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PHYSICS
A. M. BRODSKII and Yu. A. KOLBANOVSKII
ON THE MECHANISM OF INHIBITION OF RADIOLYSIS
(Presented by Academician V. N. Kondrat’ev, March 29, 1961)
In the present work the effects of inhibition of the radiolysis of organic systems by small additions of impurities are considered. This effect is interpreted here as a consequence of direct transfer of excitation to the impurity molecule*. According to an analysis of the work \((^2)\), the inhibition of radiolysis in a liquid at very small additive concentrations \((10^{-3}—10^{-4}\ \text{mol/l})\) is in a number of cases proportional to \(C_i^{2/3}\), where \(C_i\) is the inhibitor concentration. It follows from this that the rate of inhibition decreases no faster than \(1/R^2\), where \(R\) is the mean distance to inhibitor molecules.
Let us first examine the following model problem. There is a molecule \(II\) excited by radiation (an ion, radical, etc.) (see Fig. 1). At a distance \(R = |R|\) from it there is an inhibitor molecule \(I\). Let us take a coordinate system with its origin near \(I\). We shall assume that
\[ r_I \ll r_{II} \simeq R, \tag{1} \]
where \(r_I\) and \(r_{II}\) are the current radius vectors of \(I\) and \(II\). In accordance with the features of the particular radiolysis problem under consideration, we shall suppose that \(II\) is in an excited electronic state with excitation energy \(\omega\), equal to several electron volts. At the same time the inhibitor molecule \(I\) has a practically continuous system of levels in the region of excitation energy \(\omega\), with density \(\rho(\omega)=dN(\omega)/d\omega\). The further discussion will be carried out in the system of units \(\hbar=c=1\), in which \(\omega R\) may be either greater or less than unity. At the same time
\[ \omega r \ll 1, \tag{2} \]
where \(r\) is the extent of the excitation regions of \(I\) or \(II\).
Fig. 1. \(I\) — inhibitor, \(II\) — excited system
The matrix element of the effective perturbation energy, corresponding to the transition of \(II\) to the ground level and the excitation of \(I\) without emission of photons, will be taken in the form
\[ U_{i\to f} = \int (j_{\alpha II}(r_{II}))_{fi} \frac{e^{i\omega(|r_I-r_{II}|)}}{|r_I-r_{II}|} (j_{\alpha I}(r_I))_{fi} \, (dr_I)(dr_{II}), \tag{3} \]
where the integration is over all space; \((j_{\alpha i})_{fi}\) \((i=I, II)\) are the matrix elements of the components of the current vector corresponding to the transitions under consideration and different from zero in the regions \(I\) and \(II\), respectively. A formula of the form (3) is given, for example, in \((^3)\), p. 296, in Heaviside units. In (3) summation over the four components of the current vector is implied. However, since in the problem under consideration \(j_i \sim 10^{-2} j_0\) \((i=1,2,3)\), and since below we restrict ourselves to the dipole approximation, one may retain only the components \(j_0\), i.e. the density of dis—
* A qualitative suggestion of such a path of inhibition was made earlier (see, for example, \((^1)\)).
