Physics 42, Fall 1996
Lab 2 - Moseley's Law
Theory
X-rays were discovered and first investigated in 1885 by Wilhelm Roentgen, who assigned this
name because the true nature of the radiation was at first unknown. X-rays are now known to be
electromagnetic waves, or photons, having wavelengths of about 10-10 meter. The essential parts
of a simple x-ray tube are shown in figure 1.
Figure 1
Electric current through the filament heats the cathode to a high temperature so that the electrons
in the cathode surface have enough kinetic energy to overcome their binding to the surface and
escape into the evacuated tube. There they are accelerated down the tube by a (Figure 1) large
potential, typically several thousand volts and strike the anode, or target. While going from the
cathode to the anode and before striking the target, each electron gains a kinetic energy, Ek given by
E k = eV
.
(1)
where e is the charge on the electron and V is the accelerating potential. Upon striking the target,
the electrons are decelerated and brought essentially to rest by a series of collisions with the atoms
of the target, thus losing the kinetic energy they gained from being accelerated down the tube.
For a majority of the electrons, the collisions are glancing and they lose their energy a little at a
time through two processes. The first process is the direct transfer of kinetic energy to the atom
with which they collide. The net effect of this process is to increase the average kinetic energy of
the target; the target gets hot. The second process occurs when the fast moving electrons come
close enough to a positively charged nucleus of a target atom to be deflected (i.e. accelerated) from
their straight line path by the Coulomb force. In accordance with quantum electromagnetic theory,
these accelerated electrons radiate one or more photons and leave the site of the collision with less
kinetic energy than they had. The radiation produced in such a collision is called bremsstrahlung,
or braking radiation. Any electron striking the target can undergo many bremsstrahlung collisions,
thereby producing many photons. It is this process that is the source of most of the x-rays emitted
from the tube. The energy of such a photon is given by
E k1 - E k2 = hν
.
(2)
where Ek1 is the kinetic energy of the electron before the radiation of the photon and Ek2 is the kinetic
energy of the electron after the radiation of the photon.
Most of the electrons undergoing the bremsstrahlung process produce two or more photons.
The energy lost with each photon will depend on the amount the electron had to lose and on the
closeness of approach of the electron to the nucleus. The closer the approach, the more severe the
deflection, the larger the energy loss. Occasionally, an electron will be brought to rest by a single
collision and its entire kinetic energy converted into a single photon. For this most energetic
photon, equation (2) becomes
eV = E k = hν max
.
(3)
where υmax is the maximum frequency of the x-ray photons. Therefore, from a given tube we
would expect x-rays with a continuous range of energies to be emitted with a well defined
maximum frequency (or minimum wavelength) given by
E k = hν max = hc/λ min = eV
.
(4)
Figure 2 shows a typical x-ray spectrum.
Figure 2
Superimposed on the continuous spectrum are sharp increases in the intensity whose
wavelengths are characteristics of the target material. When the accelerating voltage V is changed
but the target material is not, the high-frequency limit of the continuous x-ray spectrum changes but
the characteristic x-ray frequencies do not. Conversely, when the target material is changed but the
accelerating potential is not, the characteristic x-ray ( Figure 2 ) spectrum changes but the limit of
the continuous x-ray spectrum does not. The explanation of these characteristic x-ray lines is found
in the quantum description of the atomic structure of the target atoms.
Figure 3
Figure 3 is a schematic diagram of the electron configuration and energy levels of copper in the
ground state. The K, L and M shells are completely filled with their respective quotas of electrons
(2, 8 and 18), specified by the exclusion principle. There is a single electron, the valence electron,
in the N shell. Near the top of the diagram are the unfilled shells. Transitions of the outer electron
to these levels gives rise to the characteristic optical spectrum of copper. The difference in energy
between any of the unfilled levels and E = 0 (i.e. a free electron) is very small when compared to
the difference between the K shell and E = 0.
