Bessel function terms

What (xvii) shows is that the Bessel function terms provide single-crystal-like diffraction and all other Bessel terms continuous intensity. Although this kind of array is not the only one for which Bragg and continuous intensity both occur in the same pattern, it is mainly for this that the unfortunate term "semi-crystalline" has sometimes been used. Examples are C-DNA (9,10), ribosomal RNA fragments (11). Fig. 4 shows an example of the diffraction from such an array of molecules. The helices in this case are 12-fold therefore has large values only for small values of Si. It follows that the Bragg diffractid is confined to the center of the pattern. 

Variations of this method can, in some cases, provide other kinds of information about the diffracting particles. For instance, TMV is a helical virus with 49.02 0.01 subunits in the three turns of its helix in its 69 R axial repeat. This deviation from an integral number of units in three turns causes the layer lines to split slightly (6). The first two Bessel function terms that contribute to the first layer line of TMV do not fall at a spacing of 69 R above the equator, but rather, at 67.6 and 71.5 R, which corresponds to a splitting of about 0.15 mm on the film at typical specimen-to-film distances (11 cm). Figure 1... 

Figure 1. The (n,l) plot for TMV showing the positions of the different order Bessel function terms.

The horizontal lines indicate the position on which the layer lines would fall if there were exactly 49 subunits in three turns in the TMV axial repeat. The circles show the positions the Bessel function terms take assuming 49.02 subunits in three turns. The vertical deviations from the solid lines give an indication of the magnitude of the layer line splitting. The horizontal positions indicate the relative positions of the Bessel function terms along the layer lines each term can only contribute to diffraction further from the meridian than the positions marked. 

The positions of the other layer lines must be corrected for this error before they can be used to calculate the relative contributions of the Bessel function terms. 

The intensities and angular positions of layer lines 1 and 2 are also shown in Figure 2. Again, the noise in the positions calculated is lowest at points corresponding to the peaks of intensity. For these layer lines there are also systematic variations in the position with subsequent peaks falling at different positions relative to that which would be expected if there were exactly 49 subunits in the 69 X axial repeat. This layer line splitting was used to calculate the relative contributions of the Bessel function terms on layer lines 1 and 2. 

The separation of Bessel function terms using layer line splitting is confined to regions where only two terms contribute. At higher diffraction angles, where more terms contribute, the separation of Bessel function terms should, ideally, utilize both heavy atom derivative data and the apparent positions and widths of the layer lines. The combined use of both types of information may be possible using a linear relationship between the layer line position and the relative intensities of the Bessel function terms. 

Figure 3. Intensities for separated Bessel function terms on a portion of layer lines 1 and 2.

Tlie application of helical diffraction theory in combination with experiments on suitably oriented molecules allows us to introduce a step wise fitting procedure with which we first can use knowledge of the basic cylindrical structure of the molecule, then add the Bessel function terms relating to the modulations on the cylindrical surface. The next step is to compare these data with solution scattering patterns in order to be able to deconvolute the equatorial and layer line intensities. Once this model is completed the positions of the dimers in the microtubule wall will be known and the scattering parameters obtained from the known dimer structure can... 

The derivation of the discontinuity expressions for conical shells is similar to that for cylinders. The resulting moment and force equations for conical shells are expressed in the more complicated Bessel function terms. However, approximate solutions for various edge loading conditions can be expressed in simple form as shown in Table 6.4. In this table. 

The series converge for all x. Much of the importance of Bessel s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. 

The analytical solution of this equation Is Known (9) (10) In terms of modified Bessel functions of the first kind. AccorxUngly, the dlstrltutlon of the active chains In the particles with volume V, fn(v)/f(v), and the average nunher of active chains In the same — 00... 

This expression can be put in terms of modified Bessel functions. Applying the method of steepest descent to Eq. (3.52) yields... 

Here, a represents the set of all combinations of 1,2, possessing w elements and Kv(x) is the Bessel function of the third kind. This is an analytical extension of the multivariable Epstein-Hurwitz eta-function to the whole complex /x-plane (A.P.C. Malbouisson et.al., 2002). The first term in Eq. (71) leads to a contribution for Ed which is divergent for even dimensions D >2 due to the pole of the T-function. We renormalize Ed by subtracting this contribution, corresponding to a finite renormalization when D is odd. 

Therefore the Casimir energy for the two spherical cavities inside a non-relativistic non-interacting fermion background can be approximated in terms of a spherical Bessel function j as... 

Equation (8.7) may be written more explicitly in terms of spherical Bessel functions ... 

Here, Jo(x) is the Bessel function of zero order, r7- kf-ki is the momentum transfer, which depends on the scattering angle 6, and 6 b) is the semiclassical phase shift, which is given in terms of the deflection function as b) = dd b)Hk , db. 

The functions PJT(cos 9) are associated Legendre functions of the first kind of degree n and order m, and z (kr) denotes any of four spherical Bessel functions. The choice of the spherical Bessel function depends on the domain of interest, that is, on whether we are looking for the solution inside the sphere (r < a) or outside the sphere (r > a). For the internal field we choose z (kr) = j (kr), where j (kr) is the spherical Bessel function of the first kind of order n. The solution for the external field can be written in terms of spherical Bessel functions j kr) and y kr), where the latter is the spherical Bessel function of the second kind, but it is more convenient to introduce the spherical Hankel function /i / (kr) to determine tj/ for the outer field. 

Denoting p = r — ro, the Green s function can be written in terms of a spherical modified Bessel function of the second kind. 

Similarly, for the sample wavefunction, up to the lowest significant term in the power expansion of the spherical modified Bessel function of the first kind, ,(kp),... 

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