of charge redistribution. Bearing in mind the spatial separation of \(I\) and \(II\), we expand the factor
\[ \frac{e^{i\omega(|\mathbf r_I-\mathbf r_{II}|)}}{|\mathbf r_I-\mathbf r_{II}|} \]
as follows (see \((3)\), p. 317):
\[ \frac{e^{i\omega(|\mathbf r_I-\mathbf r_{II}|)}}{|\mathbf r_I-\mathbf r_{II}|} = \frac{i\omega}{4\pi}\sum_{lm}\varphi_{lm}^{*}(\omega r_I)\Phi_{lm}(\omega r_{II}), \tag{4} \]
where
\[ \varphi_{lm}(\omega r_I) = (2\pi)^{3/2} i^l \frac{J_{l+1/2}(\omega r_I)}{\sqrt{\omega r_I}} Y_{lm}\!\left(\frac{\mathbf r_I}{r_I}\right), \]
\[ \Phi_{lm}(\omega r_2) = (2\pi)^{3/2} i^l \frac{H^{(1)}_{l+1/2}(\omega r_2)}{\sqrt{\omega r_2}} Y_{lm}\!\left(\frac{\mathbf r_2}{r_2}\right). \]
Here \(J\) is the Bessel function, \(H^{(1)}_{l+1/2}\) is the Hankel function of the first kind; \(Y_{lm}\) are spherical functions; \(m\) runs through values from \(-l\) to \(l\), and \(r_I=|\mathbf r_i|\). By virtue of (1),
\[ \varphi_{lm}(\omega r_1)\cong 4\pi \frac{(i\omega r_i)^l}{(2l+1)!!} Y_{lm}\!\left(\frac{\mathbf r_1}{r_1}\right). \tag{5} \]
Integral (3) will now be decomposed into a sum of products of integrals over regions \(I\) and \(II\), with each subsequent term of the sum in \(l\), according to (5), differing from the preceding one by an additional factor \(\sim \omega r\). In consequence of (2), we shall retain only the first nonvanishing term of this sum; for it we take the term corresponding to an electric dipole transition \(l=1\) in the inhibitor molecule. As a result, taking into account (5) and the approximations mentioned above, we obtain
\[ U_{i\to f} = \frac{\omega^2}{3} \sum_{m=-1}^{1} \int r_I(j_{0I}(r_I))_{fi} Y_{1m}\!\left(\frac{\mathbf r_I}{r_I}\right)(d\mathbf r_I) \times \]
\[ \times \int \frac{i(2\pi)^{3/2}H^{(1)}_{3/2}(\omega r_{II})}{\sqrt{\omega r_{II}}} Y_{1m}\!\left(\frac{\mathbf r_{II}}{r_{II}}\right) (j_{0II}(r_{II}))_{fi}(d\mathbf r_{II}) = \frac{\omega^2}{3} \sum_{m=-1}^{1}J_m^{(1)}J_m^{(2)}. \tag{6} \]
Let us consider separately the integrals \(J_m^{(1)}\) and \(J_m^{(2)}\) entering into (6). The first of them is proportional to the off-diagonal matrix element of the operators of the components of the electric dipole moment \(D_{im}\),
\[ J_m^{(1)} = \int r_I(j_{0I}(r_I))_{fi} Y_{1m}\!\left(\frac{\mathbf r_I}{r_I}\right)(d\mathbf r_I) = \sqrt{\frac{3}{4\pi}}(D_{Im})_{fi}. \tag{7} \]
Under the integral (7), let us carry out the transformation corresponding to passage to a coordinate system with origin near \(II\) (see Fig. 1):
\[ \mathbf r'_{II}=\mathbf r_{II}-\mathbf R, \tag{8} \]
and direct one of the axes (the third) along \(\mathbf R\). Then, substituting the explicit expression for the Hankel function appearing here, we obtain
\[ J_m^{(2)} = -\frac{4\pi i}{\omega} \int \frac{1}{|\mathbf r'_{II}+\mathbf R|} e^{i\omega|\mathbf r'_{II}+\mathbf R|} \left[ 1-\frac{1}{2i\omega|\mathbf r'_{II}+\mathbf R|} \right] \times \]
\[ \times (j_{0II}(\mathbf r'_{II}+\mathbf R))_{fi} Y_{1m}\!\left( \frac{\mathbf r'_{II}+\mathbf R}{|\mathbf r'_{II}+\mathbf R|} \right) (d\mathbf r'_{II}). \tag{9} \]
Expanding (4) in the small parameter \(\omega r'_{II}\cong r'_{II}/R\) and retaining only terms of zero and first order, we obtain
