5 . J)

d t ill C

Since v == d s/ d t, where s denotes distance along

the electron trajectory,

dlf eAX

==

-d S ill C V

TI1us

e A clx eA

- - - -

= - - -- _{cos If ,}

d S ill C V

and so

(::)

2 e A

== sin W + constant. (5.6) ill c v

From (5.4) and ( 5.6)

;t

Jx~

2 ill C V

X _{== + sin W}

,

_(5.7)

e A

where x

Ch.1 11

H::::.o

_ _ O~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~j

Ma..jne..-t"'G

+ieLL

uFwa..rcLs

X,.-

-

- -

HI

X1.- -

~-- ~-- ~-- ~-- ~-- ~-- ~-- ~-- H

_{I 0}

D.j

- - -

-

-

--

- -

-

- - H

1.

x - - -_o

-:O"lrec. t,"OV"l...-

of

d..l"ift v

-x

Ch.1-5 12

For x positive,

(

5

.

])

gives

m c d'¥

( 5 .8)

t

-e t.. 2 _{2 ill C V}

Xo + sin '¥

e t..

Also , since

dy

sin '¥ ( 5 .9)

=

ds

( 5 • 4) gives

m c v sin '¥ d '¥

(5.10)

y=

e A. _x2

0 +

2 m c v _{sin '¥}

e A.

(b) Comparison with Alfven' s Result .

For motion which does not cross the line H

=

0,

as in Fig. 2, (5.10) gives the y-drift per cycle

exactly as 271:

sin'¥ d '¥ _(5.11)

o

where Ho == t.. _{x o '} and ~o== m c v i s the orbitys

eHo

radius of curvature for field strength H _o'

When Po « x_o' which implies electron orbital

motion far from the y-axis, a first-or~er result for

the drift velocity in the y-direction can be readily

obtained. After expansion of the denominator in

(5.11),

integration gives approxlinately

!:. Y

=

2

71: Po ==

2

7C t.. Po

Ch,l-5

13

Similarly, from (5.8) the periodic time in this

case is

r

r

=

2 7C Po _{and so the corresponding drift}

v

velocity in the y-direction is given by

6 y

T

=

(5.12)

This is the result obtained by Alfven, and

discussed by Spitzer and Post (p.354).

When Ho has a fixed value and A approaches

zero, we obtain the circular orbit result for a

homogeneous magnetic field, i.e. _va

2 7C _{Po 27C e}

T _{= = - -}

,

where w _{c =}

v Wc _m

called cyclotron angular frequency.

(c) Exact Solution in Terms of

Elliptic Integrals.

= 0 and

Ho _{i s the} so-c

To complete the exact solution, two cases require

consideration:

Case 1. Electron does not enter Regiqn of

Reversed Magnetic Field.

For an electron motion which does not cross the

line H

=

0, as shown in Fig. 2, the limits of x

ar e, from (5.7)

2 m c v

e A

for '¥ = 3 7C /2,

(5.13 )

for 'If = 7C/2 ,

so that

°

~ x < x < X

1 0 2

Using the sUbstitution '¥

=

7C _ 2

¢,

we 0 btain

Ch.1

-

5

14

from

(5

.

10)

the drift per cycle in the y-direction as

'lC

2m c

J

(1

= e H2 v

J

1 o

!::'y - 2 sin

2

¢)

_d

¢

, 2 . 2 r / .

- K Sln '{J 1

where A x₂ = H2 and k~ = .L~~:

::

~

,

if P;2 is

e A x₂ x₂

the orbit' s radius of curvature

r::

:

'

Jd:s:r:n

g

:hJx~2~~

,

\iV'hen x

1 ~ 0,

(5

.

13)

gives

so that the upper l imit of is unity.

Reduct ion of

(

5

.1

4)

to standard form for elliptic

integrals results in

t

~

/

2

7C/2

2

J

1

-

k~

• 2

¢

d¢-

(2

-

k~)J

d ¢

l1y=x Sln

2

J1

-k~

. 2

¢

Sl!l

a

a

(5

.

15)

where K and E are complete elliptic integrals of the

first and second kind respectively, of modulus k1

(Dwi ght

1957)

.

The periodic time is derived from

( 5

.

8)

as

'lC/

2

4 ill C

f

J

1

d

¢

k

2

1

(

5

.

1

6

)

T =

₌

x -₂ - _{K ·}

e H2 v

-

k₁ 2 sin2 _¢

The exact drift velocity i s, therefore, from

(5

.

