X-ray Spectroscopy. Danny Bennett and Maeve Madigan. October 12, 2015
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1 X-ray Spectroscopy Danny Bennett and Maeve Madigan October 12, 2015 Abstract Various X-ray spectra were obtained, and their properties were investigated. The characteristic peaks were identified for a molybdenum anode and the energies corresponding to these peaks were found to be E β = (19.06 ± 0.23) kev and E α = (17.01 ± 0.29) kev, respectively. It was shown that the quantity Uλ min was constant, and a value of (1.22 ± 0.03) 10 6 Vm was obtained for it. Using this, a value of h = (6.51 ± 0.16) Js was obtained for Planck s constant. The K-edges for filters of different materials were found, which lead to values of R = (0.99 ± 0.08) 10 7 m 1 and σ K = (1.96 ± 0.08) for the Rydberg constant and the K-shell screening parameter, respectively. Unfortunately, due to experimental errors, the dependence on the absorption cross section, σ a, and Z was not observed. By placing the Zr filter in front of the anode, the K β peak was filtered out, and a spectrum with one characteristic peak was obtained. In an attempt to further investigate the effects of the Zr filter, the n = 2 peaks were obtained, and their energies were found to be E β = (19.66 ± 0.18) kev and E α = (17.44 ± 0.12) kev, which agreed with the values found for the n = 1 peaks. 1 Introduction and Theory This experiment was split into two parts. For the first part of the experiment, the aims were to observe X-ray spectra for a molybdenum anode for varying accelerating voltages, U, of electrons, to determine the centre of the peaks of the spectra in terms of angle of rotation, β, and hence λ, to determine λ min (and hence E max ), E α and E β of the peaks and to check if the quantity Uλ min was a constant in each case and to compare the obtained values to the theoretical values. For the second part of the experiment, the aims were to determine E K (K-absorption edge) for various elements (Zr, Mo, Ag and In), to show that E K (Z σ K ) 2 and hence obtain values for R and σ K, to determine the mass attenuation coefficient and absorption cross section, σ a, for various materials at a fixed wavelength and to determine the dependence of σ a on Z. 1.1 X-ray Spectrum The X-ray spectrum is composed of sharp peaks (characteristic spectrum) as well as an underlying smooth continuous curve (continuous spectrum) which has a minimum wavelength λ min. The continuous spectrum is due to bremsstrahlung ( braking radiation ) caused by the deceleration of electrons upon striking the target. To satisfy the law of conservation of energy, a photon is emitted from the electron, accounting for the kinetic energy lost during deceleration. If the electron is decelerated through a voltage, U, then since E = hc for the photon, the minimum wavelength is determined by λ λ min = hc eu, (1) where e is the charge of the electron. This is known as the Duane Hunt law. The characteristic spectrum represents characteristic X-rays, which are emitted when outer-shell electrons fill a vacancy in the inner shell of 1
2 an atom. When an incident electron strikes a bound electron in the target atom, the bound electron is ejected and the vacancy is filled by an electron from one of the outer shells. If electrons are ejected from the K-shell, they can be replaced by electrons from the outer M- and L-shells, for example. This results in the emission of a photon whose energy usually lies in the X-ray range, which, when detected, gives rise to the characteristic peaks in the X-ray spectrum. The peaks K α and K β are caused by transitions of electrons from the L- and M-shells, respectively. Since the M-shell electrons have higher energy, the emitted photons will have a lower wavelength than the ones emitted due to L-shell transitions. This allows the K α and K β to be easily identified on the X-ray spectrum. Figure 1: Defining the peaks K α and K β. For X-ray diffraction, the wavelength, λ, of the incident X-rays can be related to the incident angle β using Bragg s law: 2d sin (θ) = nλ. (2) Therefore, if the X-ray spectrum is known in terms of the angle β, it is also known in terms of λ, and hence the wavelengths and energies of the peaks in the X-ray spectrum can be determined, as well as λ min /E max. By determining λ min for each voltage U, the quantity Uλ min should be a constant, according to (1). Thus, a value for Planck s constant can be obtained using h = (Uλ min ) e c, (3) taking the velocity of light, c, and the charge of an electron, e, to be known. Theoretical values for the peak energies can be obtained using the modified Bohr model of the atom: E n = Rhc n 2 Z2 eff, (4) where R is the Rydberg constant, Z eff = Z σ m and σ m is the screening constant representing a screening effect due to the inner-shell and other electrons. Since the energy of the peaks K α and K β are due to transitions between the different shells in the target atom it is clear that E α = E K E L and E β = E K E M, so according to (4) the theoretical values are given by 1.2 X-ray Attenuation E α = 3Rhc 4 Zeff 2, E β = 8Rhc Zeff 2. (5) 9 For radiation of intensity, I, incident on a thin slab (in our case, a filter) of thickness dx which n atoms per unit volume, each of removal cross section, σ (any incident photon in the area σ is removed from the beam by absorption or scattering), the change in intensity, di is related to these quantities as follows: di (x) = Inσdx. (6) 2
