Hational Bureau of Standards

library, H.W. Bldg

APR 8 19E

^ecknical v2©te 164

ON PLASMA COLLISION FREQUENCIES

PROPORTIONAL TO ENERGY IN

THE RADIO WAVE REFLECTION

AND TRANSMISSION PROCESS

J. R. JOHLER AND JOHN D. HARPER, JR.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

^scUnlcal Cf\ot& 164

ISSUED MARCH 1, 1963

ON PLASMA COLLISION FREQUENCIES

PROPORTIONAL TO ENERGY IN

THE RADIO WAVE REFLECTION

AND TRANSMISSION PROCESS

J. R. Johler and John D. Harper, Jr. Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. Price 40 cents

CONTENTS

Page

ABSTRACT iv

1. INTRODUCTION 1

2. THE COMPLEX INDEX OF REFRACTION OF THE PLASMA 2

3. BOUNDARY CONDITIONS 6

4. CLASSICAL MAGNETO-IONIC THEORY 12

5. ELECTRON COLLISION FREQUENCIES PROPORTIONAL

TO ENERGY 13

6. COMPUTATIONS AND DISCUSSION 15

7. CONCLUSIONS 23

8. REFERENCES 24

APPENDDC 1 30

APPENDDC II 33

APPENDIX III 3 9

FIGURES 44

FOREWORD

This work was sponsored by the U.S. Air Force as part of Tasks

4 and 5 of the Rome Air Development Center D.O. AF 30 (602) -2488,

March 1961.

in
ABSTRACT

The high conductivity of the ionosphere in large measure determines the propagation of terrestrial radio waves. The ionosphere can

be treated theoretically as a magneto-ionic plasma composed of electrons, ions, and neutral particles. Certain macroscopic properties of

these particles, such as the complex index of refraction, can be deduced from the microscopic particle statistics of ionized gases.

These properties can then be applied directly to Maxwell's equations

to determine the reflection and transmission process, provided a suitable composition for the model plasma has been found.

The composition of the real ionosphere is not uniform, especially in the vertical direction. Such nonuniformity can be treated theoretically with tne aid of a flexible model ionosphere employing the notion

of a continuously stratified plasma.

An extensive survey of both the index of refraction of typical

electron-ion plasmas as models of the ionosphere together with the

reflection coefficients is made in this paper to determine the significance of the notion of heavy-particle electron- collision frequencies

proportional to energy.

The introduction of electron-ion collision frequencies with a linear dependence upon the average value of the Maxwellian particle

velocity distribution results in some interesting changes in the detail

of the reflection and transmission coefficients at low frequencies. It can be noted, however, that the reflection coefficients determined by

application of the continuously stratified notion to particular models

of the ionosphere are not drastically changed by the introduction of

such electron collisions.

IV
ON PLASMA COLLISION FREQUENCIES PROPORTIONAL TO ENERGY

IN THE RADIO WAVE REFLECTION AND TRANSMISSION PROCESS

by

J. Ralph Johler

and

John D. Harper, Jr.

1. INTRODUCTION

Both the reflection and transmission process of a radio

wave at an electron-ion plasma have considerable theoretical and

practical importance. Thus, theoretical treatment of the reflection process is necessary to describe the field of a radio wave transmitted around the earth by reflections from the ionosphere. Also,

the field of a radio wave at a space vehicle either in or above the

ionosphere is a timely consideration for the theoretician.

The treatment of a continuously varying medium (varying in one

direction, such as the vertical z-direction, Fig, 1) by approximating

the medium with one or more slabs of uniform composition has been

utilized by many authors, such as Hines [1951], Wait [1953], Ferraro

and Gibbons [1959], Brekhovskikh [I960], Johler and Harper [1962],

Johler [ 1962] . This paper applies such techniques to a magneto-ionic
-2-

plasma carrying the process to the limit until the increase in the

number of slabs and the decrease in the thickness of such slabs result

in a convergence to a stable reflection or transmission coefficient.

Recently the work of Phelps [ I960] has developed the notion of

electron- ion collision frequencies with a linear dependence on the

energy exponent of the Maxwellian particle-velocity distribution.

This has raised grave questions as to the validity of the use of a constant collision frequency in the classical magneto- ionic theory, especially at low frequencies. This paper introduces the monoenergetic

electron collisions (electron collisions proportional to energy) into

the analysis of nonuniform plasma reflections and transmissions so

that a comparison can be made with the classical magneto-ionic

theory, especially at low frequencies. An extensive survey of the

index of refraction of an electron-heavy particle magneto-plasma is

made together with appropriate reflection coefficients to determine

the significance of electron-ion collision frequencies proportional to

energy.

2. THE COMPLEX INDEX OF REFRACTION OF THE PLASMA

The complex index of refraction of a non-uniform medium

varying in one direction, such as the vertical z-direction, Fig. 2,

will be examined in some detail in this paper. This is accomplished

by subdividing the medium into slabs of uniform composition. It is
-3-

therefore necessary initially to determine the complex index of

refraction of each such slab.

The index of refraction of a uniform magneto-ionic plasma can

be examined in detail with the aid of the plane coordinate system

illustrated, Fig„ 1„ The xy-^plane is situated at the lower boundary

of the model plasma defined as a position in space at which the electron

density is nil (N = 0) and above which(z > 0) the ionization varies as

some arbitrary function (or profile), N(z), N(h), where h in practice

is the height above the terrestrial sphere in radio wave propagation

problems. Also, at all z < 0, the ionization is assumed to be negligible.

The mathematical description'of the propagation in a magnetoionic medium can be formulated with the aid of a plane wave, E.,

i incident from the region below the xy-plane (z< 0, Fig. 1) at an angle

of incidence with the vertical z-direction, <j>.,

E. = |E. I exp [i (cot - - y] D)] (1)

1 ' l

c

where the field is assumed to be a time-harmonic wave oscillating in

o time, t, at a frequency, f= co/ 2tt, with c the speed of light, c ~ 3(10 )

meter/ second, and r\ the index of refraction, which for a vacuum;

Ti = r\ =1, and D = ti D = x sin d>. sin <j> + y sin <t>. cos <t> + z cos <t>..

' ' o 'o i a ' l a i The magnetic azimuth, $ , is reckoned clockwise from magnetic

ct north, Fig. 1, i.e., from the yz-plane, which contains the static

t Rationalized MKS units are used throughout this paper.
-4-

magnetic field vector, W , oriented at a dip of inclination angle

relative to the horizontal, I. A resultant wave, E , transmitted into

the model ionosphere magneto-ionic plasma (z > 0) is then assumed

to have the form,

E

t = |Ej exp[i(cot -^tj'D)], (3)

where in the plasma,

T) D = x sin <t> . sin <t> + y sin $ . cos <t> + z £,, (4)

i a l a

in which t, is in general a complex number, the value of which will

depend upon the altitude, z, and on £, = C,(z), since the complex index

of refraction, t|,

2 2 2

r) = C + sin $., (5)

varies with the electron-ion composition which in turn changes with

altitude.

The complex index of refraction, T), can be determined locally at a

particular altitude by a simultaneous solution of Maxwell's equations,

VX E + H

Q ff =0,

VX H - J - € |p-= 0, (6) o at

where E and H are the electric (volts /meter) and magnetic (ampereturns/meter) fields, J = -NeV" the vector convection current (amperes/

square meter),
-5-

and the Langevin equation of motion of an electron,

dV — — -. m— + mg V + |i e (V X W)+ e E = 0, (7)

in which m is the electronic mass, e the electronic charge, p. the

permeability of space (jj = 4n(10 ) henry/meter), & the vector static

magnetic field (ampere-turns/meter), V is the vector velocity, and g

is the collision frequency parameter (seconds ). The elimination of

the vectors J and H from the simultaneous solution of equations (6, 7)

results in the matrix equation, using the Cartesian coordinates of Fig. 1,

a

il al2 al3

a

21 a22

a

23

a

31 a32

a

33

x

= 0, (8)

where the coefficients will be defined in Appendix I.

If the field E, equation (8), exists the determinant of the matrix must

vanish. Expansion of the determinant yields a quartic equation in

£, [Booker, 1939], [Johler and Harper, 1962],

a4

l

4 + a

3

C

3 + a

2

£

2 + a^ + a

Q

= 0, (9)

with coefficients also defined in Appendix I.
The Booker [1939] quartic equation (9) is quite significant,

since the four complex roots of such an equation describe four indexes

i r of refraction, r\ = r\ '

' corresponding to upgoing (i) with negative

imaginary part (Im t\ < 0) and downgoing (r) with positive imaginary

part (Im r\ > 0) together with ordinary (o) and extraordinary (e) for

each upgoing or downgoing magneto- ionic propagation component. The

intensity and phase of these propagation components are dependent

upon direction of propagation relative to the direction of the staticmagnetic field, &, (Fig. 1) which acts on the convection currents, J,

as the Lorentz force, equation (7),

|i

-^(Jxl) = ne (VX U).

IN O

Such a medium of propagation is said to be anisotropic. Indeed, the

ionosphere which surrounds the earth is one such plasma medium.

3. BOUNDARY CONDITIONS

The detailed structure of the flexible plasma model is illustrated,

Fig. 2, as a stack of plasma slabs of arbitrary thickness (except

for the topmost slab of thickness z = °°). The number of such slabs,

P

p, and the thickness is quite flexible, since the idea of this analysis is that the measured electron density- altitude and collision

frequency- altitude (N(z) and v(z), respectively) profiles can be
approximated to any desired accuracy by decreasing z and increasing

p simultaneously until a stable reflection process is obtained.

