The magneto-ionic theory for bound electrons

Journal of Atmospheric and Terrestrial Physics, 1963, Vol. 25, pp. 281 to 286. Pergamon Press Ltd. Printed in Northern Ireland

The magneto-ionic theory for bound electrons* H. UNzt Antenna Laboratory, Department of Electrical Engineering, The Ohio State University, Columbus, Ohio, U.S.A. (Received 26 December 1962)

Abatra~t--The magneto-ionic theory for bound electrons is developed. The result includes the Appleton-Haxtree equation as a particular case. The case when the Lorentz term is included in the above theory is discussed in the Appendix. l. MAXWELL'S EQUATIONS ASSUMING w a v e p r o p a g a t i o n ei(~t-kz) in t h e z d i r e c t i o n in a g y r o - e l e c t r i c m e d i u m , t h e first M a x w e l l ' s e q u a t i o n V × /~ = a/Ot(~oE ~- t5) m a y be w r i t t e n as follows in t e r m s o f its c o m p o n e n t s : i k H ~ = i~O[eoE ~ -~ P~] (la) --ikH~ : 0 :

i w [ e o E ~ ÷ P~]

(lb)

i ~ [ e o E z ~- P z ] ;

(lc)

a/7 a n d t h e s e c o n d M a x w e l l ' s e q u a t i o n V × E = --/~o-a-t- m a y be w r i t t e n as: ikEy = --i(ottoH ~

(2a)

--i~OttoH ~

(2b)

0 = - - i ~ t ~ o H ~.

(2c)

--ikE~ ~

S u b s t i t u t i n g e q u a t i o n s (2) i n t o e q u a t i o n s (1) a n d d e f i n i n g t h e r e f r a c t i v e i n d e x n = c/v, w h e r e v ---- eo/k a n d c --~ 1/~/~oE o, o n e o b t a i n s : %(1 -- n2)E~ ~- P~ = 0

(3a)

Co(1 -- n2)E~ + P,j = 0

(ab)

EoE ~ + P : = 0.

(3c)

2. THE CONSTITUTIVE I~ELATIONS I n t h e case o f b o u n d e l e c t r o n s one c a n a s s u m e t h a t t h e r e is a r e s t o r i n g f o r c e in t h e d i r e c t i o n o f t h e e q u i l i b r i u m p o s i t i o n : Frest :

--q~

(4)

* The research reported in this paper was sponsored in part by the Detection Physics Laboratory, Electronics Research Directorate, Air Force Cambridge Research Laboratories, Office of Aerospace Research, United States Air Force, Bedford, Massachusetts, U.S.A., under Contract No. AF 19(604)-7270. On leave from the Electrical Engineering Department, University of Kansas, Lawrence, Kansas, U.S.A. 28l

282

H. U ~ z

where q is a constant. This term could be the dominant term of a power series expansion of the force for small oscillations. I t has also been shown by quantum mechanics theory t h a t for small perturbations the behavior of the bound electron is to a very close approximation identical with the classical harmonic oscillator. The equation of motiol~ in this ease will be then:

,t~R m all"

)

dR [dR ~ "'~Vdi 47 ql? = --el'? -- e / % t d t

>" _Ho

(5)

where v is the collision frequency, .m is the mass of tile electron, (--e) is its charge, and /Jo is the static imposed magnetic field. Assuming harmonic time variation e ~i,,,q equation (5) m a y be rewritten as follows: --',,(o 2 R + ' , n . i e ) R + , t R =

--e£

-- e # o i o , ( R

:. H o ) .

(6)

I f there are N active eleetr(ms per unit volume, and if they all are displaced through equal distance /~, the equivMent dipole moment per unit volume, which is the polarization _P, is given by (RATCIAFFE. 1959): P = -NeR.

(7)

Substituting equation (7) into equation (6) and defining (RATCLIFFE, 1959) X-

~')x2-(,) 2

(sa)

Ne" E O';'l~"( 0 2

i= _ &H __ /~oeBo _ (,)

Z

Yx[. +

Y,i, +

Y j,

(8b)

'D~ ( 0

(So)

= (o

~+

--

(')0 2 __

(,) 2

(/

(8d)

'Db(O 2

for the rationalized system of units, where o)o = v q / m is the resonant frequency of the bound electrons, one obtains (]~VDDEX, 1961: UYz, 1962), from equation (6): %.X21`7 = --(1 -- T -- i Z ) P

+ i(P

x

Y).

(9)

Taking U = 1 -- 5/' -- i Z . equation (9) can be rewritten in terms of components as follows: eoXR:~ -- _ [ T p . __ i y z p ~ _ i Y ~ P = (10a) eoXE ~ = --UP~

+ i Y x P . " -- i Y z P .

(10b)

%XE z =

47 i Y . P ~

(lOc)

-- U P z

-- i Y ~ P ~ .

Equations (9) and (10) represent the eonstitutive relations of the gyro-electrie medium with the bound electrons.

