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Electric charge in the volume of β′-radioactive dielectrics
Radiat. Phys. Chem. Vol. 23, No. 3, pp. 307-317, 1984 Printed in Great Britain.
0146-5724/84 $3.00 + .00 Pergamon Press Ltd.
ELECTRIC CHARGE IN THE VOLUME OF fl'-RADIOACTIVE DIELECTRICS V. V. GROMOV and A. G. SAra-IAROV Institute of Physical Chemistry of the Academy of Sciences of the USSR, Leninsky Prospect 3 l, Moscow, U.S.S.R.
(Received 6 February 1983) Abstract--The mechanism of formation and distribution of electric charge in the volume of dielectric solids, containing a fl--radioactive isotope is discussed in the present paper. Experimental data of measurements of charge in different materials are given. Examples illustrating the influence of the electric volume charge on kinetics of dissolution,.sorption, evaporation of solids are given here. It is shown that the results otained are applied for external irradiation of dielectrics by the beam of accelerated electrons.
INTRODUCTION Tim rORMAaaONof electric charge in radioactive solids as a physical phenomenon was discovered in 1900. o) During the experiments the current of charged partides emitted by radium in samples of Ba(Ra)Cl2 was measured qualitatively. Further studies in this field had technical objectives--formation of the nuclear battery or radioactive current sources. (2) In works °'2) the possibility of accumulation of electric charge in the volume of radioactive material(dielectric) was not in the question, as attention was paid to the process of "outer charging" and thermal effects, connected with break-down in collectors with different constructions of charging particles, emitted by radioactive source.
Nowadays it has been confirmed that any radioactive solid accumulates electric charge, the sign of which is opposite to the charge of the particles which leave the surface of the sample. However, under normal conditions (for example, in a water or in an air-conducting surrounding medium) the charge in radioactive preparations is rather small because of its neutralization by carriers outside of the surface. It was considered that this behaviour was true for any solid: metals, oxides, salts, ceramics, etc. However, for dielectrics this phenomenon turned out to be more complex. Detailed study of formation of charge in radioactive and irradiated solids of high ohmic materials (which has been performed lately) shows that in radioactive dielectrics, even in conducting media, one can observe the accumulation of electric charge within large volume. Its value depends on the level of radioactivity and electric properties of the sample. A peculiarity of such preparations is the following: at total electronic neutralization of the sample a noticeable gradient of the charge appears
within the specimen as on the surface there is only compensation of volume charge by carriers from surrounding space. DESCRIPTION OF THE C H A R G E PROCESS Let us discuss the following model: the sample has isotropic properties, the sources of charged particles are distributed in the volume arbitrarily, the process of neutralization of charge of opposite sign proceed only on the surface. To simplify this fact let us consider that electrophysical properties of the sample are characterized by electroconductivity a and dielectric constant E, Ohm's law is obeyed and only one type of carriers is present. In the common case to solve this task it is necessary to use the equation of continuity 3) dp (l)
--
~t
=
-- V "j
where p is the volume point density of the charge, j is the vector of current density in the sample and V the Hamilton's operator. In the common case dielectric penetration E and electroconductivity of the sample a as a function of coordinates-depend also on intensity of irradiation. For homogeneous dielectrics the current density can be represented as
(2)
j--~8+ji
where ae is the current component, which is determined by its conductivity of dielectric under the influence of the field e of the volume charge andji the current density of injected carriers; in this case t~ corresponds to elcctroconductivity under the given conditions of irradiation. 307
V. V. OROMOVand A. G. SAKHAROV
308
If we substitute expression 2 into expression 1 and take into account that Iz • s = p/EEo, after some transformation we obtain the following expression
Op (3)
- -
=
(F~r)8
--
Ot
tr
- --p
--
Vj,.
EEo
This expression can be considered to be an equation determining p if in some definite case it is possible to express 8 using the volume density of charge. For example, if in an infinite dielectric there is an area in the form of a sphere equation (3) has the following form
(7)
r
(4)
Op O~ =
do 4~ f tr 1 O Or r-Y-~EEo P~2 d~ ----PEEo --~Orr (r2"j')" 0
Expression (4) is written in the spherical system of coordinates (r, 0, ~ ) the centre of which coincides with the centre of radioactive area. It is supposed that tr and Ji do not depend on 0 and tp. If j~ (projection ji onto r) is a known function of coordinates and time, and t r a known function of coordinates, then equation (4) is an integro-differential equation with reference to p. Even in such a simple case the task is to solve integrodifferential equation, that is why only simplified models of practical importance are discussed under specific experimental conditions. If (a, E) is supposed to be independent of the coordinate (which are stable in the sample) and it does not change during the process of interaction of ionizing irradiation, equation (3) leads to the simple differential equation (5)
dp(R, t) _ dt
tr eeo p (R, t) + f(R, t)
where R is a coordinate, t is time, f = - 17ji. As it is seen from equation (5) the solution of p will depend on R as the parameter, and the coordinate part of the volume density of charge is determined by the functionf. The particular solution tS) under condition when p(R, 0 ) = 0 will run as follows t
(6) p (R, t) = exp
--
ers (radioactive isotope) as well as the formation of a volume electric charge in dielectrics, irradiated by an outside source (for example, an electron accelerator). It should be noted that in equation (6) it is supposed that not only cr but E as well are constant and correspond to the values of E under the given conditions of irradiation (the sample should be a high ohmic dielectric). Let us analyse some properties of equation (6) and particular solutions. Equation (6) shows that after irradiation the volume density of charge p relaxes to zero with characteristic time z0 equal to
-eEo -' t
R, 1:) exp
~ dz.
0
As during the derivation of equation (6) there was no limitation of "surface distribution" f then equation (6) describes the process of charging a radioactive dielectric, which has an inner source of charge carri-
Zo=EEo/a.
The simplest solution of equation (6) can be obtained if we assume that f d o e s not depend on time, i.e. constant irradiation with an outside source or "self irradiated" using a long-lived radio-isotope distributed in the volume of dielectric. Then formula (6) runs as follows
(8)
P(R,t)=~II-exp(-Et)]f(R).
Let us discuss two particular cases of the solution of equation (8) when a is small (dielectric) and a is large (conductor).
1. The irradiated sample is a dielectric. In this case, expanding exponent (8) in terms of tr we obtain (9)
p(R, t) = f ( R ) ' t.
That means that up to the moment when t ~ EEoltrthe charging process should obey a linear law. For large values of t the volume density of charge will lead to an equilibrium value: (t >> EE0/a). (10)
££0
p(R) = - - f ( R ) . O"
2. The irradiated samples is a conductor. In this case the time of relaxation of charges, determined from ratio (7) is small. Then p will be determined by expression (10). To find the form of the function f(R, t) let us consider a radioactive sample (V) of arbitrary form (Fig. 1). Let us calculate the amount of charged particles, forming in the volume dV 2 in unit time, as appearing in d V2 and coming to d V2 from the volume (V). According to Fig. 1 the increase of the charge dq
309
Electric charge in volume of fl'-radioactive dielectrics
(v}
X
FIG. I. The scheme of charging volume (IF), I and R-radiusvectors of the volumes d V, and d 112,respectively.
in the volume d V2 per time unit will run as follows
function f(R,t) it is necessary to know Q(R, t), oh(r, E) and I(E). All these formulae correspond to the charging process of dielectric samples with arbitrary shape, subjected to the influence of ionizing irradiation from outer sources or from inner ones (radioactive isotope, distributed within the dielectric). The form of the final expression is determined by the function f(R, t) Let us discuss the charging process of a radioactive dielectric sample with a 13- irradiated radionuclide (evenly distributed in the volume) (i.e. Q does not depend on R and is a function of t). It is obvious that in the volume of such samples a positive electric charge accumulates, appearing as the result of emission of fl- particles into the surrounding medium. Let us assume that the dependence of Q on t is given by the known law of radioactive decay Q = Q 0 e x p ( - 2 0 , where 2 is a decay constant. Moreover, I(E)= 6(E- Eo), where 6(E -Eo) is the 6 function. The latter shows that the source of irradiation emits the particles only with energy E0 (monoenergetic source). After transformation of formula (12) using spherical coordinates from the marked point of volume, we obtain oo
oo
o
0
(19 o0
-- edv2
2~
n
M
o
o
0
f Q(t,R)I(E) dE
where r = Irl e is the electron charge, Q(t, R) is the quantity of charged particles leaving unity volume of coordinate R per time unit, ~ (r, E) is the probability of charged particles covering the distance (r, r + dr) with energy E; in this case the function ¢(r, E) is isotropic by reference to r and I(E) dE is the probability of charged particles having energy in the range of (E, E + dE). The first term of (11) in this case corresponds to the current of charged particles, coming to volume dV2, the second term leaving dV> For the results of d V2 we obtain the function of density of the souce of the charge. This function runs as follows:
}. From (14) it follows that there is not any change in the volume of the sample, at the distance (from its surface) which is longer than the path length of charged particles L, emitted by radioactive isotope because f(0, t) = 0 at any M 1>L (represents an integral number). Let us assume, that the function .q~ looks like 4~ - - ~ ( r - L). It corresponds to the fact that all particles, leaving dV1, are captured in a super thin sphere layer at a distance L from d VI (Fig. 2). Formula (13) looks like
,,4, (e)
(12)
f(R, t)
Let us analyze the integral of expression (14), i.e. oo
= e fj Q(t'l) ~ dv , f dp(r,E)l(E)dE-eQ(R,t). (v)
05)
o
From (12) it is seen that for the calculation of the
It is obvious that the result of integration will depend
310'
V.V. GROMOVand A. G. SAKHAROV (a)
(b)
FIG. 2. The scheme of charging radioactive sphere (a) and flat layer (b) (formulae (14) and (15): o'm -- L) (the length of the path of fl-particles); OK-the radius of the sphere in case (a) and the thickness of the layer in case (b); O'K-parameter of the same point of the sample; M O ' K - t h e angle 0 for the sphere (a) and 01 for the flat layer (b) M O ' P is the angle 02 in case (b).
