Modern Physics Laboratory
Bragg Diffraction with Microwaves
In this experiment, Bragg diffraction of microwaves is observed using a macroscopic "crystal" of
metallic spheres as "atoms". This simulates Bragg diffraction of X rays in crystalline materials.
REFERENCE Tipler and Llewellyn, Modern Physics, 5th ed., pp. 134-135
EQUIPMENT PASCO Microwave Bragg diffraction apparatus—including microwave
transmitter and receiver, mounts, linear and angular scales.
THEORY
Bragg Diffraction
In 1912 Bragg derived a relation which explained x-ray diffraction effects in terms of reflections
from a family of' atomic planes within a crystal. Several conditions are necessary for Bragg
diffraction
1. No matter what the value of the incident angle, each individual plane consisting of an ordered
array of diffracting centers acts as a partially-reflecting mirror. The reflected waves reinforce
each other to produce maximum intensity when the angle of incidence equals the angle of.
ref1ection—as in ordinary light reflection. In the case of reflections from atomic planes, the
angle  is taken to be the angle between the incident or reflected beam and the plane rather than
the angle between the beam and the normal to the plane, as is customary in optics.
2. When a beam of radiation strikes a family of parallel planes , each plane will reflect part of
the energy. If the reflected waves from points O and Q , as indicated in Fig. 1, are to be in phase
(interfere constructively) the path difference
PQ + QR = 2dsin
must equal an integral number of wavelengths, or:
2dsin = n , n = 1, 2, 3…
(1)
Note that the path length NQT is an integral number of wavelengths longer than the length MOS
when Eq. 1 holds.
Incident wave
Reflected wave
Figure 1. Incident waves on a family of crystal planes
separated by distance d. Constructive interference occurs between waves
reflecting from adjacent planes when the Bragg condition [Eq. (1)] is met.
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Eq. (1) is Bragg's law. When this condition holds, the waves reflected from the all the parallel
planes in the family are in phase and the overall reflected beam has relatively high intensity. At
other angles, the reflected beam is very weak due to destructive interference.
Families of Planes in a Simple Cubic Crystal
A simple cubic crystal has atoms at the corners of an elementary cube, and this structure is
repeated through space to form the crystal. The side of the cube, denoted by a, is the distance
between nearest-neighbor atoms in the crystal.
In Figures 2a and 2b below, several families of planes are shown for a simple cubic crystal. In
the figures, the planes are all perpendicular to the plane of the paper. The lines in the figure are
the intersections of the planes with the plane of the paper. Each family of planes is labeled by
three integers known as the Miller indices [e.g. (100), (110), (210), etc.]. These indices (which
are assumed to have no common factor) are a means of specifying the orientation of the planes
and can be constructed by a simple process we shall not describe here. (One may note that a
vector with x,y & z components equal, respectively to the three Miller indices is perpendicular
the family of planes labeled by those indices.) The key quantity determining Bragg diffraction is
the spacing d between the planes in a family. That spacing will differ for each family, as is seen
in Figures 2a and 2b:
Figure 2a. (100) and (110) families of planes
for a simple cubic crystal.
Figure 2b. (210) family of planes for
for a simple cubic crystal.
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When we analyze a crystal at a specific orientation with monochromatic radiation, a spectrum of
reflections is obtained as a function of the angle. We arrange the apparatus so that the incident
and reflected beams make equal angles (both equal to  with a given family of planes, as in
Fig. 1. As  is increased, we expect peaks in the reflected intensity when is equal to one of
the values satisfying the Bragg condition, Eq. (1). For a given family of planes the smallest 
giving a Bragg reflection occurs for the order n = 1 (the 1st order peak). Peaks may occur at 
values corresponding to n = 2 ( 2nd order) and larger values of n. However, examination of Eq.
(1) for shows that for n sufficiently large, Eq. (1) can no longer be satisfied. Thus, for each
family of planes, only a limited number of orders of Bragg diffraction can be observed.
