This experiment studies x-rays in a number of different contexts including both using x-rays as a probe and measuring emitted x-rays. In particular, the goals of this experiment include the following:

• to produce x-rays by bombarding a copper target with electrons accelerated through a several kV potential;
• to use Bragg scattering to achieve a fine angle (and therefore energy) resolution measurement of the x-rays emitted by the copper target;
• to observe bremsstrahlung radiation and the high energy cutoff of this emission spectrum and to measure the wavelengths of the Kα and Kβ emission lines unique to copper;
• to use high energy x-rays incident on copper, nickel and cobalt metals to produce absorption spectra and measure absorption edges; and
• to induce x-ray fluorescence in several metals and to test Moseley's law.

When a metal target in an x-ray tube is struck by a beam of electrons accelerated through a voltage V, two concurrent processes give rise to an x-ray emission spectrum.

First, the electrons lose kinetic energy in Coulomb interactions with nuclei in the target. This lost energy produces a continuous spectrum of photons called bremsstrahlung (braking radiation) covering a range of energies between 0 and _E_max = eV - Φ. This maximum energy is the kinetic energy of a single electron (eV) minus the work function, (Φ), or electron binding energy, of the metal. As the work function is typically on the order of a few eV and the kinetic energy due to the accelerating voltage is of the order of several keV, we can neglect Φ. Therefore, this electron high energy cutoff can be related to an x-ray short wavelength cuttoff by

Second, beam electrons knock atomic electrons in the target out of inner subshells, and giving the liberated electrons kinetic energy. Atoms with missing inner electrons are unstable. When electrons from outer shells of that same atom fall into the vacant inner shells, they radiate discrete energies, characteristic of the atomic species of the target. In this experiment we are particularly interested in K lines, which appear when electrons fall into vacancies in the K (_n _= 1) shell. The most prominent are the Kαline (from _n _= 2 to _n _= 1 transitions) and the Kβ line (from n = 3 to _n _= 1 transitions). See Fig. 1 for a schematic of K and L x-ray emission lines.

{ ${/download/attachments/131007215/Fig_1.png?version=3&modificationDate=1446221012000&api=v2}$ Figure 1: Atomic shell model showing the origins of K (ending on n = 1) and L (ending on n = 2) emission lines

Thus, the x-ray spectrum produced is the superposition of the continuous (bremsstrahlung) and discrete (K, etc.) components as illustrated in Fig. 2.

{ ${/download/attachments/131007215/Fig_2.png?version=1&modificationDate=1439325054000&api=v2}$ Figure 2: X-ray spectrum produced when electrons are accelerated into a metal target

The photoelectric cross-section decreases with increasing photon energy. However, as the photon energy approaches the binding energy of an atomic electron, a new mechanism for photon absorption becomes possible. Above this energy there is an abrupt increase in absorption, called an absorption edge. There is a distinct absorption edge for each distinct atomic electron binding energy. (See, e.g. Fig. 3).

The energy at which the absorption edges occur increases with atomic number of the absorber.

{ ${/download/attachments/131007215/copper_electron_levels.png?version=1&modificationDate=1446222342000&api=v2}$ Figure 3: For copper, the electron energy levels are displayed with gamma emissions highlighted on the left and gamma absorptions on the right. (Note that the vertical energy axis is plotted on a logarithmic scale.) Importantly, gamma emissions occur when an electron falls from a higher energy level into a lower energy level (releasing a photon equal to the difference in energy levels), while a gamma absorption knocks a bound electron completely out of the atom (with the electron carrying away any excess energy from the absorbed photon in the form of kinetic energy.)

When a material is exposed to x-rays with energy larger than an atomic electron’s binding energy, the photon may be absorbed and the electron will be ejected from the atom. The vacancy created leaves the atom in an unstable state. An electron from a higher energy state will fall into the lower state, and a photon will be emitted. The emission of this secondary photon is called fluorescence. Since the energy levels of each atomic species is unique, so is the energy of the fluorescent X-rays.