on the geometry of the sample. Let us discuss two cases which frequently happen: 1. When the charged body has the form of a sphere, 2. When the charged body is an infinite sided shape plate. (see Fig. 2). Let us calculate integral (15) if the charged body has a form of a sphere of radius R. Transforming to the sphere system of coordinates in expression (15) after integration we obtain: (16)
1
y = ~ (1 + cos 0)
where cos 0 = ( 2 R h -- h 2 - L 2)/(2L(R -- h )). Further substituting (16) and cos 0 into (14) and then into (6), after some transformations we obtain (17)
eQoEeo ( 2 R h + 2 L h - 2 L R - - h 2 - L 2 ) 4L(R-h)
P = ~- -----~E~
x(exp exp( )) t7 l
As it follows from (18) the volume density of charge depends exponentially on time and is determined by the third factor of expression (17). The two first factors determine the charge amplitude at the given value h. If their product is marked as P0, formula (17) is like (6). (18)
tr t P = p o ( e x p ( - 2 t ) - e x p ( - ~Eo ))"
For the radionuclide with a large period of half decay (i.e. 2 ~ 0) expression (18) is simplified (19)
exp( ))
From (19) it follows that the kinetics of accumulation of the volume charge in radioactive samples depends on its electrophysical properties as the large value of a and the increase of t cause the trend of p towards P0. The accumulation of the volume charge during irradiation of dielectrics with the use of the outside sources of irradiation obeys the same law. e) This fact is quite natural as the mechanism of tile given process is similar in both cases. All these reasons were taken into account during the composition of the discussed theoretical model. The analysis of the distribution of the volume charge (as of the radioactive sample) is rather interesting. It is necessary to discuss the dependence of P0 on the parameters of the sample R, the value L of the path of fl-particles in the given material and the coordinate h of the analysed element (of the point to be more exact) of the sample. It is better not to have to deal with absolute values of p0 but with the relative value p~ = (eQoEE0/(EE02--a)). Then P6 (taking into account the sign of the volume charge) will be expressed as (20)
p6 --
h 2 + L 2 + 2RL - 2Rh - 2Lh 4L(R -
h)
Let us discuss three interesting cases of distribution of the volume charge, taking into account that (20) in accordance with (14) has sense only when h .<
tr t
P6=
2R + L 4R
Electric charge in volume of fl'-radioactive dielectrics 2. R ~< h < 2R. In this case, as before, at h = 0 the value of P6 > 0. However, when h increases to the value h = 2R - L the charge also increases gradually. Beginning from the values when h i> 2 R - L, the volume density of charge will be constant and equal, according to the value of P6 = 1. 3. L >/2R. At such ratios of radius and path the total volume of spheres is charged uniformily, and P6 = 1 (Fig. 3). Distribution of the volume charge (positive in the case of beta radioactive samples) for different ratio h and L is shown in Fig. 3. A quantitatively analogous distribution of the charge is obtained if the radionuclide has a continuous energetic spectrum of beta decay. However, quantitative calculations for such real systems are difficult to do, as the path of fl-particles is characterized by a continuous set of values of L and 0 to Lu,x. (s) For this purpose the function of distribution ~b(r, E) can be presented as ~b = 3(E -- (KL + 1))
(21)
where K represents a constant proportional to the electron density in the given case. The argument of 6-function is taken on the basis of work. O) The form of the t-spectrum is given by the function with the following form
t-spectrum, E is energy of electrons, Co is the normalizing constant. The energy of electrons in formula (22) is expressed in the units too&. °°) For charging radioactive samples which have the form of a sphere the stationary volume charge (in relative units) can be calculated using the following expression h
(23)
P6 = - K f I(KL + 1) dL 0
t._,,
q
2L(R - h )
]
h
+ Z(KL + 1) dL + 1. The calculations according to formula (23) to determine the values K, E = , and corresponding values of Lm,x are done by quantitative integration. (8) If a radioactive sample has the form of an infinite sided plate of thickness d, it is possible to obtain the following common expression to calculate the values of P6 h
(24)
P6 =
-- K
I(E) = CoE(Em,x-- E)2(E 2 -- 1)'/2
(22)
31 !
f Z(KL +
1)
dL
0 /-max
where Era,x is maximal energy of the electrons in the
K
2 f(h+l)
I(KL+I)dL+I
h
1.0 h
f
(25)
d-h
o6 = - K f I(KL +
1) dL - - 5K
0
1) h
/,max
xI(KL+I)dL-K
0.5
z I ( K L + 1)dL + 1 d-h
2R-L
L
I R
Formula (24) can be used at d > 2Lm,x, formula (25) d ~< 2Lm,x. If the radionuclide emits monoenergetic beta particles, the amplitude values p, as function h, can be represented the same as in (20) according to (24) the form:
h
FIG. 3. Distribution of the space charge in radioactive sphere of the radius R with ~-irradiator uniformly distributed in the volume, L is the length of the path of t-particles (~-particles are monoenergetic). The surface of the sphere coincides with the axis of ordinate. 1 - L
2--R <~h <2R.
(26)
1
P6 = ~ (cos 0, -- cos 02) + 1.
The angles 0= and 02 can be determined using the scheme in Fig. 2. Corresponding meanings of cosines
312
V. V. GROMOVand A. G. SAKHAROV
are given by the equations, involving parameters h, L, d.
I
h, if L2 _ h2 > 0
cos O1 = ~ 1, i f L 2 - -
h 2
~< 0
(27)
I
~-~--~, i f L 2 -- (h - d) 2 > 0
cos02=[
_l, ifL_(h_d)
2~<0
Calculations with the use of formulae (24)-(27) have been done for three cases: d = 1.1; 0.7 and 0.4cm. Value of k equal to 10, maximal energy of beta particles was 3 MeV which corresponds to a maximal length of the path L ~ = 0.5 cm (Fig. 4). The form of the dependence of p~ on h is determined mainly by the//-spectrum (Fig. 4). An uninterrupted //-spectrum is characterised by a gradual change of curves of distribution of the volume charge (compare curves 1 and 6 on Fig. 4). Conclusions obtained on the distribution of charges in radioactive samples correspond to compounds with different a. However, in conductors it makes no sense to speak about the volume charge, as it quickly appears on the surface. These questions are discussed in work. °~) In the dependence on electroconductivity of the surrounding medium the volume charge of radioactive sample of dielectric can be compensated partially or totally by the current carder, attracted to the surface (Fig. 5). A hatched region corresponds to the
positively charged above-surface layer, the thickness of which L0 is equal to the path length of the t-particles, emissed by the given radionuclide. As it is seen from the Fig. 5 it is better to pay attention to the total charge q +, distributed in the layer L0, and residual surface charge r/~ (residual), equal to the difference between q + and the absolute value r/. It is obvious that such conditions can be realised when r/~ = 0, i.e. the sample is compensated totally and the outside field is absent. The meaning of rT~sand q+s can be calculated precisely, if the electrophysical properties of radioactive material, its geometry and properties of surrounding medium are known. If distribution of 4 ÷ in the volume is not taken into account and accumulation of charge obeying the law (18) or (19) is taken into account we can determine the mean meaning of fi~+sin the charged area and also the value of rT~s. It is necessary to mark that accumulation of charge in a dielectric does not proceed infinitely: some balance is reached, when the leakage currents (because of the conductivity of the sample) compensate accumulation of the charge reaches the value, at which electrical breakdown is possible. Let us take only the "before breakdown" values of the charge*, p will be used to mark symbols, relating to
/q
(+)
/77
o7I
3
0.6 0.5
0.4 0.3 0.2 0.1 0i
0.1
0.2
0.3' 0.4
0.5
0.6
0.7
0.8
0.9
1.0
i=
L
,.
1
h
FIG. 4. The dependence of relative density of the charge of //-radioactive sample in the form of the plate of the fiat infinite type; the axis of ordinates coincides with one of its side surface: curves 1-3 correspond to the thickness of the plate 1.1; 0.7 and 0.4cm (monoenergetic source of //-particles); curves 4.-6--the same for uninterrupted spectrum of the source of irradiation.
FIG. 5. The scheme of distribution of common volume charge in the above surface layer L of radioactive dielectricsample, containing homogeneously-distributed fl radioactive isotope. On the surface compensating carders with the opposite sign q accumulate. The change of density of the volume charge is given by uninterrupted curve in the layer L.