A constant electron density, collision frequency, and static magnetic field with respect to altitude, z, is of course assumed for each

slab, z , and associated with each such slab a set of four roots, Q = Q , n n

is found to exist. Two of the roots will exhibit a negative imaginary

part (Im C, negative), corresponding to an upgoing propagation component (+ z direction, Figs. 1 and 2). Also, two of these roots will exhibit a positive imaginary part (Im Q positive), corresponding to a

downgoing propagation component (-z direction, Figs. 1 and 2). Except for the topmost slab, it is necessary to consider both upgoing and

downgoing components in this analysis.

The treatment of a continuously varying medium, i. e. , one which

varies in the vertical z-direction, Figs. 1 and 2, with one or more

slabs of uniform composition was presented in a previous paper [Johler

and Harper, 1962] . Indeed, the technique was carried to the limit at

which the number of slabs, p, for each calculation depends upon the

computation precision required and the particular values of the electric

and geometric parameters. Reflection and transmission coefficients

were calculated, and the stratification process was continued for

model electron density and collision frequency profiles until a

stable reflection or transmission was obtained. It was therefore
-8-

demonstrated, that subject to the provision that a convergence could

be demonstrated numerically, the stratified model could be employed

to represent a continuum, hence the notion of continuous stratification.

The distinction between an ordinary and an extraordinary propagation component is ambiguous and obscure in the nonuniform ionosphere. Indeed, if only the reflection coefficient is of interest the

distinction is immaterial. However, if the progress of the wave upward through the ionosphere is to be examined in detail, the distinction should be maintained to avoid a confusing analysis. This

is accomplished by an examination of the form of the index of

refraction function with respect to frequency and altitude (or electron

density and collision frequency which vary with respect to altitude).

Thus, the complex index of refraction, equation (5), is detailed for

each frequency and slab, z , r\ =T\ , and the upgoing (i) ordinary and

n n

extraordinary r\ . and the downgoing (r) ordinary and extra1, o, e ordinary ri function continuity is examined in detail as a funcr, o, e

tion of frequency to determine the crossover point of the function

for each slab or electron density under consideration. The Re r\

was employed in this analysis. Thus, above the crossover point

the ordinary wave was considered to be the greater of the two roots,

Re T\ >Re r\ , and below the crossover point the ordinary wave was
-9-

considered to be the lesser of the two roots, Re ti < Re r\ . At

o e precisely the crossover point, the two, of course, are identical,

-n = r\ . This point, for examples with low electron density examined

o e

in this analysis, was found to be below the plasma frequency,

f = go /2tt. The absolute distinction between the two propagation

components remains quite arbitrary, but the analysis must be consistent between each slab and consistent within each slab for upgoing

and downgoing propagation components to simplify the detailed study

of the wave progress in the ionosphere. The almost random possibility as to order in which the four components emerge in the numerical solution [Johler and Walters, I960] of the quartic equation (9)

(the Muller [1956] iterative procedure was employed) makes necessary

this logical procedure for sorting out the roots thus obtained.

The boundaries of each slab are introduced as an expression

for the continuity of the tangential E and H fields and the normal H

fields at each boundary, Fig. 2, of the model plasma. This is

accomplished by equating the field immediately above and immediately

below each boundary which, after considerable ado, can be expressed

in concise form as the matrix equation (10),
-10.

"1.1 'I. 2 1.

2 d

l.

*2, b

2.

3*1.4 a

l. 5*1.6

,3 b

l,4 b

l,S b

1.6

3

c 1.4 c 1.5 c

l. 6

3 "2. 4 *2. 5 2. 6 2. 7 2. 8 2. 9 2,10

3 b2. 4 b2, 5 b2. 6 b2, 7 b2, 8 b2. 9 b2. 10

"2. 3 "-2. 4 ^2. 5 "-2. 6 "-2, 7 "-2. 8 "-2. 9 2. 10

*3, 7 a

3, 8*3. 9 *3, 10 *3. 11 *3. 12 *3. 13*3, 14

b

3. 7 b3. 8

b

3. 9 b

3. 10 b3. 11 b3. 12 b3. 13 b

3. 14

c 3.7 c

3. 8

c

3. 9

C

3. 10 c3. 11

c

3. 12

c

3, 13 c

3. 14 d

3. 7 d3. 8 d

3. 9 d

3. 10 d3, 1 1 d

3. 1 2 d3. 1 3 d3. 14

P. P + « V P

+ *

C

P. P + * V P

+ 4

P. P + 9

b

p. d + 9

°P. P + 9 V P + 9

T , T em mm oe om

T . T

ee me b , boe om

eio mio c , c oe om

(D ud)

eie mie d . doe om

(i) >0(i) ero mro u'".u« l » ere mre

u(2) (2) eio mio

u(2) <2> eie mie

U<2) „U)

ero mro

u

(2 > u<2> ere mre

-tU

<p-Du(p-D

eio' mio

jjfp-Dulp-ll

eie' mie

ulp-Dulp-l) ero mro

u<p-i>u<p-i>

ere' mre

u(p) u(p) eio mio

utp) u(p) eie mie

=

(10)

where the elements a. can be deduced as a consequence of

11 ' '

these boundary conditions with results as shown in Appendix II,

provided the reflection coefficients, T, and the transmission

coefficients, U, T , are defined,

ee

em

me

y'r

y'i

x'r

y'i

ll

E

i

; mm E

x'r

X'l

ee

em

E

y't

Y'i

E

x't

y'i

Vt

me E

X'l

E

x't

mm E

u

(n) =

U

U

(n)

eie

U

E

(n)

yio

eio E

E

(n)

mio E

y'i

(n)

jdo

E

(n)

JLle

E

(n)

mie E

y'i

(n)

yie

X'l

U

(n)

ero

U

(n)

mro

U

(n)

ere

U

(n)

mre

E

(n)

y.ro

y'i

E

(n)

yro

E

E

E

(n)

vre

y'i

E

(n)

yre

E

(ID
-11-

where, Hg. 2, n = 1, 2, 3. . . , p-1, p, and the four T" are

defined as a transmission coefficient into a topmost slab of infinite

extent, z, and zero electron density, N = 0, f] = n = 1. p o The reflection coefficients, T , T , T , T , describe

ee em me mm

the reflection into the region below the model plasma, z < 0. Thus,

T refers to vertical electric polarization of the incident wave and

ee

a corresponding vertical electric polarization of the reflected wave.

T refers to the generation of the abnormal component (horizontal em

electric polarization) by the incident vertical electric wave.

Similarly, T refers to vertical magnetic incident wave and mm °

vertical magnetic reflected wave, and T refers to a correspond- me

ing abnormal component (horizontal magnetic).

The transmission coefficients, U, refer to transmission at

the particular point in the ionosphere under investigation to whiGh

the wave has penetrated. There are four propagation components

and two polarizations and, hence, eight types of transmission coefficients: upgoing and downgoing, ordinary and extraordinary, vertical

electric and vertical magnetic.

Thus, the nature of the wave reflected from the plasma, the

nature of the wave transmitted through the plasma, and indeed the

nature of the wave progressing within the plasma are completely

described by the coefficients determined by this analysis.
-12-

4. CLASSICAL MAGNETO -IONIC THEORY

The Booker [1939] quartic equation C9) is satisfied by the

famous Appleton-Hartree formula [Appleton, 1932], [Hartree, 1929,

1931],

n =«i

/ 2

S 1 2

< 12 > 2(1-—) sm w ± —- sin \\> + 4 —r- (1 • —) cos oi. s 2 \ 2 2 s

s

v s s

2 2 2

where, £> s "H - sin <j>., (13)

provided,

t 1 cos di = - — sin I + — sin <j>. cos <t> cos I, (14)

r| 7] i a • 1

or,

sin d/ = —

2 „ 2 2 . 2

ri - £, sin I - sin <j>. cos <j> (sin <t>. cos cos I

1 a 1 a

- Zt, sin I cos I)

This formula, employing real v|j, has become identified with the

classical magneto-ionic theory- This solution becomes identical

with the Booker quartic with complex \\i [Johler and Walters, I960],

i. e. , the propagation direction relative to the earth's magnetic field

is in general complex, i|i = Re ip + i Irn i|j.

(15)
-13-

5. ELECTRON COLLISION FREQUENCIES PROPORTIONAL

TO ENERGY

The particle statistics of the electron-ion gas with superposed

electrodynamic and magnetostatic fields has been investigated in

some detail since the classical magneto -ionic theory was introduced (1932). Most of these studies have arrived at macroscopic properties from the well known Maxwell-Boltzmann theory by means of

approximations of the Boltzmann equation. The Langevin equation (7)

for constant collision, g ~ v, is one such approximation. The reexamination of the classical magneto-ionic theory in recent years

can be traced to the work of Huxley [1937, 1938, 1940], Pfister [1955],

Jancel and Kahan [1955], Dingle, Arndt and Roy [1956], Molmud [1959],

Allis [1956] , and Sen and Wyller [I960] . Particular emphasis,

based on experimental work with gases in the laboratory, has been

placed by Phelps [I960] on the dependence of the collision process

upon the energy, au , where u = kT/e (k = Boltzmann's constant,

T = temperature, degrees kelvin, and e = electronic charge), with

special emphasis on the linear dependence, j = 2, on the collision

frequency, V = v(u) = au, where a is a constant. This linear dependence of the collision frequency, v(u) , upon energy has been called a

monoenergetic electron collision process. Phelps [ I960] anticipated

grave consequences as to the general validity of the classical magnetoionic theory. This paper, therefore, has introduced collision
-14-

frequencies proportional to energy into the analysis of the reflection

coefficient, and the consequences of such a theory are examined.