3 Integrating (6) leads to I (x) = I 0 exp ( µx), (7) ρ where µ = nσ is the linear attenuation coefficient. Since n = N A A, where N A is Avagadro s number, ρ is the density of the material and A is the atomic weight, the mass attenuation coefficient, µ ρ, can be determined using µ (T ) = ln ρ ρx, (8) using the following relation between transmittance, intensity and count rate: T = I I 0 = R R 0. (9) R and R 0 can be measured experimentally and hence so can the transmittance. The values of the thicknesses of the absorbers and the densities are given, allowing for the mass attenuation coefficients to be determined experimentally. Since the filtered photons are removed from the beam by either scattering or absorption, σ can be written as σ = σ s + σ a, (10) where σ s and σ a are the cross sections pertaining to scattering and absorption, respectively. Since the X-rays in this experiment are mainly in the energy range of 10 40keV absorption via the photoelectric effect is the main contributor to attenuation, meaning σ s < σ a, and σ s is approximately given in SI units by σ s = 0.02 A N A. (11) Since σ = µ n, and both µ and n can be determined, and since σ s is known for any given material, then the absorption cross section, σ a, can be determined experimentally by σ a = A N A ( µ ρ 0.02 ). (12) The main contributor to attenuation in this experiment is absorption via the photoelectric effect; the photon energy is used to eject electrons from atoms in the absorber. For the ejection of an electron from the K-shell, the energy of the photon must be greater than the binding energy of the K-shell electron, so there will be a jump discontinuity in absorption and hence attenuation when the photons reach this binding energy. This is discontinuity is called the K-edge and here, according to (4), the energy is given by E K = Rhc (Z σ K ) 2. (13) If the energy at the K-edge of a material lies between the peak energies E β and E α, it is possible to filter out most of the higher energy peak, K β, and obtain an X-ray spectrum with only the K α peak (this peak will also be slightly filtered, but still recognisable). 2 Experimental Method 2.1 Apparatus The apparatus is set up as shown in the diagram; incident X-rays are emitted from the molybdenum anode towards a monocrystal which is mounted on a support, and are then diffracted towards the detector. The support and the detector are both rotated automatically through angles β and 2β by the apparatus in order for (2) to take effect. The upper and lower limits of the angle of rotation, the measuring time interval, t, and the step angle, β, as well as the voltage, U, and the current, I, can also be set. Pressing SCAN on the equipment rotates the support and the detector through their respective angles with step β in time 3
4 intervals of t, starting with the lower limit of β and ending with the upper limit. A spectrum of count rate, R (proportional to intensity), is recorded as a function of β and is displayed on the computer. This can be converted to a spectrum as a function of λ using (2), and can be calibrated automatically on the computer since for the NaCl crystal target, 2d is given to be 563pm. A range of filters were available which could be mounted on the molybdenum anode. Figure 2: Bragg geometry. 2.2 Procedure For the first part of the experiment, the apparatus was set up as previously described. The NaCl cystal was carefully mounted on the support and clamped in place. The voltage was initally set to 35kV and the current was maintained at 1mA. By setting t = 5s, β = 0.1 and the upper and lower limits of β to 3 and 10 respectively, measurements were taken for 5 second intervals for angles of rotation, starting at 3 and increasing in increments of 0.1 up to 10. The spectrum was recorded and saved, the voltage was decreased by 5kV and the process was repeated. For each of the spectra, a Gaussian curve was fit to each peak to determine the angle / wavelength corresponding to it, and hence the energy. The results were tabulated and, after taking errors and uncertainties into account, were compared to the expected values. For the second part of the experiment, the energies at the K-absorption edge for various filters were determined. The apparatus was similar to the one in the first part of the experiment, the only difference being the filters placed in front of the molybdenum anode. A spectrum was recorded and displaced for each filter, and was calibrated on the computer to obtain a spectrum of transmittance, T, as a function of wavelength, λ, in order to determine the