15)

and

(

5

.

16)

l1y

_V(

~(1

-

E

)

-)

(5

.

17)

v --_.-_{a - =}

T _k2 _Ie

Ch.1-5 15

Since(~

for the drift

k2

1

velocity is always in the negative y-direction.

2

v

1;2

k «1, Alfven's result va =

-1 8

lifhen is

again obtained.

l~lfhen k~ = 1 , which occurs when x₁ == 0, we have

va = - v, and the electron moves along the line H == O.

For x < 0, the electron drift pattern is precisely the mirror b"Uage in the y-axis of that shown in Fig. 2.

Again the drift is always in the negative y-direction.

Case 2, Electron enters Region of Reversed

Magnetic Field.

Consider now a motion in which the electron crosses

the line H

=

0, and tllUS enters a reversed magnetic

field, as shown in Fig.

3.

when x

=

0 and Vx > O.

Let ¥o be the value of ¥ Then in place of

(5.7)

v .

(sln ¥ - sin ¥o )

and so the limits of x are now

~

J

2 In C v (1 - sin ¥o ) for ¥= 71./2. (5.19)

e f..

In this case various drift patterns may occur, as

shown in Fig. 4, but each pattern possesses s~llinetry about the ,V-axis, on which H = O.

¥

o =

Introducing as before W = 71. 2

2

2 rI.

'P o' From Figs. 3 and 4,

2

¢,

then

71. 71.

< ¥ «

+-2 " 0 " 2

0<0₀ < 71. _and

the above limits thus reduce to

,

"

"

2

x

Ch.1

H==-o

x

Mo...Jne.tic..

fie.Lc:L.

d..owYlWo..rd....s

:Dire.c.tio\'"\

of

eLY-'l f t

v

-

- -

x

~

16

Ch.1-5

17

Utilising the properties of symmetry exhibited by

the drift pattern of, say, Fig. 3, then for positive x

in

(5.1

8

)

we obtain from

(

5

.4)

and

(5.

9

)

/:;y= AC

=

2 ... \B - 2 TIl C V sin '1' d '1'

e f...

'1'0

J

2

ill C v (sin 'l' - sin 'l'ol

e t..

0

₀

₀

0

cos2

\1

d

0

-

~

h Sin2

d0

o

-

sin2

0

₀

0

-0

0

If another variable of integration G is defined by

sin

¢

=

sin

0

₀ sin 8,

(5.21)

may be transformed to

:::

8

~

e A.

where k2 sin

₀

0

=

=

x

2

Similarly

T

=

4

(

E

-

~

K)

2

F

4

ill C V 1

~

J~

K

using

and the drift velocity in this case is

E

(2 _ - 1 ) K

( 5 • 20) •

(5.21)

Ch.1-5

E k2

When _k2 «1, ~ 1

-

2 and

K ~ 2

18

( 1 2

Va ~ V

-k2 ) N V

N (5.25)

as expected from Fig. 3.

For ElK

==~,

corresponding to

0

₀

~

650, the

drift ve10ci ty becomes zero. As ¢o increases

beyond 65 0 (ElK < ~) the drift veloci ty becomes

negative, as in Case 1. Fig.

4

gives typical drift

pa tterns for ElK < ~, ElK == ~ and ElK > ~. Since in

this Case va becomes + v and - v for k2 == 0

and k2 == 1 respectively, it follows that at the limit

¢

=

0 the electron moves along the line H == 0 in

o

the positive y-direction, while at the limit ~o== ~/2

it moves along the same line in the negative y-direction,

Thus the drift patterns of Cases 1 and 2 coincide in

the limits k == 1

1 and k2 == 1 •

For positively-charged particles, the principal

results are that the formulae for k1 and k2 relilain

unchanged, if the particle charge is + elc, whereas

the va are changed only in sign.

Since dx

ds

=

cos '¥ ,

it can be readily

ascertained that the drift velocity in the x-direction

is zero in all cases.

(d) Dis cussion of Results.

The drift velocity results obtained in Cases 1

and 2 for an electron are shown plotted against the

in Fig.

5.

This

[image:38.571.12.554.14.734.2]

x

E

I

- <

K

2.,

M

~ n~

t,·e,..

fl·e.LL

cLow'I\wof"J..s

A

rrovvs sho""

cL i ,.. e.c

t

"0 If\. of

Fart icLe cLrift

-E _

I

---"v

" E I

->"2-

_K

- -

-K

2..

