## Procedure for Calculating the Relaxation Component of the Drift Velocity

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From the familiar relation [cf. Eq. (4.149)], it is clear that the relaxation component v^ of the drift velocity of an ion can be obtained by substituting for the relaxation force FR in

The problem therefore is to evaluate the relaxation force.

Since the latter arises from the distortion of the ionic cloud, one must derive a relation between the relaxation force and a quantity characterizing the distortion. It will be seen that the straightforward measure of the asymmetry of the cloud is the distance d through which the center of charge of the ion and the center of charge of the cloud are displaced.

However, the distortion d of the cloud itself depends on a relaxation process in which the part of the cloud in front of the moving ion is being built up_and the part at the back is decaying. Hence, the distortion d and the relaxation force FR must depend on the time taken by a cloud to relax, or decay.

Thus, it is necessary first to calculate how long an atmosphere would take to decay, then to compute the distortion parameter d, and finally to obtain an expression for the relaxation force Once this force is evaluated, it can be introduced into Eq. (4.30) for the relaxation component vR of the drift velocity.

### 4.6.8. Decay Time of an Ion Atmosphere

An idea of the time involved in the readjustment of the ionic cloud around the moving central ion can be obtained by a thought experiment suggested by Debye (Fig. 4.92). Consider a static central ion with an equilibrium, spherical ionic cloud around it. Let the central ion suddenly be discharged. This perturbation of the ion-ionic cloud system sets up a relaxation process. The ionic cloud is now at the mercy of the thermal

Velocity = Absolute mobility x force

CHARGED CENTRAL ION

-IONIC CLOUD

CENTRAL ION DISCHARGED AT t =0

IONIC CLOUD

IONIC CLOUD DISPERSED AFTER TIME Tf)

DISCHARGED CENTRAL ION

Fig. 4.92. Debye's thought experiment to calculate the time for the ion atmosphere to relax: (a) the ionic cloud around a central ion;

(b) at t=0, the central ion is discharged; and

(c) after time rfl, the ion atmosphere has relaxed or dispersed.

forces, which try to destroy the ordering effect previously maintained by the central ion and responsible for the creation of the cloud.

The actual mechanism by which the ions constituting the ionic atmosphere are dispersed is none other than the random-walk process described in Section 4.2. Hence, the time taken by the ionic cloud to relax or disperse may be estimated by the use of the Einstein-Smoluchowski relation (Section 4.2.6)

What distance x is to be used? In other words, when can the ionic cloud be declared to have dispersed or relaxed? These questions may be answered by recalling the description of the ionic atmosphere where it was stated that the charge density in a dr-thick spherical shell in the cloud declines rapidly at distances greater than the Debye-Huckel length k-1. Hence, if the ions diffuse to a distance k-1, the central ion

CHARGED CENTRAL ION

-IONIC CLOUD

CENTRAL ION DISCHARGED AT t =0

IONIC CLOUD

IONIC CLOUD DISPERSED AFTER TIME Tf)

DISCHARGED CENTRAL ION

Fig. 4.92. Debye's thought experiment to calculate the time for the ion atmosphere to relax: (a) the ionic cloud around a central ion;

(b) at t=0, the central ion is discharged; and

(c) after time rfl, the ion atmosphere has relaxed or dispersed.

can be stated to have lost its cloud, and the time taken for this diffusion provides an estimate of the relaxation time tr. One has, by substituting k~x in the Einstein-Smoluchowski relation [Eq. (4.27)]

which, with the aid of the Einstein relation D = nabsW[Eq. (4.172)], can be transformed into the expression