You already have a lot of experience in this class working with photons in what is normally considered the gamma energy range. Such gamma particles (which can span from a few keV up to a few MeV in energy) are typically produced from nuclear transitions. (For example, when a radioactive isotope decays into a new isotope in an excited nuclear energy state, which then relaxes to the nuclear ground state.)

In this experiment, we will instead focus on photons in the x-ray range. These photons are typically produced from electronic transitions. (For example, when an electron moves from a higher electron orbital into an unoccupied lower electronic orbital.) X-rays range in energy from 10s of eV up to 10s or 100s of keV.

Experimentally, we will study x-ray spectra in two ways:

• by using a proportional counter to directly measure the energy of the photons (in analogy to the way we used a NaI+PMT detector to measure gammas); and
• by using Bragg scattering to obtain finer energy resolution through diffraction of x-rays off of a crystal lattice.

Pedagogically, we will study x-ray creation from two sources:

• x-rays produced when we strike a target with high energy electrons; and
• x-rays produced when we strike a target with high energy x-rays.

After completing the experiment, students will be able to do the following:

• Explain the origin of atomic x-rays and their interactions with other atoms. In particular…
  • …how the bombardment of a target by high energy electrons can produce x-rays (i.e. x-ray emission); and
  • …how the bombardment of a target by a broad band of x-ray energies can result in absorption (i.e. x-ray absorption) and/or the emission of new x-ray energies (i.e. x-ray fluorescence).
• Use a proportional counter detector to measure x-ray energy and intensity.
• Use Bragg scattering to obtain finer wavelength (or energy) resolution on x-ray intensity measurements.
• Identify (and measure the energy of) features of a “characteristic” emission spectrum including the following:
  • Bremsstrahlung;
  • “characteristic” x-rays, such as the $K_\alpha$ and $K_\beta$ emission lines; and
  • the high energy cut-off.

Before you come to lab, familiarize yourself with the following concepts. Much of the relevant material can be found in this wiki page and in the supplemental materials section of the course wiki home page. (You are also welcome to search the internet for more information.)

Characteristic radiation: When the electrons surrounding an atom are excited and subsequently relax, that atom emits so-called “characteristic radiation”.

• Why do different elements have different characteristic emissions?
• What are the $K$-, $L$-, $M$-lines, etc.?
• What are typical energies for these types of emissions? (e.g. are we talking about eV, keV, MeV, GeV?)
• What is Bremsstrahlung and when do we expect to observe it?

Proportional counter: For this experiment we will use a type of detector known as a proportional counter.

• How is a proportional counter similar to a NaI detector and how is it different?
• Why do we need to use a different type of detector for this range of energies?

Bragg diffraction: The energy resolution of the proportional counter is not sufficient to resolve the spectral features we are interested in, particularly the $K_{\alpha}$ and $K_{\beta}$ emission lines. Bragg diffraction of x-rays from a lithium fluoride (LiF) crystal will be used to achieve the required degree of energy resolution.

• What is Bragg diffraction?
• Assume you have a source which emits x-rays over a wide range of energies, a crystal suitable for Bragg scattering, and an x-ray detector which can record the number of x-rays which strike it. How would you use these three elements to measure x-ray intensity at different energies?

The following links provide some of the theoretical background related to the emission and absorption of x-ray energy photons.

X-ray emission

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

$E = eV = hc/\lambda.$

(1)

Note that we can express $hc$ in units of electron-volt nanometers as $hc \approx 1240 \textrm{eV}\cdot\textrm{nm}$

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.

X-ray absorption

Absorption of radiation may be considered as any mechanism which removes some radiation from a directed beam. For x-rays – those photons with energies from about 100 eV to 100 keV (higher energy than ultraviolet light, but lower than what are typically termed gamma rays), – the two most common interaction modes in the absorber for removing x-rays from a beam are the photoelectric effect and Rayleigh scattering, of which the photoelectric dominates. Since these mechanisms are energy-dependent, the effect on an absorber is also energy-dependent. (See Interactions of Photons with Matter.)

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.)

For an idea of where absorption edges occur, see the NIST list of K-edge energies. You can also look at the energy levels by element by use of the more general search form.