M

0-5

VI e.

tic

.f

I'e.

L

d.- ()

F

\IV Gl. r"' c::L S

"""-- """--

-F

i

9

.

'

+_-,

j

tic

a.

L

E

L

e

c.

t

ro

n. ] ) (' ;

f

t

P

a.

t

t-

e

r

V)

s

i

VL.

-re

r

VV' S 0

f

V VI

for Ca..se

~.

J

I

E

lK

(")

~

f-'

f-'

Ch.l 20

~

--d

- - - - _c

0 d

~ ~ V) Q) V> d

C>

U

u;

~

0

L\->

0

(.

>

~ _{~ ~}

r<-eV \I) d U 0

("r) .-C

+l

_.

-$

>

()

~

N ::> Y-O

C

0

.-N

_-t.J

CV

<n _d

d

U _i..

.t.

_~

lJ) lI) It)

~ ~

0 _r- It, N

...,

_r- 0

0 0 0 0 0 0 0

-

.

_lr)

I

,

I

Ch.1-6 21

the region of Case 1. For 1 /k1'~ 00 , we have the

well-known region of Alfven drift velocities. In spite

of the smallness of Alfven drift velocities r elative

to particle velocities, it has been suggested that

charge separation effects could be obtained in a plasma ,

and that these might lead to motion of the plasma

towards the region of weakest magnetic field (Post,

p. 354). Larger drift velocities, of the order of the

particle velocities, are expected, however, in the

region _x

J

_eX

2 4 m c v < 1 •

Thus if the magnetic

field within a plasma varies in a direction normal

to the field, and somewhere changes sign, the motion

of the charged particles in the neighbourhood where

it changes sign may lead to larger charge separation

effects than in the Alfv~n region.

Recalling the briGf comments given in Section 3

on the physical mechanism of instability in a simple

plasma geometry, we see that if charge separation is

produced in a plasma by non-uniformity of the

magnetic field, instability could similarly be

initiated or au@aented. In fact , if larger charge

separation effects than produced by the Alfven drift

are encountered, it is likely that the instability

growth rate will be increasod.

6. References.

Albares, D. J. , Krall, N. A., Oxley, C. L. (1961).

-Rayleigh-Taylor Instability in a Stabilized

Linear Pinch Tube. General Atomic Report

Ch.1-6

Alfv~n, H. (1940) . - On the Motion of a Charged

Particle in a Magnetic Field. Ark. Mat. Astr.

Fys. 27A, No. 22: 1-20.

Alfven,

n

.

(1950). - (ICosmical Electrodynamics."

pp. 13-23. (Clarendon Press: Oxford.)

22

Allis, vv. P. (1956). - "Handbuch der l)hysik.1i _{pp. 38}

4-392, Vol. 21. Edited by S. Flugge. (Springer:

Berlin. )

Brueckner, K. A., ~ivatson, K. M. (1956). - Use of the

Boltzmann Equation for the Study of Ionized Gases

of Low Density. II. Phys. Rev. 102 : 19-27.

Chandrasekhar, S., Kaufman, A. N., vl/atson, K. M.

(1957). - Properties of an Ionized Gas of Low

Density in a Magnetic Field. III. Ann. Phys. 2:

435-470.

Chandrasekhar, S., Trehan, S. K. (1960). - "Plasma

Physics. I? pp. 14-36. ('rhe University of Chicago

Press. )

Chapman, S., Cowling, T. G. (1953). - "The Ivlathematica1

Theory of Non-uniform Gases." pp. 107-133, 134-150,

319-358. (Cambridge University Press: London.)

Chew, G. F., Goldberger, M. L., Low, F. E. (1956).

-The Boltzmann Equation and the One-fluid Hyd

ro-magnetic Equations in the Absence of Particle

Collisions. Proc. Royal Soc. A, 236: 112-118.

Cowling, T. G. (1957). - !?I'Jlagnetohydrodynamics.1! p. 3.

(Interscience Publishers: New York and London.)

Delcroix, J . L. (1960). - "Introduction to the Theory

of Ionized Gases. 11 pp. 42-56. (Interscience

cnol-6 23

Dwight, Ho B. (1957). - "Tables of Integrals and Other

:Mathematical Data.11 _{pp. 272-274. (The Macmillan}

Company: New York.)

Francis, G. F. (1960). - :'Ioniza tion Phenomena in

Gases .l i _{pp. 253-254. (Butterworths Scientific}

Publications: London.)

