The energy separations between electron levels in atoms range typically from eV to keV meaning that the photons emitted or absorbed in transitions between these states range from infrared light to x-rays. The radiation emitted when electrons accelerate due to Coulomb interactions with heavy nuclei in solids can range from 0 eV up to the initial kinetic energy of the electron. The separation between atoms in crystalline solids is comparable to x-ray wavelengths. For all these reasons, x-ray photons are used extensively for probing material structure and characterizing atomic processes.

This experiment will study x-rays in a number of different contexts including both using x-rays as a probe and measuring emitted x-rays. In particular, the objectives 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 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.

• [1] J. H. Sparrow and C. E. Dick, "The development and application of monoenergetic x-ray sources", National Bureau of Standards Special Publication 456, 41-45, (Nov. 1976).
  • This paper details the intensity of x-ray emissions as a function of atomic number, Z, and electron accelerating voltage for thick targets
• [2] S. Fine and C. F. Hendee, (Philips Laboratories, Irvington on Hudson, New York), "X-Ray Critical-Absorption and Emission Energies in keV", Ortec publication.
  • This paper lists x-ray emission lines and absorption edges for all elements.
• [3] Electron Binding Energies, from the X-Ray Data Booklet, “Section 1-1: X-Ray Properties of the Elements”, Lawrence Berkeley National Laboratory, http://xdb.lbl.gov/Section1/
• [4] Reuter-Stokes Proportional Counters Performance and Specifications Guide.
  • This is the information guide for the proportional counter detector used in this experiment.

Read over the theory section below and consider the following prompts. In particular note the similarities and differences between x-ray emission and absorption. (It may help to look 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.

Question 1: Given the above information, do the following:

(a) Predict the shape of the absorption spectrum as a function of energy and sketch this shape (roughly). (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.)

(b) Look up the binding energy for nickel and use this as an estimate of the energy at which the nickel absorption edge will occur. (Ref. [3] above Electron Binding Energies, may be useful.)

(c) Compare this value to the K-line emission energy for nickel. (Will absorption occur at the same energy as emission? At a lower energy? At a higher energy?)

(d) From your energy, use Eq. (4) and the fact that energy is related to wavelength as $E = hc/\lambda$  to estimate the angle at which at which these absorption edges will appear.

Question 2: The activity of a radioactive source is the number of decays per unit time of that source and is therefore a measure of how much radiation is being released. A common unit of measure for activity is the curie, where $1 \textrm{Ci} = 3.7 \times 10^{10}$ disintegrations/sec. This is a very large unit! For comparison, the small button sources used in the Gamma Cross Section experiment are typically 1-10 microcuries, whereas the sources used in the Compton Scattering or Mossbauer Spectroscopy experiments are on the order of 1-10 millicuries.

In this experiment, you are not working with a naturally radioactive source, but you are working with a source that produces radiation. We can therefore calculate an equivalent strength of the source by determining the number of x-rays produced per unit time.

Your device produces x-rays by bombarding a copper (Z=29) target with electrons. Assume that you produce a current of 10 microamps of electrons and that you accelerate them through a potential of 25 kV before they hit the target. The efficiency of converting electrons into x-rays can be determined from the information plotted in Fig. A. Calculate the total intensity of the x-ray source (emitted in all 4π sr) and express this strength in units of curies. (Don't know what a steradian (sr) is? See here.)

Figure A: Dependence of the K x-ray yield on incident electron energy. (Source: [1])  The figure has been modified to show dependency for copper in red.

Note that the yield is in terms of number of x-rays per steradian (sr) per electron.

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 $E_{max} = eV - \Phi$. This maximum energy is the kinetic energy of a single electron (eV) minus the work function, $\Phi$), 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 $\Phi$. Therefore, this electron high energy cutoff can be related to an x-ray short wavelength cut-off 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 $K_\alpha$ line (from $n = 2$ to $n = 1$ transitions) and the $K_\beta$ line (from $n = 3$ to $n = 1$ transitions). See Fig. 1 for a schematic of $K$ and $L$ x-ray emission lines.

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.