The magneto-ionic theory for bound electrons

283

3. THE REFRACTIVE INDEX S u b s t i t u t i n g e q u a t i o n (10) into e q u a t i o n (3) and writing (RATCLIFFE, 1959) A ---

X I

,n 2

one obtains: = 0

(lla)

= 0

(llb)

X ) P ~ = O.

(llc)

( A -- U ) P . + i Y ~ P ~ -- i Y ~ P : - - i Y ~ P + + (A iY,,P.

-

-

-

U)P, + iY~P:

-

iY~P~

-

(U

-

-

-

F o r a n o n t r i v i a l solution the d e t e r m i n a n t of the coefficients in e q u a t i o n s (11) will be zero: A -

U

--iY:

+iY.

-iY~

A -- U

+iY.

iY~

--iY:

=0.

--(U--X)

Developing the d e t e r m i n a n t ill e q u a t i o n (1 ")) and defining (U~:z, 1962): YL

=

Y:

= e)L

=

=

e1Y T 2

S __

~:

(13a)

(r)

(13b)

11"T2

X

1 -- X

(13C)

-- T -- i Z '

one obtains the following q u a d r a t i c equation: ( A -- U) 2 + 2 S ( A

(14)

U) -- Y L " = O.

T h e solution of the q u a d r a t i c e q u a t i o n (14) is given by: A = ~ --S

(15)

z~ ~ 'zS2 _1_ y L 2,

a n d since A = X / 1 -- n 2 and U = 1 -- T -- i Z , e q u a t i o n (15) m a y be solved to give: X 1

--

T

--

iZ

--

1

--

X

--

T

--

iZ

±

1

--

X

--

T

--

iZ

-~

YL2

(16) E q u a t i o n (16) gives the r e f r a c t i v e i n d e x of the gyro-electric m e d i u m for b o u n d electrons when the L o r e n t z t e r m is neglected (see A p p e n d i x A). F o r the p a r t i c u l a r case of no binding force, q = 0; a n d as a result, oJ0 ---- 0 a n d T = 0. S u b s t i t u t i n g T = 0 in e q u a t i o n (16) one obtains the classical A p p l e t o n H a r t r e e e q u a t i o n (RATCLIFFE, 1959; :BUDDEN, 1961). F o r a more c o m p l e t e discussion see A p p e n d i x A.

284

H. U~z

For the particular case of no collisions, Z ---- 0, and no magnetic field, Y = 0, equation (16) becomes: i

n2 = 1

1

--

O)N2

T -- 1 -+-

(O0

2

--

o)2

(17)

where equation (17) m a y be found in several references (LoRENTZ, 1952; BORN and WOLF, 1959), discussing bound electrons with no magnetic field. For the particular case of no collisions, Z ---- 0, and no transverse magnetic field, YT = 0, one obtains from equation (16): X

n 2 = 1 -1 --

T

~

YL

----- 1 -+-

50N 2

~°02 - - ( ° 2 : ~

(18) ~°LC°

where equation (18) may be found in several references (LoRENTZ, 1952; ROSSI, 1957), discussing the Zeeman effect due to magnetic field from the classical point of view. One may use equation (16) in a more general form for bound electrons with no collisions, Z = 0, taking YL ~ 0 and Y T ~ O. 4. THE

DISPERSION

RELATIONSHIP

So far we have assumed t h a t the system has only one resonant frequency. In general there will be m a n y such frequencies, even with a system with the same kind of molecules. For gases (LoRENTZ, 1952; ]3ORN and ~TOLF, 1959) (n ~-~ 1) the Lorentz term m a y be neglected (see Appendix A) and equation (16) m a y be rewritten in the form: n 2

1 +~

X~ k Tk _ 1 -+-iZ k -

~. Y T Tk + X k _ 1 + i Z k

•

1YT2 ÷ X~ -- 1 ÷ i Z k

+ YL 2

(19) where X~, T k represent the values of X, T corresponding to a particular resonance frequency (ook of a particular group of electrons. Equation (19) agrees for particular cases with results derived previously (LORENTZ, 1952; BORN and WOLF, 1959, RossI, 1957). In the above result the Lorentz term in the Lorentz-Lorenz formula has been neglected. For the case of gases this could be done (LoRENTZ, 1952; ]~ORN and WOLF, 1957), if n ,~ 1, as is usually the case. In the case of the ionosphere and unbound electrons it has been found experimentally t h a t this term should be ignored; a complete discussion of the Lorentz term m a y be found elsewhere ( R A T C L I F F E . 1959; ~ B U D D E N , 1961; GrNZBURG, 1961), and in Appendix A. For the case of no magnetic field, Y L ~ Y T ~ O, equation (19) may be rewritten as:

n2 = 1 +

T~

--

1 g-iZ k

1 +

~0,~ 2

_ w2

÷ i ~ovk"

(20)

Equation (20) agrees with the results derived previously (LoRENTZ, 1952; BORN and WOLF, 1959).