Phelps [i960] has developed integrals of the electron velocity

or conductivity tensor. The integration over a Maxwellian energy

2

distribution expressed in terms of energy, eu = i m V , or, f =

-i. [e/ir k T] exp [-eu/k T], involves momentum transfer collisions with

gas molecules proportional to the energy u, v = v(u) and can be

accomplished for a continuous electromagnetic wave of angular frequency, oj = 2tt f.

The notion of a complex parameter, g, in the Langevin equation

(7) was introduced by Molmud [1959]. Thus, in the classical theory,

g ~ v. In the monoenergetic theory the parameter g = f(v(u)) is a

complex and frequency-dependent number. The implication of the

classical magneto-ionic theory that such a quantity can be represented by a real effective value without a frequency dependence is

therefore not in general valid concept.

The derivation of the parameter, g, and its introduction into the

analysis is given in Appendix III.
-15-

6. COMPUTATIONS AND DISCUSSION

The precise model for the lower ionosphere has been the subject

of extensive investigation. Such a model can be presented as an electron density-altitude, N(h) or N(z), Fig. 2, profile together with a

collision frequency- altitude profile. Two such models, N(h) or N(z)

profiles, are illustrated, Fig. 3. The Houston Composite [ Waynick,

1957] represents the approximate condition of the ionosphere at

daytime-noon. A profile which was measured with a rocket sounding

by Sedden and Jackson [ 1958] is also illustrated, Fig. 3. This model

is representative of the condition of the lower ionosphere during a

local auroral-zone blackout of the type confined to the auroral zone

or some segment thereof. Measurements were not made to establish

this model below 1000 electron/cm3

. However, various methods

have been proposed to extrapolate such models to lower electron densities. The Chapman theory of ionized -layer formation can be written,

Chapman [ 1931, 1939] , Budden [ 1961] ,

r z " z -\ n N (50) = N exp i

o ^2

z - z z - z

i ° o

1 - —^ - sec x exp

. 1 where N = N /(cos y) 2 o max ' A

z = z - S In (sec y) o max A

S = RT/Mg
-16-

R is the gas constant, M is the mean molecular weight, T is the

absolute temperature, g is the acceleration of gravity, and x is

the zenith angle of the sun. The scale height, S, can in practice be

adjusted to fit measured profiles provided N and z are

max max

determined by such profiles. The gaussian extrapolation formula is

often employed, [Budden, 1955], for the daytime-noon profiles, in

which N = N , z = z o max o max

N = N exp [- (z - z ) 2

A'l, max max ' J

where the constant k' is adjusted for best fit of the measured profile.

This latter procedure was followed in the ionosonde blackout model

profile illustrated, Fig. 3 . Obviously, a variety of other measured

or theoretical profiles could be selected. These profiles were

employed primarily to demonstrate the techniques described in this

paper. The analysis of a variety of other profiles is reserved for

future work.

Numerous v(z) profiles, both theoretical and measured, based

upon the classical magneto- ionic theory, are also available in the

literature. Some of these illustrated, Fig. 4, have been deduced

by Nicolet [ 1958] , Fejer [ 1955] , Crompton, Huxley, and Sutton

[ 1953] , Gardner and Pawsey [ 1953] , and Sedden and Jackson

[ 1958] . The results of Sedden and Jackson [ 1958] indicate the
-17-

theoretical values of Nicolet should be divided by 3, hence the

Nicolet/3 v (z) profile. The Nicolet/3 v(z) profile was selected for

the analysis presented in this paper. The collision frequency can be

deduced from experimental measurements at high frequencies, (HF).

Thus, for example, Kane [i960] employed a frequency 7. 75 Mc/s

together with the sixth harmonic as a reference and measured the

relative attenuation of the ordinary and extraordinary waves, from

transmitters in a rocket traveling upward through the D-region.

These attenuations are dependent on Im r\ , or

o, e

exp - — Im Ti D

c o, e

The theoretical calculations of Im ti at 7.75 Mc/s and 2.0 Mc/s

o, e

'

employing the roots £, of the quartic equation (9) in equation (5) to

determine -n and using an arbitrary value, N = 500, is illustrated,

o, e Figs. 5 and 6, for classical and monoenergetic theory. This

implies that the numbers v for the abscissa of the graphs are constant

average values of v for the classical curve and proportional to

energy, u, or v = v(u) for the monoenergetic curve. Upon selecting

a value on the monoenergetic curves for ordinary and extraordinary,

and keeping Im r\ constant and moving over to the classical curve,

v . 4

one notes a change —;—r- ~ 2. 5 for collision frequency values 10 to

v(u)

7

10 approximately, which is in agreement with the deductions of
-18-

Kane [I960] . The values of the Nicolet/3 curve, Fig. 4, based on the

classical theory were therefore reduced by a factor of 2. 5 when used

in the monoenergetic theory, V = v(u). It is interesting to note that the

Nicolet/3 curve then agrees quite well with one of the curves of Kane

[ I960] deduced with the aid of monoenergetic theory from rocket

measurements.

Reflection coefficients, T , T , T , T , which can be

ee em me mm

employed to determine the field strength of a radio wave at the surface

of the earth [ Johler, 1961c] were evaluated with the aid of the models,

Figs. 3, 4, and the continuously stratified technique, Fig. 2.

The results of the calculation of the reflection coefficients based

on the monoenergetic theory are compared with the results based on

the classical theory as a function of frequency for grazing incidence

on the ionosphere, $ . ~ 82°, (d ~ 1000 statute miles for a single

ionosphere reflection [ Johler, 1961c] s using the Houston composite

model, Fig. 3. A magnetic inclination or dip, I = 60 , was

employed, and the results for propagation into the east, magnetic

azimuth, <\> = 90°, Fig. 7, and propagation into the west, <t> = 270°

a a Fig. 8, are illustrated. In view of the grave consequences anticipated by the introduction of the electron collisions proportional to
-19-

energy it is most remarkable that the curves of the classical theorydiffer only in detail from those of the monoenergetic theory. Indeed,

for the models employed in this analysis no new gross phenomenon

in the propagation were described by the monoenergetic theory.

However, some of the detailed changes are of considerable interest.

Thus, for example, at 100 kc/s there is a noteworthy difference

between the abnormal components, T *» T f based on classical

theory as compared with monoenergetic theory. This suggests

comparison of loop and vertical antenna measurements at grazing

incidence for first ionospheric reflected wave from a vertical

radiator type transmitter, for example, to check the monoenergetic

theory experimentally.

Similarily, the reflection coefficients, T, based on monoenergetic

and classical theories were compared with the blackout model profile

N(z), Fig. 3, employing the same (Nicolet/3) model v(h) profile,

Fig. 4, as shown in Figs. 9 and 10 for propagation into the east

(cj> = 90 ) and propagation into the west (<t> = 270 ). The geometrica a

optical angle of incidence, <t>., was, of course, adjusted slightly to

take account of the lowering of the ionosphere. Here again, the

difference between monoenergetic and classical are only those of

small detail. The greatest effect seems again to be in the abnormal

components, T = T

em me
-20-

The reflection coefficient as a function of angle of incidence, $.,

1 which is related to geometric-optical ray distance subtended along

surface of the earth between transmitter and receiver [Johler, 1961c]

d, statute miles as shown by the additional abscissa, is illustrated for

the quiescent model using the monoenergetic theory, Figs. 11, 12.

The results for propagation into the east, $ =90°, Fig. 11, can be

Si compared with the results for propagation into the west, <j> = 270°,

3,

Fig. 12. The particular case of a reflection coefficient for vertical

polarization of transmitter and receiver, T (vertical electric inci- ee dent wave and corresponding vertical electric reflected wave), is illustrated as a function of distance, d, or angle of incidence, $., in

Fig. 13 as a comparison, employing the Houston Composite model,

Fig. 3, of east-west, <t> = 270 with west-east, <t> =90 propagaa a

tion together with a comparison of monoenergetic with classical

3

magneto-ionic theories. Thus, at grazing incidence, $ . ^ 82°, d ~ 10

miles, I T I monoenergetic and classical results exhibit a greater field

1 ee 1

intensity for propagation into the east as compared with propagation

into the west. This is consistent with previous conclusions based on

classical theory, only [Johler and Harper, 1962], [Johler, 1961a],

[Johler and Walters, I960], [Johler, 1961c]. For both monoenergetic

theory and classical theory both curves cross in the vicinity of

Brewster's angle (d ^ 250), where at short distances or near vertical
-21-

incidence the propagation into the west is more favorable than the

propagation into the east for this particular orientation of the earth's

magnetic field, I = 60°, $ =90°, 270°. The fields predicted by the

cL classical theory are less than those predicted by the monoenergetic

theory for propagation into the west. However, for propagation into

the east there is a cross-over in the curves as a function of angle of

incidence, <{>., or distance, d. A distinguishing feature of these curves

when compared with ordinary sharp-boundary-type Fresnel reflection

coefficients is the multiple Brewster angles instead of a single Brewster

angle (between d = 90 and 300). This can be interpreted physically as

multiple regions within the ionosphere of large contribution to the total

reflection coefficient. Thus, each such region would exhibit a slightly

different Brewster angle since the electron-ion composition of the

ionosphere varies with altitude, z.