Figure 2: X-ray spectrum produced when electrons are accelerated into a metal target. Note that a scale is not given on the x-axis; it will be quite different than the PHA.

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.

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 $K_\alpha$ line of an element to its atomic number $Z$:

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

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, $\lambda$ is the wavelength of the x-ray, $d$ is the distance between neighboring planes of atoms in the crystal, and $\theta$ is the angle between the incident x-rays and the surface of the crystal. (See Fig. 4).

If you need to propagate uncertainties in Eq. (4), make sure that $\Delta \theta$ is in radians.

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.

Figure 5: The Tel-X-Ometer and associated electronics.

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.

The LiF crystal should already be mounted in the central post as shown in Fig. 6. Verify that it is as shown.

Figure 6: Mounting the LiF crystal in the post

Open the dome and rotate the carriage arm to the $2\theta = 0^\circ$ 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 $\theta$ 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. 7.)

Figure 7: 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. For the first part of the experiment (head-on beam), make sure that this slit is VERTICAL. It is likely that the previous group of students left it horizontal.
• 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. Note that the sensor may be rotated or moved from side to side to a degree, and that misalignment may result in a very low count rate (on the order of 10 counts/s).

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.
• Turn on the power supply for the detector, amplifier, and frequency counter
• 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.
• 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.

• Adjust the electron beam current to about 10 $\mu$A. 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.

Let us now look at pulse signals on the oscilloscope.

Move slowly across the full range of scattering angles and observe the amplitude and intensity of the pulses on the scope. (If the detector is locked at the 0° location, open the dome and move the proportional counter away from the center. Re-close the dome and restart the x-rays.) Remember that as you move the carriage arm, you are changing the Bragg scattering condition and therefore changing the wavelength of the scattered x-rays you detect. You should observe much greater count rates at angles corresponding to the diffracted $K_\alpha$ and $K_\beta$ 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 $K_\alpha$ and which is the $K_\beta$?

When you do observe high intensity at certain angles, check for detector saturation. (See note below.) 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.

With the detector positioned at 0$^{\circ}$, 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 $K_{\alpha}$ and $K_{\beta} 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.

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

• 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.

In the experiments which follow, you will use Bragg scattering to take narrow “energy slices” of the PHA spectrum you observed above. Rather than collect at all energies simultaneously as we do when we observe the head-on beam, we will observe the scattered beam ($\theta>0$) 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.

• 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 $2\theta$ scale.
• Collecting data for this part of the experiment will be time consuming. Pick points judiciously.
  • Concentrate first on the $K_\alpha$ and $K_\beta$ 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 $\sqrt{N}/N$. 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 choosing 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. 

ANALYSIS: 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.

ANALYSIS: Determine the wavelengths of the copper $K_\alpha$ and $K_\beta$ lines, estimating their uncertainties. (If you have a sufficient number of points, you should consider a fit. But if you have too little data, it may be necessary to use some other by-eye estimate to determine the center and uncertainty of each peak.) Compare with literature values.

ANALYSIS: 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).

Part II of the experiment is intentionally left open-ended. You may need to consult outside resources for more theory information or equipment manuals for details about the apparatus. Given your experience collecting data in Part I, use your judgement to determine effective collection and analysis strategies, and budget your remaining time in lab appropriately.

The information given below is only a suggestion for how to proceed. Based on your interests and the quality of your work after Part I, you and the faculty instructor may discuss alternate goals. You are encouraged to bring ideas and propose ideas at your meeting, but you should also be prepared to defend approaches which stray far from the outline below.

In part I, you looked the x-ray emission spectrum from a copper target. In this part, you will use that spectrum as the incident light on several materials (copper, nickel, and cobalt) and measure the absorption spectrum. Predict the shape of the spectrum and collect data appropriately. Discuss the relationship of the features seen on the three spectra given that the three materials are adjacent in the periodic table.

Beyond emission and absorption, one can also study x-ray fluorescence. A rotary dial with thin films of metals from vanadium to zinc on the periodic table is available. Study the fluorescence emissions of these materials and test Moseley's law.