The magneto-ionic theory for bound electrons

285

5. DISCUSSION In the above analysis the magnetization M has been neglected since the effect of the magnetic dipole produced by the electron circulating in orbit is negligible compared to the effect of the electric dipole produced by the displacement of the electron from its stationary position. In the case of bound electrons where X ~ Y, one will obtain equation (18) for the case of no losses. In case T ~ "~ Y one will obtain equations (17) or (20). For the cases of most regular gases and standard radio frequencies, T ~ X, Y, Z and ~ 1. Equations (16) and (19) are applicable in particular for gases with very sparse population or for very strong magnetic fields. A c k n o w l e d g e m e n t - - - T h e author is grateful to Dr. L. PETERS, JR. of the A n t e n n a

Laboratory, The Ohio State University. for several discussions and valuable comments. APPENDIX A T h e L o r e n t z term

In his book LORENTZ (1952) considered the effect of passing an electromagnetic wave through a medium in which an electron could perform free damped oscillations. The equation of motion of one of these electrons under the action of the wave is of the form (RATCLIFFE,1959: ]3UDDEN, 1961): m - d f i + m v - [ [ + qR =

e E + l

P

-- e#o ~ [

.~. O o

(A-l)

E0

where the term 1-1 P is called the Lorentz polarization term.

LORENTZ (1952)

Eo

found t h a t for spherical distribution of discrete molecules when they are distributed in a cubic lattice (or at random (RATCLIFFE, 1959)), l = 1/3. He suggested t h a t this result is correct for fluids and gases only for a certain degree of approximation, and in general (LoRENTZ, 1952) l = 1/3 -4- S where, for each body, s is a constant which it will be difficult to determine. The Lorentz term is neglected if one takes l = 0. Assuming harmonic time variation e +i~t. equation (A-I) m a y be rewritten as follows:

--mco21~ ~- r e v i s o r + q R = --e~E -- e l P -- e~oiO)(R ><, Ro).

(A-2)

Eo

Taking (RATCLIFEE, 1959) P = --XeR

(A-a)

and using the definitions of equation (8), one m a y obtain (BUDDE~, 1961; UNZ, 1962), from equation (A-2):

EoXE = --(1

T -- i Z + I X ) P + i ( P ><, ~ ) .

(A-4)

Comparing equation (A-4) with equation (9) one can see t h a t in the present case U = 1 --iZ--

T+lX.

(A-5)

286

H. UNz

From Section 3 one obtains:

X 1

--

S =

1Yr2 I r --X

(A-6a)

~"

N i

(A-6b)

~ / S 2 -~ Y L 2.

,~2

Using equations (A-6) and (A-5) one obtains the final result: X n2~1 1 -- i Z -- T --~ 1X --

(A-7a)

-1"YT2 1 -

iZ

-- T --

(1 -- 1 ) X ~ K

where K =

1-iZ

--T-

(1 --1

+ YL2"

(A-7b)

Equation (A-7a) is identical with equation (16) when the Lorentz term is neglected, 1 = 0. Equation (A-7a) will give the classical Appleton-Hartree equation with the LORE~TZ term when 1 = 1/3 and T = 0, and it is identical then with the result given previously (RATCLIFFE, 1959). LO~E~TZ (1952) seems to have obtained the result in equation (A-7a) for the case of no magnetic field, Y T = O. It has been shown experimentally (RATCLIFFE 1959; ~BuDDEN, 1961), that the Appleton-Hartree equation of the magneto-ionic theory should not include the Lorentz term. Therefore, in equations (A-7) one should take T = 0 1= 0 (A-8) in order to obtain the classical Appleton-Hartree equation. Another possibility of obtaining the Appleton-Hartree equation is to take in equations (A-7): T = lX. (A-9) This possibility has been suggested originally by GINZBURG (1961), who gave a quasi-quantitative proof for his suggestion. One could also assume that 1 >~ I T - - I X ] ~ 0 (A-10) where equation (A-10) will account for the experimental results (RATcLIFFE, 1959; BVDDE~, 1961) based on the classical Appleton-Hartree equation. REFERENCES BORN M. a n d WOLF E .

1959

Principles of Optics, p. 83. Pergamon

BUDDEN K . (~.

1961

Radio Waves in the Ionosphere, Cambridge

GINZBURG V . L .

1961

Propagation of Electromagnetic Waves in Plasma, p. 27. Gordon and Breach

Press, London. University Press. Publishers, New York (Translated from Russian). Theory of Electrons, Chap. 4. Dover Publishers.

LOREIVTZ H . A .

1952

RATCLIFFE J . A .

1959

The 3iagneto-Ionic Theory and Its Applications to the Ionosphere, Cambridge

ROSSI B . B . UNz H.

1957 1962

Optics, p. 428. Addison-Wesley. Trans. I.R.E. A.P-10, 459.

University Press.