Figures 14-25 illustrate the complex indexes of refraction representative of low, medium, and high elect ron densities. The collisionfrequency parameter also affects the shape of the curves. Those chosen

were associated with the Houston Composite N(h) profile in the reflection coefficient analyses. Grazing incidence, <t>. ^ 82° is assumed,

and the earth's magnetic field, Jt = 0. 5 gauss, is oriented such that

<D =0°, and I = 60°.
-22-

In making a comparison of the indexes derived from the

monoenergetic theory with those derived from the classical theory, perhaps

the most noteworthy observation is the lack of major differences. In

fact, as the electron density is increased the two cannot be separated

graphically. This would, in large measure, account for the only

minor changes in the reflection coefficients exhibited by the monoenergetic

theory.

An interesting feature of both the monoenergetic and classical

index curves is the tendency, as the electron density is increased,

for one or both of the extraordinary propagation components to evanesce as a

function of frequency (i.e., the point of evanescence, Re r\ = 0) although

not at the same place.
-23-

7. CONCLUSIONS

The continuously stratified concept together with measured or

theoretical N(h), u(h) profiles of the ionosphere can be employed to

analyze a variety of conditions of the ionosphere at LF. The introduction of electron collision frequencies proportional to energy into

the magneto- ionic theory introduces some interesting changes in the

detail of the reflection coefficients and index of refraction of the

lower ionosphere. However, the classical theory demonstrates the

principal features of the reflection process with remarkable accuracy

in view of the grave consequences anticipated by the introduction of

electron collisions proportional to energy.

The introduction of electron collision frequencies proportional to

energy provides a more accurate description of the macroscopic

properties of a magneto- ionic piasma. The comparative simplicity

with which such collision frequencies can be introduced into the fullwave solution for the ionosphere warrants their use in describing the

detailed reflection or transmission process.

Boulder, Colorado, March 29, 1962.

(Revised August 8, 1962).
-24-

8. REFERENCES

Allis, W. P. (1956) Motions of ions and electrons, Handbuch der

Physik, edited by S. Flugge (Springer -Berlin), 21, Electron emission,

383-444.

Appleton, E. V. (1932) Wireless studies of the ionosphere, J. Inst.

Elec. Engrs. (London), 71, 642.

Barron, D. W. (March 1961) The numerical solution of differential

equations governing the reflection of long radio waves from the

ionosphere, IV, Proc. Roy. Soc. , 260 , Ser. A. , No. 302.

Booker, H. G. (November 1934) Some general properties of the formulae of the magneto-ionic theory, Proc. Roy. Soc. (London), A 147,

352-382.

Booker, H. G. (1939) Propagation of wave packets incident obliquely

upon a stratified doubly refracting ionosphere, Phil. Trans. Roy.

Soc. , London, Ser. A, 237 , 411-451.

Bremmer, H. (1949) Terrestrial radio waves -- theory of propagation, Elsevier Publ. Co., Inc., N.Y., N.Y.

Brekhovskikh, L. M. (I960) Waves in layered media, Academic Press,

Inc. , New York, N.Y.

Budden, K. G. (1955) The numerical solution of the differential

equations governing the reflection of long radio waves from the

ionosphere, II, Phil. Trans. Roy. Soc. , London, A 248, 45-72.

Budden, K. G. (February 1955) The numerical solution of differential

equations governing reflexion of long radio waves from the ionosphere,

Proc. Roy. Soc., Ser. A, 227, No. 1171, 516-537.

Budden, K. G. (1961) Radio Waves in the Ionosphere, Cambridge at

the University Press, Cambridge, England.
-25-

Chapman, S. (1931) The absorption and dissociative or ionizing effect

of monochromatic radiation in an atmosphere on a rotating earth,

Proc. Phys. Soc. 42, 20-45.

Chapman, S. (1931) The production of ionization by monochromatic

radiation incident upon a rotating atmosphere, Part I, Proc. Phys.

Soc. 43, 26.

Chapman, S. (1931) The production of ionization by monochromatic

radiation incident upon a rotating atmosphere, Part II, Grazing

Incidence, Proc. Phys. Soc. , 43, 483.

Chapman, S. (1939) The atmospheric height distribution of band

absorbed solar radiation, Proc. Phys. Soc. , 51, 93.

Crompton, R. W. , L. C. H. Huxley, and D. J. Sutton (July 1953)

Experimental studies of the motions of slow electrons in air with

applications to the ionosphere, Proc. Roy. So . (London) A 218,

507-519.

Dingle, R. B. , D. Arndt, and S. K. Roy (1956) The integrals § (x)

and D (x) and their tabulation, Appl. Sci. Research, 6B, 155-T.64.

P

Fejer, J. A. (Dec. 1955) The interaction of pulsed radio waves in the

ionosphere, J. Atmospheric and Terrest. Phys., 7, 322-332.

Ferraro, A. J. and J. J. Gibbons (1959) Polarization computations

by means of multislab approximation, J. Atmospheric and Terrest.

Phy. , 16, 131.

Gardner, F. F. and J. L. Pawsey (July 1953) Study of the ionospheric

D-region using partial reflections, J. Atmospheric and Terrest.

Physics, 3, 321-344.

Hartree, D. R. (1929) The propagation of electromagnetic waves in a stratified medium, Proc. Cambridge Phil. Soc. 25, 97.

Hartree, D. R. (1931) The propagation of electromagnetic waves in a refracting medium in a magnetic field, Proc. Cambridge Phil. Soc.

27, 143.
26-

Hines, C. O. (Mar. , June - Dec. 1951) Wave packets, the Poynting

vector and evergy flow, Pts. I-IV, J. Geophys. Research, 56 .

Huxley, L. G. H. (1937) Motions of electrons in gases in electric and

magnetic fields, Phil. Mag., 7-Ser. , 23, 210-230.

Huxley, L. G. H. (1937) Motions of electrons in magnetic fields andalternating electric fields, Phil. Mag., 7-Ser. , 23^, 442-464.

Huxley, L. G. H. (1940) The propagation of electromagnetic waves in

an ionized atmosphere, Phil. Mag., 7-Ser., 2^, 313-329.

Jancel, R. and T. Kahan (February 1955) Theorie du couplage des

ondes electromagnetiques ordinaire et extraordinaire dans un plasma

inhomogene et anisotrope et conditions de reflexion applications a L'ionosphere, JLe Journal de Physique et Le Radium, 16_> 136-145.

Johler, J. R. (Jan. - Feb. 1961a) Magneto-ionic propagation phenomena in low- and very- low -radiofrequency waves reflected by the ionosphere, J. Research, NBS, 6_5D, (Radio Prop. ) No. 1, 53-65.

Johler, J. R. (1961b) On LF ionospheric phenomena in radio navigation

systems, Avionics Panel Meeting of the Advisory Group for Aeronautical Research and Development (AGARD), 3-8 October I960,

Istanbul, Turkey, or Oranisation du Traite de Atlantique Nord, Pergamon Press, 64, rue de Varenne, Paris 7eme.

Johler, J. R. (Sept. - Oct. 1961c) On the analysis of LF ionospheric

phenomena, J. Research NBS, 65D, (Radio Prop.), No. 5, 507.

Johler, J. R. and J. D. Harper, Jr. (Jan. - Feb. 1962) Reflection and

transmission of radio waves at a continuously stratified plasma with

arbitrary magnetic induction, J. Research NBS, 66D, (Radio Prop.)

No. 1, .81-99.

Johler, J. R. and L. C. Walters (May-June I960) On the theory of

reflection of low- and very- low -radiofrequency waves from the ionoshpere, J. Research, NBS, No. 3, 269-285.
-27-

Johler, J. R. (2-6 July 1962) On radio wave reflections at a continuously stratified plasma with collisions proportional to energy and

arbitrary magnetic induction; International Conference on the Ionosphere, London, England, Physical Society - Institute of Physics.

Kane, J. A. (Feb. 1959) Arctic measurements of electron collision

frequencies in the D-region of the ionosphere, J. Geophys. Research,

64, No. 2, 133-139.

Kane, J. A. (Nov. I960) Re- evaluation of ionospheric electron densities and collision frequencies derived from rocket measurements of

refractive index and attenuation, IGY Rocket Report No. 6, 135,

National Acad. Sci. , Washington, D. C.

Molmud, P. (Apr. 1959) Langevin equation and the ac conductivity of

non-Maxwellian plasmas, J. Phys. Res., 114, No. 1, 29-32.

Muller, D. E„ (1956) A method for solving algebraic equations using

an automatic computer, Math Tables and Other Aids to Computation,

10, No. 53-56, 208.

Nicolet, M. (Apr. 1958) Aeronomic conditions in the mesosphere and

lower ionosphere, Science Report No. 102, Pennsylvania State Univ. , Univ. Park, Pennsylvania.

Pfister, W. (1955) Studies of the refractive index in the ionosphere:

the effect of the collision frequency and of ions, Phys. Soc. of London,

Physics of the Ionosphere, Cavendish Lab. , Cambridge, England,"

394-401.

Phelps, A. V. (Oct. I960) Propagation constants for electromagnetic

waves in weakly ionized, dry air, J. of Applied Physics, 21, No. 10,

1723-1729.