While investigating the emission spectra of the elements, Moseley discovered the following empirical formula, relating the energy E of the Kα line of an element to its atomic number Z:

{ ${/download/attachments/131007215/eqn_2.png?version=1&modificationDate=1439911799000&api=v2}$

(2)

where h is Planck's constant, c is the speed of light and R is the Rydberg constant. Though Moseley didn't know it at the time, this formula was later justified by the early quantum mechanical model of the atom due to Bohr where the energy of the electron level n is given by

{ ${/download/attachments/131007215/eqn_3.png?version=5&modificationDate=1439911985000&api=v2}$

(3)

Although the proportional counter used to detect the X-rays in this experiment can distinguish different energies, its resolution is limited. Much better energy resolution can be obtained by diffracting x-rays with a crystal of known lattice spacing. Bragg reflection from a single crystal is analogous to the diffraction of visible light from an optical diffraction grating. As formulated by Bragg and von Laue, and as explained, for example, in Kittel's Introduction to Solid State Physics, the condition for constructive interference of diffracted rays is two-fold.

First, we must satisfy the equation

where n is an integer called the order number, λ is the wavelength of the x-ray, d is the distance between neighboring planes of atoms in the crystal, and θ is the angle between the incident x-rays and the surface of the crystal. (See Fig. 4).

{ ${/download/attachments/131007215/Fig_2a.png?version=1&modificationDate=1440167317000&api=v2}$ Figure 4: The geometry of Bragg scattering. (Source: Wikipedia).

Second, since the crystal planes form a three-dimensional “grating”, in order for phases to add constructively, the angle of incidence must equal the angle of diffraction. (This constraint is not present for an optical diffraction grating.)

If a parallel, polychromatic beam of x-rays is incident on a crystal, the only wavelengths that will be diffracted constructively will be the wavelengths satisfying the Bragg condition (both parts!) for that angle of incidence. Thus, diffraction can be used to separate different wavelengths into different angles for quantitative analysis.

You will use a Tel-X-Ometer x-ray spectrometer. It provides several features described below. In the following, all identifying letters refer to Fig. 5.

{ ${/download/attachments/131007215/Fig_3.png?version=1&modificationDate=1439327186000&api=v2}$ Figure 5: The Tel-X-Ometer and associated electronics.

The X-ray tube is a glass vacuum tube (a) containing an electron gun and a copper target. The electron gun accelerates a beam of electrons upward toward the target, through a potential difference in the range of about 8 kV to 30 kV. An exit port (b) with a lead collimator produces a collimated beam of X-rays directed towards the crystal post holder ©.

At the center of the unit, a LiF crystal may be mounted on a crystal post. A θ:2_θ_ table maintains the Bragg condition of equal angles of incidence and reflection. The carriage arm has slots for holding collimators or for some of the experiments.

The LiF crystal acts as the diffraction grating to give the spectrometer high wavelength resolution. LiF has a lattice spacing of

d = 0.2008 ± 0.0001 nm

(source: F. W. C. Boswell, _Proc. Phys. Soc. A _64, 465-476 (1951).)

The proportional counter is a Xe-CO2 filled tube with a central wire held at about +2100 V. X-rays which enter the counter, through a delicate beryllium side window, will ionize the gas in the tube, releasing excited electrons. These electrons are in turn accelerated towards the positively charged central wire.  As they pass through the Xe-CO2 gas, these electrons liberate more electrons thus creating a cascade.  The more energetic the incident x-ray, the more electrons which will be liberated in the cascade.  The resulting pulse of electrons striking the central wire produces a dip in the voltage which is proportional to the energy of the incident x-ray. These pulses, after passing through an amplifier, can be viewed on a scope or sent into a pulse height analyzer for further analysis.

The plastic dome contains lead and is a good shield for x-rays. The spectrometer is interlocked so that the electron accelerating voltage should not turn on (therefore, no x-rays can be generated) unless the dome is closed and centered.

CAUTION: If you are able to turn on the HV with the dome open, TURN THE MACHINE OFF IMMEDIATELY AND CONTACT THE LAB STAFF.

On the side of the Tel-X-Ometer are dials to control the electron accelerating voltage and the electron current. The accelerating voltage (which is on the order of kV) can be safely read by a digital multimeter (which measures on the order of V) by using a 1/1,000 voltage divider (f) mounted on the back of the unit. The current can be read on an ammeter which is plugged into the side of the unit.

The charge-sensitive pre-amp collects the total charge and shapes the pulses from the proportional counter. The amplifier provides further shaping and variable gain.

Use the oscilloscope to observe pulses from the amplifier. It will be helpful in determining if the proportional counter is saturated or if the amplifier is clipping.