313
Electric charge in volume of fl'-radioactive dielectrics the surrounding medium, the sign m radioactive material. Residual stationary charge 0H depends on specific radioactivity and the processes of recombination of carriers in surrounding space. (6,n,~3)Its value is determined from the conditions of equality of the current of emission of beta particles from the sample is and the current of carriers to the sample ip, appearing under the influence of the field of the volume charge 8 of the surface. Equating these points and taking into account that I .1 = I~:,,/,.o,,, and ip = eup(nop + n,)l .l and i,, = K2mQo we obtain (28)
~
= KtEOEm"mQo/eup(nop +
np)
where Kt, K2, K3 are constants, depending on the geometry of the sample and dimensions of quantities in formula. (28) m is the mass of radioactive material; Q0 its specific radioactivity; up is the mobility of carriers, nop, np their concentration without irradiation and during the process of irradiation respectively, E0, EI are the dielectric constants of x'acuum and the material of the samples.3" The constant km determines the fraction of beta particles, leaving the radioactive sample (the form of the sample are taken into account). To find np let us suppose that the rate of ionization V~ of the compound surrounding the surface (gas for example) is proportional to Q0, and the rate of recombination of carriers V2 is a bimolecular process. Then under stationary conditions when/:1 = / : 2 (but I:1 = consh Q0 and V~ = consh n~) and (29)
np = const
Q1/2
substituting (29) into (28) we obtain (30)
ff~ =
(31)
(32)
~l,~ =
KoeoemQ~/2[nom~ll(#, r)/S~] I/2
• uf(~Lk: F(E:) dEdx(AE)ffW'COnst21 '/2" Here k 0 is constant, depending on the dimensions of the values of the values in formula (32); nQ is the quantity of t-particles, emitted in time unit by unit of radioactivity of the given radioisotope; S m is the surface of the sample; w is the energy which is spent in the formation of a pair of ions in surrounding medium; const2 the coefficient of the ions recombination in surrounding gas medium, F(E) is the distribution function of beta particles in the spectrum of irradiation; dE/dx is the mean loss of the irradiation energy in the surrounding medium; the sign Y. represents the amount along the whole spectrum of energy, ~(~, r) is the function, determining the fraction of particles, leaving m, at different thicknesses of the sample r and with a coefficient of irradiation absorption/~. For example, for the sphere the function g,(g, r) runs as follows (16): (33)
@(#, r) = [1 - e x p ( -
#r)]/#r.
The meaning of dE/dx in a gas medium near the surface m can be determined using the formulae from works. (17,1s)Formula (32) describes the change of ff~s of radioactive samples, being in a gaseous medium at a pressure 1-103 mm of mercury. F o r the same intervals of pressure, using (31) and (32) it is possible to find a simple dependence between ffr~ and pressure p.(14,15)
KIEoEmmQo/eUp(nop+ const Q 5/2)
From (30) and (28) it follows that at fl~ = aQo and at hop ~ nr
the calculations of the residual stationary charge, relating to the unit of surface of radioactive sample r/.s, is rather complex (6:4:5)
(34)
rT,~ = const P - 1/2.
nop >>%,
~,~ = f "Q~/2
where a and b are constant. As follows from (31), the residual charge ~r~ of radioactive preparation under all other conditions increases proportionally to the square root of specific radioactivity. A similar result in the particular case of a flat radioactive sample in air atmosphere was obtained in work. t13) The final form of expression (31), allowing us to do
1"Deriving(28) ip is supposed to be determined by the drift of carriers under the influence of the charge field.
From (34) it follows that with a decrease of pressure the residual charge should increase. This fact is observed experimentally.(6,n,~4) In the dependence on P the time of accumulation of F/~ will also change• It is easy to show that with the decrease of pressure the equilibrium between medium m and P is reached m o r e slowly. (6'~5)
It should be noted that formulae (28), (34) can be applied for electric materials and conductors. However in case of conductors ff~ corresponds to change, localized on the surface but the volume change is absent. Stationary volume change P~+s can be calculated easily, using formula (18) at 2 = 0 (the radionuclide has a large half decay time) t ~ ~ (stationary condi-
314
V. V. GROMOVand A. G. SAKI-IAROV
tion). After elemental transformations for sphere samples with radius r we obtain (it is supposed that the processes of discharge on the boundary of media II and I do not influence the kinetics of accumulation (35)
t
-15
+ = KonQmQoEoE,,~k (It, r ) / a m " V . P- ~s
where a m is electroconductivity of radioactive (self irradiating) material and K 0 is constant depending on the values of equation (35). (6) In formula (35) P,+s corresponds to mean charge in volume unit of charged layer V. The great difficulties in calculations are caused in the necessity of evaluation of am, which usually differs from corresponding values of electroconductivity of non-irradiated material. Here two approaches are possible: 1. am=anoni~ad-this fact causes calculated values of P~s to b e higher. 2. Different empiric formulae, giving the bond between electroconductivity of irradiated compound and density of the energy stream of irradiation (intensity of irradiation). ~5) Direct experimental determination of as is possible of course. M E A S U R E M E N T S O F ELECTRIC C H A R G E O F R A D I O A C T I V E SAMPLES It is easy to change the charge in case of radioactive conductors and low ohmic semiconductors in the gas phase. °5'19) Such determinations have been done electrochemically (the method of charge accumulation) for plates made of nickel, tungsten, containing radioisotopes of 63Ni, 65Zn, 185W, respectively, and for germanium alloyed with radioactive antimony (Fig. 6). Experimental data are testified by the linear dependence of the surface charge of radioactive samples on the square root of specific radioactivity; (coefficient of slope of the direct line in Fig. 6 is equal to 1/2). Using direct "current methods", it is difficult to determine the volume electric charge of a radioactive dielectric sample as all surrounding surface is ionized and the self conductivity of the material is low. As a result the currents of discharge, giving information about ~+,, are small in comparison with the leak currents of measuring techniques. Nevertheless, such measurements have been done. (2°)To the charged layer (its thickness was "-"3 mm) of radioactive samples the current contacts (pressure-in and sealing-in) have been introduced and the difference between them has been determined; the value of this difference has been used for evaluation of ~+,. Substance
-14
I
I
I
-1
-2
-3
IgO (mCu/g)
FIG. 6. The dependence of the charge of the surface ~/~,of germanium on its specific radioactivity by reference to 124Sb (fl, ~,-irradiator) in the presence of air at 2--6mm of mercury.
As is seen from the given data, in radioactive dielectrics the formation of a charged layer proceeds. However, in this case, the results otained are qualitative but not exact. The difference between experimental and calculated values of/~+, can be explained by inaccuracies in the experimental technique, and errors in calculations. It has been remarked earlier, that the volume charge in radioactive dielectrics can accumulate to such values and lead to the appearance of an immediate inner electric discharge. F o r example, according to calculations using (35) in samples with a = 10-2°(f2 • m ) - 1 at Em= 5 -- 10 and specific radioactivity 0.1-1 Cu/g in charged layer Lo, the intensity of field reaches 105 - 106 V/cm. This fact corresponds to the break-down voltage for the most part of dielectrics. To test the formation of self discharges lower-conductive glasses (a = 10 - 2 2 1 0 - 2 4 ( ~ ' ~ " m ) - i, containing up to 30 mCu/g) of radivactive strontium 9° in equilibrium with yttrium 9° have been prepared. In such glasses 2 h after their production the intensity of field in a charging layer can reach 106 V/cm and self electric charges appear. This
Qo[mcu~g]
p[tor]
BaSO4; a = 3-10 - 16[o.m]- J
I0
200
1.10 - "
2.10 - io
Glass, cr = 10-25[o-m]- l
30
50
1-10-9
1"10-6
~+dc'cm-~]exper.
~+~[c'cm- 3]calcul.
Electric charge in volume of/]'-radioactive dielectrics phenomenon has been observed though to stimulate the process of discharge in glass a mechanic voltage had to be formed by pressing the glass with metal or glass stick. (2~) These experiments testified to the idea of volume charge of radioactive dielectrics. To measure the charge accumulating in radioactive samples the method of dynamic condensor was used. (22)The experiments were done with radioactive neodimium oxide (147Nd, /l-irradiator, 7 mCl/g) at pressure 10-s-10 -~ mm of mercury. The difference of potentials between the neodimium oxide and the comparison electrode (gold plate) reached 100 V. However at heating it decreased linearly in the range 300-400°C and was close to the contact difference of potentials of nonradioactive oxide. This fact can be explained by higher electroconductivity of samples at heating, accompanying by the charge leak of radioactive oxide. Practically this method for the charge measurement allows us to evaluate only the values of Pr,, but not the volume charge. That is why at normal air pressure the difference of potentials being measured decreased to values corresponding to nonradioactive samples. Unfortunately the experiments mentioned do not give any information about charge distribution in the layer L0. The most exact results in this ratio were obtained by the method of sound zoning based on the display of a running sound wave through the sample and electric signal in measuring electrode. The value of this signal corresponds to the intensity of electric
tThis term is used to underline the first reason for the appearance of charge, i.e., the influence of irradiation (irradiation or self-irradiation).
field in the area being deformed/23) The sensitivity of the method in charge was 5 x 10 -3 c/m 3, distortion was 0.1 mm. Experiments with the crystals of yttrium-aluminium garnet (Y3AIsO2) being irradiated by slow neutrons have been done. In the result of an irradiation radioisotope 9oy (10 mcu/g) ~-irradiator, E = 2.25 MeV, Tt/2 = 64.2 h) formed in granate. The calculations and experimental determinations showed that the residual charge of the granate crystals (flat parallel plane) is practically absent in air under normal conditions but being compensated it reaches high values (Fig. 7). From Fig. 7 it follows that in an electrononeutral sample Y2AI50~2 in the subsurface layer a positive charge accumulates, the distribution of which corresponds to the theoretically predicted one for the case when L < d (thickness of plate) (compare with Fig.
