  Fresnel theory is used to derive the complex electric-fields above and below
an X-ray reflecting interface that separates two materials with differing
indices of refraction. The interference between the incident and reflected
waves produces an X-ray standing wave (XSW) above the reflecting interface. The
XSW intensity modulation is strongly enhanced by the total external reflection
(TR) condition, which occurs at incident angles less than the critical angle.
At these small milliradian incident angles the XSW period (lambda/2theta)
becomes very large, which makes the TR-XSW an ideal probe for studying
low-density structures that extend 1 to 1000 nm above the reflecting interface.
Fourier inversion of the XSW induced modulation in the X-ray fluorescence (XRF)
yield from a specific atomic distribution within the overlayer directly
produces a model-independent 1-D atomic density profile. The modulation can
also be used to analyze the degree of coherence in the incident X-ray beam.