\[ J_m^{(2)} = -\frac{4\pi i}{\omega R} \sqrt{\frac{3}{4\pi}} e^{i\omega R}\delta_{m0} \left\{ \left[ 1-\frac{1}{2i\omega R} \right] \int j_{0II}(\mathbf r'_{II}+\mathbf R)_{fi}(d\mathbf r'_{II}) + \right. \]
\[ \left. + \left[ 1-\frac{3}{2i\omega R} + \frac{1}{(i\omega R)^2} \right] \int i\omega r'_{II}\cos\varphi\, (j_{0II}(\mathbf r'_{II}+\mathbf R))_{fi}(d\mathbf r'_{II}) \right\}, \tag{10} \]
where \(\varphi\) is the angle between the directions \(\mathbf r'_{II}\) and \(\mathbf R\), and the Kronecker symbol \(\delta_{m0}\), arising from
\[ Y_{1m}\!\left( \frac{\mathbf r'_{II}+\mathbf R}{|\mathbf r'_{II}+\mathbf R|} \right), \]
corresponds to retaining only the transverse-
…to \(R\) waves. The second integral in (10) is proportional to the off-diagonal matrix element of the component \(m=0\) of the operator of the electric dipole moment of system \(II\):
\[ \int r'_{II}\cos\varphi\,(j_{0II}(r'_{II}+R))_{fi}\,dr'_{II}=(D_{IIm})_{fi}. \tag{11} \]
The first integral on the right-hand side of (10) for molecules and radicals is equal to zero for orthogonal wave functions \(i\) and \(f\). In this case
\[ J_m^{(2)}=\frac{(4\pi)^{1/2}\sqrt{3}}{R}\,e^{i\omega R}\delta_{m0} \left[1-\frac{2}{i\omega R}+\frac{3}{2(i\omega R)^2}\right](P_{I0})_{fi}. \tag{12} \]
Substituting (7) and (12) into (6), we obtain
\[ U_{i\to f}=\frac{\omega^2 e^{i\omega R}}{R} \left[1-\frac{2}{i\omega R}+\frac{3}{2(i\omega R)^2}\right] (D_{I0})_{fi}(D_{II0})_{fi}. \tag{13} \]
It follows from (13) that the probability of the process under consideration per unit time, summed over the final states of system \(I\), is finally equal to
\[ W_{if}=\int 2\pi |U_{i\to f}|^2\delta(E_{fI}-\omega)\rho(E_{fI})\,dE_{fI} \]
\[ =2\pi\frac{\omega^4}{R^2} \left(1+\frac{1}{(\omega R)^2}+\frac{9}{4(\omega R)^4}\right) \rho(\omega)(D_{I0})_{fi}^{2}(D_{II0})_{fi}^{2}, \tag{14} \]
where the energy of the ground state of the inhibitor, \(E_{iI}\), has been taken as zero. In passing to the specific problem of radiolysis under consideration, it may prove necessary to integrate (14) over the width of the excited level \(II\), equal to \(\Delta\), and arising because of the possibility of chemical decomposition and luminescence. Since \(\Delta\sim \hbar/\tau \ll 10^{-27}/10^{-13}\simeq 10^{-14}\,\mathrm{erg}\simeq 10^{-2}\,\mathrm{eV}\ll \omega\), the indicated integration reduces to interpreting the energy \(\omega\) as the mean value over the width \(\Delta\) and replacing \(\rho(\omega)\) by \(\displaystyle \int_{\omega-\Delta/2}^{\omega+\Delta/2}\rho(x)\,dx\). To estimate the magnitude of \(W_{if}\), it is useful to express (14) in terms of the probabilities of dipole radiation of the excited molecules \(I\) and \(II\), \(W_I\) and \(W_{II}\). Starting from the known formula for polarized dipole radiation, we obtain, in the usual system of units,
\[ W_{if}=\frac{9}{2^{7}\pi}\frac{\hbar^{3}c^{2}}{\omega^{2}R^{2}} \left[1+\frac{\hbar^{2}c^{2}}{(\omega R)^{2}}+\frac{9\hbar^{4}c^{4}}{4(\omega R)^{4}}\right] \rho(\omega)W_IW_{II}. \tag{15} \]
It is clear from (15) that the ratio \(W_{if}/W_I\) becomes sufficiently large if the density of levels \(\rho(\omega)\) is large and \(W_{II}\) is small.