Ratcliffe, J. A. (1959) The Magneto-Ionic Theory and Its Applications

to the Ionosphere, A Monograph, Cambridge at the University Press,

Cambridge, England.

Seddon, J. C. and J. E. Jackson (July 1958) Rocket Arctic ionosphere

measurements, IGY Rocket Report Series, No. 1, 140 148.

Wait, J. R. (1953) Fields of a line current source over a stratified

conductor, Applied Science Research, 33.
28

Wait, J. R. (Mar. - Apr. I960) Terrestrial propagation of VLF radio

waves, J. Research NBS, 64D (Radio Prop.), No. 2.

Wait, J. R. (Apr. - June 1961) Some boundary value problems involving plasma media, J. Research NBS, 65B, (Math and Math Phys.),

No. 2, 137-150.

Waynick, A. H. (June 1957) The present state of knowledge concerning

the lower ionosphere, Proc. IRE, 45, No. 6, 741-749.
-29-

APPENDIX I

The elimination of the vectors, J and H, from the simultaneous

solution of Maxwell's equations and the Langevin equation of motion of

the electron gives

11 12

a a

21 22

a a

31 32

13

'23

33

y

= 0,

where*

,22s L a

il = X " aL - 6

" ~2~2 ' a

21 = aL

aT

+ X 1—2 > s -h

al2" aLaT

-1

2 2'

s -h

s -h

2

-u 2

i o s -hm

i r 2 2 T

a22" i " *• " aT "2 2.'

s(s -h )

a

i3

=a

T C+ i 1~2'

s -h

hT h^

a23~ al> +

; 2 2/

s(s -h )

, . h

T

a

31 " V _1 1~1'

s -h

a32=a^ + -7172-'

s(s -h ) 2 u

2

-1 2 L

s(s -h )
-30-

If the field, E,, exists the determinant of the matrix must vanish,

which determinant upon expansion yields a quartic equation in C

[Booker, 1939] , [ Johler and Harper, 1962] ,

a

4

£

4 + a

3

C

3 + a^ 2 + a^ + a

Q = 0,

where,

aV

a

o = < a2 -1)2 ^ - "t4 1 + <a2 - 1) [ i

+ -TT2 + T^1 +

s -h s -h s(s -h ) T^TV s(s -h )

, hL

h

T

aL , 2,. 3

1 = 7^7^ (a -1 '' s(s -h ) .2 2 2

f r s -, L V, 2 1% . T

aL , s-2 , 1 •z-f^-a-T 1 + TT-r} ( a -1)+ -T-z + T-2 + 7'

^ s -h s(s -h ) J s(s -h J s -h

hL

h

T

aL 2

a = 2 -—— = - a sec <|> , s(s -h )

2

T 2

- 1 L

a , = 1 -

s ( s -n )

2 - w n • v

1

CO CO h s

2 ' CO N
-31-

h - - h sin I,

JL/ h

T

= h cos I,

aT = sin <t>. cos $ , a^ - sm <t> . sm <j> L. 1 a T 1 a

a = s m <t>

.

l 2 2

,

00,. = Ne /€ m N ' o

the plasma frequency,

cj= u. e Jf/m, M = Ijt I, ri o

the gyro frequency,

e = l/c [i , o o and where N = N the median value of electron density in the

n

interval z corresponding to the uniform slab number n, Fig. 2

n

provided the approximation g ^ v or g ^ v for each slab of

ordered number n, can be made, i. e. , the classical magneto-ionic

theory is assumed by a real constant value of collision frequency, v .

Model electron density profiles, N(z), and collision frequency profiles,

v(z), are shown in Figs. 3 and 4. Such profiles are considered to be

representative of daytime noon and local auroral zone blackout conditions of the earth's ionosphere, and therefore have considerable

practical significance to radio wave propagation science.
-32-

APPENDIX II

The coefficients of the matrix equation (10) can be written:

a. . = cos <{> 1,1 a bn , » - sin $

1,1 a

1,2

1.3

1,4

1,5

1,6

cos <\>. sin

= - Q.

(1)

1 o a

Q

Q

(1)

i e

(1) r o

(1) r e

b. _ = - cos <t>. cos $

1, 2 i a h.s— 1 V'- 1

1,5

'1,6

= - 1

= - 1

2,3

a

l, 3

6XP

C 1 1 o 2,3

= exp -1 - Z L,. C 1 1 o

a2,4

= - a

l,4 6Xp

' . w (I)"

-l - z L,. c 1 i e b

2, 4

= 6XP

".co (1) -i- z t,. c 1 l e

a

2, 5

=

" a

l, 5

6XP

"

. co (1) -i - z t, c 1 r o b

2, 5 = exp -i^ Zl C

(1) c 1 ro

2,6 - a

l, 6

6XP

'

. co (1) -1— Z L, c 1 re

b

2, 6 = 6XP

'

. co

r (l)

-1 - z £ c 1 re

'2,7

"2,8

'2,9

- Q

- Q

- Q

- Q

(2)

i o

(2)

i e

(2)

ro

(2)

'2,10 re

2,7

2, 8

2,9

2,10"

- 1

- 1

- 1

- 1
33-

3,7

a

2, 7

GXP

co (2) c 2 i o b

3, 7 = 6XP

' . co (2)'

-i — z £. c 2 i o

a = - a exp

3, 8 2, 8 P

".co

r (2) -i- z £,. c 2 i e b

3,8 =6Xp ".co (2T

-i - z £. c 2 i e

^3,9 " a

2, 9

6XP —c 2^2 ro b

3,9 =6XP

-co (2)

-l — z C

c 2 ro

3,10"

' a 2,10 eXp

-co (2) -i- z t, c 2 re b

3,10=exp

-co (2) -1 — Z L, c 2 re

U 3,ll

= - Q

(3)

1 o V3 - 1

a3,12

= " Q

(3)

i e V2- 1

= - Q

(3)

3,13 ro

*>, ,„= - 1 3,13

= - Q

(3)

3,14 re b

3, 14= "

*

a A' ~ a 1 . ^ eXP

p, p+ 4 p-1, p+ 4

-1*. ,&-»

c p-1 1 o

c p-1 10

= - Q

( P»

p, p+ 9 1 e b IQ^ 1 P, P+9

c, , = - cos d> . sm * 1,1 1 a

cL , = - cos (j). cos <\> 1,1 1 a

C n _ = - COS <J) 1, 2 ; 1,2

= sin

1,3

[a

T P

(1) - CI" L 1 o 10 1,3 Ul

1

' Q

a)

- aP(1)

] 10 10 T10
-34-

1,4

- [a, P?) - ^>] Lie 1 e 1,4 [^

(1) Q(1) - a P

(1)

Lb ] ie ie T ieJ

1,5

- [a pW - C

(1)

] L ro ro 1,5 [l

(1) Q

(1) - a/]

ro ro T ro

1,6

- [aP(1

> - ;

(1)

] Li re re d

1,6

[t

(1) Q

(1) - aP(1)

] re re T re

C

2,3

= - C

l,3 exp

".<•> ,(1)'

-l - z i.

C 1 1 o d

2, 3 =

- d

l, 3

6XP

C 1 1 o

C

2,4

= - C

l,4 eXp 1—25, b. c 1 i e d

2, 4

=

" d

l, 4 6XP

c 1 i e

C

2,5

= - C

1,5

6XP

'•« ,(1)' -l — z C c 1 ro 2,5

d

l, 5

6XP

'.co (1) -1 - z t, c 1 ro

'2,6" C

l, 6

6XP

• . co (1) -1 - Z L, c 1 re 2,6

d

l, 6

6XP

c 1 re

2,7 M»!

2,

-l!

2)

] L 10 i o 2,7

[t; (z) Q

(2) ~a

T P^]

io 10 Tio

'2,8 [aL

P

(2) -; (2)

] L i e le 2, 8

[C

(2) Qi

2) -aT

P<

2

>] ie le Tie

'2,9 [a p< 2)

-c

(2)

] L ro ro '2,9

(2) Q

(2). ,2)

ro ro T ro

2,10" [aP(2)

- &

(2)

] L re re 2,10"^

2' Q

' 2

>-aT

p( 2

)] re re T re

C

3,7

= - C

2,7 6XP

" w (2) c 2 i o d

3, 7 =

" d

2, 7

6XP

".-«

r (2) -i - z C. c 2 i o

C

3,8

= - C

2,8 eXp

' ."" ,(2)

-l - z £. c 2 i e d

3, 8 =

" d

2, 8

6XP -i — z £. c 2 i e
35-

'3,9

c

2, 9

exp

c 2 ro 3,9

d

2, 9

6XP

co (2) -i-z & c 2 ro

C

3,10

= - C

2,10 exp

co (2) -i-z ^ ' c 2 re d

3,l(f

- d

2,10 exp

".« ,(2)

c 2 re

'3,11" [aL

P

(3)

- ^ (3)

] L i o i o 3,11 U

(3) Q(3) -aT

p' 3

»l io io Tio

'3,12 [a P<

3) -;< 3)

] L ie ie J 3,12 U

(3 >Q< 3

>-aT

p!