Bragg scattering

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

$n\lambda = 2d \sin\theta$,

(4)

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 do not need to fully understand the details of how to operate the apparatus prior to coming to lab, but it will be helpful if you look over the schematics and explanations. Lab staff and TAs will walk you through the details in the lab.

Overview of equipment

The image above shows a typical setup for the x-ray experiment. From left to right you have:

• A digital multimeter for monitoring the Tel-X-Ometer accelerating voltage.
• A Tel-X-Ometer apparatus which comprises the X-ray source and detector.
• An old fashioned galvanometer for measuring the current in the electron tube.
• A NIM bin, which is simply a power supply and chassis with three modules mounted inside it, displayed below. The modules include:
  • A ratemeter which displays the rate of pulses on its input.
  • An amplifier which is used to amplify the pulses from the detector in the Tel-X-Ometer.
  • A high voltage supply which powers the detector in the Tel-X-Ometer.

The main device is the 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.

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 (c).

At the center of the unit, a LiF crystal may be mounted on the crystal post holder (c). A $\theta$:$2\theta$ table maintains the Bragg condition of equal angles of incidence and reflection. The carriage arm (d) 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

$d = 0.2008 \pm 0.0001$  nm (source: F. W. C. Boswell, Proc. Phys. Soc. A 64, 465-476 (1951).)

The proportional counter (e) is a xenon and carbon dioxide 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 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 notify 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 1/1000 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.

As you collect data, you will need to complete a number of specific tasks, each of which is focused on a skill or technique which you need to understand in order to complete the experiment. Successfully completing these exercises, as determined by the instructors during the lab, will count for a total of 25% of the grade of this lab.

For the first part of this experiment, you will explore how the proportional counter detector can be used to make direct energy measurements. There is a lot you can observe this way, but later in the experiment we will switch to using Bragg scattering to achieve finer energy resolution.

You will prepare the apparatus and test it by looking at pulses on the oscilloscope. You will then switch to using the PHA, and perform a rough 2-point calibration using a ${}^{57}$Co radioactive source.

Observing the signal on the oscilloscope

When the detector is powered-up and the x-ray source is on, you can check the detector output to verify that everything is operating properly. To do so, look at the output of the amplifier in the NIM bin on the oscilloscope with the detector set to an angle somewhere near zero degrees. (You do not need to be directly in the beam. The x-ray output is strong and you get more than enough intensity off-axis at 20 or 30 degrees.)

If everything is working, the proportional counter should be detecting x-rays coming out of the source. The pulses produced by the detector (and subsequently amplified by the amplifier) should look something like what follows.

The pulses should have positive polarity and be a few microseconds long. Their amplitude will vary with the gain setting on the amplifier. You should be able to adjust settings so that pulse amplitudes are in the range of a couple volts.

Do the following:

• Sketch some representative pulses in your lab notebook. Make sure to label axes and note the x- and y-scales (or add numbers to your axes) so that your sketches are to scale.
• Note the detector high voltage value and the amplifier gain settings.

We will now put away the calibration source and switch back to studying the copper emission spectrum emitted by the x-ray source itself.

With the detector positioned slightly outside the direct beam (at maybe 15$^{\circ}$ or 20$^{\circ}$, so that we don't have too great an x-ray intensity), collect a spectrum. What you should observe is a very rough version of the spectrum shown in Fig. 2, (As mentioned in the introduction, the energy resolution of the detector is not sufficient to distinguish between the $K_{\alpha}$ and $K_{\beta}$ lines, so you will not see two distinct lines. Instead, you should find one broad peak above a low, asymmetric background.)

• Adjust the accelerating voltage and comment on the change in the shape of the background (if any) and the position of the main peak (if any). (Do not exceed 30 kV).
• Adjust the beam current and note the effect on count rate. (Do not exceed 30 $\mu$A).

Once you have a feeling for what you are looking at, we want to make more careful measurements of the spectrum.

We will investigate the x-ray emission spectrum in more detail later, but for now let's put these x-rays to use!

In Part 1, we saw that bombarding a material with high energy electrons creates x-rays (for example, by knocking electrons out of an inner orbital, allowing a higher energy electron to fall down into the newly available state). However, we can also create x-rays by bombarding a material with other x-rays (which can likewise knock electrons out of an inner orbital, allowing a higher energy electron to fall down into the newly available state). This is called fluorescence.