The PHA provides a histogram of pulse heights. Since the pulse height is proportional to total charge in each pulse, and since the charge is proportional to x-ray energy, it follows that the x-axis on the PHA is proportional to x-ray energy. Thus the PHA plots intensity vs. energy. The PHA should be set to “Direct Input” mode.

To open or close the dome, slide the absorber holder on top of the dome to its uppermost position. Slide the plastic dome to the same side as the carriage arm and lift the front of the cover.

Mount a LiF crystal in the central post as shown in Fig. 7.

{ ${/download/attachments/131007215/Fig_5.png?version=1&modificationDate=1439828617000&api=v2}$ Figure 7: Mounting the LiF crystal in the post

Open the dome and rotate the carriage arm to the 2_θ_ = 0º position. Loosen the knurled clutch plate beneath the crystal post. Move the slave plate holding the crystal post until the two scribed lines are as close as possible to the zeros on the θ scale. If the scribed lines cannot be exactly aligned with the zeros, the lines should both be displaced to the same side (i.e., a slight centering offset is ok, but a rotation offset is bad). This adjustment is critical. Carefully retighten the clutch plate. (See Fig. 8.)

{ ${/download/attachments/131007215/Fig_6.png?version=1&modificationDate=1439828675000&api=v2}$ Figure 8: Aligning the θ:2θ table

Important

• There should be a primary beam collimator (with a 1 mm slit) on the exit port of the glass dome housing the x-ray tube. 
• To adjust the slit to be vertical, sight past a vertical edge of the crystal. 
• Place the 3 mm slit vertically in the carriage arm slot closest to the X-ray tube. 
• Place the 1 mm slit vertically in the slot closest to the proportional counter.

Check that the proportional counter is mounted so that the beryllium window faces the crystal holder.

CAUTION: Do not touch the beryllium window! It is very fragile.

NOTE: There is no reason that these connections should not be already in place as they do not change during the course of the experiment. You just want to confirm that everything is hooked up correctly.

Check to make sure that the apparatus has the appropriate electrical connections as shown in Fig. 5. In particular, double-check the following:

• Trail the thin high voltage cable from the proportional counter under the plastic dome and connect the cable to the charge-sensitive preamp’s detector input.
• Use a high voltage cable to connect the power supply to the preamp’s high voltage (“bias”) input.
• Connect the preamp output to the amplifier and connect the amplifier output to the direct input of the PHA.

Next, we will set voltages and turn the apparatus on.

• The proportional counter voltage should be set to about +2100 V. Note that the coarse knob moves in 200 V increments and the fine knob can set an additional range of 0-200 V. The fine gain knob, however, is a standard dial ranging from 0 to 10 representing the low and high ends of the range. Set the fine gain to 5.0 for a value in the middle of the range, or 100 V.
• Set the x-ray tube’s electron accelerating voltage to 15 kV. Remember that you are reading the value over a 1/1000 voltage divider, so your meter should read only 15 V.
• Move the detector to 0º and close the cover shield. Slide the dome to the center position until it locks.
• Turn the timer knob away from zero. This feature prevents the machine from running indefinitely. Remember to periodically add time to the timer as you work through the lab period.
• Turn the power key to “ON”. A white light should now be illuminated. You may hear a buzzing. 
• Press the red “X-RAYS ON” button. A red light should now be illuminated.

NOTE: If the red light under the shield fails to go/stay on, open and close the dome to re-center and satisfy the interlock switches. This may have to be repeated several times. 

• Adjust the electron beam current to about 10 µA. At the lowest settings, the beam current may be unstable and you may find that it drifts. If so, increase the beam current slightly until the reading remains steady.

With the detector positioned at 0º, start the PHA software and make sure that the mode is set to “Direct Input.” (You may need to shut the apparatus off and open the dome to move the detector back to its central position if it is away from zero.) Collect a spectrum. What you should observe is a very rough version of the spectrum shown in Fig. 2, possibly with the edges pushed off screen. As mentioned above, the energy resolution of the detector is not sufficient to distinguish between the Kα and Kβ lines, so you should find one broad peak above a low, asymmetric background.

You may observe many pulses at the lowest heights. These pulses are due to electronic noise. They should be eliminated by dragging the lower level discriminator to approximately 5% of full scale. Drag the upper level discriminator to 100%.

NOTEBOOK: Adjust the amplifier gain and note the how the peak and background shift. Adjust the accelerating voltage and comment on the change in the shape of the background and (possibly) the position of the main peak. Adjust the beam current and note the effect on count rate.