Figure 4
The characteristic spectra are produced when the bombarding electrons have sufficient energy to
penetrate to the inner levels of the atom and eject an electron from the K shell. The removed electron
cannot be accepted in the filled L or M shells. It must go either to one of the higher unfilled shells
or leave the atom entirely. In any case, there is now a vacancy in the K shell which can be filled by
one of the electrons in the L shell. Because the difference in energy between the K and the L shell
is typically on the order of thousands of electron volts, the photon created in such a transition is an
x-ray photon. It is this transition that gives rise to the Kα x-ray line. Other transitions can (Figure 4)
also occur. The vacancy in the K shell can be filled by the somewhat less probable transition of an
electron from the M shell. This transition results in the Kβ x-ray line. The group of transitions from
successively higher energy states, all terminating at the K shell, compose the K series of x-ray
lines. Other series of x-ray lines having smaller energies also occur. For example, when an
electron from the L shell jumps to the K shell, there is a vacancy in the L shell which when filled by
electrons from higher energy levels results in the L series (see figure 4).
One striking difference between x-ray spectra and optical spectra is that x-ray spectra vary
smoothly from element to element. This regularity was first observed by Henry Moseley in 1913
when he made a survey of x-ray spectra and obtained data for about forty different target elements
on the wavelengths of two prominent lines, Kα and Lα. Earlier that same year Bohr had published
his theory of the structure of the atom. Moseley used Bohr's theory to interpreted his results. His
simple and successful application of the Bohr theory to x-ray line spectra provided one of the
earliest confirmations of that theory.
Moseley surmised that the regular variation occurred because the characteristic x-ray spectra
were due to transitions involving the innermost electrons. Because of the shielding of the outer
electrons, the inner electron energies do not depend on the complex interactions of the outer
electrons which are responsible for the optical spectra. According to the Bohr theory, the energy of
an electron in the innermost shell is proportional to the square of the nuclear (Figure 5 ) charge.
With that in mind, Moseley reasoned that the energy, and therefore frequency, of a characteristic xray photon should vary as the square of the atomic number of the target atom. When he plotted the
square root of the frequencies of a particular x-ray line versus the atomic number Z of the element,
he got the graphs shown in figure 5.
He found that these curves could be fitted within experimental accuracy by the simple empirical
formula
ν 1/2 = An Z - b
. (5)
where An and b are constants for each line. He found that for the K series b = 1 and for the L series
b = 7.4. To give this empirical formula a theoretical footing he reasoned as follows.
According to the Bohr theory, the energy of an electron in the shell of quantum number n is
Figure 5
2 4 2
E n = - mk e2 z
2h n2
When an inner electron undergoes a transition from the ni shell to the nf shell, the energy of the
resulting x-ray photon is
E = E ni - E nf
This energy is related to the frequency of the photon by the equation E = hυ, so
2 4
ν = - mk e3 Z - 1 2 12 - 12
nf
ni
4πh
If we let nf = 1 and using (Z - 1) in place of Z, we obtain the frequencies for the K series
2 4
ν = - mk e3 Z - 1 2 12 - 12
n
4πh
1
ν = cR Z - 1 2 1 - 12
n
(6)
where R is the Rydberg constant and n = 2, 3, 4,... . Comparing this with equation (5), we see
that An is given by
A2n = cR 1 - 12
n
Likewise, if we let nf = 2 and using (Z - 7.4) in place of Z, we obtain the frequencies for the L
series
= cR 12 - 12 Z - 7.4 2
(7)
n
2
where n = 3, 4, 5 .... .
Moseley justified the use of (Z - 1) as the effective nuclear charge rather than Z by reference to
the shielding effect of the one electron remaining in the K shell. The one electron remaining in the
K shell shields the nucleus so that the effective Z is reduced to (Z - 1) for the emission of the K
line. Similarly, the nine electrons remaining in the K and L shells shield the nucleus during the
transition of an electron from the M shell to the L shell. This shielding is not perfect, however,
and the effective Z for the emission of the Lα line is reduced to only (Z - 7.4).
At that time, all of the known elements were placed in the periodic table according to their
atomic weights. Moseley postulated that a more fundamental ordering was obtained using the
atomic number:
".... we have here a proof that there is in the atom a fundamental quantity which increases by
regular steps as we pass from one element to the next. This quantity can only be the charge
associated with the central positive nucleus."
