REFERENCES

1.

E. O. Johnson and L. Malter, Phys. Rev., 80, 58, 1950. Yu. M. Kagan, Vesmik LGU, no. 4, 63, 1967. Yu. M. Kagan, R. I. Lyagushchenko, and A. D. Khakhaev, Optika i spektroskopiya, 14, 599, 1968. 4. A. A. Zaitsev and E. I. Yankovskaya, DAN SSSR, 29, 562, 1940; O. P. Bochkova, L. P. Razumovskaya, and S. E. Frish, Optika i spektroskopiya, 11, 697, 1961. 5. A. A. Zaitsev, ZhETF, 8, 569, 1988; H. B. Dorgelo, H. Alting, and C. J.' Boers, Physica, 2, 959, 1986. 6. F. Penning, Electrical Discharges in Gases [Russian translation], IL, 1960. 7. S. E. Frish and O. P. Bochkova, Vesmik LGU, no. 16, 70, 1961. 8. V. V. Kokhanenko, N. A. Prilezhaeva, and L. A. Chernenko, Izv. VUZ. Fizika (in press). 2. 3.

22 April 1964

THEORY

Knznetsov Siberian Institute of Technical Physics

OF T H E E L E C T R I C A L

CONDUCTIVITY

OF F E R R I T E S

P. T. Oreshkin Izvestiya VUZ. Fizika, No. 8, pp. 170-171, 1965 It is usual to represent a polycrystalline ferrite as an assembly of highly conducting grains separated by poorly conducting films in order to explain the frequency and thermal characteristics of ferrites [1]; but the analogous relationships for single crystals cannot be interpreted in that way. This has led to the explanation of relaxation effects in terms of local displacement of conduction electrons [2] and of polarization of exchange electrons [8]. The following treatment is based on my proposed domains-in-shells m o d e l I assume that there.is a narrow region along the boundary of a domain that differs in conductivity from the body of the domain; the difference may lie in magnitude or type of conduction. Rectification theory [4] then indicates that depleted layers exist at the domain boundaries; an external electric field causes part of the shell to operate in the forward direction and part in the reverse (barrier) direction. These parts change places when the field reverses. This model has a general resemblance to the grains model, but it is more general, because it is applicable to single crystals. In addition, it provides an explanation for some observations that have not previously been satisfactorily interpreted. Let e, no, and v being the charge, concentration, and mobility of the carriers (electrons or holes) in the domains; n is the carrier concentration in the barrier layer (thickness d), p is the mean density of the space charge, No is the donor concentration (e. g., FEZ+), n 1 is the mean concentration of ionized donors, and NO- nl is the mean concentration of unionized donors in the barrier layer.

Following [5, 6], we have for no << N0 that ?$~

p = e 9~

--

n

122

,

(1)

and also t

p=eno}o(l_e-'~-~)

(2)

and

n(~ (

n=n0--~-t0

9

l-e

I

- 4/T, )

'

(8)

in which f is the frequency of the external field, r 0 is the relaxation time of the space charge in the barrier layer, and defines the maximal mean density of that charge. We assume each domain to be a cube of side a, these being fitted together to give l/a per length l and s/a z per area s. Then the total resistance R of a specimen of that length and area is _

a - - d

t {-g-J

a.s

d.

+

I'

ev 120

no-- 2 ~~

I (4)

and the effective specific resistance is 1

Pelf -- a

'a-___.d!+ l

enoV

ev

no---x-%

1-e

(5)

Thus Oeff decreases as f increases, while d becomes about 10-a cm; Peff falls steeply when d equals the m e a n free path of a carrier (about 3 ~ [1]). The specific resistance and dielectric constant of a ferrite are [1] dependent on the field strength E above 1 V/cm; this is explained if the mean critical field for the barrier layers is 1 V/cm. It seems unlikely that the grains and insulating layers would have the same activation energy for conduction [1], but this is entirely reasonable for a barrier layer and a domain. The thickness of the insulating layers is c a l c u l a t e d [1] as several angstroms, which is also improbable; but it is a c c e p t a b l e for barrier layers. The space charge in the barrier layers contributes to the polarization. These layers are principally causes of relaxation if they are sufficiently deep and iride; standing polarization waves are set up at them [5, 6], or waves of carrier drift. Formula (1) gives only the m e a n space charge in a barrier layer, whereas the field moves the carriers and produces concentration nodes at the barrier layers, with antinodes in the middle of the cross-section. From [6], the relaxation time of a barrier layer is t~$ o

d "~ 9 =

-

-

v*Ud

9 e

~T

(6)

Here v* is the carrier m o b i l i t y corresponding to an external field Ud/d alone and As 0 is the activation energy for conduction; (6) gives r of 10 -3 to 10 "4 sec for d ~ 10 -a cm and v*= v in fields of about 1 V/era. The m a x i m u m tan 8 occurs at a frequency of the order of 1/% or 103 to 104 cps [8]. This m a x i m u m increases with temperature because (6) shows that r falls (from reduction in d and increase in kT). A fuller development of this theory can cIearly be based on current rectification theory; it can also explain the galvanomagnetic effects observed in ferrites. The model can be extended to grain boundaries, dislocations, and so on. A similar m o d e l can be used for ferroeleetrics and the like. REFERENC ES 1. J. Smit and H. Wijn, Ferrites: Physical Properties of Ferrimagnetic Oxides in Relation to Their Technical Applications [Russian translation], IL, Moscow, 1962. 2. V. A. Ioffe, G. I. Khvostenko, and Z. N. Zonn, Zh. Tekh. F i z . , 27, no. 9, 1958, 1967. 3. L. I. Rabkin, High-Frequency Ferromagnetics [in Russian], Moscow; 1960. 4. A. t. aubanov, Theory of Rectification by Semiconductors [in Russian], Moscow, 1958. 5. P. T. Oreshkin, Izv. VUZ. Chernaya Metallurgiya, no. 2, 163-169, 1962. 6. P. T. Oreshkin, Izv. VUZ. Chernaya Metallurgiya, no. 10, 162-168, 1963. Ryazan Radio Engineering Institute

10 February 1964

ROZHDESTVENSKII'S

METHOD

FOR ANOMALOUS

DISPERSION

B. Sh. Perkal'skis and V. L. Larin

Izvestiya VUZ. Fizika, No. 3, pp. 171-173, 1965 Anomalous dispersion is one of the most important and striking effects in optics, but it is rather difficult to demonstrate directly to students. Rozhdestvenskii's method [1] is often presented in a purely descriptive fashion, because the Mach-Rozhdestvenskii interferometer is e x c e p t i o n a l l y expensive and difficult to adjust. The experiment is therefore very rarely seen in the teaching laboratory and appears to be in common use only in the physics courses at Moscow University [2], where a custom-built instrument is used. The principal advantage of this instrument is that heating of the plates (and consequent drift in the pattern) are m i n i m i z e d by the use of thin plates; the interfering beams are then widely separated. A sensitive film (e. g., RF-.3 for fluorography, sensitivity 1200 units) greatly reduces the exposure time (to a fraction of a second if a prism is used), so it becomes possible to use the very much more convenient standard Jamin interferometer (the IZK-453) with thick glass plates, which gives beams .4 cm apart. The light source can be an STs-62-G f i l a m e n t lamp, which causes much less heating of t h e p l a t e s than does Rozhdestvenskii's are. We have used this at Tomsk University to observe anomalous dispersion (Fig. 1). The carriages on the heavy rails of the interferometer are fitted with wood blocks having two quartz 121