CAUTION: The x-rays may shut off or not turn on if either the accelerating voltage or beam current is too low. If the dome is closed and locked, but x-rays do not start, increase the high voltage and/or current and try again.

NOTE: With the detector at 0º, the x-ray intensity on the detector is very high. You should find that the PHA suffers from a large dead time and the proportional counter is likely to saturate.

In its normal state, the central wire returns quickly to its full bias voltage after each x-ray detection. However if the intensity is too great, the time between pulses may be shorter than the recovery time and subsequent electrons kicked off by ionizing x-rays will be collected by a central wire at a lower-than-normal voltage. The resulting pulses are therefore smaller than they should be and the pulse will ultimately be recorded at a lower channel on the PHA spectrum than expected. In saturation, therefore, all spectrum features shift to the left.

When measuring energy, we want to avoid saturation, but it is impossible when looking at the full beam, even at the lowest beam current and accelerating voltages.

Once you understand what you are looking at, return the beam current to 10 μA and the accelerating voltage to 15 kV

NOTEBOOK: Adjust the gain until the peak is about 25% of the way across the screen. The high energy cutoff should be visible. Sketch the spectrum (to scale) in your notebook.

REPORT: Save the spectrum to file (in both *.tsv and *.spu formats). You will need to plot this for the final report.

NOTEBOOK: Increase the accelerating voltage to 25 kV and repeat the sketch. Again, the high energy cutoff should remain on screen, but be located nearer to the rightmost side. Record the final value of the amplifier gain in your notebook.

REPORT: Again, save the spectrum to file (in both *.tsv and *.spu formats). You will need to plot this for the final report.

NOTE: For the what follows, we will operate at 10 μA with an accelerating voltage of 25 kV.

• Higher voltages will place the high energy end of the spectrum out of angle range of the detector’s carriage arm.
• Lower voltages will reduce the rate of x-ray production.
• Higher beam current will lead to saturation when measuring K lines.
• Lower beam current will be unstable.

Let us now look at pulse signals on the oscilloscope. Open the dome and move the proportional counter away from the center. Re-close the dome and restart the x-rays.

Move slowly across the full range of scattering angles and observe the amplitude and intensity of the pulses on the scope. Remember that as you move the carriage arm, you are changing the Bragg scattering condition and therefore changing the energy of the scattered x-rays you detect. You should observe much greater count rates at angles corresponding to the diffracted Kα and Kβ wavelengths. If you do not see these features then the crystal alignment needs more work! 

NOTEBOOK: Sketch typical scope pulses (to scale). Are the pulses positive or negative? Does the pulse height change as the angle changes? Does the intensity change? Note (roughly) the angles where you believe you observe the two K lines. Which is the Kα and which is the Kβ?

When you _do _observe high intensity at certain angles, check again for detector saturation. To do so, move the detector from side to side around this high intensity region. If the pulse heights suddenly become smaller where the intensity is greatest, the detector is saturated by excessive charge collection. You may wish to temporarily turn the beam current up to observe this phenomenon, remembering to return to approximately 10 μA when you are finished.

In the experiments which follow, you will use Bragg scattering to take narrow “energy slices” of the PHA spectrum you observed in Sec. 4.1. Rather than collect at all energies simultaneously as we do when we observe the head-on beam, we will observe the scattered beam (θ > 0) which will contain only energies which satisfy the Bragg conditions at that angle. By collecting the count rate at many angles (energies) we will be able to map out the spectrum with finer resolution.

In what follows, concentrate on count rate, not raw counts. At each angle, reset the region of interest around the peak – the peak center will change as a function of energy – and use the net counts and live time to compute the appropriate rate with uncertainties. When turning the device on and off, make sure to allow some time for the count rate to stabilize as the electron gun warms up.

The x-ray wavelength can be calculated from the detector angle using Bragg's law, Eq. (4). Note that for larger angles more than one energy will be diffracted into the detector due to higher order (_n _= 2 or _n _= 3) diffraction. All data should be taken using only the first order diffracted x-rays.

