4. BRAGG SCATTERING WITH MICROWAVES
Ken Cheney
June 21, 2006
PICTURES
http://www.paccd.cc.ca.us/instadmn/physcidv/physics/teachers/cheney/lab%20manuals/Category.htm#Qu
antum
ABSTRACT
Microwaves are scattered from a cubic crystal made up of ball bearings held in rigid foam. The angular
dependence of the scattering intensity is compared with theory.
HISTORY
Bragg received the Nobel Prize for using X-rays to determine the properties of crystals. This was the
first direct measurement of crystal arrangements and spacing.
The structure of the DNA molecule was determined by Crick and Watson (working in Bragg’s lab!) using
X-ray scattering pictures done by Roseland Franklin.
EQUIPMENT
The nominal frequency of our microwaves is 10 GHz (1010Hz); no precision is given.
The atoms are represented by ball bearings imbedded in rigid foam blocks. You will have to figure out
the spacing. The ball bearings make up a cubic crystal. No real crystal is this easy!
NOTATION
Angles are measured from the velocity vector to the normal of the plane of atoms being considered
i: angle of incidence
r: angle of reflection
spectral reflection: the angle of incidence equals the angle of reflection
Velocity vector: a vector along the direction of propagation of the microwave
Ray: A beam of microwaves moving along the velocity vector.
Plane wave: microwaves where the wave fronts are planes. This is approximated if the waves are
observed at a considerable distance from the source so the curvature of the wave front is large. The
velocity vector is perpendicular to the wave front.
n: integer
: lambda, the wave length
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a: the atom-to-atom spacing along the sides of a cubic crystal
SPECTRAL REFLECTION
Normals to the top plane
of atoms
Incoming rays
Plane wave
front
Outgoing rays
D
C
r
i
a
B
E
Atoms
THEORY, SPECTRAL REFLECTION FROM A PLANE OF “ATOMS”
Angles are measured from the normal to the plane under consideration.
It will be shown below that these special cases are really generally applicable!
Consider a plane through the “top” side of the cubic crystal; let this be the surface of our crystal. This is
the plane reflecting the microwaves in this step.
Assume the microwaves (or X-rays) are approaching with a velocity vector parallel to another side of the
crystal cube.
Take a view of a plane through the third side of the crystal cube.
We see the atoms making up a lattice of squares, say with the microwaves approaching from the top left
of the top surface.
Presumably the microwaves will scatter in many directions from each “atom” but there will only be
intense scattering in directions that produce “in phase” contributions from scattering from many atoms.
To be in phase the path differences between the rays must be n, where n is an integer and  is the
wavelength of the microwaves.
The simplest, and most effective case, is when the angle of incidence (from the direction of the incoming
microwaves to the normal to the surface) is equal to the angle of reflection (normal to outgoing
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microwaves). A little sketching will show that for equal angle, spectral, reflection all parts of an
incoming plane wave travel equal distances to make up an outgoing plane wave.
In the figure “SPECTRAL REFLECTION” the incoming wave front BC is reflected into the outgoing
wave front DE. For all parts of the wave front to be in phase the incoming distance CE must equal the
outgoing distance BD, which implies that i equals r.
From now on we will assume spectral reflection.
Notice that the arguments so far don’t restrict the incoming angle, all angles will reflect equally well. But
the outgoing angle will always equal the incoming angle.
THEORY, REFLECTION FROM PARALLEL PLANES OF ATOMS
For the cubic crystal considered above consider reflection from the surface plane of atoms and also from
the next parallel plane down below the surface.
See the figure "SCATTERING FROM PARALLEL PLANES OF ATOMS"
Consider an atom from each plane, one directly above the other. We want to find out what
incidence/reflection angle will result in path differences of n between reflections from the top atom and
the bottom atom. We know from the considerations above that the reflection must be specular to be in
phase with other atoms in the same plane.
Light striking the bottom atom travels farther than light striking the top atom both before and after the
reflection. The symmetry of the reflection means that the total path difference is just twice the incoming
path difference.
We draw a right triangle: one side is between the atoms, the other sides are determined by the incoming
velocity vector of the wave striking the bottom atom and the wave front of this velocity vector when the
wave front passes the top atom. Call the intersection of the velocity vector and the wave front W, the top
atom T and the bottom atom B.
WB is the incoming path difference, half the total path difference, or n/2.
