Shorter Contributions 4. C O N C L U S I O N The results of measurements indicate that the change of the zero position of the gravimeter caused by variations of atmospheric pressure cannot be neglected. If we calibrated the instrument at some vertical gravity bases, the corrections may reach as much as several hundredths or even tenths of a miligal in dependence on the magnitude of the barometric coefficient of the instrument a n d on the magnitude of the variation of pressure. Thus, e.g., the corrections for the checked gravimeters at the base Je~t~d (Ah ~ 400 m) vary from 0-019 reGal to 0-176 reGal, at the base Lomnick3~ gtlt (Ah ~ 1740 m) the corrections reach values between 0.086 and 0'766 reGal. The barometric coefficients determined are equal in order of magnitude. As for the Sharpe gravimeter, the quality of the instruments with a higher production n u m b e r is better; this fact was ascertained earlier. The accuracy of the barometric coefficients determined for the Sharpe gravimeters is approximately the same for both instruments. The Worden gravimeter is more sensitive to vibration, which results in a larger error in determining coefficient b. The proper value of the coefficient b is substantially larger than for the Sharpe instrument. Therefore, the question arises whether it is at all expedient to calibrate the Worden gravimeter at vertical gravity bases, and if it is advisible to use it for gravity measurements in mountains. The authors would like to t h a n k Prof. M U D r . J. Koles~ir, DrSc., director of the Research Institute of H u m a n Bioclimatology in Bratislava and Assist.-prof. M U D r . M. M i k u l eck3~, CSc., for letting them experiment in the pressure chamber of this institute.

Reviewer: Z. ~'mon

Received 1. 3. 1977

References [1] M. P i c k : The Tidal Corrections for the Gravity Measurements. Zfiv. zpr., Geofysika, n. p., Brno 1974 (not published). [2] J. Rygav3~: Ni~gi geodesie. CMT, Praha 1949. [3] L. K u b ~ e k , A. P ~ i z m a n : Guide to Statistics in Measurement. Acta metronomica, 8 (1972), 1.

CHANGES OF

THE

OF

ELECTRICAL

CURIE

CONDUCTIVITY

TEMPERATURE THEORETICAL

Dedicated

to

RNDr.

Jan

Picha,

OF

IN THE

BASALTIC

VICINITY

ROCKS

-

MODEL CSc.,

on his

60th

Birthday

MARCELA LA~TOVI(~KOV.&

Geophysical Institute, Czechosl. Acad. Sci., Prague*) S u m m a r y : Samples of basalts, haematites and magnetites display either temporary or permanent (magnetite) weakening of the increase of the electrical conductivity with increasing tempera*) Address: Bo~ni II, 141 31 Praha 4 - Spo~ilov.

98

Studio geoph, ot good. 22 [1978~]

KpamKae eoo6tqenu~l ture in the vicinity of their Curie temperature. Using the second quantization, this paper explains the observed pattern of the electrical conductivity adequately for magnetite and approximately for the other rocks by means of a quantum theory model of ferromagnetic minerals. This theory describes only the electron component of the electrical conductivity, which is responsible for the Curie temperature effect.

Our measurements [1, 2] indicate that the electrical conductivity of rocks changes in the neighb o u r h o o d of the Curie temperature. These changes can be explained by means of the multielectron model of ferromagnetic minerals (e.g., [3]). In this model the magnetic and electrical properties of a crystal are described as the properties of a single system of interacting electrons [4]. The exchange energy, generating an ordered orientation of the magnetic moments of electron spins, vanishes at the Curie temperature and, at temperatures T > O, an unordered orientation of spins is established, which leads to changes in the temperature dependence of the electrical conductivity. Ferromagnetic semiconductors have a very complex electron structure which complicates their study considerably. As regards quantum-mechanical computations by means of the multielectron theory, a simplified model of a ferromagnetic semiconductor is used [5]. It is assumed that in its basic state a crystal has, apart from two "external" (valence) electrons which form a closed s-shell with respect to spins, also one "internal" electron with an uncompensated spin, corresponding to the unfilled d-shell of an isolated atom, in each node of the grid. Moreover, in the lowest energetic state the spins of all "internal" electrons of a crystal are parallel. The transition of an s-electron into an excited state (usually the p-state) of a given or another node, together with the switching of the d-electron spin will represent the elementary excitations of the system (exciton and ferromagnon). Having introduced this division of the electron systems into internal and external electrons, we also assume that the internal electrons are essentially responsible for the magnetic and the external for the electrical properties of the crystal. The association of the magnetic and electrical properties follows from the mutual interaction of the external and internal electrons. A suitable method for studying a set of identical particles is second quantization. This method is based on selecting the n u m b e r of particles in the individual states as the independent variables instead of the complete set of mechanical quantities characterizing the individual states of the individual particles. We thus establish the Schr6dinger equation in which the independent variables are numbers which give the n u m b e r of particles in the individual states. This equation will have a particularly convenient form, if we introduce the creation operator d and the annihilation operator d +, which increase (dO or decrease (d +) the n u m b e r of particles in a given state by unity [3], because in this way we obtain a more general Schr6dinger equation wich holds true for any n u m b e r of particles. The energy operator of our set of particles, using the excitation number, then reads [4]:

(1)

O = Ho + Z L(cq c() d+d..