4). R A D I A T I O N ELECTRIC C H A R G E r A N D PROPERTIES OF SOLIDS A number of physical-chemical processes involving the participation of solid radioactive or irradiated dielectric samples are affected by radiation charge. These processes are: kinetics of dissolution of solids, ¢24) adsorption and catalytic processes. 03,22.25,26) evaporation, electrochemical processes, etc. Let us discuss the most typical examples of influence of volume charge on kinetics of heterogenic processes. The first link between the volume charge of a radioactive sample and the rate of dissolution of solids was discovered in the dissolution in water of radioactive crystals of strontium sylphate, containing a pure/~-irradiator-sulphur-35 (Fig. 8)/27,2s) As follows from Fig. 8 the rate constant of dissolution
3 E 2 7
O X LO
0 2
315
3 x (ram)
Fro. 7. The distribution of the charge in the monocrystal of yttrium-aluminium garnet. Specific radioactivity according to ~Y, 10 mCi/g.
V. V. GROMOVand A. G. SAKrtAROV
316
b
olI 10-2 ~ -4t 0
J
10-~ IgQ (rnCu/g) I
I 10
10 -~
I
I
10
10 2
t 10
Ige (V/cm)
Fro. 8. The change of the rate constant of dissolution k~ of radioactive crystals. SrSO4 in the dependence on its specific radioactivity Q and the intensity of field ~p in the above surface charged layer. The calculations of the charge (and its corresponding value of n.) have been done using formu[a.(~)
(specific rate of dissolution) decreases when Q0 and the corresponding field 8 of the volume charge increase. Additional experiments, performed with nonradioactive but electron-irradiated (or simply outer field polarized initially) of monocrystals of LiF, K2SO4, SrSO4 and NaF showed the similarity of this phenomenon327-29~ It turned out that under certain conditions the rate of disolution not only decreases when e increases but can even increase as well33°~ Theoretical analysis of the given process permits the determination of the relation between the rate constant of dissolution ko, crystal polarization, properties of carrier and charged sample in the form: (36)
In(kJk*) = A( h'(ez~k'~ k S \4kT) • ~:
4ezCo
ez~,
where A is a constant, a is the net polarization of the molecular, dissolving compound, Em is statistic dielectric, k is the Boltzmann constant, z is the ion charge in the solution, Co is dissolution, x is an inverse length, equal to (8ne2z2Co/Ep'kT)m, Epis the mean dielectric penetrability of solution and ~k~is the mean potential of the outer plane, where the ion centres of solutions are localized (the nearest to the
surface of dissolving crystal); p is the crystal polyrization.(2s'29) As follows from formula °6) at mean and high values of dissolution and E,~< Ep the inner electric field causing polarization of the dielectric, make the rate of dissolution slower. Besides, with a decrease of dissolution and dielectric constant of the liquid phase, the break-down influence of the electric field becomes weaker. At small values of ep(Ee ~
Electric charge in volume of fl'-radioactive dielectrics one micron (i.e. activity for a particle 10-3Cu)). 04-41) The mechanism of formation o f the charge of hot particles is practically the same as in the case of charging o f big radioactive bodies. It has been determined that hot particles, in spite o f the nature of irradiator, (a or/~) have a positive charge. This effect is the case of x-radioactive particles can be explained by the main role o f secondary 6-electrons (their number, emitted from the sample, is larger than the number of emitted a particles) during the process of the charge formation. F o r example from a particle of dibutylphalate with the radius 1 mkm, containing as irradiator 222Rn 5-40 secondary electrons were emitted during ~-decay. °s) The positive charge o f hot particles reached big values in spite o f the conductivity of the surrounding air medium because of its intensive ionization in hot particles. Practically if the particle emits more than 1 electron per sec (i.e. activity is equal 10 - n Cu/particle) at atmosphere pressur e it will have a stationary positive charge. 0'36) This p h e n o m e n o n is not only scientifically interesting as the charge determines the laws of travel of " h o t " airsol particles within the air stream but also allows us to work out the methods o f electrostatic volatilization o f such particles. N o w a d a y s the facts obtained show that the influence o f volume charge (it forms not only in radioactive and irradiated materials, but also under the influence of light mechanic loads, heating, etc) on the process of different heterogeneous processes. This fact has physical-chemical meaning. The given formulae allow us to evaluate the volume and residual charge under different experimental conditions. This fact provides an opportunity to take into account the influence o f "the charge effects" during the study o f the influence of irradiation, light and other factors on solid and heterogeneous systems. REFERENCES 1. P. Ctmm and M. C-MRm,Comp. Rend. 1900, 130, p. 647. 2. U. KORLISand D. HARVY,Energetic Sources on Radioactive Isotopes. (Translations from English), M., Mir, 1967. 3. I. E. TAMM, Osnovy teorff elektrichestva. Nanka, Moscow 1966 (in Russian). 4. A. G. SAKrlAROVand V. V. GROMOV,Zh. Fiz. khimii 1976, 50(6), 1513 (in Russian). 5. B. GROSS, G. M. SESSLER,J. E. WEST, J. Appl. Phys. 1974, 45, 4724. 6. V. I. SPITSYNand V. V. GROMOV,Fiziko-khimicheskiye svoystva radioaktivnykh tel Atomizdat, Moscow 1973 (in Russian). 7. B. GROSS, d. Appl. Phys. 1965, 36, 1635. 8. A. G. SAKI-IAROV, V. V. GRoraov and K. M. MERZHANOV,Zh. fiz. khimii, 1978, 52(2), 433 (in Russian).
317
9. S.V. STARODUBTSEV,Prokhozhdeniye zar. chastits cherez veshestvo. FAN Tashkent, 1962 (in Russian). 10. M. BORN, Atomn. fizika. M.: Mir. 1965 (in Russian). 11. A. A. VononYEV and O. B. EVDOKIMOV,Khimiya vysokikh energii 1970, 4, 356 (in Russian). 12. V. V. GROMOV and V. I. SPITSYN, Radiokhimiya 1972, 14(4), 795 (in Russian). 13. V. G. BARYand F. F. VOLKENSrr~IN,Iz. A N S S S R , Ser. khim. 1964, No. 11, 1935 (in Russian). 14. V. V. GgoMov and V. I. SI'ITSYN,J. Res. Inst. Catalysis, Univ Hokkaido 1971, 19(1), 1. 15. V. V. GROMOV, Atomn. energiya 1968, 26, 250 (in Russian). 16. L. M. EFIMOV,In Tr. Vsesoyuzn. nauch.-tehn, konf M.: Izd. A N SSSR 1958, 382 (in Russian). 17. L. KATZ and A. S. PEr~OLD, Rev. Mod. Phys. 1952, 24 (1), 28. 18. YA. C'mJDASand I. TAURE,lZV. ANLatv. SSR 1959, 3, 33 (in Russian). 19. V. I. SPITSYN,V. V. GROMOV,V. S. AREKELYANand N. G. LVSENKO,Dokl. A N SSSR 1968, t. 182, s, 390. 20. V. V. GROMOV, S. N. OZmANER, V. I. SVITSYNand A. A. MII-L~YEV,Zh. fiz. khimii 1971, 45, 2804. 21. V.V. GROMOVand V. V. StmlKOV,Atom. energiya 1972, 32, 172. 22. V. I. SPITSYN, G. N. PmOGOVA, A. A. SOPINA and E. KH. YENEKEEV, Dokl. A N SSSR 1969, 186, 1358. 23. A. G. ROZNOand V. V. GROMOV,Pisma v ZhTF 1979, 5, 648. 24. V. V. GROMOV, Vliyaniye ioniz, izlucheniya, tomizdat, Moscow, 1976 (in Russian). 25. V. V. GROMOV,V. I. TROrlMOV, V. M. LUKYANOVICH and V. I. SPITSYN, DokL A N S S S R 1968, 178, 1307. 26. V. I. SPITSYN,IZV. A N SSSR, OKhN, 1958, 11, 1296. 27. V. V. GROMOVand T. N. BESPALOVA,In Radiatsionnaya fisika, ionnye cristally. Riga Zinatne, 1966, 4, 531 (in Russian). 28. V. V. GROVOM and V. S. KRYLOV, Dokl. A N SSSR 1970, 192, 123. 29. V. V. GROMOV,T. N. PARKHAEVAand V. S. KRYLOV, Zh. fiz. khim. 1973, 47, 107. 30. V. V. GROMOV,V. S. KRYLOVand V. N. SHULYATYEVA, Zh. fiz. khim. 1978, 52, 2861. 31. L. M. IVANKIVand Z. V. SOLVANIK, Ukr.fiz. zh. 1968, No. 13, 852. 32. G. F. MALINaN,Yu. A. PRISELKOVand P. P. KLIMENKO, Dokl. A N SSSR 1969, t. 188, s. 846. 33. R. WILLIAMSand M. J. CAMPOS,J. Appl. Phys. 1970, 41, 4138. 34. G. L. NATANSON,Uspekhi khimii 1956, 25, 1429. 35. V. D. IVANOV,V. P. KmlcnEm¢o and I. V. PETRYANOV, Dokl. A N S S S R 1968, 182, 307. 36. V. D. IV^NOVand V. N. IORICHENKO,Dokl. A N S S S R 1969, 188, 65. 37. V. D. IVANOV,V. N. KIRICrmSKO, V. M. BER~ZI-INOY and I. V. PETRYANOV,Dokl. A N SSSR 1972, 203, 806. 38. V. D. IVANOV,V. N. KmlCttENKO and B. V. SHAN'GIN, Kolloidn. zh. 1974, 36(3), 468. 39. V. N. KIRICHENKOand V. D. IVANOV,Dokl. A N S S S R 1969, 188, 315. 40. V. N. KlmCrmNKO, V. M. BERZHNOY and B. V. SHAN'GIN, Dokl. A N S S S R 1972, 205, 78. 41. V. N. KImCHENKOand N. N. SUVRUN,Dokl. A N SSSR 1974, 215, 325.