With his theory, Moseley predicted that not only cobalt, but also argon and tellurium were
misplaced in the periodic table and that there were elements as yet undiscovered having atomic
numbers 43, 61, 72 and 75. In this lab we will simulate Moseley's work.
References
Besides the discussion given in your text, the following references will be helpful in answering
the questions at the end of the lab. All are on reserve in Kresge library.
1. Eisberg, Fundamentals of Modern Physics (John Wiley and Sons Inc., NY, 1967)
2. Lonsdale, Crystals and X-rays (G. Bell and Sons Ltd, London, 1948)
3. Nuffield, X-ray Diffraction Methods (John Wiley and Sons, Inc., NY, 1966)
4. Wilson, Elements of X-ray Crystallography (Addison-Wesley, Reading, MA, 1970)
Procedure
When Moseley did his survey of x-ray line spectra, he used a wide range of x-ray tubes each
with a different target anodes. To reproduce his work using the same method would be very costly
and very time consuming. Therefore, you are going to simulate his work through an analysis of
absorption edges, which are slightly different from emission wavelengths but just as discreet.
Consider an experimental setup which consists of a source of monochromatic x-rays the
wavelengths of which are variable, a foil of some element through which the x-rays are passed and
a device capable of detecting x-rays arranged as shown in figure 6a. With such an arrangement we
can measure the fractional part of the incident beam which is transmitted as a function of the
wavelength (i.e energy) of the incident beam. When this is done a plot such as the one shown in
figure 6b is obtained. Note that at a certain wavelength there is a sharp increase in the absorption
of the incident beam. This sharp increase in absorption marks the absorption edge of the absorbing
material. If the absorbing material is changed, the wavelength at which the absorption edge occurs
changes, but the overall shape of the graph remains the same.
(a)
(b)
Figure 6
What is occurring at the absorption edge wavelength is the beginning of the classic
photoelectric effect. At that critical wavelength, the incident x-ray photons have sufficient energy
to eject electrons from the K shell. Most of the energy of the incident radiation is therefore utilized
in ejecting the K electrons and consequently the x-rays are heavily absorbed. X-rays of shorter
wavelengths (higher energy) also have enough energy to expel K electrons, but the probability of
collision decreases rapidly with decreasing wavelength and the absorption effect falls off sharply
resulting in an increase in the fractional amount of the incident beam transmitted. X-rays of longer
wavelength lack the energy to eject the electrons and are readily transmitted.
Note that both the characteristic line x-ray spectra and the absorption edge phenomena result
from the same process - the ejection of an inner electron from an atom of an element. In the former
case the ejection is due to a collision with a high energy electron and in the latter case the ejection is
due to a collision with a high energy photon. In both cases the energy needed by the ejecting
particle is determined by the binding energies of the inner electrons which in turn are dependent on
the nuclear charge of the atom. Because both phenomena are the result of the same fundamental
process and both depend on the same atomic structure, we can use a study of absorption edges to
simulate Moseley's work.
The actual experimental apparatus is shown in figures 7a and 7b. Figure 7a shows the general
features of the x-ray machine and gives the names and locations of the various parts referred to in
the procedure. Figure 7b shows the actual experimental setup that will be used with this lab. It
consists of the primary x-ray tube (copper target anode), a round black plastic cylinder called the
rotary radiator which contains two windows, and a tray with a Geiger tube. The tray with the
Geiger tube has slots into which filters of various elements can be placed. The function of the
Geiger tube is to detect the x-rays transmitted through the filters.
The function of the plastic cylinder is to provide the source of x-rays of varying energies. It
does this in the following way. Inside the plastic drum are foils of vanadium, chromium,
manganese, iron, nickel, cobalt, copper, and zinc affixed to a molded plastic octagonal drum.