Geneva Papers. (1958). - "Proceedings of the Second

United Nations International Conference on the

Peaceful Uses of Atomic Energy.1I P/2300 (pp.

99

-Ill), p/1307 (pp. 134-136), P/365 (pp. 137-143),

P/349 (pp. 144-150), P/2214 (pp. 151-156), Vol.31.

(United Nations Publication: Geneva.)

Green, H. S. (1959). - Ionic Theory of Plasmas and

Magnetohydrodynamics. Phys. Fluids 2: 341-349.

Kruskal, M., Schwarzs child, M. (1954). - Some

Instabilities of a Completely Ionized Plasma.

Proc. Royal Soc. A, ,223: 348-354.

Lamb, H. (1945). - lIHydrodynamics.!1 pp. 370-371.

(Dover Publications: New York.)

Linhart, J . G. (1960) . - ItPlasma Physics.?? pp. 6-53.

(North-Holland Publishing Company: Amsterdam.)

McRea, W. H. (1954). - l?Relativity Physics. it _See

particularly pp. 50-51, Ch. VI. (Methuen and

Company: London.)

Post, R. F. (1956) . - Controlled Fusion Research

-An Application of the Physics of High

Tempera ture Plasmas. Rev . .lvlod. Phys. 28:

353-355.

Rosenbluth, M. N. , Longmire, C. L. (1957).

-Stability of Plasmas Confined by Magnetic

Ch.1-6 24

Rosenfeld, L. (1951). - "Theory of Electrons.1i

pp.

13-27. (No~th-Holland Publishing Company:

Ams t erdam. )

Seymour, P. W. (1959) . - Drift of a Charged Particle

in a Magnetic Field of Constant Gradient. Aust.

J. Phys. 12: 309-314.

Simon, A. (1959) . - IIAn Introduction to Thermonuclear

Research.?? pp. 24-26. (Pergamon Press: London.)

Spitzer, L., Jr. (1952). - Equations of Motion for an

Ideal Plasrr-a. Astrophys. J. 116: 299-316.

Spi tzor, L" Jr 0 (1956). - l1Physi cs of Fully Ionized

Gases,n pp. 1-7. (Interscience Publishers:

New York and London.)

Stratton, J. A. (1941). - l1Electromagnetic Theory."

pp. 78-81. (McGraw-Hill Book Company: New York

Chapter 2

ESTU1ATION OF THE MAXJJVIUM TEMPERATURE IN A RADIALLY CONSTRICTED PLASMA BETWEEN ELECTRODES

1. Introduction.

In this chapter the flows of heat and electricity

in a steady non-equilibrium state gas discharge between

a pair of electrodes are examined, with the principal

object of determining theoretically the maximum plasma

temperature Tm' For simplicity, the treatment is

confined to discharges of highly-ionized gases having

low bremsstrahlung loss and free boundary surfaces

perfectly insulated thermally by a high vacuwn. Non-constricted and Non-constricted discharges are treated.

2. Application of the Saha Equation.

The Saha equation (Saha

1920,

Saha and Saha

1934,

Allis

1956,

Cobine

1

9

58)

applies to a gaseous system in

thermal equilibrium, v:hereas we shall consider sys tems which depart from this condition by virtue of an excess

of the electron temperature over the ion temperature, as discussed by Allis in the above reference, and

treated in greater detail by Alfven

(1950).

However,

as Allis points out, the temperature can never become

less than that given by the Saha equation, and

accordingly we apply it here to obtain some idea of

the minimum. temperatures at which a gas may be regarded as substantially ionized as the particle density varies

over an appropriate range. Thus, applying Cobine's form of the Saha equation to a plasma, having equal concentrations of singly ionized atoms and electrons

(ne = _{n i )} we obtain

2

5,050

V.

log (_(1_)

=

1.5 log T - (N- 1

5.385) _.

l ,

(2.1)

1-(1 T

Ch.2-J

26

where, if na atoms/cm~ is the original concentration

of the gas,

Q ~ ni/na is the degree of ionization,

T is the gas temperature in degrees Kelvin,

N ~ _{log n a ,}

v· is the ionization potential of the gas in volts,

l

Here our main interest will be in the hydrogen

isotope, deuterium, and so the temperature dependence

of Q for hydrogen has been calculated from equation

(2.1) for N-values varying from 14 to 20, and presented

in Fig. 1.