3

>]

ie ie Tie

'3,13

[aP< 3 > - ;'

3

»]

1j ro ro 3,13

<3) Q

<3) (3) ro ro T ro

•3,14 [aP(3,

-.< 3)

] Li re re 3,14

[(3) Q

(3). (3, re re T re

p, p+ 4 p-1, p+ 4 c p-1 1 o p, p+ 4 p-1, p+ 4

"

- co (p-1)" c p-1 1

C

p,P+ 9=-[aL

p[P>-^>] d . =-[^p) Q< p) -aT P

( P>]

p, p+9 l e ie Tie

a = cos <t> . sm $ oe i aa = cos <j) om a

b = cos <(>. cos <t> oe i a

b = - sin $ om a

c = - COS <J> oe a

c = cos (j) . sin

om i a

d = sin $

oe a d = cos $. cos $ . om l a
36-

Field ratios in the plasma are of considerable interest.

Defining,

P = E

z E~ ' y

Q = E

y

equation (8) can be written,

b

ll b12

b

21

b

22

Q

10

20

=

where, t> u= ^ ^ a^

b - a

12 13 a a13 33

b

21 = a21 °

r a

31 °

r a

31

b = a a a 22 23 33 33

b

10 " ai2 a

i2 a 22

20 22 32 32 respectively,

or, for example,
-37-

Q =

1-a

2

s2 -V

2

s(s V) a, a„ -l

L

L T 2 2

s -h J s -h -1 L.

hT h

L T

L

s(s -h

2 )J

1-a

2

», 2

2

S

" hL

2 2

s(s -h )•

i 2r 2

1-a -C, L ' 2 2

s -h

V +i

sTT-h

V ^ ~2-T

s -h

P =

_

hA_

s(s -h

. 2 2 s

s -h -I + L T

L

s -h

a^ -i

|_ T^

s -h -J 2

i, 2

• 2

S

" hL

1-a - 2 2

s(s -h ) J L

', 2 2 s_

" aL-^ "TTz s -h -J s -h

a^-i

T 2

i 2

s -h

where the particular slab (Fig. 1), n = 1, 2, 3, . . . p-1, p, under

consideration is designated by the notation, C = ^ ,P=P , Q = Q

Since there are four roots to the quartic, the field ratios equations

(27, 28) are P W, P

(n)

, P

(n)

, P

(n)

, Q

(n)

, Q

(n)

, Q

(n)

, Q

(n)

, corresio le ro re 10 le ro re ponding to the zetas (£,) for upgoing and downgoing (i, r), ordinary and

extraordinary (o, e) propagation components in the particular region,

z , of the electron-ion magnetic plasma under consideration.
38-

APPENDIX III

The velocity, V, tensor can be written,

J = _ NeV =

A

l A

Z

A

3

B

l B

2

B

3

C

l °2 C3

E

y

(i)

where

s(iwe

Q

) A

l =

2 2 ' s -h

B

l =

- ihL

(i(V

2 2

s -h

ih

T

(ia)€

Q

)

2 , 2 ' s -h

2 2

ihL

(iwe

Q

) -(h - s )(iwc ) -h h (iwc ) A

2 = 2~~2 ' B

2 = .2.2. ' C

2 = .2.2.

s -h s(s -h ) s(s -h )

-ihm(iwe.) -hT hm (iooe^)

3 "

2 2' B

3 " .2.2.

s -h s(s -h )

2 2

(s -h ) (iwe ) C3~

( Zh2)

°

- < 2 > s(s -h )
-39-

The coefficients of the tensor matrix can be reduced to the form

A

i =

(-j) <-A>

g

(n

1

) + in

x g(n2

) + in

2 _ '"n'o1 -

A

2 = .(— sin I) (- sin I)

g^) +

^

+

g

(n

2

) + i^

2 j

( - UN

£ ] '

A„ = (— cos I) (~ COS I) -, +

3 [g^) + i^ g(J2

2

) + i^

2

( - WN

e h

B

l =

r (f sinl) (^sinl)

g^) + jsi1 g(fi

2

) + ift 2

( "UN

6 )j

r (-cos I)

B„ =

(-i sin

2

I) (-4 sin

2

I) +

2 |_g(n

Q

) + i^

Q g^) + ^

g

(n

2

) + i^

z

( -wN

€ , «

r (sin I cos I) (-i sin I cos I) (-A sin I cos I) _, B

3 =

g

(n

Q

) + in

Q

) g(^) + in

x g

(n

2

) + *2

2

( "*N e ) '

C

l = (y cos I) (-— COS I) + g^) + iflj g

(n

2

) + i»

2

("'Vo*'

r (sin I cos I) (-i sin I cos I) (-i sin I cos I) _. C

2 =

g(« ) + ^

g^) + ^ g(fi

2

) + ifl 2

^n'o*'

.(-sin I)

2 2 (-i cos I) (-i cos I)

3 [g(£2 Q

) + i^

Q g^) + iS2 1 g(fi

2

) + ift 2

^"N^1 ' (3)
^40-

where B = -A , C = -A , C

2 = B , and where Q

Q

= u, Q^ = co + wR

,

£2 =00-00 . 2 H

The complex, frequency-dependent parameter, g, in the equation of

motion (7) is found by integrating over the velocity distribution for

electron collision frequencies, v(u) = (constant)(u), at each frequency,

o 1 Z

g

=

,

- SP 3/2 * /• , \3/2 - _-l

4tt P u M^t) 6XP ( ' GU/kT) dU

l

iO. (4) 3 j [ V (u) + i n ] a o which can be written in terms of Dingle, Arndt, and Roz[1956]

tabulated functions, £c/ 2 M> £3/? M:

«-{5[!(?)^fi)-'@«S/ l (T)}5 4 < 5 >

where T = v(u)/ ft, u = kT/e, and v(u) = akT/e = au. Thus, the implication of the classical magneto-ionic theory, g ~ v, a constant, real

number, is not in general a valid notion, if an energy dependence

v = v(u) is admitted.
-41-

The complex collision frequency, g, can be introduced into the

analysis by evaluating the coefficients of the quartic (Appendix I)

follows:

2 2 a

4

= ** w e o (C

3 + ^O*' (6)

a

3 = -2c

2 ^aL

B

3

, (7)

a

2 = J €q

2 [A

x + B2 + 2C3

- a

2

(B

2 + C

3

) -a* (^ + CJ ]

+ iu*

Q

[-A

1 C

3 - B

2

C

3 - A

2 + B

2 + 2cA 2

(l- a

2

)] (8)

\ = -21**^ (A

2

A

3 + AlB

3

) .2. 2

6 2 aL

B

3

(a

2

.l) (9)

a = A

1B

2

C3" 2A

2

A

3

B

3 + ^2 - A

1B3

2 + A2°3 + ^0(A

1 C

3 + B

Z°3

+ A

1B 2+ A

2 + A2

- B

2

) - iu, a

2

( AlC3+ A^* A* +A2

)

+ ^0aL

( -B

2

C

3

" A

1B

2

+B

3

2 - A

2

) " w

" < A

1 +B2

+C3> "

+ co 2e 2a

2

: (2A1 + B 2+ C

3

) + uA 2 a

2

( V 2B 2+ C

3

)

2 Z 2 2 / a -d \ 2 2 4 a 2 2 4 t, 3 3 X1 2

2

" w 6 aT

aL

{ A

l + B

2

) " w 6 aT

A

l " w e aL

B

2

" TM 6 (1 ~ a )

'(10),
42-

/ 5 3\ The functions t, (x) ( p = —, — I can be evaluated by an application of

quadrature techniques to the integrals [Johler and Walters, I960], or

other methods described by Dingle, and Roy [1956],

00 -1

e 5/2

(x) = (f •') f

P 5/2

(x

2 + p Z

) exp (-p) dp, (11)

*3/2(x) "'(2 !

oo . -1

p (x + p ) exp (-p) dp, (12)

The quantities P and Q are rewritten ;

Li iooe

-I

2 2

A

] L, 1006 J

A„ ., a a + :

L T iooe J

A

3

-I (13)

1-a +

1006 0-IL

f

i 2r 2 + Al

1-a - £

L. 1006

-J

A,

T 1006 J L

am & - — T* iooe J

Q =

1 - a +

1006 o J

A .,

, 2_ a a +

L, T ltoe n T iooe

-I

B

aT C + — L. 1006

-I

1 - a +

1006 JL

2 2 1 l-aT - t, + t^ L 1006 o -i aT

C "" io3T aT

&

A

3

1006 J

(14)

The constant e or the factor icoe can be eliminated from eq (1 - 3,

6 - 10, 13 and 14) for convenience. The factor was retained so that

eq (37), would be correct dimensionally, ia. t J = current, amperes

per square meter.
-43-

Fig. 1 - Coordinate systems, illustrating angle of incidence, §., magnetic azimuth, § , magnetic dip, Ijof the magnetic field.

St
• 44-

>•

<r<Q2

Z>O

GOX

*"^ o

'*—*' <

UJ oz hCD oo < 2 Q >- £ UJ

,_^ cr. o _l o < Q CL 2 =>

>-

cr.

Oo <

-z. o< < UJ

oc ^ DC

Q h- »—

< O X

UJ

(/) o UJ o >

Q_ <t

Z> ^ ua

<n . "J ^b

^* § O O

iJ

W «>

X 3 » • S o h ft

fi 2 ° 3 « t< ~» fi3

to

w rt

.s «>

a

O ni nj

fi <u

w

fi o o

GO

to nj

en o p< 6H fio

nj

u S fi to O —

ni £i

^J ^ S,bo

(U CL £ °

•H ^

is *

0)s

fi

co-2 m O AH »

.-i fi

fiO

0) ^ BE

O o mi • (Jo win ^ £ * o a 11 m

fi fi

u

OjO •H
45-

100 1,000

ELECTRON DENSITY, EL/CC

100,000

Fig. 3 - Electron density-altitude N(z), profiles of the lower ionosphere

illustrating approximate daytime-noon (Houston) and local

ionosonde blackout conditions.
46

I0 !