Each of the foils can be placed in the window which faces the primary x-ray tube by pushing in on
the plunger at the end of a cable attached to the cylinder. The identity of the foil in the window
facing the x-ray tube is shown in the second window located at the back of the cylinder above the
spot where the cable is attached. As the primary x-ray beam from the x-ray tube impinges on the
foil in the front window of the rotary radiator, it causes inner electrons in the atoms of the foil to be
ejected via the photoelectric effect as described above. When other inner electrons fall into the hole
left by the ejected electron, an x-ray photon is emitted. The energy of the x-ray photon depends on
the binding energy of the atom and hence is characteristic of the element of which the foil is
composed. In a sense then, the rotary radiator is a second x-ray tube with the x-ray photons from
the primary beam taking the place of the bombarding electrons. To change the characteristic line
spectra emitted, we have to merely change the target of the primary beam by changing the foil in
the front window of the radiator. Because the eight foils consist of eight consecutive elements as
listed in the periodic table, the energies of their line spectra increase regularly from vanadium
through zinc.(Figure 7)
In addition to the apparatus described above, there should also be a stopwatch for measuring
the counting times, a tray containing eight filters and one blank slide, and a scaler for counting the
x-rays detected by the Geiger tube. The high voltage needed by the Geiger tube is also provided
by the scaler. The filters are all of the same thickness and diameter. Each one contains one of the
elements contained in the rotary radiator. Five are labeled with a number and the identity of the
element in the filter. Three are labeled only with a number. Your task in this lab is to use
Moseley's theory and the measured absorption edges to identify the elements in the three unlabeled
filters.
'tilt' adjust grummet
hinged cover plate
drive cord termination
cable port
cable port
skave plate
dome locking screw
basic port
crystal port
carriage arm
knurled clutch
plate
ES spring clip
manual control
thumb wheel
ball ended spigot
aluminum/lead back stop
(a)
(b)
Figure 7
1. The radiation emitted from the Tel-X-Ometer when it is run at an accelerating potential of 30 kV
and a current of 80 A is 0.1 mR/hour at 10 cm from the shielding. Such a dose rate presents
no health hazard. However, it is a requirement of the college that the dose actually received be
monitored. Therefore, the first thing you must do is to sign out a film badge - the monitoring
device. To do this, see Jan Largent. This badge should be worn at all times while in the lab
and should be returned to Jan Largent when you are done. He will also supply you with the
keys needed to start the x-ray machine.
2. Open the radiation scattering shield by displacing it sideways with respect to its hinge until the
ball ended spigot lines up with the left or right hand release port and then lift. The cover is
self-supporting when it is lifted beyond the vertical line of the hinge. Around the outside of
main the platform of the x-ray machine are angle markings labeled (2 θ). Position the carriage
arm containing the Geiger tube at the 90˚| mark. Insert the blank slide in slot 13 in front of the
Geiger tube and carefully close the cover and center it. (Note: the x-ray machine cannot be
started if the cover is not correctly centered. This is part of the safety interlock system built
into the machine.)
3. Turn on the scaler via the toggle switch on the back panel. Do the following to turn on the xray machine:
a. Turn the TIMER control to the position marked 55. (Note: The x-ray machine cannot be
activated when the timer is in the 0 position. Throughout the lab this timer will count
down from 55 minutes to 0 minutes. When it gets to 0 minutes the machine will shut off.
Unless you reset the timer each time you turn on the machine, this may happen in the
middle of one of your counts which can be very annoying.)
b. Place the key in the POWER ON slot and turn the key to the horizontal position. The white
power light near the x-ray tube should light.
c.
On the initial start, allow 5 minutes for the machine to warm up. When the machine is
warmed up, press the X-RAYS ON button on the front control panel. If all is well the red
x-rays light near the x-ray tube should light and a buzzing sound should be heard. If this
does not happen, recheck the timer and the centering of the radiation shield and try again.
If it still does not start, see Jan Largent.
4. Place the vanadium foil in the window of the rotary radiator facing the primary x-ray beam.
Push down the START button on the scaler and then hold down on the RESET button. Reset
the stopwatch. Simultaneously, release the RESET button and start the stopwatch. You
should see counts accumulating on the scaler. At an elapsed time of one minute,
simultaneously push down the STOP button on the scaler and stop the stopwatch. Record the
number of counts and the counting time. Compute the incident counting rate, Io , in
counts/second. Repeat this three times and compute an average incident counting rate for
vanadium.