For substantially-ionized hydrogen gas (Q - 0.9 say)

the minimum temperature for a given value of N can now

be readily obtained. It is of interest to note from

each of these curves that this minimum temperature lies

well below the temperature corresponding to the

ionization potential of the gas concerned, particularly

for the smaller values of N.

J.

Prelim~narz Di.scussion of thELQurrent Density and Heat Flux Vectors.

In the absence of an external magnetic field H,

the flows of electricity and heat in a system mutually

interfere, and as Callen (1948) has pointed out, the

thermoelectric effects of Peltier and Seebeck may be

viewed as the result of this interference. Further,

if an external magnetic field is impressed on the

system, thermomagnetic and galvanomagnetic effects

(Ettingshausen, Hall, Nernst and Righi-Leduc effects)

appear. To account for these effects, we shall now

follow the procedure adopted by Marshall in his

\·0

0·9

o·s

0·7

cC 0·61

_://;/

:z

0·;

0'4

0·3

0·2,

0·\

o

104

I

0

'\I.

2,..104 3-xI04- 4)(10'"

h~ o..tow\~ /c.t'f\~

(

Vl"

=

ION)

n

~ 5'''~\'J ~Ol'\i~e.cl ()..'corn'>/c.~~

_/ ni. j r . . L '

' " " - - L h Q . ) o~,re(... 01' lOnl,C).t'IOI'\

T}

~o..S kMpU-e"lUI'l-- ~ dC-jree.s Ke.lvI·~.

5~104- '"./0'" 7)(.104 8}( /04- 9,clo4 _/05

T°k. ~

Fill

Sa.ha... Curves -For

HJcLroJ-en...-.

o P'

N

Ch.2-3

anisotropic by the presence of a magnetic field, write

the conduction current density vector, j, and the

heat flux vector, q, as

j==OI~" + OIIE.L + OIIIh~.L + ¢I (9T)lt +0'11 (9T}.L +¢IIIhx( 9T).L,

\ I \ \ ---J

No special Hall Seebeck Nernst

28

name

().l) where

and

E is the electric field,

Ell _{is the component of E parallel to H,}

E.L _{is the component of E perpendicular to H,}

h = H/H is the unit vector in the direction

of H,

VT is the temperature gradient,

( VT)II _{is the component of 9T parallel to H}

-'

(VT)J. is the component of VT perpendicular

to H,

cj I, cj II, and (J III are coeffi cients of

electrical conductivity,

¢ I, ¢ II, and ¢III are thermal diffusion

coefficients,

q=_r...I(9T)"_r...II(9T).L_rtIIhx(9rr).L+1 jll+~I j.L+ _{'1'IIIhxj J.} ,

.

-

-().2)

where

jll is the component of j parallel to H,

ji is the component of j perpendicular to H,

f...I, f...II and r...III are thermal conductivity

coefficients, '1'1, 'I'll, and '1'111 are coefficients

giving the heat flux due to electric currents.

Ch.2-J 29

q = _KI (VT)" _ KII(VT).L _KIII£x (VT) .L+-f:IEIl + r;IIE.L+ t;IIIhxE.L ,

-

_\

.

-

-

_{I \}

-

_{I \ -1 \ I}

No special name Righi-Leduc Peltier

Ettings-hausen

( J. J)

where are thermal conductivity

coefficients when electric currents are allowed to

flow, given by

KI =

,,.1 _

\][1 ¢I ,

KII= All _ \][11 ¢II + \][111 ¢III ,

KIII = AlII _ \][11 ¢III _ \][111 ¢II

and r;I, r;II, and r;III are coefficients accounting

for the contribution to the heat flux from the electric

field E, given by

1;;1 = \][1 0 I ,

r;II \][11

6'

II _ ,¥III ~III

(J .5)

=

o

,

t;III = \][11

0

111 + rII (j II

The results (J.4) and (J.5) are not given explicitly

by Marshall: they can alternatively be obtained from

results given in Part J of his report (pp. 28-29) by

neglecting the electron mass me relative to the ion

mass mi' and by noting that the K's here correspond

to Marshall's 8's.

Further information on the terms representing the

effects discovered by Ettingshausen, Hall, Nernst,

Peltier, Righi and Leduc, and Seebeck in the expressions

(J.l) and (J.J) for j and q is provided by Chapman

and Cowling (195J), Hix and Alley (1958), Linhart (1960)

and the above article by Callen.

An attempt to base the followin~ analysis on