0146-5724/84 $3.00 + .00 Pergamon Press Ltd.
ELECTRIC CHARGE IN THE VOLUME OF fl'-RADIOACTIVE DIELECTRICS V. V. GROMOV and A. G. SAra-IAROV Institute of Physical Chemistry of the Academy of Sciences of the USSR, Leninsky Prospect 3 l, Moscow, U.S.S.R.
(Received 6 February 1983) Abstract--The mechanism of formation and distribution of electric charge in the volume of dielectric solids, containing a fl--radioactive isotope is discussed in the present paper. Experimental data of measurements of charge in different materials are given. Examples illustrating the influence of the electric volume charge on kinetics of dissolution,.sorption, evaporation of solids are given here. It is shown that the results otained are applied for external irradiation of dielectrics by the beam of accelerated electrons.
INTRODUCTION Tim rORMAaaONof electric charge in radioactive solids as a physical phenomenon was discovered in 1900. o) During the experiments the current of charged partides emitted by radium in samples of Ba(Ra)Cl2 was measured qualitatively. Further studies in this field had technical objectives--formation of the nuclear battery or radioactive current sources. (2) In works °'2) the possibility of accumulation of electric charge in the volume of radioactive material(dielectric) was not in the question, as attention was paid to the process of "outer charging" and thermal effects, connected with break-down in collectors with different constructions of charging particles, emitted by radioactive source.
Nowadays it has been confirmed that any radioactive solid accumulates electric charge, the sign of which is opposite to the charge of the particles which leave the surface of the sample. However, under normal conditions (for example, in a water or in an air-conducting surrounding medium) the charge in radioactive preparations is rather small because of its neutralization by carriers outside of the surface. It was considered that this behaviour was true for any solid: metals, oxides, salts, ceramics, etc. However, for dielectrics this phenomenon turned out to be more complex. Detailed study of formation of charge in radioactive and irradiated solids of high ohmic materials (which has been performed lately) shows that in radioactive dielectrics, even in conducting media, one can observe the accumulation of electric charge within large volume. Its value depends on the level of radioactivity and electric properties of the sample. A peculiarity of such preparations is the following: at total electronic neutralization of the sample a noticeable gradient of the charge appears
within the specimen as on the surface there is only compensation of volume charge by carriers from surrounding space. DESCRIPTION OF THE C H A R G E PROCESS Let us discuss the following model: the sample has isotropic properties, the sources of charged particles are distributed in the volume arbitrarily, the process of neutralization of charge of opposite sign proceed only on the surface. To simplify this fact let us consider that electrophysical properties of the sample are characterized by electroconductivity a and dielectric constant E, Ohm's law is obeyed and only one type of carriers is present. In the common case to solve this task it is necessary to use the equation of continuity 3) dp (l)
--
~t
=
-- V "j
where p is the volume point density of the charge, j is the vector of current density in the sample and V the Hamilton's operator. In the common case dielectric penetration E and electroconductivity of the sample a as a function of coordinates-depend also on intensity of irradiation. For homogeneous dielectrics the current density can be represented as
(2)
j--~8+ji
where ae is the current component, which is determined by its conductivity of dielectric under the influence of the field e of the volume charge andji the current density of injected carriers; in this case t~ corresponds to elcctroconductivity under the given conditions of irradiation. 307
V. V. OROMOVand A. G. SAKHAROV
308
If we substitute expression 2 into expression 1 and take into account that Iz • s = p/EEo, after some transformation we obtain the following expression
Op (3)
- -
=
(F~r)8
--
Ot
tr
- --p
--
Vj,.
EEo
This expression can be considered to be an equation determining p if in some definite case it is possible to express 8 using the volume density of charge. For example, if in an infinite dielectric there is an area in the form of a sphere equation (3) has the following form
(7)
r
(4)
Op O~ =
do 4~ f tr 1 O Or r-Y-~EEo P~2 d~ ----PEEo --~Orr (r2"j')" 0
Expression (4) is written in the spherical system of coordinates (r, 0, ~ ) the centre of which coincides with the centre of radioactive area. It is supposed that tr and Ji do not depend on 0 and tp. If j~ (projection ji onto r) is a known function of coordinates and time, and t r a known function of coordinates, then equation (4) is an integro-differential equation with reference to p. Even in such a simple case the task is to solve integrodifferential equation, that is why only simplified models of practical importance are discussed under specific experimental conditions. If (a, E) is supposed to be independent of the coordinate (which are stable in the sample) and it does not change during the process of interaction of ionizing irradiation, equation (3) leads to the simple differential equation (5)
dp(R, t) _ dt
tr eeo p (R, t) + f(R, t)
where R is a coordinate, t is time, f = - 17ji. As it is seen from equation (5) the solution of p will depend on R as the parameter, and the coordinate part of the volume density of charge is determined by the functionf. The particular solution tS) under condition when p(R, 0 ) = 0 will run as follows t
(6) p (R, t) = exp
--
ers (radioactive isotope) as well as the formation of a volume electric charge in dielectrics, irradiated by an outside source (for example, an electron accelerator). It should be noted that in equation (6) it is supposed that not only cr but E as well are constant and correspond to the values of E under the given conditions of irradiation (the sample should be a high ohmic dielectric). Let us analyse some properties of equation (6) and particular solutions. Equation (6) shows that after irradiation the volume density of charge p relaxes to zero with characteristic time z0 equal to
-eEo -' t
R, 1:) exp
~ dz.
0
As during the derivation of equation (6) there was no limitation of "surface distribution" f then equation (6) describes the process of charging a radioactive dielectric, which has an inner source of charge carri-
Zo=EEo/a.
The simplest solution of equation (6) can be obtained if we assume that f d o e s not depend on time, i.e. constant irradiation with an outside source or "self irradiated" using a long-lived radio-isotope distributed in the volume of dielectric. Then formula (6) runs as follows
(8)
P(R,t)=~II-exp(-Et)]f(R).
Let us discuss two particular cases of the solution of equation (8) when a is small (dielectric) and a is large (conductor).
1. The irradiated sample is a dielectric. In this case, expanding exponent (8) in terms of tr we obtain (9)
p(R, t) = f ( R ) ' t.
That means that up to the moment when t ~ EEoltrthe charging process should obey a linear law. For large values of t the volume density of charge will lead to an equilibrium value: (t >> EE0/a). (10)
££0
p(R) = - - f ( R ) . O"
2. The irradiated samples is a conductor. In this case the time of relaxation of charges, determined from ratio (7) is small. Then p will be determined by expression (10). To find the form of the function f(R, t) let us consider a radioactive sample (V) of arbitrary form (Fig. 1). Let us calculate the amount of charged particles, forming in the volume dV 2 in unit time, as appearing in d V2 and coming to d V2 from the volume (V). According to Fig. 1 the increase of the charge dq
309
Electric charge in volume of fl'-radioactive dielectrics
(v}
X
FIG. I. The scheme of charging volume (IF), I and R-radiusvectors of the volumes d V, and d 112,respectively.
in the volume d V2 per time unit will run as follows
function f(R,t) it is necessary to know Q(R, t), oh(r, E) and I(E). All these formulae correspond to the charging process of dielectric samples with arbitrary shape, subjected to the influence of ionizing irradiation from outer sources or from inner ones (radioactive isotope, distributed within the dielectric). The form of the final expression is determined by the function f(R, t) Let us discuss the charging process of a radioactive dielectric sample with a 13- irradiated radionuclide (evenly distributed in the volume) (i.e. Q does not depend on R and is a function of t). It is obvious that in the volume of such samples a positive electric charge accumulates, appearing as the result of emission of fl- particles into the surrounding medium. Let us assume that the dependence of Q on t is given by the known law of radioactive decay Q = Q 0 e x p ( - 2 0 , where 2 is a decay constant. Moreover, I(E)= 6(E- Eo), where 6(E -Eo) is the 6 function. The latter shows that the source of irradiation emits the particles only with energy E0 (monoenergetic source). After transformation of formula (12) using spherical coordinates from the marked point of volume, we obtain oo
oo
o
0
(19 o0
-- edv2
2~
n
M
o
o
0
f Q(t,R)I(E) dE
where r = Irl e is the electron charge, Q(t, R) is the quantity of charged particles leaving unity volume of coordinate R per time unit, ~ (r, E) is the probability of charged particles covering the distance (r, r + dr) with energy E; in this case the function ¢(r, E) is isotropic by reference to r and I(E) dE is the probability of charged particles having energy in the range of (E, E + dE). The first term of (11) in this case corresponds to the current of charged particles, coming to volume dV2, the second term leaving dV> For the results of d V2 we obtain the function of density of the souce of the charge. This function runs as follows:
}. From (14) it follows that there is not any change in the volume of the sample, at the distance (from its surface) which is longer than the path length of charged particles L, emitted by radioactive isotope because f(0, t) = 0 at any M 1>L (represents an integral number). Let us assume, that the function .q~ looks like 4~ - - ~ ( r - L). It corresponds to the fact that all particles, leaving dV1, are captured in a super thin sphere layer at a distance L from d VI (Fig. 2). Formula (13) looks like
,,4, (e)
(12)
f(R, t)
Let us analyze the integral of expression (14), i.e. oo
= e fj Q(t'l) ~ dv , f dp(r,E)l(E)dE-eQ(R,t). (v)
05)
o
From (12) it is seen that for the calculation of the
It is obvious that the result of integration will depend
310'
V.V. GROMOVand A. G. SAKHAROV (a)
(b)
FIG. 2. The scheme of charging radioactive sphere (a) and flat layer (b) (formulae (14) and (15): o'm -- L) (the length of the path of fl-particles); OK-the radius of the sphere in case (a) and the thickness of the layer in case (b); O'K-parameter of the same point of the sample; M O ' K - t h e angle 0 for the sphere (a) and 01 for the flat layer (b) M O ' P is the angle 02 in case (b).
