====== X-Ray Studies ======
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 several different contexts, including Bragg scattering, emission and absorption spectra, and x-ray fluorescence.
====== References ======
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* [1] {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:sparrow_and_dick_1976.pdf |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] {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:fine_and_hendee_x_ray_energies.pdf |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] {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:electron_binding_energies.pdf |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] {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:r_s_proportional_counter.pdf |Reuter-Stokes Proportional Counters Performance and Specifications Guide}}.
* This is the information guide for the proportional counter detector used in this experiment.
====== Objectives ======
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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 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.
====== Related Applications ======
* Energy-dispersive x-ray fluorescence (EDXRF) was used as a non-invasive method of characterizing surface sulfur-based corrosion on Michelangelo’s David. This guided restoration efforts to do the least harm when cleaning the statue. [[https://doi.org/10.1002/xrs.907|Portable EDXRF surface mapping of sulfate concentration on Michelangelo's David]]
====== Preparation meeting ======
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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 {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:electron_binding_energies.pdf |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.
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**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 [[https://en.wikipedia.org/wiki/Steradian|here]].)
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:yieldchart.png?400 |}} |
| **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.
====== Theory ======
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===== X-ray production in a metal target =====
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) |
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.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:electron_shells.png?400 |}} |
| **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.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:rough_xray_spectrum.png?600 |}} |
| **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 [[phylabs:lab_courses:supplemental-material:physics-and-mathematics-references:interactions_of_photons_with_matter:start|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.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:copper_electron_levels_v2.png |}} |
| **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.) |
===== X-ray fluorescence =====
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.
===== Moseley's Law =====
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 $Z$:
| $E=0.75hcR\left(Z-1\right)^2$ | (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
| $E_n = \dfrac{hcRZ^2}{n^2}$. | (3) |
|We see that Moseley's law is just the energy difference between the $n = 2$ and $n = 1$ states for an atom with atomic number $Z-1$. (The reason that the formula is for $Z-1$ and not $Z$ is a very subtle one. See the [[phylabs:lab_courses:archive:phys-211-wiki-home:x-ray-studies-absorption-edges#appendixmoseley_s_law_and_the_z_z-1_correction|appendix]] at the end of this write-up for more information.) |
===== 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.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:plane_spacing.png?400 |}} |
| **Figure 4**: The geometry of Bragg scattering. (Source: [[https://en.wikipedia.org/wiki/File:DiffractionPlanes.png|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.
====== Apparatus and Operation ======
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===== Apparatus =====
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.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:xray_apparatus_diagram.png?600 |}} |
| **Figure 5**: The Tel-X-Ometer and associated electronics. |
=== X-ray tube ===
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)**.
=== Crystal post, carriage arm ===
At the center of the unit, a LiF crystal may be mounted on a crystal post. A $\theta:2\theta$ 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.
=== LiF crystal ===
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).)
=== Proportional counter ===
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.
=== Leaded plastic dome shield ===
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.
=== Tel-X-Ometer controls ===
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.
=== Pre-amp/amplifier ===
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.
=== Oscilloscope ===
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.
=== Pulse height analyzer (PHA) ===
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.
===== Preparing the spectrometer =====
=== Opening and closing the leaded plastic dome shield ===
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.
=== Mounting the crystal ===
The LiF crystal should already be mounted in the central post as shown in Fig. 6. Verify that it is as shown.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:xray_crystal_holder.png?400 |}} |
| **Figure 6**: Mounting the LiF crystal in the post |
=== Align the θ:2θ table ===
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.)
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:xray_detector.png?400 |}} |
| **Figure 7**: Aligning the θ:2θ table |
=== Collimators ===
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.
=== 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).
**CAUTION**: Do not touch the beryllium window! It is very fragile.
=== Electrical connections ===
**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.
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 $\mu$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.
**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.
===== Observing the signal on the oscilloscope =====
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.
**NOTE:** 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 (as viewed on the scope) 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.
====== Experimental procedure ======
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==== X-ray emission spectrum =====
**NOTE**: For what follows, we will operate at a current of 10 μA with an accelerating voltage of 20 kV. Make sure the slit on the exit port of the x-ray tube housing is set to **vertical**. Make sure two collimator slits on the carriage are **vertical** (with the 3 mm slit in the slot closest to the crystal and the 1 mm slit in the slot closest to the detector).
* 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.
==== Taking data ====
In the experiments which follow, you will use Bragg scattering to take narrow "energy slices" of the PHA spectrum you observed in Sec. 3.4. 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.
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 $2\theta$ 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.
==== Analysis ====
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).
===== K-lines and Moseley's law with x-ray fluorescence =====
==== Collecting data ====
In this next portion of the 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 $K_\alpha$ and $K_\beta$ lines individually, but will instead observe one peak.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:xray_emission_setup.png?400 |}} |
| **Figure 8**: 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:
* Make sure the primary beam collimator on the x-ray tube is horizontal. (See Fig. 8.)
* Remove the 1 mm and 3 mm collimators from the carriage.
* Make sure the crystal, screw, clamp, and the half-post have been removed and are sitting inside the dome (so you don't lose them).
* Gently place the slot in the Rotary Radiator over the crystal post. This device contains samples of V, Cr, Mn, Fe, Co, Ni, Cu and Zn.
* 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 $2\theta = 90^\circ$ and check that the Rotary Radiator presents the foil (not plastic) to the incident beam at an angle of about $45^\circ$.
* Turn on the x-rays. Set the tube voltage to 30 kV and the beam current to 10 $\mu$A.