5. Repeat step 4 for each of the other foils in the rotary radiator.
6. Open the radiation scattering shield as described in step 2 and replace the blank slide with one
of the filters. (Note: The x-rays will automatically be shut off when you displace the shield in
preparation for opening the cover. When this happens a bright flash of light will be seen in the
x-ray tube. This is normal. As a safety feature, the capacitors inside the machine are
discharged each time the machine is shut off. The bright flash is the discharge of the
capacitors.) Make certain the carriage arm is still at the 90˚ mark and carefully close the cover
and center it. Restart the x-rays. (If you did not turn the power to the x-ray machine off by
turning the power on key to the vertical position, you can restart the x-rays by merely
depressing the X-RAY ON button. Otherwise, you will have to repeat the start up procedure in
step 3.)
7. Place the vanadium foil in the window of the rotary radiator facing the primary x-ray beam.
Push down the START button on the scaler and then hold down on the RESET button. Reset
the stopwatch. Simultaneously, release the RESET button and start the stopwatch. You
should see counts accumulating on the scaler. At an elapsed time of two minutes, push down
the STOP button on the scaler. Record the number of counts and the counting time. Compute
the counting rate, I, in counts/second. Repeat this three times and compute an average
counting rate for the filter when exposed to the secondary radiation from the vanadium foil.
8. Repeat step 7 for each of the foils in the rotary radiator.
9. Repeat steps 6, 7, and 8 for each of the remaining seven filters. When you are done turn off
the x-ray machine, the scaler, the stopwatch and return your film badge and the key to the x-ray
machine to Jan Largent.
Lab Report
Your lab report will be your lab notebook. It should be a complete record of what you did in
the lab. In recording your data, think in terms of perhaps having to return to the book some
months or years hence, and being able to understand what you were doing, what you observed and
what your conclusions were. All the data for the lab, all the calculations, tabulations and
comments should be written in the notebook. If you make an incorrect calculation, don't erase it mark through it and redo it. Ideally, your notebook should be a clear record of what you did in the
lab and what was done later.
Include the following in your analysis:
1. plots of I/Io vs. the elements in the rotary radiator listed in order of increasing atomic weight for
the zinc, copper, cobalt and nickel filters (use the specially labeled graph paper available in the
lab);
2. plots of I/Io vs. the elements in the rotary radiator listed in order of increasing atomic number
for the zinc, copper, cobalt and nickel filters;
3. plots of I/Io vs. the elements in the rotary radiator listed in order of increasing atomic number
for the remainder of the eight filters;
4. an explanation of the differences between the plots of numbers 1 and 2 above;
5. an approximate calculation of the wavelength of the K absorption edge for each of the filters as
computed from the transmission plots in numbers 2 and 3 above (Hint: There is a relationship
between the K absorption wavelength for a particular element and the wavelengths of the Kα
and Kβ emission lines of the neighboring element in the direction of increasing atomic number.
The wavelengths of the Kα and Kβ lines for any element can be found in the Handbook of
Chemistry and Physics and the relationship mentioned can be discovered in the references.);
6. an identification of the element in the each of the three unlabeled filters along with an
explanation of how you arrived at those identifications; and
7. the answers to these questions:
a. The characteristic x-ray spectra of any target anode consists of a number of x-ray lines.
How could the filters above be used to make the beam from a given tube monochromatic
(or nearly so)? For example, if we had an x-ray tube with a copper target anode, how
could we isolate just the Kα line for use in diffraction studies?
b. Why is it important that the filters all had the same thickness and diameter?
c. Why did we have to determine an incident counting rate, Io, for each foil in the rotary
radiator? Since the primary beam came from only one source, should the incident counting
rate be a constant?
d. When doing the experiment, why did we concern ourselves with counting rates and
fractional rates of transmission rather than absolute counts or simply the number of counts
transmitted?
e. When an element is being bombarded by x-rays such as the foils in the rotary radiator, they
are said to be fluorescing. Explain what is meant by that.