To observe this, you will irradiate foils of different metals with the broad spectrum of x-rays from by your copper target. You can use the proportional counter to measure the energy of the x-ray emissions from each foil to see how the “characteristic” emission of sources change. In particular, you are provided a “rotary radiator” containing all the metals between vanadium (V, $Z = 23$) and zinc (Zn, $Z = 30$).

Set up the apparatus as follows:

• Place the rotary radiator on the central post as shown in figure 8.
• Move the detector carriage arm to 90 degrees (so that you are out of the direct beam from the source, but within view of the foil emissions).
• Make sure the output slit on the x-ray tube is still oriented vertical and that both collimating slits on the carriage arm have been removed.

Figure 8: Tel-X-Ometer with x-ray fluorescence sample holder (rotary radiator) mounted on crystal post.

The proportional counter is useful for energy measurement, but it does not have great energy resolution. We know, for example, that the peaks we have been seeing are not just one energy, but are instead several closely spaced energies (usually $K_{\alpha}$, $K_{\beta, 1}$, $K_{\beta, 2}$, etc.)

In order to improve our energy resolution, we can use a clever technique called Bragg scattering. As discussed in the theory section, when x-rays are incident upon a crystal (with an appropriate lattice spacing), there is a diffraction effect; as we move our source (or detector) through different angles of incidence (or reflection) off the crystal, we will be sensitive to different constructive interference conditions which obey the Bragg formula (Eq. 3 from the theory section, reproduced here for convenience)

where $\theta$ is the angle of incidence (which is also the angle of reflection), $d$ is the lattice spacing, $\lambda$ is the wavelength where constructive interference occurs and $n$ is the “order”. For the crystal we will be using here (LiF), the lattice spacing is

Preparing the apparatus for Bragg Scattering

Mounting the crystal and aligning the θ:2θ table

The LiF crystal should be mounted in the central post as shown in Fig. 6.

Figure 6: Mounting the LiF crystal in the post

In order to use Bragg scattering to select x-ray energy, the detector and crystal are attached to a $2\theta$ table. Notice how rotating the carriage arm containing the detector through an angle $\theta$ causes the crystal post to rotate through $\frac{\theta}{2}$. When properly aligned, this insures that the angle of incidence of the x-ray beam and the angle of the detector with respect to the normal to the crystal face meet the Bragg condition. The following procedure will allow you to align the two theta table if necessary.

To check the alignment, open the dome and rotate the carriage arm to the $2\theta = 0^\circ$ position. When the carriage arm is set to $2\theta = 0^\circ$ the scribe line scribe lines on both sides of the crystal post should be aligned as shown.

If the alignment needs to be adjusted, 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

Collimators

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. This collimator can be rotated to be either horizontal or vertical. For Bragg scattering, rotate it to vertical.

Additionally there are a pair of collimating slits which can be placed in the carriage arm (in between the detector and the LiF crystal).Place the 3 mm slit placed nearest to the crystal and the 1 mm slit closest to the detector.

For your final analysis, you will need to map out the full emission spectrum using Bragg scattering. However, to get oriented, you should first make a quick scan of the range of data to identify where the more important features are. You do not want to just collect evenly spaced data from X degrees to Y degrees in Z degree increments and then wait to look at the results until after the data have been collected. This is both inefficient and (likely) insufficient.

After your TA approves the quick scan results, continue collecting your full data set. Some additional points to keep in mind are the following:

• Plot your data as you take it so that you can see what your spectrum looks like as it builds up. There really is no other way to know when you have collected enough data.
• Move the detector in as small an increment as you can around the $K_{\alpha}$ and $K_{\beta}$ lines to capture the fine detail.
• On the other hand, since the Bremsstrahlung changes intensity gradually, you can sample that space more sparsely.
• As you approach the high energy cutoff, the rates will fall significantly. Leave time to collect as much data there as you can.
• Keep in mind that at some angles you may not just see the first order diffraction peak, but you may also see the second order peak. Make sure you are always monitoring the correct peak!

With the remainder of your time, select one of the following extension activities. Discuss your activity with the TA.

In addition to the rotary radiator, we have other interesting materials which can be used as either calibration or fluorescence sources. Data collected from these sources may be combined with your previous data (from the rotary radiator above) to allow for a more wide-ranging fluorescence study.

