The Modified Bessel Functions

The Modified Bessel Functions. By an argument similar to that employed in 1 we can readily show that Laplace s equation in cylindrical coordinates d2yi, 1 dy 1 d tp d2yi 

If wc proceed in exactly the same way as in 30 we can show that if v is neither zero nor an integer the solution of this equation is 

If v is an integer, say, then I n x) is a multiple of I (x) so that the solution (33.3) in effect contains only one arbitrary constant. By a process, similar lo that outlined in 30 we can show that in these circumstances the general solution of equation (31.2) is 

The result (03.5) is very useful for deducing properties of the modified Bessel function In x) from those of the Bessel function For instance, when n is an integer 

All of these relations can, ol course, be derived directly from the definition (33.4) of I, (.r) and it is suggested ns an exercise to the render to derive them in this way. 

---

In Eq. (77), x = h(o/2kg T is the reduced internal frequency, q = EJhoi the reduced solvent reorganization energy, p = hElha> the reduced electronic energy gap and / (z) the modified Bessel function of order m. The quantity S is a coupling parameter which defines the contribution of the change in the internal normal mode ... 

In this equation, Jv is the Bessel function of the first kind, and Kv is the modified Bessel function of the second kind, U = aikfnf—fi1)112, W = a(f21—konf)if2,... 

The solutions of Eq. (C.l) are defined through the modified Bessel functions. Those are spherical modified Bessel functions of the first kind... 

Here r]0n = w0nb/v, and /f,(% ) and K0(rf0n) are the modified Bessel functions.146 At large values of their arguments, K0(r/0n) and Ki(ri0n) behave as exp (-rj0n), so P0n(b) decreases exponentially with... 

In Section 3 we derive that for the vacancy-mediated diffusion mechanism, one expects the shape of the jump length distribution to be that of a modified Bessel function of order zero. Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probability prec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps. The effect of steps will be discussed in more detail in Section 4. 

An imaginary argument is also possible for Bessel functions. When this occurs, they become the modified Bessel functions I and K. This substitution changes them from oscillatory to monotonic, as in the analogous case of the trigonometric functions. The modified Bessel function of the first is defined as... 

With proper adjustments due to the factor i, these functions follow recurrence relations similar to those for J Io(0) = 1, In (0) = 0, and for n >0 the modified Bessel function is monotonically increasing. The second solution, K, does not follow the same recurrence relations as I. Macdonald s definition of the modified Bessel function of the second kind is... 

The modified Bessel function, 70, is always positive increasing, the free energy is therefore always negative. That is, the free energy of interacting surfaces is always attractive irrespective of whether the surfaces are held at constant charge or constant potential. 

---