In a simple cubic lattice with the distance between adjacent atoms equal to a, it can be shown
that the family of planes labeled by Miller indices (hkl) has spacing d between adjacent planes
given by :
a
d  d hkl 
(2) 
2
h  k2  l2
PRELIMINARY QUESTIONS
1. Explain why it is necessary to have  ~ d in order to observe Bragg diffraction?
HINT: Examine Eq. (1) and consider what would happen (a) if >> d, and (b) if  << d.
2. A major application of Bragg diffraction is in determining the crystal structure of materials by
observing Bragg diffraction ofX rays from planes of atoms in the crystal. The family of planes
o
with the largest spacing in an iron crystal has spacing d = 1.44 A . Assuming X rays of
o
wavelength  = 1.25 A , determine the angle  for each of the possible Bragg reflections from
this family of planes.
NOTES ON EQUIPMENT
Instead of x-rays as are employed in diffraction studies of crystalline solids, our Bragg diffraction
apparatus uses microwaves -- electromagnetic waves with wavelength on the order of a
centimeter. Because the spacing of the planes producing Bragg diffraction must be comparable
to the wavelength, the “crystal” planes must have spacing of macroscopic size. In fact the
“crystal” in this case consists of metallic spheres embedded in a Styrofoam block. The spheres
are arranged at corners of a cube in a repeated pattern, thus simulating the atoms of a simple
cubic crystal.
The incident microwaves are produced by a microwave transmitter. The diffracted wave is
detected by a microwave receiver. The arrangement of the apparatus is shown schematically in
Figure 3 below.
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Figure 3. Bragg diffraction with
Microwaves apparatus, with microwave
transmitter and receiver in position to
measure Bragg diffraction at angle .
The receiver has several different scales, which can be selected by a knob on the receiver.
Always begin taking measurements with the 30X scale, which is the least sensitive scale.
If the readings are small, then switch to a more sensitive scale, such as the 10X scale. Be sure to
specify the scale used as well as the reading when you record the microwave receiver readings.
PROCEDURE
A. Determination of the wavelength of the microwaves
1. Remove the "crystal" from the apparatus and set off to one side. Set up the transmitter and
receiver on their mounts so that they face each other along the same straight line. This
corresponds to the angle 0 in Figure 3 above.
2. Set the microwave receiver to read on the least sensitive scale (the one marked 30X). Turn
the variable sensitivity setting to its lowest setting. The receiver reading gives the intensity of
the microwaves. Move the receiver mount slowly toward the transmitter (which must stay in a
fixed position). What do you observe for the behavior of the microwave receiver reading? Can
you explain this result?
3. If your explanation was "standing waves", you are correct. At a particular position of
transmitter and receiver, the emitted and reflected waves add up constructively and the intensity
is at a maximum. If the receiver is then moved toward the transmitter, the intensity decreases
and then increases and reaches another maximum. Using the distance scale on the arm that the
receiver mount slides on, measure the position of at least 5 successive maxima of microwave
intensity that occur as you move the receiver toward the transmitter.
4. Compute the distance between the positions where successive maxima occur. You should
find that these distances are all approximately the same. Draw a standing wave pattern and
explain why the distance between the positions of successive maxima must equal /2, where is
the wavelength of the microwaves. Using your data, calculate at least 4 values of  and find the
mean value of .
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Bragg Diffraction
1. Before beginning measurements, the transmitter and receiver must be set to point toward
each other along the same straight line ( = 0 in Figure 3 above). Note that the apparatus has two
arms attached to an angular scale. One arm is fixed, while the other can rotate, and its angular
position can be read on the angular scale. The transmitter should be mounted on the fixed arm of
the apparatus. The instructor will place the stryrofoam block "crystal" on the mount at the center
of the apparatus. The block will be oriented so that one particular family of planes will be
parallel to the line between the microwave transmitter and receiver. The transmitter should be
placed near the crystal, but far enough away so that the crystal can be rotated without bumping
the transmitter. The microwave receiver must be set initially at the least sensitive scale (30X).
