Myelin: A One-Dimensional Biological “Crystal” for X-Ray and Neutron Scattering - Molecules: Nucleation, Aggregation and Crystallization - page 80

Because of the apposition of like surfaces to one another (see above
and Fig. 1C), the electron densities ρ ( r ) and ∆ρ ( r ) are centrosymmetric,

and r ). Assuming that a cylindrically sym-

metric structure is equivalent to the fully rotated flat structure, Fourier
transforms of

ρ ( r ) = ρ ( − r ) and ∆ρ ( r ) =∆ρ ( −

ρ ( r ) and ∆ρ ( r ) are given by

d 2

Ú

FR () = 2 r ()co ( r 2 p rRdr ) ,

0

d 2

Ú

D FR () = 2 D r ()co ( r 2 p rRdr ) = FR () - fd s in cdR ( p ). (1)

0

Here, R is the reciprocal coordinate in the same direction as real coordinate r
with is wavelength

(1.542 Å for CuK α ). The structure factor contains only the real term as the
phase is either 0 or

| R | = 2sin θ / λ , where 2 θ is the scattering angle and λ

. With the 1D periodic lattice of period d (corresponding
to the myelin repeat of membrane pairs), F ( R ) gives values at discrete recip-
rocal R

π

h/d , where h is an integer. With slit collimation or using a line-
focused X-ray beam, the observed intensity I obs ( h/d ) (e.g. Fig. 2) can be
measured as the average intensity over the height of the beam as measured by
a position-sensitive detector (Gabriel, 1977; Boulin et al ., 1988) or by digi-
tizing the exposed X-ray film. These methods integrate the scattered intensity
due to disorientation of the myelinated nerve fibers. For the line-collimated
beam, the Lorentz type correction factor is C ( h )

=

h , whereas for the point-
focused beam it is h 2 . The structure factor F ( h/d ) is related to the observed
intensity by the scale factor K and Lorentz correction C ( h ) according to

=

h
d

h
d

h
d

Ê
Á

ˆ Ê

Á

ˆ Ê

Á

ˆ
˜

F

˜ =

D F

˜ =

KF ,

obs

where

Ê
Á

h
d

ˆ
˜ =±

Ê
Á

h
d

ˆ
˜

F ChI () .

(2)

obs obs