K-edge. The K-edge wavelength, λ K, was determined using the computer, and the corresponding energy, E K, for each filter was then obtained. The results were recorded in a table, and the K-edge energies obtained were compared to the accepted values. By plotting a graph of Z vs E K for the filters, the Rydberg constant, R, and the K-shell screening parameter, σ K, were determined. The mass attenuation coefficient and absorption cross section were then determined for various elements. Transmittance was measured at a fixed wavelength (this was done by fixing the angle of rotation on the apparatus), and the desired quantities were obtained for each filter. Since the machine gave the count rate for each filter, the unfiltered count rate, R 0, was measured, and the transmittance was then obtained. The dependence of σ a on Z was determined by plotting the two sets of values. 4
5 3 Results and Analysis 3.1 Error Analysis The uncertainty in β was taken to be β = 0.1, since the quantity was measured by the apparatus to one decimal place. The uncertainty in λ min was determined using (2) by Using the standard formula, λ min = 2d (sin (β)). (14) f (x) = f x, (15) x lead to very large uncertainties in λ min because of a cos (β) term and small values of β. So (sin (β)) was taken to be 1 2 (sin (β + β) sin (β β)). Since n = 1, and 2d = 563pm was given, an average value of λ min = 0.98pm was obtained. The uncertainty in the voltage was taken to be U = 0.1kV since the values of U were measured by the apparatus up to an accuracy of 0.1kV. A value for λ min was obtained, so the uncertainty in (Uλ min ) could be determined. Using the formula f (x, y) f (x, y) ( x ) 2 = + x ( ) 2 y, (16) y an average value of (Uλ min ) = Vm was obtained. Since h is given by (3), then h is simply determined by h = e c (Uλ min), (17) which lead to a value of h = Js. Using the method described above, the uncertainties in the peak wavelengths were determined, although they were roughly the same as before, so they were taken to be λ α = λ β = 0.98pm. To find the uncertainty in energy, (15) was used given that E = hc λ, and the following expression was obtained: E = hc λ. (18) λ2 Thus, uncertainties of E α = 0.29keV and E β = 0.23keV were obtained. This method was also used in the second part of the experiment anywhere it was needed to determine the uncertainty in energy when given the wavelength. 5
6 3.2 Results Figure 3: Different spectra were recorded for various voltage values. From largest to smallest, the spectra correspond to U = 35, 30, 25, 20kV, respectively. A count rate R was recorded against different angles of rotation β in intervals of 0.1 in time intervals of 5s. One possible observation is that on the spectrum for U = 20kV, the peaks are not recognisable. This is expected, as lowering the voltage lowers the photon energy, meaning less electrons will be ejected from the K-shells. The spectra were recorded for different values of U. The peaks were observed and the value of β which corresponded to λ min was recorded; λ min was then obtained using (2). The results obtained are tabulated below. Average values of E β = (19.06 ± 0.23) kev and E α = (17.01 ± 0.29) kev for the K β and K α peaks. Using (5), the expected values are E β = 20.36keV and E α = 17.18keV. σ m was assumed to be 1. Since the K-shell only has 2 electrons, so when one is ejected, there is only one electron left to screen the nucleus, making it a reasonable assumption. From the table, an average value of Uλ min = (1.22 ± 0.03) 10 6 Vm was obtained. This gave a value of h = (6.51 ± 0.16) Js for Planck s constant. U (kv) β min ( ) λ min (pm) E max (kev) Uλ min (Vm) ± ± ± ± A Gaussian curve was fit to the peaks of each spectrum and the mean values obtained were taken to be the centres of the peaks. The wavelengths and hence the energies corresponding to the peaks were then obtained and are recorded in the table below. U (kv) β (K β ) ( ) λ β (pm) E β (kev) β (K α ) ( ) λ α (pm) E α (kev)
7 (a) U = 35kV. (b) U = 30kV. (c) U = 25kV. Figure 4: Spectra as a function of λ for different voltages. 7
8 Filters of different materials were placed in front of the molybdenum anode in order to determine the K-edge for each material. Spectra were recorded and used to determine transmittance as a function of wavelength (see below). The K-edges were determined using the computer, and the results were recorded. Here, λ K was given by the computer when determining λ K and E K was obtained using E K = hc λ K. Element Z λ K (pm) E K (kev) Zr ± ± 0.18 Mo ± ± 0.21 Ag ± ± 0.35 In ± ± 0.42 Figure 5: The K-edge for Zr. Since the energy at the K-edge is given by (13), a graph of Z vs E K will allow R, the Rydberg constant, and σ K, the K-shell screening parameter, to be determined. The slope and intercept of this graph are m = 1 Rhc and σ K, respectively. The data was plotted and the line of best fit was obtained, the equation of which was Z = (9.02 ± 0.36) E K + (1.96 ± 0.08). This gave values of R = (0.99 ± 0.08) 10 7 m 1 and σ K = (1.96 ± 0.08) for the Rydberg constant and the K-shell screening parameter, respectively. 8