on the geometry of the sample. Let us discuss two cases which frequently happen: 1. When the charged body has the form of a sphere, 2. When the charged body is an infinite sided shape plate. (see Fig. 2). Let us calculate integral (15) if the charged body has a form of a sphere of radius R. Transforming to the sphere system of coordinates in expression (15) after integration we obtain: (16)
1
y = ~ (1 + cos 0)
where cos 0 = ( 2 R h -- h 2 - L 2)/(2L(R -- h )). Further substituting (16) and cos 0 into (14) and then into (6), after some transformations we obtain (17)
eQoEeo ( 2 R h + 2 L h - 2 L R - - h 2 - L 2 ) 4L(R-h)
P = ~- -----~E~
x(exp exp( )) t7 l
As it follows from (18) the volume density of charge depends exponentially on time and is determined by the third factor of expression (17). The two first factors determine the charge amplitude at the given value h. If their product is marked as P0, formula (17) is like (6). (18)
tr t P = p o ( e x p ( - 2 t ) - e x p ( - ~Eo ))"
For the radionuclide with a large period of half decay (i.e. 2 ~ 0) expression (18) is simplified (19)
exp( ))
From (19) it follows that the kinetics of accumulation of the volume charge in radioactive samples depends on its electrophysical properties as the large value of a and the increase of t cause the trend of p towards P0. The accumulation of the volume charge during irradiation of dielectrics with the use of the outside sources of irradiation obeys the same law. e) This fact is quite natural as the mechanism of tile given process is similar in both cases. All these reasons were taken into account during the composition of the discussed theoretical model. The analysis of the distribution of the volume charge (as of the radioactive sample) is rather interesting. It is necessary to discuss the dependence of P0 on the parameters of the sample R, the value L of the path of fl-particles in the given material and the coordinate h of the analysed element (of the point to be more exact) of the sample. It is better not to have to deal with absolute values of p0 but with the relative value p~ = (eQoEE0/(EE02--a)). Then P6 (taking into account the sign of the volume charge) will be expressed as (20)
p6 --
h 2 + L 2 + 2RL - 2Rh - 2Lh 4L(R -
h)
Let us discuss three interesting cases of distribution of the volume charge, taking into account that (20) in accordance with (14) has sense only when h .<
tr t
P6=
2R + L 4R
Electric charge in volume of fl'-radioactive dielectrics 2. R ~< h < 2R. In this case, as before, at h = 0 the value of P6 > 0. However, when h increases to the value h = 2R - L the charge also increases gradually. Beginning from the values when h i> 2 R - L, the volume density of charge will be constant and equal, according to the value of P6 = 1. 3. L >/2R. At such ratios of radius and path the total volume of spheres is charged uniformily, and P6 = 1 (Fig. 3). Distribution of the volume charge (positive in the case of beta radioactive samples) for different ratio h and L is shown in Fig. 3. A quantitatively analogous distribution of the charge is obtained if the radionuclide has a continuous energetic spectrum of beta decay. However, quantitative calculations for such real systems are difficult to do, as the path of fl-particles is characterized by a continuous set of values of L and 0 to Lu,x. (s) For this purpose the function of distribution ~b(r, E) can be presented as ~b = 3(E -- (KL + 1))
(21)
where K represents a constant proportional to the electron density in the given case. The argument of 6-function is taken on the basis of work. O) The form of the t-spectrum is given by the function with the following form
t-spectrum, E is energy of electrons, Co is the normalizing constant. The energy of electrons in formula (22) is expressed in the units too&. °°) For charging radioactive samples which have the form of a sphere the stationary volume charge (in relative units) can be calculated using the following expression h
(23)
P6 = - K f I(KL + 1) dL 0
t._,,
q
2L(R - h )
]
h
+ Z(KL + 1) dL + 1. The calculations according to formula (23) to determine the values K, E = , and corresponding values of Lm,x are done by quantitative integration. (8) If a radioactive sample has the form of an infinite sided plate of thickness d, it is possible to obtain the following common expression to calculate the values of P6 h
(24)
P6 =
-- K
I(E) = CoE(Em,x-- E)2(E 2 -- 1)'/2
(22)
31 !
f Z(KL +
1)
dL
0 /-max
where Era,x is maximal energy of the electrons in the
K
2 f(h+l)
I(KL+I)dL+I
h
1.0 h
f
(25)
d-h
o6 = - K f I(KL +
1) dL - - 5K
0
1) h
/,max
xI(KL+I)dL-K
0.5
z I ( K L + 1)dL + 1 d-h
2R-L
L
I R
Formula (24) can be used at d > 2Lm,x, formula (25) d ~< 2Lm,x. If the radionuclide emits monoenergetic beta particles, the amplitude values p, as function h, can be represented the same as in (20) according to (24) the form:
h
FIG. 3. Distribution of the space charge in radioactive sphere of the radius R with ~-irradiator uniformly distributed in the volume, L is the length of the path of t-particles (~-particles are monoenergetic). The surface of the sphere coincides with the axis of ordinate. 1 - L
2--R <~h <2R.
(26)
1
P6 = ~ (cos 0, -- cos 02) + 1.
The angles 0= and 02 can be determined using the scheme in Fig. 2. Corresponding meanings of cosines
312
V. V. GROMOVand A. G. SAKHAROV
are given by the equations, involving parameters h, L, d.
I
h, if L2 _ h2 > 0
cos O1 = ~ 1, i f L 2 - -
h 2
~< 0
(27)
I
~-~--~, i f L 2 -- (h - d) 2 > 0
cos02=[
_l, ifL_(h_d)
2~<0
Calculations with the use of formulae (24)-(27) have been done for three cases: d = 1.1; 0.7 and 0.4cm. Value of k equal to 10, maximal energy of beta particles was 3 MeV which corresponds to a maximal length of the path L ~ = 0.5 cm (Fig. 4). The form of the dependence of p~ on h is determined mainly by the//-spectrum (Fig. 4). An uninterrupted //-spectrum is characterised by a gradual change of curves of distribution of the volume charge (compare curves 1 and 6 on Fig. 4). Conclusions obtained on the distribution of charges in radioactive samples correspond to compounds with different a. However, in conductors it makes no sense to speak about the volume charge, as it quickly appears on the surface. These questions are discussed in work. °~) In the dependence on electroconductivity of the surrounding medium the volume charge of radioactive sample of dielectric can be compensated partially or totally by the current carder, attracted to the surface (Fig. 5). A hatched region corresponds to the
positively charged above-surface layer, the thickness of which L0 is equal to the path length of the t-particles, emissed by the given radionuclide. As it is seen from the Fig. 5 it is better to pay attention to the total charge q +, distributed in the layer L0, and residual surface charge r/~ (residual), equal to the difference between q + and the absolute value r/. It is obvious that such conditions can be realised when r/~ = 0, i.e. the sample is compensated totally and the outside field is absent. The meaning of rT~sand q+s can be calculated precisely, if the electrophysical properties of radioactive material, its geometry and properties of surrounding medium are known. If distribution of 4 ÷ in the volume is not taken into account and accumulation of charge obeying the law (18) or (19) is taken into account we can determine the mean meaning of fi~+sin the charged area and also the value of rT~s. It is necessary to mark that accumulation of charge in a dielectric does not proceed infinitely: some balance is reached, when the leakage currents (because of the conductivity of the sample) compensate accumulation of the charge reaches the value, at which electrical breakdown is possible. Let us take only the "before breakdown" values of the charge*, p will be used to mark symbols, relating to
/q
(+)
/77
o7I
3
0.6 0.5
0.4 0.3 0.2 0.1 0i
0.1
0.2
0.3' 0.4
0.5
0.6
0.7
0.8
0.9
1.0
i=
L
,.
1
h
FIG. 4. The dependence of relative density of the charge of //-radioactive sample in the form of the plate of the fiat infinite type; the axis of ordinates coincides with one of its side surface: curves 1-3 correspond to the thickness of the plate 1.1; 0.7 and 0.4cm (monoenergetic source of //-particles); curves 4.-6--the same for uninterrupted spectrum of the source of irradiation.
FIG. 5. The scheme of distribution of common volume charge in the above surface layer L of radioactive dielectricsample, containing homogeneously-distributed fl radioactive isotope. On the surface compensating carders with the opposite sign q accumulate. The change of density of the volume charge is given by uninterrupted curve in the layer L.
