The Samuel-Magee Model

Samuel and Magee (1953) employed a 1-radical model to find the relative forward yield in water radiolysis as a function of radiation quality. In such models, no distinction is made between reactive radicals or molecular products. The products of radiolysis are called forward (F) to denote observable molecular yield or radical (R), denoting yield of scavenger reaction at small concentration. The aim of the theory is to calculate the relative forward yield G(F)/[G(F) + G(R)], where the G values refer to the respective yields for 100 eV energy absorbed in 

The second assumption has been effectively invalidated by the discovery of the hydrated electron. However, the effects of LET and solute concentration on molecular yields indicate that some kind of radical diffusion model is indeed required. Kuppermann (1967) and Schwarz (1969) have demonstrated that the hydrated electron can be included in such a model. Schwarz (1964) remarked that Magee s estimate of the distance traveled by the electron at thermalization (on the order of a few nanometers) was correct, but his conjecture about its fate was wrong. On the other hand, Platzman was correct about its fate—namely, solvation—but wrong about the distance traveled (tens of nanometers). 

In the spirit of prescribed diffusion, Samuel and Magee write the normalized distribution of radicals at time t as 

Here fi2 = 1/Lv(t + r), L is the mean free path of radicals at thermal velocity v, and the initial spur radius r0 and the fictitious time T are related by r2 = Lvr. On random scattering, the probability per unit time of any two radicals colliding in volume dv will be ov/dv, where 7 is the collision cross section. The probability of finding these radicals in dv at the same time t is N(N - 1 )p2 dv2, giving the rate of reaction in that volume as crvN(n - 1 )p2 dv. Thus, 

The final result is obtained by using Eq. (7.2) and the definition of f. Integrating (7.3) with the initial condition N = N0 at t = 0 gives the surviving number of radicals at infinite time as 

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In summary, the Samuel-Magee model of low-LET tracks consists of isolated spherical spurs distributed exponentially in energy. No distinction is made between primary and secondary tracks inherent slowing down of the particle is also ignored. 

The Samuel-Magee model can be extended to a-particle tracks, considered as cylindrical columns formed by excessive spur overlap due to high LET. To a good approximation, the length I of the cylinder remains constant while its radius grows by diffusion. In this geometry, the normalized radical distribution is given by... 

The numerical and approximate analytical treatments using the Samuel-Magee model and an intercomparison of the results have been presented in an excellent review of diffusion kinetics by Kuppermann (19). He has pointed out that there is a quantitative inconsistency between the model and experiment although the agreement can be... 

The applicability of homogeneous kinetics is attributed to first-order disappearance of H20, excited water, as the rate-determining step for H2 formation, instead of the combination of reducing species as commonly assumed when using the Samuel-Magee model. Two alternative physical models of H20 are proposed. In one, H20 is the HsO + OH radical pair which is assumed to undergo geminate recombination with... 

Notwithstanding Platzman s theory, most calculations of radiation-chemical yields in water and aqueous solutions were performed using the free-radical model (see Magee, 1953 Samuel and Magee, 1953 Ganguly and Magee, 1956). The hypothesis was that the recapture time of the electron would be shorter than the dielectric relaxation time. Therefore, recombination would outcompete solvation. 

In early models (Samuel and Magee [12]), only glancing collisions were considered. Bethe [13] has shown that for non-relativistic electrons, the differential cross-section for glancing collisions is given by... 

The string of beads model has been proposed by Samuel and Magee [12] and has been widely used for the discussion of diffusion-controlled reactions in water. Radicals are supposed to be formed in spherical volumes called spurs . About 40 eV energy is deposited in each spur which are equidistant. The distance between spurs is about 3000 A for a 450 eV electron in water. The initial spur radius is 10—15 A. The picture of an electron track according to this model is given in Fig. 4(a). 

The diffusion model drawn up by Samuel and Magee (59) and later modified in the light of the knowledge that the reducing radical is a hydrated electron rather than a H atom, tried to explain the yield of all entities in the following manner (53, 54, 55, 63). 

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