+ ½

Z

F(cq, c~2;c@~i) d+¢a=~a~2,a~,~+` ~ ,

where H 0 is the Hamiltonian of the original undisturbed system, ~ is a complete set of quantum numbers, characterizing the given individual state of the electron: c~ -- f, 2, ¢, where f i s the index of the grid node, 2 = 0, 1, 2 is the orbital q u a n t u m n u m b e r (corresponding to the s- p- d-states), and q / = ~ } is the spin q u a n t u m number; L(~, c~') are matrix elements corresponding to the kinetic energy and the interaction energy of the electrons with the periodic field of the grid; F(cq, ez; c~, c~) are matrix eIements, corresponding to the mutual interaction of the electrons. We shall also assume that the energy of the ferromagnetic spin orientation kO (where O is the Curie temperature) is much smaller than the energy required to the sub-system of s-electrons. Studia geoph, et geod. 22 [1978]

99

Shorter Contributions In other words, temperatures close to O, which are still too high for exciting the system of internal electrons, will be considered as insufficient for exciting the s-sub-system of the external electrons. Under this assumption we may then apply the method of quasi-particles to the system of external electrons and the method of the energy centre to the internal sub-sYstem. The method of the energy centre is based on computing the mean value of the complete Hamiltonian (1) of the system for all states with given values of spontaneous magnetization of the internal electrons. The result is an expression for the energy of the whole system [4]:

(2)

e =

+

+

+

where EBo, EBL and E F are the complete energies of Bose excitons with spin 0, Bose excitons with spin 1 and Fermi excitons with spin = -~, including terms which describe the mutual interaction with d-electrons; E ° is a constant corresponding to the part of the energy which is independent of the number and distribution of the excitons in various states and includes the mean energy of the sub-system of the d-electrons; p is the relative magnetization of the sub-system of d-electrons in Bohr magnetons per one atom. Since we are still considering just the electrical conductivity of our system, we shall only take the term E F (energy of the flux carriers) into account,

(3)

E,. =

E

(i,k,~)

i

E(i)( k, ]~) Hk~,

where E(O(k, It) is the energy of the Fermi exciton of the #th type (i = 1, 2; coresponds to "electrons" and "holes" in the single-electron theory) in a state with the quasi-impulse k and spin ~, under magnetization of the d-electrons/1; n~0 is the occupancy of these states. By using the method of effective mass (and f o r / t < I) we arrive at (4)

E~i)(k, #) = o~, -- 2~i0~1~ + (fli - 2fl'iO#) (ak) 2

where ctoi is the activation energy of the exciton and fli is a quantity indirectly proportional to the effective masses of the quasi-particles f o r / t = 0; ct}0 and ,8~ define their change in dependence on ~, a n d / t and they are associated with the exchange interaction between internal and external electrons. As is well-known [6, 7], the electrical conductivity for electronic meehanismus is computed using equation (5)

tr = o"o e x p

(-Eo/2kT),

~o = qv 2(2~trn*kT)3/Z/h 3 ,

where E o is the overall activation energy and k is Boltzmann's constant, the parameter cro, which is usually considered as a material constant within the limited temperature interval with regard to the rather weak dependence on temperature (q is the charge of the carrier, m* the effective mass, h the Planck constant and v the mobility). From Eq. (4) it follows that the activation energy of Fermi exciton is given by

(6)

0

E(oi)(~O = cxi -

r

2ai~,~b #

because the second part of Eq. (4), dependent on the vector k, is not associated with the activation energy of the Fermi exciton. Assume that the activation energy of the Fermi excitons changes in the same way during magnetization for both types of spins (~t = :E½). After substituting (6) into (5) and summing over ~t we then arrive at (7) 100

th = 2tro e x p [ - ( c t ° + ~x'i#)(2kT) - 1

nl], Studia geoph, et geod. 22 [1978]

Kpam#eue coo6ulenusl where n' is the overall n u m b e r of excitons of t h e / - t y p e with spin ± ½ and a~ = [ai~0[, because one may expect that 10~iO[is the same for excitons of b o t h spins, since the Fermi excitons only differ in the sign of the spin. For T > O Eq. (7), after taking the logarithm, yields

