Lab Manual: Determination of Planck s constant with x-rays 1. Purpose: To obtain a better understanding on the production of X-rays, the bremsstrahlung radiation and the characteristic radiation of a Molybdenum anode, and to investigate how they can be used to determine Planck s constant. 2. Apparatus: Klinger X-Ray Apparatus 3. Introduction: X-rays are created when fast-moving electrons are rapidly decelerated as they pass through the magnetic and electric fields of atomic nuclei in a material, the anode. According to the laws of classical electrodynamics, this deceleration gives rise to electromagnetic radiation, which is mainly radiated perpendicular to the direction of acceleration, i.e. in this case perpendicular to the direction of the electrons striking the anode (see Fig. 1-15 p. 20 ref. [1] for a schematic of an x-ray tube). For historical reasons, this x-ray component is referred to as bremsstrahlung (braking radiation) after the German word for the deceleration process by which it occurs [1, Chap.1]. The bremsstrahlung radiation has a continuous spectrum, which extends to a certain maximum frequency ν max or a minimum wavelength λ min i.e., it displays a peak and cutoff due to the limited energy of the incoming electrons. In addition to the continuous spectrum, characteristic radiation is generated, which appears in the spectrum as individual lines or peaks. These peaks are generated when highenergy electrons penetrate deep into the atomic shells of the anode material and eject electrons from the innermost shells by collision. The holes (i.e., missing electrons) created in this process are filled by electrons from the outer shells under emission of x-rays. The resulting peaks in the spectrum are characteristic of that anode material. As electrons transition between the shells, in accordance with the laws of quantum mechanics, these transitions entail either the absorption or emission of radiation. A series of sequential lines designated K α, K β, K γ, etc. (Fig. 1) occur when electrons transition from outer shells to the K-shell. Starting from K α, the energy of the transitions increases and the Fig. 1: Simplified diagram of the energy levels of an atom, showing the relaxation of electrons in the outer shells to the holes created in the inner shells. Relaxations to the K-shell are designated K α, K β, K γ, etc. These relaxations are accompanied by an emission of radiation. 1
corresponding wavelength decreases. You can roughly compare this to the optical line spectrum of a material in a gaseous or vapor state. This experiment records the energy spectrum of an x-ray tube with a molybdenum anode. We use a spectrometer to obtain the wavelength, frequency or energy spectrum of the radiation, depending on the selected representation mode. One of the main components of the spectrometer is a crystal, i.e. regularly spaced atoms that act as scattering centers for x-rays. The interatomic distance in crystals is close to the wavelength of x-rays leading to diffraction. Hence, a goniometer with a NaCl crystal allows variation of the angle of incidence of the x-ray beam, and a Geiger-Müller counter tube records the intensity of the x-rays diffracted from the crystal. The crystal and counter tube are pivoted with respect to the incident x-ray beam in 2θ coupling (Fig. 2). In accordance with Bragg s law of reflection, peaks resulting from constructive interference as the x-rays reflect from a family of crystal lattice planes, occur at specific angles, depending on the wavelength of the produced x-rays as well as the interatomic spacing of the crystal. The scattering angle θ in the first order of diffraction corresponds to the wavelength λλ = 2 dd sin θθ, where dd = 282.01 pm [2] is the lattice plane spacing of NaCl (Fig. 3). Note: the xray apparatus software uses β for the scattering angle, i.e., β = θ. Together with the relationships valid for electromagnetic radiation, λλ = cc νν (1) EE = h νν (2) we can find the energy of the x-rays. A typical spectrum of an x-ray tube diffracted by a single crystal sample is shown in Fig. 4. Fig. 2: Schematic diagram of diffraction of collimated x-rays (1) at the NaCl crystal (2). The counter tube (3) is positioned at an angle of 2θ with respect to the x-ray beam. θ is the angle between the x-ray beam and the plane of the NaCl crystal. Fig. 3: Schematic diagram of Bragg s law. Parallel x-rays interfere constructively when 2d*sinθ = nλ, where n=1,2,3, is a positive integer and d is the lattice spacing of the NaCl crystal. 2