The angle between WB and TB is the incoming angle i. Since the velocity vector is perpendicular to the
wave front, the angle between WB and WT is a right angle. Finally (!):
WB=TB*cos (i)
Remembering that TB is a, WB = n/2 we get for the total path difference
Path Difference = 2WB=n=2a*cos (i)
cos (i) =n/2a
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SCATTERING FROM PARALLEL PLANES OF ATOMS
Outgoing
Wave Front
T
W
Atoms
i
Outgoing
Ray –
Velocity
Vector
r
B
OTHER PLANES
The next most interesting planes are those you obtain by rotating the “crystal” by 45 degrees and taking
planes separated by the diagonals of the “atomic” cubes.
In experimental X-ray diffraction we know, measure i for many angles depending on the crystal structure
and deduce the interatomic spacing and geometry from the pattern of reflections.
MIXED CRYSTALS
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Generally real crystals are not one perfect “single crystal” like our ball bearings in foam but are random
mixes of perfect crystals oriented every which way. You might well wonder then what application our
analysis and experiment have in the real world??
Imagine a beam of X-rays striking a real, mixed crystal and scattering in all directions. In most directions
the waves will not be in phase and there will be only a small intensity. Some crystals however will be
oriented just right so the incoming waves make the proper angle with the crystal planes and constructive
interference can occur between the rays. This can happen for any plane whose normal makes the proper
angle with the incoming ray. The result will be a cone of in phase rays centered on the incoming ray. If
the detector is a film normal to the incoming ray then there will be circles exposed on the film.
Naturally since real crystals are not simple cubic the resulting pattern will be more complex!!
We will do a matter wave experiment (electrons providing the waves and carbon providing the random
crystals) which will indeed let us see such circles on a CRT type tube!
EXPERIMENTAL PROCEDURE
The plan is to map out the intensity of the reflected microwaves by moving the microwave source and
detector around so they are always making equal angles to the normal to the plane of atoms of interest.
Both source and detector need to be moved to keep i and r equal.
If you draw lines from the center of the “crystal” plane to the receiver you can make a polar plot of angle
verses intensity as you take your data.
To keep the distance from the crystal constant draw an arc a half meter to a meter from the crystal and
move the source and detector along this arc.
There is a trade off between sensitivity and angular resolution (the farther away you are the better the
angular resolution). Experiment and decide what works best for you. Report your findings in your report
to help others.
You are only interested in the location of the maximum so don’t waste your time carefully measuring
away from the maximum.
Equipment
You will probably receive an amp/power supply with the microwave source plugged into it. The only
switch that affects the microwaves is the on/off switch.
The detector has a small blue diode you can see at the back of the “antenna”. Presumably the diode will
work best if oriented along the electric fields of the microwaves.
This diode changes the 1010Hz AC microwaves into 2 1010 Hz pulsed DC. Be sure the voltmeter is set on
DC!
EXPERIMENTAL FINE POINTS!
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The microwaves are polarized.
Check that the detector is rotated to get the largest signal when pointed directly at the microwave source.
Remember that the velocity of electromagnetic waves is perpendicular to the electric field.
For significant reflection the electric field must be suitable both for the incoming and reflected wave, i.e.
parallel to the reflecting surface.
Try rotating the source and detector (together) to find what orientation reflects the best!
The microwave source and detector are quite directional.
If the source or detector get twisted (around an axis through the source or detector) accidentally while
they are being moved from one angle of incidence to another the intensity will change (perhaps lots) just
due to this twisting.
You can carefully line up the source and detector with a string from the center of the crystal.
And/or you can twist the source and detector to obtain the maximum reading for each angle of incidence.
The frequency is not well known.
It might be worthwhile to measure  by facing the microwave source toward the detector and carefully
moving the detector away while you record the location of maximum. There should be a maximum every
half wavelength.
ANALYSIS
Graphical:
Plot the locations of the theoretical maximum on top of your experimental plot.
Percent:
Find the percent difference between the locations of the maximum.
More?
Do you see evidence of maximum from other planes? Sketch the planes, calculate the
theoretical angles, and show on your plot if they look interesting.
Discussion:
Do your results seem reasonable? Why (%)?
Do you have any explanation of missing or extra maxima?
THANKS:
My thanks to Bill Schramm for many discussions as to the how to clarify the english and physics.
Version: Wednesday, June 21, 2006, 5:06 PM
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