Collect enough counts at enough angles to map out the source spectrum with adequate energy resolution. Some hints:

• Angle increments should be dictated by the rate at which the intensity changes with angle; some regions change slowly and need only a few points while some regions change quickly and need many points.
  • Do not choose arbitrary, fixed angle increments, but change your spacing as the features change.
  • When greater angular resolution is required, the angle can be varied in 10 minute increments using the thumb wheel on the carriage arm, to interpolate between degree marks on the 2_θ_ scale.
• Collecting data for this part of the experiment will be time consuming. Pick points judiciously.
  • Concentrate first on the Kα and Kβ lines since we wish to make a quantitative measure of their wavelength.
  • Next, collect data near the high energy cutoff. Count rate will dip and eventually go to zero, so collect enough data so you can eventually extrapolate to zero.
  • Finally, use the remaining time to fill in the more slowly varying bremsstrahlung background. 
• Statistical uncertainty depends on both the uncertainty in time and the uncertainty in counts. Do not choose arbitrary, fixed counting times.
  • In most cases, the dominant uncertainty will be in the count.
    • The relative count uncertainty is √(N)/N. How many counts are needed for 10% uncertainty? For 3% uncertainty? for 1% uncertainty?
    • Note, however, that you have a finite time to collect data. At some points, 1% uncertainty may not be attainable.
    • Note also that there is uncertainty due to region of interest choice and background removal. Is it worth pursuing 1% count uncertainty when you find a 5% shift by changing the ROI end-points?
  • When the x-ray intensity is high – e.g. near the K peaks – it may require only a few seconds to get tens of thousands of counts. In this case, the dominant uncertainty will be in time.
    • This can be mitigated by chosing some minimum collection time so that d_t_/t is small. 
• Plot each data point as you go in order to catch mistakes early. Since there are large variations in the intensity, it will be helpful to plot the count rate on a logarithmic scale.

For the report, you will need to produce a plot of the copper x-ray emission spectrum and identify all the important features. Extract numerical values where possible and make comparisons to literature or expectations as appropriate. 

REPORT: Use the Bragg scattering formula, Eq. (4), to convert your measured angles to wavelength. Plot your count rate data (with uncertainties) versus wavelength and identify the features seen in the emission spectrum.

REPORT: From a fit of the peaks in the spectrum, calculate the wavelengths of the copper Kα and Kβ lines, estimating their uncertainties. Compare with literature values.

REPORT: Estimate the low-wavelength (high-energy) cutoff (with uncertainties) at which the spectrum goes to zero. Since the count rate becomes so low, you may need to do a linear fit/extrapolation to zero. Using Eq. (1) and your known accelerating voltage, determine a value for Planck's constant h (with uncertainties).

Although the X-ray machine is not a radioactive source, it does emit photons at a range of energies similar to Co-57 and other sources used in this course. It is instructive, therefore, to determine the effective strength as though it were a radioactive source. Do so by using the following data:

• the x-ray machine accelerates electrons into its copper target with a current of about 40 μA;
• the electron accelerating potential is typically about 30 kV;
• the efficiency for converting electrons to x-rays is given by Fig. 6; and
• the definition of the Curie is 1 Ci = 3.7 x 1010 disintegrations/sec.

NOTEBOOK and REPORT: Compute the x-ray source strength and compare it to the small button sources commonly used in the elementary labs (about 1 μCi).

{ ${/download/attachments/131007215/Fig_4.png?version=1&modificationDate=1439328776000&api=v2}$ Figure 6: Dependence of the K x-ray yield on incident electron energy. (Source: [1])

You saw the x-ray emission spectrum for copper in your Day 1 experiment. In the following experiment, we will measure absorption spectra for copper, nickel, and cobalt by bombarding each metal with x-rays and observing how much the beam is attenuated as a function of energy.

Recall the similarities and differences between x-ray emission and absorption. (It may help to look again at Fig. 3.)

• In emission, an electron in a high energy state falls into a hole in a lower energy state; a photon equal to the energy level difference is released.
• In absorption, a high energy photon strikes an atom and gives its energy to a bound electron; that electron is kicked out of the orbit of the nucleus and carries away any excess energy in the form of kinetic energy.
• While emission occurs only at discrete energies (those corresponding to energy level transitions), absorption can occur for photons at or above the electron binding energy.

<blockquote> NOTEBOOK: Compete the following in preparation for Day 2: </HTML>

• Given the above information, predict the shape of the absorption spectrum as a function of energy and sketch this shape (roughly) in your notebook. (You do not, for example, need to estimate percent absorption on the y-axis. Instead, simply show where photons will be absorbed and where they will not.)

| * Look up the binding energies for copper, nickel and cobalt and use those as an estimate of the energies at which each absorption edge will occur. (Ref. [3] above, Electron Binding Energies, may be useful.) |

• Compare these values to the K line emission energies. (Will absorption occur at the same energy as emission? At a lower energy? At a higher energy?)
• From your energies, estimate the detector positions 2_θ_ at which these absorption edges will appear. 