(1) Fig. 4: Screenshot of the spectrum of an x-ray tube diffracted by the NaCl crystal. Plotted as Intensity (counts/s) vs. diffractometer angle. The bremsstrahlung continuum in the emission spectrum of an x-ray tube is characterized by the limit wavelength λ min. In 1915, the American physicists William Duane and Franklin L. Hunt discovered an inverse proportionality between the limit wavelength and the x-ray tube high voltage [3]: λ mmmmmm ~ 1 UU. However, an x-ray quantum attains a maximum energy at precisely the moment in which it acquires the total kinetic energy EE = ee UU of an electron decelerated in the anode. It thus follows that νν mmmmmm = ee h UU (3) λ mmmmmm = h cc ee 1 (4) UU This equation corresponds to the proportionality in the Duane-Hunt law. The proportionality factor A = h cc can be used to determine Planck s constant h when the quantities ee c and e are known. 4. Setting up the system: 4.1 Prepare a PC-based measurement Connect the X-Ray Apparatus to the computer using a USB cable. Open the software X-Ray Apparatus. Delete any existing measurement data by clicking the button on the top left or by pressing F4. 3
4.2 Calibrate First, run a test scan to see what the counting rate is. In the X-Ray Apparatus software, click the settings button in the taskbar to open up the settings screen. Set the x-ray high voltage U = 35.0 kv, emission current I = 1 ma, measuring time per angular step Δt = 2 s, and angular step width Δβ = 0.1. Set the lower limit value of the target angle to 5 and the upper limit to 8. Press the COUPLED key on the device to enable 2θ coupling of the target and sensor, that is when the NaCl crystal is rotated by a number of degrees, the G-M counter is rotated by twice that. Press the SCAN key to start the measurement and data transmission to the PC, and wait for the data to come in. If the scan will not start, make sure that the door to the experiment chamber is closed. The X-Ray Apparatus has a built-in safety system that will not allow the X-Ray tube to turn on while the doors are open. If the count rate value on the screen is blinking, that means the doors are not properly closed. You should see two clear peaks in the data, one small and one large (see K α and K β in Fig. 4). The small peak should have a maximum of at least 800 counts, and the large peak at least 1700 counts. If your peaks are smaller than that, you will need to adjust the setup. Ask the TA for help. 5. Experiment: Energy Spectrum of the x-ray tube and determination of Planck s constant. Part 1: You will start the experiment by running scans with the parameters shown in the table 1, with each row representing one scan. Make sure that the dialog box for COUPLED is selected before starting the scans. U (kv) I (ma) 20 1.00 25 1.00 30 1.00 35 1.00 35 0.8 35 0.6 35 0.4 Table 1: The measuring time per angular step for these scans is Δt = 7 s, and angular step width Δβ = 0.1. Set the lower limit value of the target angle to β min = 2.5 and the upper limit to β max = 12.5. Open the Settings dialog and select the lattice plane spacing 2d = 564.0 pm for NaCl from the dropdown menu under the Crystal tab to convert the x-axis from the angle of incidence to the wavelength, and then save the data under a suitable name. Analysis Part 1: Calculate the wavelengths λ of the K α and K β peaks for all of the measurements. To do this, right click and select Calculate Peak Center, then click a point to the left and right of the peak you want to calculate. The program will display a vertical line showing the location of the center, along with the coordinates in the status line on the bottom left of the program window. Record the values in two separate tables, one for varying tube high voltage U and the other for varying emission current I. Provide an estimate for the uncertainty of your determination of the values for λ (you will use this in question 6.3). 4
U (kv) λ(k α ) (pm) Δλ ( K α ) λ(k β ) (pm) Δλ( K β ) 25 30 35 I (ma) λ(k α ) (pm) Δλ( K α ) λ(k β ) (pm) Δλ( K β ) 0.4 0.6 0.8 1.0 For the varying current data, make a separate table and record the maximum values of the K α and K β peaks as well as the maximum of the bremsstrahlung radiation. Call these values R(K α ), R(K β ), and R C, (see Fig. 4) respectively. I (ma) R(K α ) (cts s -1 ) R(K β ) (cts s -1 ) R C (cts s -1 ) 0.4 0.6 0.8 1.0 Graph these values on a plot with the y-axis as the maxima R and the x-axis as the emission current I. For each peak, fit the values using a linear fit (y =m*x+b, m = slope, b = y-intercept). Report the fit results in your write-up using the following table. Provide a plot containing the datapoints and the fit. Discuss the quality of the linear regression in your report, is a linear function appropriate? R(K α ) (cts s -1 ) R(K β ) (cts s -1 ) R C (cts s -1 ) m σ m b σ b 5