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 $K_\alpha$ 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 $K_\alpha$ 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.
==== Analysis ====
Moseley's law, Eq. (2), predicts a linear relationship between $K_\alpha$ energy and $(Z-1)^2$. For the report, plot your data and perform a fit to test this relationship.
> **ANALYSIS:** Plot $K_\alpha$ 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.
===== X-ray absorption spectrum =====
**NOTE:** For what follows, we will run with an accelerating voltage of 25 kV and a beam current of 20 $\mu$A. Make sure the slit on the exit port of the x-ray tube housing is turned to **vertical**. Make sure two collimator slits on the carriage are ** vertical ** (with the 3 mm slit in the slot closest to the crystal and the 1 mm slit in the slot closest to the detector).
==== Collecting data ====
Here you will measure the absorption of x-rays by a nickel foil. (Note that your apparatus includes foils of copper, nickel and cobalt, but will will only use nickel today.) We will use the copper x-ray emission as our //incident// beam, and we will place the foil (when needed) in the beam path //before// the crystal and detector. (See Fig. 9.) In this way, we will 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.
| {{ phylabs:lab_courses:phys-211-wiki-home:x-ray_studies:xray_bragg_setup.png?400 |}} |
| **Figure 9**: Tel-X-Ometer setup showing the location of the metal x-ray filter. |
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.
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 $2\theta$ 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.
==== Analysis ====
For the report, you will need to produce a plot of the nickel x-ray absorption spectrum and identify the absorption edge.
> **ANALYSIS:** Plot the ratio of intensity (i.e., count rate //with// absorber divided by count rate //without// absorber) as a function of x-ray wavelength. Determine (with uncertainties) the position of each edge and compare it to the K line emission wavelengths (energies). Is your edge at a higher or lower energy than the largest K-line energy? (Why?)
>
> **ANALYSIS:** 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?
>
> **ANALYSIS:** How could these absorption edges be used to build an x-ray filter?
====== Rubrics ======
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===== Analysis rubric =====
Each item below is graded on a 0-4 point scale. Full, partial and no credit correspond roughly to the following:
* **4 – Full Credit**: analysis is complete and reasonably correct; only moderate corrections are required before incorporating into the final report.
* **2 – Partial Credit**: analysis is lacking or incorrect and will need to be significantly improved prior to incorporating into the report.
* **0 – Missing or Inadequate**: the analysis is is missing or the attempt is completely inadequate.
All rubric items carry the same weight. The final grade for the analysis will be assigned based on the average (on a 4.0 scale) over all rubric items.
| Item | Description |
| X-ray emission spectrum | Plots the energy-resolved copper x-ray emission spectrum collected using Bragg scattering and identifies the features seen. Determines the energy of the $K_\alpha$ and $K_\beta$ peaks, and compares to literature values for copper. Determines (with uncertainties) the wavelength of the high-energy cutoff and uses this value (with your known accelerating voltage) to compute Planck's constant, $h$ (with uncertainty). |
| Moseley's law | Provides a plot of K-alpha energy versus $(Z-1)^2$, performs a fit to the appropriate function, and compares the extracted value of the Rydberg energy to the literature value. |
| X-ray absorption spectrum | Provides a plot of % transmission for all three metals (including some data across whole range and more closely spaced data near transitions). Determines the absorption edge energy for each metal, compares energies with literature values, and discusses (qualitatively) the trend between atomic number and absorption edge. Discusses how these absorption edges could be used to form a filter. |
===== Report rubric =====
Each item below is graded on a 0-4 point scale.
* **4 – Good (A):** completes all listed tasks and provides appropriate context; thinks carefully about data and analysis; addresses all concerns raised by the results (where appropriate).
* **3 – Adequate (B):** misses one or more minor element or lacks appropriate context; leaves a problem or ambiguity unaddressed; does not present analysis clearly enough.
* **2 – Needs improvement (C):** omits or mishandles one or more item which renders the analysis fundamentally incorrect or incomplete; presents results in an incorrect or unclear way.
* **1 – Inadequate (D):** omits or mishandles multiple items or treats them at an insufficient level; presentation is overall muddled or inaccurate; flaws in logic or process.
* **0 – Missing (F):** omits all elements or makes no meaningful attempt.
All rubric items carry the same weight. The final grade for the report will be assigned based on the average (on a 4.0 scale) over all rubric items.
| Item | Description |
| Presentation of Data | Presents plots of data as needed and uses them to support the narrative of the report. Properly labels plots, and makes presentation clean and clear. Uses error bars where appropriate. Includes captions that provide appropriate context. Presents all numerical values with appropriate units and significant figures. Clearly formats numbers, equations, tables, etc. |
| Data Handling | Describes how the raw data was processed including with uncertainties. Details fit functions and provides sample fits (if appropriate). Details other calculations/considerations and provides sample calculations or reasoning (if appropriate). |
| Discussion of Uncertainties | Identifies relevant sources of uncertainty in measured quantities, and quantifies values when possible. Describes how uncertainties were assessed and incorporated into the analysis. Identifies potential sources of systematic bias and describes how they are accounted for in the analysis or eliminated. |
| Presentation of Results | Final results are presented clearly. Data tables and plots are used where appropriate and are properly labeled and annotated. Measured and calculated quantities include units and uncertainties where appropriate. |
| Conclusions | Makes clear final conclusions that are fully supported by the experimental results and discusses the overall take-aways of the experiment appropriately. Properly accounts for or contextualizes measurement uncertainties and potential sources of systematic bias. |