</blockquote></HTML> Before you begin collecting data on the second day, your TA will check your predictions. These values will help you chose detector positions appropriately so that you collect data only where needed.  

Without this knowledge, you could very easily spend too much time collecting data at the wrong energies causing your data to be suboptimal. You will want to leave 45 minutes at the end of Day 2 to complete the final part of the experiment.

Here you will measure the absorption of x-rays by cobalt, nickel and copper foils. For our incident beam, we will use the same copper x-ray emission as in the previous experiment and we will place the foils (when needed) in the beam path before the crystal and detector. (See Fig. 9.) In this way, we will again use Bragg scattering to achieve fine angular (energy) resolution for our measurements.

The foils are positioned on a post which may be raised or lowered through the beam. Only adjust the foil heights after the plastic dome is lowered and lock them in place by tightening the nut on top of the dome.

NOTE: Each foil has a small height comparable to the x-ray aperture height. Take care that only one foil is blocking the incident x-ray beam at a time and that the foil is not twisted at an angle relative to the beam.

{ ${/download/attachments/131007215/Fig_7.png?version=1&modificationDate=1439828734000&api=v2}$ Figure 9: Tel-X-Ometer setup showing the location of the metal x-ray filter.

NOTE: For this experiment, we will run with an accelerating voltage of 25 kV and a beam current of 20 μA.

To measure absorption intensity, calculate the ratio – at a given x-ray wavelength – between count rate with the absorber foil present and the count rate with no absorber. This normalization by the “no absorber” intensity is needed because we are not using a flat spectrum, but instead the copper emission spectrum which has a wide range of intensities at different energies.

CAUTION: Measure all count rates for a particular angle at once without moving the detector. Do not, for example, measure the incident intensity at several angles, then return to measure the absorber intensities at the same angle. Because the incident intensity changes so rapidly with angle near the K peaks, (while the absorber intensity may not change much), small shifts in angle can lead to very large changes in ratio.  

In small angle steps near the positions you predicted above in Sec. 5.1, measure the intensity (count rate) with and without each absorber. It is most important to collect sufficient data near each absorption edge, but make sure that you also collect some data away from each edge on either side to establish the overall trend (where the absorption/scattering is due primarily to the photoelectric effect).

NOTE: Graph the count rate ratios against 2_θ_ as you go. Concentrate your measurements in the range of rapid variation in intensity ratio and use your remaining time to fill in the flatter regions.

For the report, you will need to produce a plot of the copper, nickel and cobalt x-ray absorption spectra and identify the absorption edges for each.

REPORT: Plot the ratio of intensities for each absorber as a function of x-ray wavelength. Determine (with uncertainties) the position of each edge and compare it to the K line wavelengths (energies) for that metal.

REPORT: Discuss the differences in the absorption edge locations for the different absorbers. Do these edge positions move in the expected way (roughly) as the atomic number Z of the metal increases?

REPORT: How could these absorption edges be used to build an x-ray filter?

In this final experiment, we wish to determine the relation between the atomic number and the K line energy for a range of elements.

One way to do this is to use X-ray tubes with targets of different elements. It is more convenient to use a single tube element and employ x-ray fluorescence to investigate K lines in other materials. If we use copper x-rays to irradiate a sample, it will fluoresce and its own K line will be emitted. We may then use the calibrated PHA measure the K energies. In these measurements we will not use Bragg scattering, but will instead measure using only the native energy resolution of the proportional counter; for this reason, we will not be able to resolve the Kα and Kβ lines individually, but will instead observe one peak.

{ ${/download/attachments/131007215/Fig_8.png?version=1&modificationDate=1439828767000&api=v2}$ Figure 10: Tel-X-Ometer with x-ray fluorescence sample holder (rotary radiator) mounted on crystal post.

To set up the spectrometer for this part of the experiment, complete the following steps:

• Twist the primary beam collimator on the x-ray tube so that its slit is horizontal. (See Fig. 10.) Remove the 1 mm and 3 mm collimators from the carriage. Remove the crystal, screw, clamp, and the half-post against which the crystal was placed.

CAUTION: Leave these small parts on the surface of the Tel-X-Ometer (under the dome) to avoid losing them!