313
Electric charge in volume of fl'-radioactive dielectrics the surrounding medium, the sign m radioactive material. Residual stationary charge 0H depends on specific radioactivity and the processes of recombination of carriers in surrounding space. (6,n,~3)Its value is determined from the conditions of equality of the current of emission of beta particles from the sample is and the current of carriers to the sample ip, appearing under the influence of the field of the volume charge 8 of the surface. Equating these points and taking into account that I .1 = I~:,,/,.o,,, and ip = eup(nop + n,)l .l and i,, = K2mQo we obtain (28)
~
= KtEOEm"mQo/eup(nop +
np)
where Kt, K2, K3 are constants, depending on the geometry of the sample and dimensions of quantities in formula. (28) m is the mass of radioactive material; Q0 its specific radioactivity; up is the mobility of carriers, nop, np their concentration without irradiation and during the process of irradiation respectively, E0, EI are the dielectric constants of x'acuum and the material of the samples.3" The constant km determines the fraction of beta particles, leaving the radioactive sample (the form of the sample are taken into account). To find np let us suppose that the rate of ionization V~ of the compound surrounding the surface (gas for example) is proportional to Q0, and the rate of recombination of carriers V2 is a bimolecular process. Then under stationary conditions when/:1 = / : 2 (but I:1 = consh Q0 and V~ = consh n~) and (29)
np = const
Q1/2
substituting (29) into (28) we obtain (30)
ff~ =
(31)
(32)
~l,~ =
KoeoemQ~/2[nom~ll(#, r)/S~] I/2
• uf(~Lk: F(E:) dEdx(AE)ffW'COnst21 '/2" Here k 0 is constant, depending on the dimensions of the values of the values in formula (32); nQ is the quantity of t-particles, emitted in time unit by unit of radioactivity of the given radioisotope; S m is the surface of the sample; w is the energy which is spent in the formation of a pair of ions in surrounding medium; const2 the coefficient of the ions recombination in surrounding gas medium, F(E) is the distribution function of beta particles in the spectrum of irradiation; dE/dx is the mean loss of the irradiation energy in the surrounding medium; the sign Y. represents the amount along the whole spectrum of energy, ~(~, r) is the function, determining the fraction of particles, leaving m, at different thicknesses of the sample r and with a coefficient of irradiation absorption/~. For example, for the sphere the function g,(g, r) runs as follows (16): (33)
@(#, r) = [1 - e x p ( -
#r)]/#r.
The meaning of dE/dx in a gas medium near the surface m can be determined using the formulae from works. (17,1s)Formula (32) describes the change of ff~s of radioactive samples, being in a gaseous medium at a pressure 1-103 mm of mercury. F o r the same intervals of pressure, using (31) and (32) it is possible to find a simple dependence between ffr~ and pressure p.(14,15)
KIEoEmmQo/eUp(nop+ const Q 5/2)
From (30) and (28) it follows that at fl~ = aQo and at hop ~ nr
the calculations of the residual stationary charge, relating to the unit of surface of radioactive sample r/.s, is rather complex (6:4:5)
(34)
rT,~ = const P - 1/2.
nop >>%,
~,~ = f "Q~/2
where a and b are constant. As follows from (31), the residual charge ~r~ of radioactive preparation under all other conditions increases proportionally to the square root of specific radioactivity. A similar result in the particular case of a flat radioactive sample in air atmosphere was obtained in work. t13) The final form of expression (31), allowing us to do
1"Deriving(28) ip is supposed to be determined by the drift of carriers under the influence of the charge field.
From (34) it follows that with a decrease of pressure the residual charge should increase. This fact is observed experimentally.(6,n,~4) In the dependence on P the time of accumulation of F/~ will also change• It is easy to show that with the decrease of pressure the equilibrium between medium m and P is reached m o r e slowly. (6'~5)
It should be noted that formulae (28), (34) can be applied for electric materials and conductors. However in case of conductors ff~ corresponds to change, localized on the surface but the volume change is absent. Stationary volume change P~+s can be calculated easily, using formula (18) at 2 = 0 (the radionuclide has a large half decay time) t ~ ~ (stationary condi-
314
V. V. GROMOVand A. G. SAKI-IAROV
tion). After elemental transformations for sphere samples with radius r we obtain (it is supposed that the processes of discharge on the boundary of media II and I do not influence the kinetics of accumulation (35)
t
-15
+ = KonQmQoEoE,,~k (It, r ) / a m " V . P- ~s
where a m is electroconductivity of radioactive (self irradiating) material and K 0 is constant depending on the values of equation (35). (6) In formula (35) P,+s corresponds to mean charge in volume unit of charged layer V. The great difficulties in calculations are caused in the necessity of evaluation of am, which usually differs from corresponding values of electroconductivity of non-irradiated material. Here two approaches are possible: 1. am=anoni~ad-this fact causes calculated values of P~s to b e higher. 2. Different empiric formulae, giving the bond between electroconductivity of irradiated compound and density of the energy stream of irradiation (intensity of irradiation). ~5) Direct experimental determination of as is possible of course. M E A S U R E M E N T S O F ELECTRIC C H A R G E O F R A D I O A C T I V E SAMPLES It is easy to change the charge in case of radioactive conductors and low ohmic semiconductors in the gas phase. °5'19) Such determinations have been done electrochemically (the method of charge accumulation) for plates made of nickel, tungsten, containing radioisotopes of 63Ni, 65Zn, 185W, respectively, and for germanium alloyed with radioactive antimony (Fig. 6). Experimental data are testified by the linear dependence of the surface charge of radioactive samples on the square root of specific radioactivity; (coefficient of slope of the direct line in Fig. 6 is equal to 1/2). Using direct "current methods", it is difficult to determine the volume electric charge of a radioactive dielectric sample as all surrounding surface is ionized and the self conductivity of the material is low. As a result the currents of discharge, giving information about ~+,, are small in comparison with the leak currents of measuring techniques. Nevertheless, such measurements have been done. (2°)To the charged layer (its thickness was "-"3 mm) of radioactive samples the current contacts (pressure-in and sealing-in) have been introduced and the difference between them has been determined; the value of this difference has been used for evaluation of ~+,. Substance
-14
I
I
I
-1
-2
-3
IgO (mCu/g)
FIG. 6. The dependence of the charge of the surface ~/~,of germanium on its specific radioactivity by reference to 124Sb (fl, ~,-irradiator) in the presence of air at 2--6mm of mercury.
As is seen from the given data, in radioactive dielectrics the formation of a charged layer proceeds. However, in this case, the results otained are qualitative but not exact. The difference between experimental and calculated values of/~+, can be explained by inaccuracies in the experimental technique, and errors in calculations. It has been remarked earlier, that the volume charge in radioactive dielectrics can accumulate to such values and lead to the appearance of an immediate inner electric discharge. F o r example, according to calculations using (35) in samples with a = 10-2°(f2 • m ) - 1 at Em= 5 -- 10 and specific radioactivity 0.1-1 Cu/g in charged layer Lo, the intensity of field reaches 105 - 106 V/cm. This fact corresponds to the break-down voltage for the most part of dielectrics. To test the formation of self discharges lower-conductive glasses (a = 10 - 2 2 1 0 - 2 4 ( ~ ' ~ " m ) - i, containing up to 30 mCu/g) of radivactive strontium 9° in equilibrium with yttrium 9° have been prepared. In such glasses 2 h after their production the intensity of field in a charging layer can reach 106 V/cm and self electric charges appear. This
Qo[mcu~g]
p[tor]
BaSO4; a = 3-10 - 16[o.m]- J
I0
200
1.10 - "
2.10 - io
Glass, cr = 10-25[o-m]- l
30
50
1-10-9
1"10-6
~+dc'cm-~]exper.
~+~[c'cm- 3]calcul.
Electric charge in volume of/]'-radioactive dielectrics phenomenon has been observed though to stimulate the process of discharge in glass a mechanic voltage had to be formed by pressing the glass with metal or glass stick. (2~) These experiments testified to the idea of volume charge of radioactive dielectrics. To measure the charge accumulating in radioactive samples the method of dynamic condensor was used. (22)The experiments were done with radioactive neodimium oxide (147Nd, /l-irradiator, 7 mCl/g) at pressure 10-s-10 -~ mm of mercury. The difference of potentials between the neodimium oxide and the comparison electrode (gold plate) reached 100 V. However at heating it decreased linearly in the range 300-400°C and was close to the contact difference of potentials of nonradioactive oxide. This fact can be explained by higher electroconductivity of samples at heating, accompanying by the charge leak of radioactive oxide. Practically this method for the charge measurement allows us to evaluate only the values of Pr,, but not the volume charge. That is why at normal air pressure the difference of potentials being measured decreased to values corresponding to nonradioactive samples. Unfortunately the experiments mentioned do not give any information about charge distribution in the layer L0. The most exact results in this ratio were obtained by the method of sound zoning based on the display of a running sound wave through the sample and electric signal in measuring electrode. The value of this signal corresponds to the intensity of electric
tThis term is used to underline the first reason for the appearance of charge, i.e., the influence of irradiation (irradiation or self-irradiation).
field in the area being deformed/23) The sensitivity of the method in charge was 5 x 10 -3 c/m 3, distortion was 0.1 mm. Experiments with the crystals of yttrium-aluminium garnet (Y3AIsO2) being irradiated by slow neutrons have been done. In the result of an irradiation radioisotope 9oy (10 mcu/g) ~-irradiator, E = 2.25 MeV, Tt/2 = 64.2 h) formed in granate. The calculations and experimental determinations showed that the residual charge of the granate crystals (flat parallel plane) is practically absent in air under normal conditions but being compensated it reaches high values (Fig. 7). From Fig. 7 it follows that in an electrononeutral sample Y2AI50~2 in the subsurface layer a positive charge accumulates, the distribution of which corresponds to the theoretically predicted one for the case when L < d (thickness of plate) (compare with Fig.