For these measurements, turn the variable sensitivity to its highest setting. Do not adjust the
variable setting thereafter. Check with the instructor if the initial readings are off scale.
2. To look for Bragg reflections, both the crystal and the receiver must be rotated. Referring to
Fig. 3, it should be clear that if the crystal is initially rotated by 1 degree, the receiver arm must
be rotated by 2 degrees in order that the microwaves can reflect off the family of planes and enter
the receiver. Since the Bragg peaks occur at angles well away from zero degrees, begin by
rotating the crystal by steps of 2 degrees. Each time the crystal is rotated by 2 degrees, rotate the
receiver by 4 degrees. At each position, record the angle the total angle that the crystal has
rotated through starting from its initial position. Also record the microwave receiver reading and
the scale used (e.g. 30X, 10X, etc.).
3. As you continue to increase the angle , the receiver readings should diminish—and may
even approach zero--and you should switch to a more sensitive scale—perhaps even the most
sensitive scale (1X). Eventually you will notice that, after being very small the receiver readings
start to rise. At this point, increase  by increments of 1 degree until the readings diminish again.
Continue recording data in this fashion until you reach the maximum angle that the arms of the
apparatus can rotate.
NOTE: It is important to measure up the maximum angle attainable, since a peak at a larger
angle may correspond to higher order Bragg reflection.
4. Return the receiver arm to its starting position in line with the transmitter arm. The instructor
will now place the crystal in an orientation so that a different family of planes is parallel to the
line connecting transmitter and receiver. Repeat the measuring process of steps 2 & 3. The
instructor will then set up the crystal for a 3rd family of planes. Repeat the measuring process of
steps 2 & 3.
5. Rescale your receiver data into readings on a common scale. For example, if you use the 1X
scale as the common scale, then each reading taken on the 30X scale must be multiplied by 30,
each reading on the 10X scale multiplied by 10, etc. This ensures that the all microwave
intensity readings are in the same units.
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6. Open the crystal and measure the distance a between adjacent metal balls. Take at least five
measurements and find the mean distance. What is the appropriate definition of the distance a?
What is the best way to measure this distance?
ANALYSIS
1. Plot a graph of your rescaled microwave receiver readings vs. the angle  for the data for each
family of planes studied. These readings are proportional to the intensity of the diffracted
microwave beam. Note the angles at which noticeable peaks occur.

2. From a knowledge of the geometry of the simple cubic lattice, determine the interplanar
spacing d for the (100), (110), and (210) families of planes. Express your results for d in terms
of a, the distance between adjacent steel balls. (HINT: Refer to Figures 2a & 2b and use
appropriate geometrical and/or trigonometric properties as needed

Also, use Equation (2) to find the interplanar spacing for the three families of planes and show
that the expressions for d obtained by two methods agree. Finally, use the measured value of a
to evaluate d numerically for each family of planes.

From the results of the previous step and your measured value of the wavelength  of the
microwaves, predict the values of the angle at which Bragg diffraction occurs for each of the
three families of planes: (100), (110) and (210). Consider all possible orders for each family of
planes.
4. Compare the predicted angles for the three families of planes (100), (110), and (210) to the
peaks in your three graphs of intensity vs. . From this comparison, determine which graph best
represents the Bragg diffraction angles expected from each particular family of planes.
NOTE: Some peaks appearing in the measured intensity may be due to effects other than Bragg
diffraction, such as electronic noise or interference effects due to the small number of "atoms"
(the metal balls) in our simulated crystal.

For those peaks identified in step 4 that correspond to actual Bragg peaks, estimate the
uncertainty in from the half-width of the peak at half-maximum. Compare quantitatively the
predicted and measured values of the Bragg angle  for those peaks you have identified, taking
into account the uncertainty in .
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