The actual "wave term" A/(R-r) * exp(i 2Pi (kR - vt)) - where R is a vector from the origin of a chosen coordinate system to the detector, r is a vector from from the origin to the electron, k is a wave vector of the scattered wave, and A is a vector proportional to the E vector of the incident wave - is not even mentioned in the expression you are referring to. Note that the scattered wave is spherical but not planar. However, compared to the phase shift term exp(i 2Pi rS), the 1/(R-r) and A terms can be considered constant over the diameter of the atom, the unit cell, and even the complete crystal (the Fraunhofer diffraction regime), and 1/(R-r) = 1/R.
A wave scattered by a crystal in the direction of vector k is: A/R * exp(i 2Pi (kR - vt)) * FT[rho(r)] where rho(r) is the complete electron density of the crystal. In principle, if we call this expression a "structure factor", it is a vector because a wave is a vector field. The Fourier transform term FT[rho(r)] of a scalar field-type function rho(r) (r^3 |-> rho) is a scalar field-type function of complex numbers F (S^3 |-> F), and calling this part of the scattered wave a structure factor makes it a "non-vector". This semantics should not distract us from the original question about the missing "wave term" on page 121 in Blundell and Johnson. I agree with BR that scattering is a single-photon process (flame suit on) and the temporal term does not matter in how diffraction is recorded by the detectors we use today and we can simply ignore it. Petr ------------ Petr Leiman EPFL Switzerland On 04/02/2014 04:39 AM, Edward A. Berry wrote: > Encouraged by recent help from the BB in filling in gaps in my > understanding, maybe I can get help with another question: > > At the top of page 121 in Blundell and Johnson, it is written: > > "The total wave scattered by a small unit of volume dv at a position r > relative to the wave scattered from the origin will therefore have an > amplitude proportional to Rho(r)dv and phase 2Pi i(r.S)dv" (OK so far) > "i.e. wave scattered = Rho(r)exp(2Pi i r.S)dv" > > How is that a wave? r and S are constant vectors. > > My best explanation so far is to say this is a complex coefficient > that will adjust the weight and phase of the wave scattered by this > point. > > Say the wave scattered from one electron at the origin will result in a > temporal cosine wave at the surface of the detector: > > E = exp(2Pi i wt) = cos(2Pi wt) > > (not sure if 2Pi is needed when w is radians/sec) > > Then the wave at the same point, scattered by dv at r, would be > the same multiplied by the quantity in question: > > E = rho(r)exp(2Pi i r.s)dv * exp(2pi i wt) > = rho(r)exp(2pi i (wt - r.S)) > > i.e. phase-shifted by 2Pi (r.S), and multiplied by Rho(r)dv > > Is that more or less it? > > (since these quantities add up to the Structure Factor F(s), > I guess I'm really asking what a structure factor is. > Rupp says a structure fator is a vector "representing" > the diffracted X-rays", which i take to be consistent > with this if vector is in the complex plane)