9 Figure 6: A plot of Z vs E K for the different materials. Next, the mass attenuation coefficients and absorption cross sections were measured for different elements. This experiment was preformed for X-rays at a fixed wavelength of 41pm, which was achieved, using (2), by setting β 4.2. The voltage and current were adjusted so that the count rates did not exceed 600s 1, eliminating the need to correct for dead time. The count rate without a filter was first recorded to obtain R 0, and then, after the count rate for each element was measured, the transmittance was obtained using (9). The mass attenuation coefficients and absorption cross sections were then obtained. The results are shown in the table below. By inspection of the results, the relationship σ a Z p cannot be observed here, which suggests that errors were made when taking the measurements. Element Z R ( s 1) T µ/ρ ( m 2 kg 1) σ a (b) Al Fe Cu Zr It was found that the K-edge for Zr was between the energies of the K α and K β peaks. The Zr filter was mounted in an attempt to obtain an X-ray spectrum with only a K α peak. It was successfully obtained alongside the unfiltered peak. The entire spectrum was shifted downwards due to absorption, and after the K-edge, the amount that the spectrum was shifted downwards began to decrease. In order to investigate if the filtered spectrum approached the unfiltered spectrum as λ increased, the upper limit of β was increased and the process was repeated. The newly obtained spectra contained two extra peaks, and when filtered, the first of the two new peaks was also filtered out. These turned out to be the n = 2 peaks, and the angles β to which they corresponded were measured to be ± 0.12 and ± 0.10 for the K β and K α peaks, respectively. These measurements lead to wavelengths of λ β = ± 0.57pm and λ α = ± 0.48pm, leading to energies of E β = ± 0.18keV and E α = ± 0.12keV. 9
10 Figure 7: X-ray spectra, both unfiltered and filtered, for a larger range of β. The smaller pair of peaks are the second order peaks. The K β peaks of both order are filtered by the Zr filter. 4 Discussion and Conclusions X-ray spectra were successfully obtained, and it was observed that as the voltage was decreased, the peaks became less distinguishable. The energies at the peaks K β and K α were found to be E β = (19.06 ± 0.23) kev and E α = (17.01 ± 0.29) kev, respectively. These answers compared very well to the calculated expected values. E α was closer to the expected value than E β, as σ m was assumed to be 1 for both calculations. This assumption is less accurate in the K β case, as electrons from the M-shell will also be screened by the electrons in the L-shell. Also, the accuracy of these values could be improved by omitting the U = 20kV spectrum, because the peaks could not be distinguished from the continuous spectrum. Values of Uλ min = (1.22 ± 0.03) 10 6 Vm and h = (6.51 ± 0.16) Js were obtained for Uλ min and h. These compare well with the expected values of Uλ min = (1.24 ± 0.03) 10 6 Vm and h = Js, respectively. The K-edges for Zr, Mo, Ag and In were determined to be (17.94 ± 0.18) kev, (19.49 ± 0.21) kev, (25.03 ± 0.35) kev and (27.17 ± 0.42) kev, respectively. These compare well to the accepted values of keV, keV, keV and keV[2]. By plotting Z against E K, a linear dependence was observed, and the Rydberg constant and K-shell screening parameter were determined to be R = (0.99 ± 0.08) 10 7 m 1 and σ K = (1.96 ± 0.08), respectively. Our value for the Rydberg constant doesn t agree with the accepted value, but is of the same order of magnitude. A relationship between σ a and Z was not observed due to experimental error. One possible reason for such error is that the dead time was not accounted for. When measuring the count rate, R, the voltage and current were lowered to ensure that values were less than 600s 1. However, our results suggest that the voltage and current would need to be lowered even further in order to obtain results that could be used to observe the relationship between σ a and Z. 10
11 The n = 2 peaks were an unexpected but interesting result. Since the computer calibrated the spectrum for the n = 1 order peaks, they appeared to be different peaks with different energies. It was even more unexpected that the second order K β peak was filtered, as it appeared to have an energy lower than the K-edge of Zr. After coming to the conclusion that they were indeed the n = 2 peaks, the energies of the peaks were determined to be E β = ± 0.18keV and E α = ± 0.12keV. At U = 35kV and I = 1mA, these energies are in agreement with the energies of the n = 1 peaks. References [1] John R Hook and Henry Edgar Hall. Solid state physics (the manchester physics series), [2] George William Clarkson Kaye and Thomas Howell Laby. Tables of physical and chemical constants: and some mathematical functions. Longmans, Green and Company,
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