Part 2: You will run through this part of the experiment with the parameters shown in the table below, with each row representing one scan. Use I = 1 ma for each scan. Make sure that the dialog box for COUPLED is selected before starting the scans. Open the Settings dialog and input the lattice plane spacing for NaCl under the Crystal tab to convert the x-axis from the angle of incidence to the wavelength, and then save the data under a suitable name. Save each of the scans separately! U (kv) Δt (s) β min (grd) β max (grd) Δβ (grd) 22 30 5.2 6.2 0.1 24 30 5.0 6.2 0.1 26 20 4.5 6.2 0.1 28 20 3.8 6.0 0.1 30 10 3.2 6.0 0.1 32 10 2.5 6.0 0.1 34 10 2.5 6.0 0.1 35 10 2.5 6.0 0.1 Analysis Part 2: First, determine the limit wavelength λ min (see Fig. 4) for each value of the tube high voltage U. Export each scan as a text file and graph it, i.e., intensity vs. wavelength. For each scan, select the points on the rising edge of bremsstrahlung radiation peak and perform a linear fit (the red line in Fig. 4 is an example of how a linear fit provides a value for λ min ), make sure that you select points such that the best fit line matches the rising edge to the best of your ability. Indicate in the plot which datapoints were used in the fit, in Excel plot the whole dataset then add to the same graph a plot of only the datapoints of interest (use a different color), then perform a linear regression to the selected datapoints. Tabulate the results of the fit values, use these results to find the x-axis intercept (λ min ), along with the uncertainty (i.e., you will have to propagate the uncertainties of the y-axis intercept and the slope). Put this in your write up using the following table. Also, provide a plot of each scan along with the fit, indicate in the plot, which data points you selected to be included in the fit. U (kv) m σ m b σ b λ min σ λ h σ h 22 24 26 28 30 32 34 35 6
Based on the values obtained for λ min determine the value of Planck s constant h and its uncertainty using equation (4) described above, using the following three methods. Method 1: Calculate and tabulate the values for Planck s constant for each value of λ min along with the uncertainty. Assume that the value U has a negligible error. For these values calculate the average (h ) and the standard deviation of the mean (σσ h ). Report how you have calculated these values in the write up. Method 2: For each value of h, provide the uncertainty by propagating the error determined for λ min (as a result of the fitting described above). Use these uncertainties to calculate the weighted average of h, and the uncertainty of the weighted average. Report these calculations in your report. Method 3: Plot λ min as a function of 1/U and fit the data to obtain a value for h and its uncertainty from the slope. 6. Questions: 6.1. Describe how the bremsstrahlung radiation changes as a function of the tube high voltage U. What can you say about the minimum wavelength λ min? Explain this behavior (see ref. [1] chap. 1). 6.2. Why do the characteristic lines K α and K β only appear above a specific voltage? 6.3 Calculate the mean, standard deviation, and standard deviation of the mean for your values of λ and put them in your write up. Using your estimates for uncertainties in these values also calculate the weighted average and weighted uncertainty of your values (see above). Discuss the possible sources of error in your measurement. 6.4. Compare your values for the characteristic wavelengths to the literature values of λ(k α ) = 71.08 pm, and λ(k β ) = 63.10 pm [4]. If these values deviate explain how you could make the measurement more accurate. 6.5. In your write up, explain how the Bragg equation is used to convert angles to wavelength. What would happen to the bremsstrahlung continuum (minimum wavelength λ min ) and what would happen to the characteristic radiation of the Mo anode if the crystal would have a smaller or larger atomic spacing? Would that alter the values for λ(k α ) and λ(k β )? Why or why not? 6.6. Compare the values for h and its uncertainty obtained in part 2 of your analysis using the three methods described above. Do they deviate significantly from one another (i.e., more than one standard deviation). Discuss. 6.7. How do the values of Planck s constant h obtained from your analysis compare with the literature value of h = 6.626 10 34 Js? How could you make the measurement more accurate. 7. References: 7
[1] Cullity, ELEMENTS OF X-RAY DIFFRACTION ADDISON-WESLEY PUBLISHING COMPANY, INC. READING, MASSACHUSETTS 1956 [2] Handbook of Chemistry and Physics, 52nd Edition (1971 72), The Chemical Rubber Company, Cleveland, Ohio, USA. Where is this reference cited? [3] W. Duane, H. H. Palmer, and C.-S. Yeh, A remeasurement of the radiation constant, h, by means of x-rays, Proceedings of the National Academy of Sciences of the United States of America, vol. 7, no. 8, pp. 237 242, 1921. [4] C. M. Lederer and V. S. Shirley, Table of Isotopes, 7 th Edition, 1978, John Wiley & Sons, Inc., New York, USA. 8