COQ

z

oc_> UJ

CO

>-oUJ

or

u_zo

CO

oCJ) oorl—

CJ>

10"

10

I0 6

I0 1

10' 50

FEJER

(1955)

NICOLET (1958)

NICOLET/3

SEDDON AND

JACKSON (1958)

CR0MPT0N, HUXLEY

AND SUTTON (1953)

\j__6ARDNER-PAWSEY

k \ (1953)

60 70 80 90 100 10 120

ALTITUDE, Z , km

Fig. 4 - Collision frequency-altitude, v(z), profiles of the lower

ionosphere, illustrating various estimates of the classical

collision frequency
-47- - 1 i i 1\\ - - O

1-

UJ oo o m o \\J - —

CD

O CD 11

—

_J tr II ii ii i\\ < LU A 1/ - o

if)

C/) <

LUO

-d?

X

/' / /'/

/'/l / /' -

_i

O ro / / / o ^ E / r I —

— 1 ^

CD

—

— i ID o ° 4^ _z / ~ / n in

II II 1 ' /A/ / CVJ ~

M— ^"^ //'/ ? // / / / / A 7 o //1 / ^ / CM - - 2 / / »it/ I - — o — e> /// —

Q_

— 3 / 7/

—

— * / / i m—

>- >-

be or / // /

in < < / // /?

c\i 2 z / A 2 - O Q /// / - tr CC o O // / _ — < nT~ -

i-X

LU

f\Jj

V

in

c\i2 -

\A 3

rf * ^ ^^ i * I I ^

'/ i I I

CO >^

lO

a

.—i o

•H

COCO

a5

COO o

CDUo

CO

M JH CD O t3 h s se G

•i-i -M CO

CD

<+-l o o

njM

CO * H X ^ • rt 2- CO o £ O o

^h CD O J2

cus

<D

JH

CO

cd6

<u X!o

CO

J-l

u

u

nj •H rt a K

>> •-I uC

•i-i nn o

+->CD •I

01) >

a)

CO CD O h

COI COI lO roI

in

ci

•i-i fa

II

U tuj
COI 10I ro & uui

CVJIcz>I

T>

flrt .—!

nJO i—1

•H

d CO

CO •H

f—i

0)

u

O

h <u

r£j

+->

•» bD «c-H

0-H

F fC s1—1

COCJ

fl

r-H

1—1 rH •H

*-> •« CJ CO

nj CD ^ r-H

<+-i

O

CD >» ^ u

bO

CD 2

T3 o CJ • H iM

X!

-i-> <v ctf 1—I a CO •H

CJ

O

m Q)

+j^ o •

rt •H lO a cJ •

.3 i

+-> a>

a

&c ci 2 s s >>

<+-l

U o

•H •H

+-> -M

CJ CD ri GO h

CO ^

•iH CD

M

J4

CD

O

ni <+-i a O CO oU

CJs

•H

COd i CD
•49-

II 1 1 1 1 1 1 II 1 1 i i j...-i ;.i i ...I.U-+-+-I—1"" I. -+"' — 1 1

1 i — '^'.^•^•'^. "

'• •- __, ^- —

o>

^---"—"—-~*^<^^^"~

——' J

-

^. — —

""

<D t_ ^^ i — __ -^^ ^

\ ^—— "T s^ s

—

\ ^^^^"** ^"^ / ^ / /

*JC^'^•^ / • s^^^ / /

S^S j^ / /

/ / - // / \ / // / / - i I I

I "Kr^^-iiL. i - - // 6

!/ E

/

/

/ ^^~"^>JD^

~~-- JS

'

' - 1 /' — t^Z' ^—

- ii/ 1 / Ss - hi / J& s

III / & / o OJ

— II / = // / 1- = o

/ / // / Ul / /

II / o

ii / / / / : / / dc -J :

/ 1 // / LJ < —

V

- V

\\ // / 2 O -

• // / w w

II 1 O CO co:

t

\ / 2 < \

\ - W/ o _i - o O

"

—1 : // 2 O ; ^£-

\\ I M 1 i •»_ 7 / i M—

>s\ /I] i »

•

^ >"

1

o

- t —

E ^ ® //

VIPOSITE 81.79° 1000

ST.

65

km

i

i

1

oLU- —z>:

oLJ-

:

- Ll_ -

~E \

a> \ Lf

E 3 CD CO O -

C3 ~ r h- \ - l- O II II ii - - X ^° E -

.. -e- ~ i -

. 1

i

i i i i i i —_

i i i i i i t—i

cvj ez> oo co xa- co o cvj o

1

1 1 i i i i i 1

i i i 1 i i i 1 I i i

SNviavy 4 i 6jv

\i i i i 1 1 1 1 1 1 i i i

o

o

»-> u

(DO

g t3 o

cti II

r

-,J H a) cn

» +-> a)

(11 w CD

O

o

a, n

c °

O

z

Ld

O

LU

0)

0)

a,

QJO o n a

o

OJ

I

lil

4 3aniiidwv

H * m

«^^ n

CJJ cn

ti> n

""^^ n

nJ o «-> ^ o

a> o 4-> (0 o 1>

rt CD fin

4)

o

O

•V o

H

^-1

II H —^ X) m rj * at

0) >,

CJ i—

•rH <l> •w

r-1 1 tn u. cr cn 1

0) rt M

<+-< O

1 t-M•Hh
-50- I ! I I I ii I i—i—r i ii i |

j~^y I p i i i |—L_ J-—

v

111!

JL - i i i i 1 1 ^.._j - > ~^.s- - V <u ^g^^ - - »-f >>r P - — E ^7

NERGE ICAL

CMto

_- UJ en - o en ~ z <

// £ / E O _l 2 o E J 1 11 1 1 o - 1

s* &> E

UJ CD ^n

*~ • — S

S5 do o o —

<=> - Q. II II ii ~ h-7\ \ 5 -A- O ~9-"D -C - E O2 « „ — l <u o ° ° H- O O lO : E en Is- to 3 w d - E O II II ii _ <U X o E t— 1 1 i i i X1 F5 o

>-- oz

UJ3:O

UJ

a:

u_ \ s^WO00<0<3-0JOPJ ^^ <=> \ SNviavd 4 i 6jv

_L_LJ I L I I I \\ I i i I I I I L

CVII o £ H O

+-> VhO

(tf -rH II * g ^

HI f 3anindwv

JH 0)

CO rC CD

4-> +-> i*

0) •Ho

•H

O CD

pCJ

<+H

<d

CO

•rH •H

u a, fl o

•rH so

+->

+->

u 00 en o aj v. a) <u a. y^ bJO o

J*. CD

C Jh

..

h

•i-i

a

*" CD rt Jh >- 4-> -t-> CO1 o z 5

UJ .

r—1

i—1 • r-1

CO

CD Z>H «t l-l UJ Jh

CO

CD or rf CJ -M u_ ^ o

<D

O 4J

CO o CD nJ CO on

a, o

CD

Cti

COo

CD

•ft o

h"

c—1

II ^ ' m Vti * aJ

cu S

4->

O

—I

CTJ CJ

•rH UJ •H

t—! 1 CO a cr co H

cu nj <

i—iU 1

oo• M
51-

o

3

Q.<

FREQUENCY, KC/S

Fig. 9 - Amplitude, |T|, of the reflection coefficient, T, as a function

of frequency (f = 10 kc/s to 1 Mc/ s) illustrating a comparison of the

amplitude for classical and monoenergetic theories, for propagation

into the east, <t> =90'

a
-52-

10 10' FREQUENCY, KC/S

Fig. 10 - Amplitude, |T|, of the reflection coefficient, T, as a function

of frequency (f - 10 kc/ s to 1 Mc/ s) illustrating a comparison of the

amplitude for classical and monoenergetic theories, for propagation

into the west, <l> = 270

a
53-

o

Q.<

10"

10'

i r

MONOENERGETIC

*a =90»

f = 100 KC/S

1 I 1 I I I

I I I l

I0

2 DISTANCE (STATUTE MILES)

I0

3

_L J I L

7 14 32 51 61 67 71 74 79 82 ANGLE OF INCIDENCE, <£ ; , DEGREES

Fig. 11 -Amplitude, |t|, of the reflection coefficient, T, as a function

of angle of incidence, <t>., for propagation into the east, <t> = 90°

i a based on monoenergetic collision frequencies.
-54-

1.0

Q.<

i-3

MONOENERGETIC

<t> Q

= 270°

f

= 100 KC/S

I I I I I

^T

io z DISTANCE (STATUTE MILES)

1 1 J I L_L

14 79

I0

3

32 51 61 67 71 74 82 ANGLE OF INCIDENCE, ^>j, DEGREES

Fig. 12 -Amplitude, | T [ , of the reflection coefficient, T, as a

function of angle of incidence, $., for propagation into the

west, <|> = 270, based on monoenergetic collision frequencies.
55-

v3

1 | I I I I

MONOENERGETIC

CLASSICAL

f = 100 KC/S

I I l _L

I0 2 DISTANCE (STATUTE MILES)