In ¢i = In 2¢o -

(8)

~°(2kT)-'

ni

because/x = 0 for this region. For T < O the general eq. (7) yields

(9)

In ~r, = In 2cr~ - (c~'i + c¢'i/~) ( 2 k T ) - 1 n ~ ,

where ¢~) @ ~0, because the material parameter Eq. (5) is not quite constant, since some of the parameters determining it depend on the magnetization. ¢i is the conductivity of the i-th type of carrier in the case of a ferromagnetic mineral e.g., of electrons. Moreover, Eqs (7)--(9) do not change of type of charge carrier (i). By comparing Eqs (8) and (9) we find that, on passing through Curie's point, the slope of the function lg ¢ = f(1/T) will change (a change in the activation energy). Its decrease is characterized by the term (~}/2k) n i, where the product c~/xi reprezents the overall change of the activation energy. Of the samples we investigated [1, 2] magnetite is closest to the ferromagnetic minerals for which this theory was intended. On passing through Curie's point the expected change really occurs with both samples of magnetite. The activation energy for magnetite 41 decreases on passing through Curie's point from 0'21 eV to 0-16 eV and for magnetite 30 from 0.17 to 0"12 eV. A similar discontinuity in the dependence of the electrical conductivity on temperature can also be observed for basalts and haematites on passing through Curie's point as illustrated Fig. 1, IgO

"1

•

3.

256

"5 ;

2~

Fig. 1. lgcr = f(1/T) for basalts: ... No 259, ©©© No 256, A~/~ No 230. Curie temperatures marked by arrows.

1.0 SO0

1.5

SO0

40O

21)

1/TxlO~lKr 1

2O0

TI°Cl

although these cannot be considered classical simple ferromagnetic minerals for which the theory was intended. That it is more difficult to apply the theory to basalts and haematites is also demonstrated by the fact that the decrease in conductivity, observed in passing through Curie's point can no longer be described by this simple theory. Perhaps a role is played here by a paraprocess, caused by the local spin orientation in the interval of temperatures close to, but higher than, the Curie temperature [8]. Studia geoph, et geod. 22 [1978]

10]

Shorter Contributions

Similar patterns of electrical conductivity in the neighbourhood of Curie's temperature in rocks and minerals were also observed in iron quartzes [9], magnetite [10], ferrites [9, 11], haematite [12] and various pure elements [13]. Our results of measuring the electrical conductivity of magnetite [1, 2] agree in absolute value with the results in [10], also pertinent to magnetite, however, in the neighbourhood of Curie's temperature the changes of the electrical conductivity are less pronounced. There are also other non-magnetic characteristics such as specific heat, the magneto-caloric and magneto-resistive phenomena [15] which display a similar anomaly as the electrical conductivity in the neighbourhood of Curie's temperature. In conclusion it can be said that the theory given in this paper explains adequately the pattern of the electrical conductivity of magnetites in the vicinity of the Curie temperature. The pattern of more complex basalts and haematites, which cannot be approximated as ferromagnetics, is also explained by the model except the small decrease observed in two samples after exceeding the Curie temperature. Received 16. 9. 1976

Reviewer: M. Hvo~dara

References

[1] V. K r o p f i ~ e k , M. L a g t o v i ~ k o v f i : Magnetic and Electric Properties of Basic Volcanic Rocks. Int. Symp. on Physical Properties of Upper Mantle Rocks to Geodynamic Processes, Liblice 1976 (not published). [2] M. L a ~ t o v i ~ k o v f i , V. K r o p f i 6 e k : Changes of Electrical Conductivity in the Neighbourhood of the Curie Temperature of Basalts. Studia geoph, et geod., 20 (1976), 265. [3] D. I. B l o c h i n c e v : Z~iklady kvantov6 mechaniky. N(~SAV, Praha 1956. [4] tO. H. HpXHH, E. A. T y p o B : K M.~oro3YleKTpOHHOfi TeoplJI4 noJIynpoBo~RHHrOB. 2. ~eppoMarHuTnb/e llOJlynpoBOJIHrlKn, q3MM, 4 (1957), 9. [5] C. B. BOHC OBc I~n~: O6 o6Mem~oM B3alelMO~Ie!~CTB~n4BaYleHTHI,IX ~I BHyTpeHHHX 3JIe~TponoB s qbeppoMarHnTa~IX (IlepexoAH~IX) MeTanax. )I(gTqb 16 (1946), 981. [6] 9, H. H a p x o M e H X O : 9~IerTprI~IecI
]02

Studia geoph, et geod. 22 [1978}