4). R A D I A T I O N ELECTRIC C H A R G E r A N D PROPERTIES OF SOLIDS A number of physical-chemical processes involving the participation of solid radioactive or irradiated dielectric samples are affected by radiation charge. These processes are: kinetics of dissolution of solids, ¢24) adsorption and catalytic processes. 03,22.25,26) evaporation, electrochemical processes, etc. Let us discuss the most typical examples of influence of volume charge on kinetics of heterogenic processes. The first link between the volume charge of a radioactive sample and the rate of dissolution of solids was discovered in the dissolution in water of radioactive crystals of strontium sylphate, containing a pure/~-irradiator-sulphur-35 (Fig. 8)/27,2s) As follows from Fig. 8 the rate constant of dissolution
3 E 2 7
O X LO
0 2
315
3 x (ram)
Fro. 7. The distribution of the charge in the monocrystal of yttrium-aluminium garnet. Specific radioactivity according to ~Y, 10 mCi/g.
V. V. GROMOVand A. G. SAKrtAROV
316
b
olI 10-2 ~ -4t 0
J
10-~ IgQ (rnCu/g) I
I 10
10 -~
I
I
10
10 2
t 10
Ige (V/cm)
Fro. 8. The change of the rate constant of dissolution k~ of radioactive crystals. SrSO4 in the dependence on its specific radioactivity Q and the intensity of field ~p in the above surface charged layer. The calculations of the charge (and its corresponding value of n.) have been done using formu[a.(~)
(specific rate of dissolution) decreases when Q0 and the corresponding field 8 of the volume charge increase. Additional experiments, performed with nonradioactive but electron-irradiated (or simply outer field polarized initially) of monocrystals of LiF, K2SO4, SrSO4 and NaF showed the similarity of this phenomenon327-29~ It turned out that under certain conditions the rate of disolution not only decreases when e increases but can even increase as well33°~ Theoretical analysis of the given process permits the determination of the relation between the rate constant of dissolution ko, crystal polarization, properties of carrier and charged sample in the form: (36)
In(kJk*) = A( h'(ez~k'~ k S \4kT) • ~:
4ezCo
ez~,
where A is a constant, a is the net polarization of the molecular, dissolving compound, Em is statistic dielectric, k is the Boltzmann constant, z is the ion charge in the solution, Co is dissolution, x is an inverse length, equal to (8ne2z2Co/Ep'kT)m, Epis the mean dielectric penetrability of solution and ~k~is the mean potential of the outer plane, where the ion centres of solutions are localized (the nearest to the
surface of dissolving crystal); p is the crystal polyrization.(2s'29) As follows from formula °6) at mean and high values of dissolution and E,~< Ep the inner electric field causing polarization of the dielectric, make the rate of dissolution slower. Besides, with a decrease of dissolution and dielectric constant of the liquid phase, the break-down influence of the electric field becomes weaker. At small values of ep(Ee ~
Electric charge in volume of fl'-radioactive dielectrics one micron (i.e. activity for a particle 10-3Cu)). 04-41) The mechanism of formation o f the charge of hot particles is practically the same as in the case of charging o f big radioactive bodies. It has been determined that hot particles, in spite o f the nature of irradiator, (a or/~) have a positive charge. This effect is the case of x-radioactive particles can be explained by the main role o f secondary 6-electrons (their number, emitted from the sample, is larger than the number of emitted a particles) during the process of the charge formation. F o r example from a particle of dibutylphalate with the radius 1 mkm, containing as irradiator 222Rn 5-40 secondary electrons were emitted during ~-decay. °s) The positive charge o f hot particles reached big values in spite o f the conductivity of the surrounding air medium because of its intensive ionization in hot particles. Practically if the particle emits more than 1 electron per sec (i.e. activity is equal 10 - n Cu/particle) at atmosphere pressur e it will have a stationary positive charge. 0'36) This p h e n o m e n o n is not only scientifically interesting as the charge determines the laws of travel of " h o t " airsol particles within the air stream but also allows us to work out the methods o f electrostatic volatilization o f such particles. N o w a d a y s the facts obtained show that the influence o f volume charge (it forms not only in radioactive and irradiated materials, but also under the influence of light mechanic loads, heating, etc) on the process of different heterogeneous processes. This fact has physical-chemical meaning. The given formulae allow us to evaluate the volume and residual charge under different experimental conditions. This fact provides an opportunity to take into account the influence o f "the charge effects" during the study o f the influence of irradiation, light and other factors on solid and heterogeneous systems. REFERENCES 1. P. Ctmm and M. C-MRm,Comp. Rend. 1900, 130, p. 647. 2. U. KORLISand D. HARVY,Energetic Sources on Radioactive Isotopes. (Translations from English), M., Mir, 1967. 3. I. E. TAMM, Osnovy teorff elektrichestva. Nanka, Moscow 1966 (in Russian). 4. A. G. SAKrlAROVand V. V. GROMOV,Zh. Fiz. khimii 1976, 50(6), 1513 (in Russian). 5. B. GROSS, G. M. SESSLER,J. E. WEST, J. Appl. Phys. 1974, 45, 4724. 6. V. I. SPITSYNand V. V. GROMOV,Fiziko-khimicheskiye svoystva radioaktivnykh tel Atomizdat, Moscow 1973 (in Russian). 7. B. GROSS, d. Appl. Phys. 1965, 36, 1635. 8. A. G. SAKI-IAROV, V. V. GRoraov and K. M. MERZHANOV,Zh. fiz. khimii, 1978, 52(2), 433 (in Russian).
317
9. S.V. STARODUBTSEV,Prokhozhdeniye zar. chastits cherez veshestvo. FAN Tashkent, 1962 (in Russian). 10. M. BORN, Atomn. fizika. M.: Mir. 1965 (in Russian). 11. A. A. VononYEV and O. B. EVDOKIMOV,Khimiya vysokikh energii 1970, 4, 356 (in Russian). 12. V. V. GROMOV and V. I. SPITSYN, Radiokhimiya 1972, 14(4), 795 (in Russian). 13. V. G. BARYand F. F. VOLKENSrr~IN,Iz. A N S S S R , Ser. khim. 1964, No. 11, 1935 (in Russian). 14. V. V. GgoMov and V. I. SI'ITSYN,J. Res. Inst. Catalysis, Univ Hokkaido 1971, 19(1), 1. 15. V. V. GROMOV, Atomn. energiya 1968, 26, 250 (in Russian). 16. L. M. EFIMOV,In Tr. Vsesoyuzn. nauch.-tehn, konf M.: Izd. A N SSSR 1958, 382 (in Russian). 17. L. KATZ and A. S. PEr~OLD, Rev. Mod. Phys. 1952, 24 (1), 28. 18. YA. C'mJDASand I. TAURE,lZV. ANLatv. SSR 1959, 3, 33 (in Russian). 19. V. I. SPITSYN,V. V. GROMOV,V. S. AREKELYANand N. G. LVSENKO,Dokl. A N SSSR 1968, t. 182, s, 390. 20. V. V. GROMOV, S. N. OZmANER, V. I. SVITSYNand A. A. MII-L~YEV,Zh. fiz. khimii 1971, 45, 2804. 21. V.V. GROMOVand V. V. StmlKOV,Atom. energiya 1972, 32, 172. 22. V. I. SPITSYN, G. N. PmOGOVA, A. A. SOPINA and E. KH. YENEKEEV, Dokl. A N SSSR 1969, 186, 1358. 23. A. G. ROZNOand V. V. GROMOV,Pisma v ZhTF 1979, 5, 648. 24. V. V. GROMOV, Vliyaniye ioniz, izlucheniya, tomizdat, Moscow, 1976 (in Russian). 25. V. V. GROMOV,V. I. TROrlMOV, V. M. LUKYANOVICH and V. I. SPITSYN, DokL A N S S S R 1968, 178, 1307. 26. V. I. SPITSYN,IZV. A N SSSR, OKhN, 1958, 11, 1296. 27. V. V. GROMOVand T. N. BESPALOVA,In Radiatsionnaya fisika, ionnye cristally. Riga Zinatne, 1966, 4, 531 (in Russian). 28. V. V. GROVOM and V. S. KRYLOV, Dokl. A N SSSR 1970, 192, 123. 29. V. V. GROMOV,T. N. PARKHAEVAand V. S. KRYLOV, Zh. fiz. khim. 1973, 47, 107. 30. V. V. GROMOV,V. S. KRYLOVand V. N. SHULYATYEVA, Zh. fiz. khim. 1978, 52, 2861. 31. L. M. IVANKIVand Z. V. SOLVANIK, Ukr.fiz. zh. 1968, No. 13, 852. 32. G. F. MALINaN,Yu. A. PRISELKOVand P. P. KLIMENKO, Dokl. A N SSSR 1969, t. 188, s. 846. 33. R. WILLIAMSand M. J. CAMPOS,J. Appl. Phys. 1970, 41, 4138. 34. G. L. NATANSON,Uspekhi khimii 1956, 25, 1429. 35. V. D. IVANOV,V. P. KmlcnEm¢o and I. V. PETRYANOV, Dokl. A N S S S R 1968, 182, 307. 36. V. D. IV^NOVand V. N. IORICHENKO,Dokl. A N S S S R 1969, 188, 65. 37. V. D. IVANOV,V. N. KIRICrmSKO, V. M. BER~ZI-INOY and I. V. PETRYANOV,Dokl. A N SSSR 1972, 203, 806. 38. V. D. IVANOV,V. N. KmlCttENKO and B. V. SHAN'GIN, Kolloidn. zh. 1974, 36(3), 468. 39. V. N. KIRICHENKOand V. D. IVANOV,Dokl. A N S S S R 1969, 188, 315. 40. V. N. KlmCrmNKO, V. M. BERZHNOY and B. V. SHAN'GIN, Dokl. A N S S S R 1972, 205, 78. 41. V. N. KImCHENKOand N. N. SUVRUN,Dokl. A N SSSR 1974, 215, 325.