Exponential integral function

Let us now briefly recall the main features of the behavior of D c °°(r) as a function of t for t > 0 [25]. First, the limiting value at infinite time of D ,c, 00(f) is, at any nonzero temperature, the usual Einstein diffusion coefficient kT/rp Above the crossover temperature Tc as defined above, ) e, 00(f) increases monotonously toward its limiting value. Below the crossover temperature (i.e., T < T( ).D"] (t) first increases, then passes through a maximum and finally slowly decreases toward its limiting value. Thus, in the quantum regime, I)" x(t) can exceed its stationary value, and the diffusive regime is only attained very slowly, namely after times t fth- At T - Q.If x(Vj can be expressed in terms of exponential integral functions ... 

The infinite continued fraction, Eq. (281), is very convenient for the purpose of calculations so that the complex dielectric susceptibility, Eq. (282), can be readily evaluated for all values of the model parameters r, p/, and a. For a = 1, the anomalous rotational diffusion solution, Eq. (281), coincides with that of Sack [40] for normal rotational diffusion. Moreover, in a few particular cases, Eqs. (281) and (282) can be considerably simplified. In the free rotation limit (( = 0), which corresponds to the continued fraction [Eq. (281)] evaluated at x = 0, that fraction can be expressed (just as for normal rotational diffusion [40]) in terms of the exponential integral function E z) [51] so that the normalized complex susceptibility is... 

Some parts of the solution in the concentration profile are imaginary, so first we need to consider the definition of the exponential integral function ... 

However, the best way to obtain the concentration profile would be the use of the inverse function of Ei(lnC,(x, y)). According to Pecina [16], by delimiting some intervals of definitions for the variable y, it is possible to obtain the inverse function of the exponential integral function, Ei 1(y). Besides, a rough approximation for Ei(y) is possible [17] and then it will be for the inverse function by means of an empirical formula valid for x > 1.6, such as... 

The integral inside the parenthesis in Equation 16.28 is known as the exponential-integral function Ei(x) ... 

Here y is Euler s constant, y = 0.57721566490.. . . The exponential integral function has a useful asymptotic form for small time scale and is given b... 

The asymptote given by equation 47 is not a normal convergent series, that is, one cannot take infinite terms. It is given here for the purpose of quick estimation of the exponential integral function at short times, say, tD < 0.2. Only a few terms are needed to reach an accurate result for Eif(tD), whereas many more terms are needed when equation 46 is used at short times. 

The exponential integral function, Eif(x), has an asymptotic form for small systems or long times, given by... 

The left-hand side of Eq. (20) is reducible to an exponential integral function of the pth order, so the exact solution of the integral transcendental equation (23) for Ox can be obtained only by numerical methods and only if the function /f Ox) is known. Nevertheless, it follows from the general form of Eq. (23) that is independent of v and cfe and is determined only by the values ofp and aox- The latter is involved in Eq. (23) through ... 

Here, S(z, p) (l//9)2r m exp -r/m[27+Ei(/92/4Az) ] ) is the layer-layer correlation function,where R2=z2+p2 is Euler s constant, Ex(x) is the exponential integral function, and A = 7(K/B). In Eq. (16), the exponential term incorporates a finite size effect because of the observed finite lamellar domain sizes typically between -2000 and 10,000 A (L is the domain volume). Also because the samples consist of randomly oriented domains, we perform an exact powder average over all solid angles in reciprocal space. The precise steps leading to Eq. (16) have been discussed elsewhere.2,3 analysis consists of... 

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