Let us pass from the model problem considered to the problem of determining the dependence of the rate of inhibition of radiolysis on the concentration of the inhibitor \(C_i\). Since the excitation levels of the organic ions formed during radiolysis lie below the first electronic excited level of the corresponding molecules, the medium between the excited ion and the inhibitor is transparent. Strong absorption with appreciable inhibition must take place on the inhibitor molecules. In this connection, on passing to the medium we would have had to add to \(\omega\) in formula (3) an imaginary term depending on the absorption probability. Instead, we shall take the final upper limit \(R_2\) in determining the averaged probability of inhibition in the medium \(W\):
\[ W\simeq\int_{R_1}^{R_2} W_{if}(R)\,C_i\,(dR). \tag{16} \]
To estimate the dependence of \(W\) on \(C_i\), the values of the mean distance to an inhibitor molecule \(R_1\) and \(R_2\) may be specified by the conditions
\[ \frac{4}{3}\pi R_1^{3}C_i\simeq 3;\qquad \frac{4}{3}\pi R_2^{3}C_i=N(C_i);\qquad 3<N(C_i)<10. \tag{17} \]
Substituting (14) into (16), we obtain
\[ W \simeq A C_i^{2/3}\left(1+\alpha_1 C_i^{2/3}/\omega^2+\alpha_2 C_i^{4/3}/\omega^4\right), \tag{18} \]
where \(A\) does not depend on \(C_i\), and \(\alpha_1,\alpha_2\) are numerical coefficients of order unity.
From consideration of formula (18) it follows directly that, in addition to the dependence of the inhibition rate on \(C_i^{2/3}\) (for small values of \(C_i\)) established in \((^2)\), one should expect that, as the concentration increases, the dependence of \(W\) on \(C_i\) will first pass through a transition region of relatively small extent, where all terms are significant, and then a region will set in where the determining term will be \(C_i^{6/3}\). Comparison of these conclusions with experiment was intended to establish whether dependences on \(C_i^{2/3}\) and \(C_i^{6/3}\) do indeed occur. This should also make it possible to approach an estimate of the magnitude of the transferred excitation energy \(\omega\), to compare this value with the electronic spectra of the corresponding inhibitors, and to determine the influence of the medium.
Fig. 2. Dependence of \(1/(G-G_i)\sim W\) (where \(G_i\) is the radiation-chemical yield of the fully inhibited radiolysis of cyclohexane in the presence of \(J_2\) according to \((^4)\)): \(a\)—on \(C_i^{2/3}\); \(b\)—on \(C_i^{6/3}\).
In Fig. 2 are shown the results of processing the experiments \((^4)\), with only the fully inhibited process being considered, i.e., \(G-G_i\) is taken. As is evident from the figure, formula (18) indeed correctly describes the course of inhibition of radiolysis over a wide range of variation of \(C_i\). Processing of the results of experiments \((^{5-8})\) led to analogous conclusions. In all cases, the values of \(\omega\) found lie in those regions of the electronic spectra of the inhibitors where absorption is very significant (as a rule, values \(\lg \varepsilon \simeq 4\)) and, in character, close to continuous.
Thus, the general form of the dependence of the inhibition rate on \(C_i\) in the experiments considered confirms the formula derived, and it is possible to give a unified description over a broad region of \(C_i\), including the so-called saturation region.* We note that the considerations developed can be applied to explain inhibition in a number of chemical reactions.
The authors express their gratitude to V. G. Levich, L. S. Polak, and the participants of the seminars conducted by them for discussions.
Institute of Heteroorganic Synthesis
Academy of Sciences of the USSR
Received
23 III 1961
CITED LITERATURE
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* It is interesting to note that the dependence of the rate of inhibition of radiolysis on \(C_i^{6/3}=C_i^2\) at large \(C_i\), cited here, was also established experimentally in an applied study \((^9)\).