Download X-ray Diffraction

Physics 290
Quantum Physics
11/2/03
X-ray Diffraction
Before coming to lab
Please read T&R 3.7 on X-ray production. Refresh your memory from Taylor section
3.2.
Introduction
X-rays are short wavelength photons (0.1Å to 10Å), usually produced when high-energy
electrons interact with solid matter. The electrons lose energy by collisions with the atoms and
much of the lost energy is radiated as photons. (The rest of the energy is dissipated as heat in the
solid material, so large x-ray generators often have to have circulating water-cooling systems.)
There are two types of collision that electrons make with the atoms in the solid target material. 1)
electrons can undergo elastic collisions, and then the electrons’ accelerations lead to photon
emission by classical E&M processes. This produces a continuous range of photon wavelengths
called Bremsstrahlung. Note that this process means that the 'elastic' collisions are only quasielastic since energy is lost from the system in the form of emitted photons. We use the term
‘elastic’ to distinguish these collisions from those that result in the atom being excited to a
different electronic state. 2) If the electrons collide inelastically with the atoms then the atoms
become excited and enter high-energy states which usually last only a few nanoseconds before
the atoms emit photons and return to the ground state. These photon have wavelengths which are
characteristic of the emitting atoms in just the same way that visible photons emitted by excited
atoms have wavelengths characteristic of the emitting atom. Thus a complete x-ray spectrum has
a continuous Bremsstrahlung background with characteristic emission peaks of the atoms
superimposed. (see T&R fig 3.17)
The wavelengths of x-rays are comparable with the dimensions of atoms and with the
spacing between atoms in a crystal. This led Max von Laue and (independently) William and
Lawrence Bragg to suggest that a crystal might serve as a diffraction grating for x-rays.
Experiments soon showed that this worked. The von Laue formulation of diffraction is quite
complex and very general. It is capable of predicting the exact angles of diffraction for a given
crystal under all conditions. The Bragg formulation is considerably simpler. Last week you
learned about the Bragg scattering configuration with microwaves and a huge ball-bearing
crystal. This week we will use x-rays and an ordinary crystal of NaCl, but the physical
arrangement is the same.
You will do three experiments with x-rays this week.
1) First you will look at the statistical error in a counting experiment. That is, you will try
to answer the question “If I just counted 275 x-rays arriving at my detector in one second, with
what accuracy do I know the rate?” Ie. what is the error in such a measurement?
2) You will use a crystal as a diffraction grating of known spacing in order to measure the
spectrum of x-rays emitted from copper bombarded with 30keV electrons. We will do a similar
thing in two weeks when we look at the hydrogen spectrum using a diffraction grating in the
visible range.
Page 1
Physics 290
Quantum Physics
11/2/03
3) Third you will look carefully at the rate of x-ray production and show that you are
seeing diffraction under conditions where there cannot be more than one x-ray photon inside the
apparatus at any instant.
Apparatus
The Tel-Atomic x-ray generator, a small shielded x-ray source with a built in
diffractometer. Inside there is a power supply and x-ray tube with a copper anode (clearly visible
through the glass envelope of the tube). The x-rays are collimated by a lead slit and emerge as a
beam toward the crystal mount. The crystal mount is coupled to a moveable detector mount in
such a way that as the detector mount turns through an angle 2 the crystal is automatically
turned through an angle , maintaining the Bragg condition for diffraction.
The x-ray detector is a small Geiger tube connected to a counter (which also supplies
power for the Geiger tube). Every time an x-ray enters the Geiger tube a current pulse is
produced which is detected and counted. You can count the number of x-rays in a period of time
T by zeroing the counter, setting it to run for time T and then reading the number of counts when
the time expires.
You will be making your measurements with a sodium chloride crystal that has been cut
so that crystal plane spacing is known quite accurately (0.282nm). This crystal must be mounted
on the specimen holder so that the special cleaved face, the matte or dull face, is against the
metal crystal holder post. It is held there by a screw that pushes a rubber-faced holder against the
crystal. NOTE the crystal MUST be removed and put in its tube at the end of each experiment
because it will rust the sample post otherwise.
Procedure
Experiment 1: Mount the NaCl crystal. Put a wide lead collimator (3mm) in slot 13 of
the detector carriage and a narrow collimator (1mm) in slot 18. Look through the slits to make
sure that you can see the crystal. If not fiddle until you can. Now place the geiger tube in slot 26,
being careful to orient it so that the x-ray beam will enter the tube. Swing the detector carriage to
the LEFT.
Now you need to make measurements at several different count rates and for several
different times. You will find that, as you move the detector carriage, the count rate will vary
from about 20 counts/minute to several thousand counts/minute. Pick three positions with
widely different count rates and at each position make several measurements of the count rate.
Make these measurements with the counter not the rate meter. Make 10 measurements of 1
second each, 10 measurements of 10 seconds each, and 10 measurements of 100 seconds each.
For each measurement set compute the average and standard deviation. How does the standard
deviation vary with a) the count rate, b) the total number of counts? You should use this
information to assign errors to your count rates in the second part of the experiment.
Page 2
Physics 290
Quantum Physics
11/2/03
Experiment 2: Investigation of the copper x-ray emission spectrum. Determine angles where you
see constructive interference and then calculate wavelengths using the Bragg diffraction formula
Move the detector carriage to its minimum angle position (about 2 = 11°). Count for 10s and
record the x-ray count. Vary the detector angle and record the counts as a function of the angle 2
to the maximum angle (about 2 = 124°). You should choose your measurement angles carefully.
Where the x-ray intensity changes very slowly you can take points a long way apart (a degree or 2
– more and you will miss the narrow peaks). When you encounter a peak, you will need to take
readings very close together; use the thumbwheel to make measurements every 10'. Use Bragg’s
law to calculate ’s and make a graph of counts vs. . Identify all peaks; some will be first-order
and others will be higher order interference maxima. Convert ’s to photon energies and,
assuming that all the transitions terminate at the ground state, draw a partial energy level diagram
for copper.
Experiment 3: Measure the total x-ray flux leaving the primary collimator. To do this you will
have to remove the 1mm and 3mm collimators and move the geiger tube forward four or five
slots so that the carriage can be placed at angle 0. The x-ray flux is more than sufficient to
saturate the geiger counter (which simply gives up at about 15,000 counts/second) so that you
will have to reduce the rate with the brass filter. This filter lets through 1/14.8±0.1 of the x-rays
entering it so that if you measure a flux of 1000 x-rays/sec the real flux would be 14,800±100 xrays/second. Use this number to calculate the average time between x-rays as they enter the
apparatus. Also calculate the amount of time a photon spends in the apparatus and thus show
that it is very unlikely for more than one photon to be in the apparatus at any instant. Discuss.
Page 3