• Ask the lab staff to help you install the Rotary Radiator on the central post. This device contains samples of  V, Cr, Mn, Fe, Co, Ni, Cu and Zn. Gently place the slot in the Rotary Radiator over the crystal post.
• Turn the Rotary Radiator such that the vanadium is showing.

NOTE: Each element in the Rotary Radiator has its name listed twice. The first instance is directly under the element; this label should not be visible when the Radiator is properly oriented for the experiment, but can be seen only if the central cylinder is removed. The second instance is opposite of the element; this is the label that shows through the window on the reverse of the Radiator when the metal is exposed on the front.

Always check that when name is visible in the back window, a metal foil is exposed on the front. If you see a name in the back window, but only black plastic in front, rotate the cylinder around again until the foil is properly exposed.

• Swing the carriage arm to rest at 2_θ_ = 90º and check that the Rotary Radiator presents the foil (not plastic) to the incident beam at an angle of about 45º.
• Turn on the x-rays. Set the tube voltage to 30 kV and the beam current to 10 μA.

Next, we wish to calibrate the x-axis of the PHA to measure energy. To do this, match the known x-ray energies for vanadium and zinc (the lowest- and highest-energy samples) to the measured peak positions.

• Find the peak for vanadium and estimate the center. Look up the known Kα energy.
• Turn off the x-rays and open the dome. Rotate the Radiator until zinc is showing, then close the dome again and turn on the x-rays.
• Find the peak for zinc and estimate the center. Look up the known Kα energy.
• Calibrate the PHA to convert channel number to energy by using the above two measurements. Your x-axis should now read energy rather than channel.
• Once you have performed the calibration, do not change the amplifier gain or the proportional counter high voltage.

Now you are ready to collect data. Expose each of the remaining elements on the “rotary radiator” in turn and record the peak centers.

• Determine the energy of the K line peak (with uncertainty) for each element.
• Here you need not wait for the count rate to stabilize as you are looking for peak positions, not intensities.
• Do not repeat the measurements of V or Zn. These have been used for calibration and consequently can not be used to test Moseley's law.

Moseley's law, Eq. (2), predicts a linear relationship between Kα energy and (Z-1)2. For the report, plot your data and perform a fit to test this relationship.

REPORT:  Plot K energy (with uncertainty) versus (Z-1)2 and fit the data to a linear function. Is your data in agreement with Moseley's law? Compare your measured slope to the predicted value.

This explanation was contributed by Michael Fedderke, a former TA for Physics 211.

Moseley's law gives the energy of the Kα x-ray in a material of atomic number Z. It was originally presented as an empirical law which fit the observed measurements of the time, but today can be derived from the Bohr model of hydrogen-like atoms; the x-ray energy is simply the difference in the energy between an L shell (2s or 2p) electron and a K shell (1s) electron where the effective nuclear charge is taken to be Z-1 instead of the normal Z. This formula, it turns out, is remarkably accurate. For example, in neon Moseley's Law predicts the x-ray to be at 0.827 keV, while experiment gives 0.822 keV.

The “-1” part of the Z-1 is often explained as being due to some sort of screening of the full nuclear charge Z by the other electrons in the atom so that the electron moving from the L to K shells sees, on average, one less unit of charge. However, this explanation is almost completely wrong. To see why, one can compute ionization energy (i.e. the energy required to completely strip an electron off the atom) for either the 1s or 2s/2p electrons. Using neon as an example again, one predicts (using Z-1 as the “screened” charge) an ionization energy of 1.102 keV for the 1s electron, but measures 0.870 keV, and one predicts 0.276 keV for a 2s/2p electron, but measures only 0.049 keV. These predictions are well outside uncertainties on the measured values and show that screening cannot be occurring. (For, if it worked to predict the x-ray energies, it would also work to predict the ionization energies.) 

It turns out that most of the effect is due to a change in the electron-electron interaction energies when an electron is in the 2s or 2p orbital compared to when it is in the 1s orbital. Consistently accounting for these N-body electron-electron interactions must be done numerically, but when they are incorporated in addition to the usual electron-nuclear-charge interactions, one correctly predicts the ionization energies of all the electrons and – to our point here – the energies of the emitted x-rays.

See the following for more information:

• K.R. Naqvi, Am. J. Phys. 64(10), Oct. 1996.
• A.M. Lesk, Am J. Phys. 48(6), June 1980.

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