The following “sources” of x-rays are available:

1. Co-57: This radioactive source has two emissions: 14.4 keV and 6.4 keV. You used this source earlier as a calibration source.
2. Am-241: This radioactive source has a number of nuclear and x-ray emissions up to about 60 keV. Only the lowest energy ones will be in the range of your detector. You may need to do a little bit of exploration to determine the energy of the lines you can see.
3. Variable x-ray source: This source functions like a miniature fluorescence machine. An Am-241 source is contained within the housing which can irradiate one of six different foils (Cu, Rb, Mo, Ag, Ba and Tb). A beam port then allows you unshielded access to the emission (both from the Am-241 and to whichever foil is currently being irradiated).
4. Additional metal sheets (various $Z$ values): There is a box of mixed metals which can be irradiated by your x-ray source (just as you irradiated the rotary radiator foils).

Also, note the following:

• “Sources 1-3” emit their own x-rays; you do not need to have the x-ray machine running (or the dome closed) while taking these spectra and instead only need to place the source in front of the detector to collect data.
• The foils of “Source 4” need to be irradiated by the x-ray machine in order to fluoresce.
• Make sure your PHA range is about 0-35 keV. The detector has reasonable efficiency out to that value (and beyond).
• Not all the peaks you see will be $K_{\alpha}$ peaks. As $Z$ increases, the $K_{\alpha}$ and $K_{\beta}$ peaks may start to widen the peaks you see on the spectrum, and at even higher $Z$, $L_{\alpha}$, $L_{\beta}$ and $L_{\gamma}$ peaks also appear.

As we saw above with the fluorescence measurements, a material can absorb x-rays that have sufficient energy to knock an electron loose. What this means is that different materials have different “absorption edges”. In a very rough sense, the following is true:

• if the incident x-rays have energy less than the energy of the absorption edge, those x-rays will be transmitted through the material without interacting.
• if the incident x-rays have energy greater than the absorption edge, the x-rays can be absorbed.

We can investigate these absorption edges by placing different metal foils in between the x-ray source and the crystal, and then using Bragg diffraction to measure count rate as a function of energy. Foils of three different metals have been mounted on a post holder which sticks out the top of the dome on the Tel-X-Ometer apparatus. By loosening and tightening the nut where the post passes through the dome, it can be raised and lowered to put the desired foil in the beam path.

Because the incident x-ray spectrum is not uniform across energies (it has the complicated spectrum measured above, with big peaks on top of a varying background), we cannot tell where the absorption edges are just by looking at the overall intensity of transmitted light. Instead, we need to look at the “percent transmission”, or equivalently the ratio of the count rate with the absorber in place to the count rate with no absorber in place:

In fact, because the incident spectrum can change so abruptly, we suggest that at each angle where you make a measurement, you determine $R_{abs}$ and $R_0$ one right after the other without moving the detector position. (If you are near a $K$ line, $R_0$ can change by an order of magnitude or more in only a fraction of a degree. It is unlikely that you can move the detector back to exactly the same spot once it changes. Don't rely on your earlier emission data for $R_0$ values and instead take all new measurements as you go.)

You have one week to perform a full and complete analysis of the data you collect in lab and submit the following assignments.

At the core of this experiment is the copper x-ray tube. You observed that when a target is bombarded by high energy electrons, it can emit a spectrum of x-rays. You studied the emission spectrum of this tube in two ways – first through a direct measurement of the spectrum using the natural energy resolution of the proportional counter, then more precisely by using Bragg scattering to achieve finer energy resolution.

Do the following:

In this experiment you also saw that when you bombard a material with high energy x-rays, that material can absorb those x-rays, and in some case, even produce different high energy x-rays (depending on the target material). You explored this through x-ray fluorescence (and possibly also through x-ray absorption/transmission).

Do the following:

Also, depending on which “extension” project you chose, also do the following:

You will need to write a complete and persuasive conclusion that includes a comparison of your results to expectations/literature and puts the results in proper context.

Do the following:

• [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.

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 4$\pi$ 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.

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

Moseley's law gives the energy of the $K_{\alpha}$ 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 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 reasonably accurate for a range of elements. 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 $Z-$ 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.