_J I 1 I I I

32 51 61 67 71 74 79 82 ANGLE OF INCIDENCE, </> it DEGREES

Fig. 13 - Comparison of the amplitude j T | of the reflection

coefficient for vertical polarization based on the classical

constant collision frequency with the same reflection coefficient

based on electron collisions proportional to energy (monoenergetic) as a function of angle of incidence, $., for propagation into the east cj> =9° and propagation into the west,

d> = 270°.
56-

1 1 1 1

1 II 1 1

1 1 1 1

1

1 1 1 1 1 1 1

(J

-

(0 H

LJ

, CD

o—lO

X • X -

«4— < UJ

II II II 2 ^3

—

S2 ^ -

en Ld

en O

< ^

-

- JO ^_, - o 2 ^

1 1 z~

1 / **-

1 / ~

/> — //

/ / i — // \

f / \

t / \

— '/ \

// \\ - '/ 1 - /J /// / //

_ / //

x /s

X //

— X //

JP X // p- / // — X //

x /s

_ X /'

s s / — X x * X // X X / — / / / — x x / «9 _ / A' p"

x X /

x x /

/ / // // // _ // i /l 1 1 1 // 1 1 1 1 1 1 1 1 1 1 1 1 1 1

CO

LO

CO

o

OL±J en

CVJ

LO

- CNJ

CVI kay

o

*" c

>* o

u

•3 to T3 ni h Xi O

CO be a) C 3

•rH .—I O rt • M > J>N Q. rH ^ O <D

rHO n

H-> vh

en o

<1) •rH -rH

rH

rH

s

4->

a. O

o a o u

iJ n u

•rH

<L> X n

rH

4-) bO s

•H 0) rC

1—

1

nj H-> V

rH

H-> c

re CO ,3 t3

1) Pi c

(1)

(UCO n u

it n rC c ex

-l-> CI •rH

+->

id

so

rC

H-> •rH

rH c t>N

IB o rH

rH rH O

aj 0J bo ** x a o

T) rH g

« a-.o

• rt . "

X <^ o

c ^ bo § S g

o fl C W rH TM

** •" ^

«1 w

(« H m

I
-57-

i-1—^r -i—i—i r

CLASSICAL MONOENERGETIC

N = 160 V = 4.5 x 10" l/(U)=lV2.5

I0 3I0 4 FREQUENCY. f C/S Fig. 15 - Imaginary part of complex index of refraction, Im r\, of the lower ionosphere for upgoing ordinary (o) and

extraordinary (e), propagation components, illustrating

the comparison of values based on the classical magnetoionic theory with those based on the monoenergetic theory.
58- - 1 1 1 1 1 1 1 1 i i i | i i i i 1 1 1 1 1 1 1 1 - o (0 X _ -

h-UJ 1° < LU o—

m

II II II - 1 cnuj 1 in o z__

l\ JO 2 ^3 M— / \ i i / \ ' ^ — ^ x \ ^ x. \ ^ X, V \ X. X <D ^OX \P* \\N \ X\ ^ \ X ^\^\ XX -

"^

x

N - - _ -

sX\ " i i i 1 t i i i 1 i i i 1 i i i i 1 N\ i 1 1 1 1 PM. k 9d l *f a

>> o s-g a <" •H CO T> rt h 43 O

CO bO cu a s

•i-i

—(

On)*

so > r a, ^ u * o g h g ^ ° O **

CO

u

cu -h -i-i " ** Q) 2 o <u S o " «

CO 00

C * .2 « o +3 ,d -* aJ +•

41

w

O

(/) \ H ° o - . rt

.,

r to 43 I+— <u £ a> rf 5

>-" - g£ o 8 PL*

-z. •2 S6 UJ3 £ o £ o i « >« UJ

of

re:

agatio theor U_ X a o

CD o -h ! "0 h S s a.2 X *4 O

<u a) -e rH •—» <U a. . a m ^ co O rt c a e! S

•H «« 73 r-J o i; «) ** 2 o h TM CO CO U m

rt " " i <U X) <u tf 5 ^ CO +J

•rH
-59-

T—I—I r—| r i r—1 i I

I i"

CLASSICAL MONOENERGETIC

N = 160 V = 4.5 x I06 I/(u) = lV2.5

I0"6 _L Jl

I 2 5 io w I0 J W 10" iu- w FREQUENCY, f, C/S Fig. 17 - Imaginary part of complex index of refraction, Im n, of the lower ionosphere for upgoing ordinary (o) and

extraordinary (e), propagation components, illustrating

the comparison of values based on the classical magnetoionic theory with those based on the monoenergetic theory,
-60-

UJ < UJ

CO UJ CO o < ^

_) o I I I I

(0o~ m

O * c\i

lO

— •*

II

|l ! I I | i—

r i i i i I x -

CO \oo

LUO

LU

0C

^9d o—

c

>. ° 2 <u H W

'-a rt M -°

to 00 « 3 o J o o ^ ^ to u

a;a

cao

CIoH

X

uo

p- to 0) Pi

O

O

TM +3 o R 3 * °one2 C

•H >« fl

H o u ao

en

+j

O

u rti ai m 5 "2 « ^ 5 * Co -t-J I 00

00 •H
-61-

— in_ 6

-5 -_

I

1 1

1 i |

i i

|

i i

|

i i i—i

-

- CLASSICAL MONOENERGETIC

-

N = 500

V = 1.6 x I0 6 I/(u)=I//2.5

-

-4 —

-10"

E

-10-2

-I02

I 2 5 I0 2 I0 3

I04

I0 5 FREQUENCY, f, C/S Fig. 19 - Imaginary part of complex index of refraction, Im r\, of the lower

ionosphere for upgoing ordinary (o) and extraordinary (e), propagation components, illustrating the comparison of values based on

the classical magneto-ionic theory with those based on the monoenergetic theory.
62-

a>

10-2 -

"i I

i i—l—

i

1—I—

i

r

CLASSICAL

- MONOENIERGETIC

N = 500

V = 1.6 x I0 6 V (U) = Z//2.5

1 1-1

CD

-in2 -

I 2 5 10 I0 2

I0

3

I0 4 FREQUENCY, f, C/S

I0 5

I06I07

Fig. 20. Real part of complex index of refraction, Re t^, of the lower

ionosphere for upgoing ordinary (o) and extraordinary (e),

propagation components, illustrating the comparison of values

based on the classical magneto-ionic theory with those based

on the monoenergetic theory.
^63-

FREQUENCY, f, C/S

,5 I

Fig. 21. Imaginary part of complex index of refraction, I'm tj, of the

lower ionosphere for upgoing ordinary (o) and extraordinary

(e), propagation components, illustrating the comparison of

values based on the classical magneto-ionic theory with those

based on the monoenergetic theory.
.64-

I0 2

I0 3 IC 105 FREQUENCY, f, C/S

Fig. 22. Real part of complex index of refraction, Re r), of the lower

ionosphere for upgoing ordinary (o) and extraordinary (e),

propagation components, illustrating the comparison of values

based on the classical magneto-ionic theory with those based

on the monoenergetic theory.
-65-

-icr 5

-io-'

— CLASSICAL — MONOENERGETIC N= IO 4 V - I.55xl0 5

l/(u) = l>/2.5

f Nl | fH

I 2 5 10 10 5

H)6

I0

T

IO2

to

3

IO4 FREOUENCY. f. C/S

Fig. 23 - Imaginary part of complex index of refraction, Im r\, of the lower

ionosphere for upgoing ordinary (o) and extraordinary (e), propagation components, illustrating the comparison of values based on

the classical magneto-ionic theory with those based on the monoenergetic theory.
-66-

FREQUENCY, f, C/S

Fig. 24. Real part of complex index of refraction, Re r], of the lower

ionosphere for upgoing ordinary (o) and extraordinary ( e ), propagation components, illustrating the comparison of values

based on the classical magneto -ionic theory with those based

on the monoenergetic theory.
67-

E

FREQUENCY, f, C/S

Fig. 25. Imaginary part of complex index of refraction, ImT], of the

lower ionosphere for upgoing ordinary (o) and extraordinary

(e), propagation components, illustrating the comparison of

values based on the classical magneto-ionic theory with those

based on the monoenergetic theory.

4 U. S. GOVERNMENT PRINTING OFFICE : 1963 O - 679589

U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineering] in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside ofthe

front cover. WASHINGTON, U. C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements

Dielectrics. High Voltage.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Scale. Volumetry and Densimetry.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Refer- ence Materials. Applied Analytical Research, Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterisation. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research.

Metallurgy. Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal Physics. Electrolysis and Metal Deposition.

Inorganic Solids. Engineering Ceramics. Class. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Bunding Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Processing Systems. Components and Techniques. Computer Technology. Measurements Automation.

Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Far Ultraviolet Physics. Solid State Physics. Electron Physics. Atomic Physics. Plasma Spectroscopy.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Ele- mentary Processes. Mass Spectrometry. Photochemistry and Radiation Chemistry.

Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering Laboratory. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic Technical Services. CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Vertical Soundings Research.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Fre- quency Utilization. Modulation Research. Antenna Research. Radiodetermination.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. High Latitude Ionosphere

Physics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Physics. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time-Interval

Standards. Radio Plasma. Millimeter-Wave Research.

Circuit Standards. High Frequency Electrical Standards. High Frequency Calibration Services. High Frequency

Impedance Standards. Microwave Calibration Services. Microwave Circuit Standards. Low Frequency Calibration

Services.
NBS