Scattering and diffraction. The Bragg's
If, after reflection, emerging waves are
phase, reflected intensity will be observed, ie, Bragg's Law
As it was shown
equations, Bragg's Law proposed by both, father and
H. Bragg and William
L. Bragg), may
also be derived graphically in an easy way...
hypothesis is to
imagine Bragg's diffraction as a reflection of X-rays on the
surface of imaginary "mirrors" formed by atomic planes in the
(shown as horizontal lines containing scattering
centers, that is, atoms shown as blue circles in the
left image). Due to the repetitive nature of the crystal,
planes would be separated by a constant distance d.
The two X-ray beams of wavelength λ, arrive
in phase onto the respective imaginary planes, with an angle of
and form a wave front (green line on the left).
In order to obtain a cooperative effect, after reflection both X-ray
beams should still be in phase (green line on the right), a
situation that will only happen if the difference of path traveled by
wave fronts OF
(wavefronts before and after reflection) corresponds with an integer
This condition is equivalent to say that the sum of the FG
segments corresponds to an integer (n)
times the wavelength (λ):
FG + GH
but FG = GH
θ = FG
= d sen
and therefore expression (1)
sin θ = n.
If the emerging reflected waves have
opposite phase, no reflected intensity will be observed, ie,
Bragg's Law is not fulfilled.
the angle of incidence of
the X-rays does not satisfy the Bragg's Law, the emergent beams are no
longer in phase (green line on the right), and cancel each other, so
that no reflected intensity will be observed.
If we consider the baseline
scenario and look carefully at the Bragg's
equation, it is easy to realize that:
- The lattice
planes behave like mirrors reflecting the 'X-light" only in certain
positions given by:
θ = sin-1
(n . λ / 2 . d)
- For given
experimental conditions (λ
and d) only
discrete values of the diffraction angle θ are
obtained, that correspond to the different values of the
= 2 . d
- There are only
a finite number of diffraction orders (as sin θ ≤ 1)
and the maximum number of them depends on the given experimental
conditions (crystal and wavelength):
- The geometry of
diffraction (ie the diffraction angle θ) depends
on the lattice geometry only.
The following animated gif gives an idea of what has been explained
above. When the orientation of the incoming waves, always in phase
meet the above mentioned geometric conditions respect to the virtual
planes , the "reflected" X-ray waves will be in phase (right part of
animation), in which case the Bragg maximum (central peak of the image)
The Bragg condition is
satisfied when the reflected waves are in phase
Readers with installed Java Runtime
tools can also play with Bragg's model
using this applet...
Taken from Philip Willmott (Paul Scherrer Institute, Switzerland)
Alternatively, if you have Flash Player installed, the following model
offered by the International Atomic Energy Agency can be used...
But let's go