Observation of relativistic corrections to Moseley's law at high atomic number
Duncan C. Wheeler, Emma Bingham, Michael Winer, Janet M. Conrad, Sean P. Robinson
OObservation of relativistic corrections to Moseley’s law at highatomic number
Duncan C. Wheeler, ∗ Emma Bingham, † MichaelWiner, Janet M. Conrad, and Sean P. Robinson
Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: December 18, 2018)
Abstract
Transitions between low-lying electron states in atoms of heavy elements lead to electromagneticradiation with discrete energies between about 0.1 keV and 100 keV (x rays) that are characteristicof the element. Moseley’s law — an empirical relation first described by Moseley in 1913 whichsupported predictions of the then-new Bohr model of atomic energy levels while simultaneouslyidentifying the integer atomic number Z as the measure of nuclear charge — predicts that theenergy of these characteristic x rays scales as Z . The foundational nature of Moseley’s experimenthas led to the popularity of Moseley’s law measurements in undergraduate advanced laboratoryphysics courses. We report here observations of deviations from Moseley’s law in the characteristic K α x-ray emission of 13 elements ranging from Z = 29 to Z = 92. While following the square-lawpredictions of the Bohr model fairly well at low Z , the deviations become larger with increasing Z (negligible probability of the Bohr model fitting data by a χ test). We find that relativistic modelsof atomic structure are necessary to fit the full range of atomic numbers observed (probability valueof 0 .
20 for the relativistic Bohr-Sommerfeld model). As has been argued by previous authors,measurements of the relativistic deviations from Moseley’s law are both pedagogically valuable atthe advanced laboratory level and accessible with modern but modest apparatus. Here, we showthat this pedagogical value can be be extended even further — to higher Z elements, where theeffects are more dramatically observable — using apparatus which is enhanced relative to moremodest versions, but nevertheless still accessible for many teaching laboratories. a r X i v : . [ phy s i c s . e d - ph ] D ec . INTRODUCTION Experimental tests of Moseley’s law using x-ray fluorescence spectroscopy appear com-monly in advanced undergraduate physics laboratory courses. Such experiments allow stu-dents to explore key supporting evidence for Bohr’s atomic theory while also introducingmodern precision spectroscopy techniques. However, experiments testing Moseley’s law havethe potential to teach undergraduates much more. Recent work by Soltis et al. shows thatwithin the precision of modern detectors, Moseley’s law becomes inaccurate at high atomicnumber and requires first-order relativistic corrections. By performing a Moseley’s law ex-periment, students receive the opportunity to identify limitations in a commonly taughtmodel, providing insight into the nature of experimental physics. In this paper, we buildupon the work of Soltis et al. by measuring elements with even higher atomic number whichdeviate even further from Moseley’s law. We show that the first-order relativistic approxi-mation used by Soltis et al. remains inaccurate for these heavier elements and find that amore exact relativistic Bohr-Sommerfeld model is required, providing more aspects of thephysics and modeling for students to explore. II. BACKGROUND
In 1913, H. G. J. Moseley experimentally measured the wavelengths of characteristicx rays from a series of elements. Using his data in conjunction with Bohr’s recent theorydescribing the hydrogen atom, Moseley proposed that the energy of the transition scalesquadratically with the atomic number Z . This quadratic relation, called Moseley’s law,formed some of the first observational evidence for a quantum theory of atomic structure.There are a number of ways to produce x rays in nature. They range from fluorescence tosynchrotron radiation to extreme blue-shifting of radio waves. In this experiment, we focuson the first of these. X-ray fluorescence typically occurs when an electron is knocked out ofa low-lying shell of a heavy element, leaving a hole in the electronic structure. This could bea consequence of bombardment by alpha rays, beta rays, gamma rays, or some more exoticprocess. The resulting hole is most often filled with an electron from a nearby higher shell,emitting a photon in the process.That electron leaves a new hole, which is then filled in much the same way, resulting2n a cascade of electrons between the quantized energy levels of the atom, each emittinga photon. The brightest line in this spectrum comes from electrons transitioning betweenenergy levels in the n = 2 and n = 1 shells, where n is the usual principal quantum number.These 2 → K α in Siegbahn notation. Bohr’s model predictsthat electrons in shell n have velocity Zαc/n , where Z is the atomic number, α is the finestructure constant (approximately 1 / ) and c is the speed of light. This implies that the K α transition has energy E Bohr K α = 38 m e c α ( Z − , (1)where Z has been replaced by Z − Z =79), the innermost electrons are moving at more than half the speed of light. Therefore, it ispossible that relativistic corrections may come into play. Soltis et al. included the first-orderperturbative relativistic corrections to Bohr’s model (that is, the power series expansion ofthe kinetic energy, spin-orbit coupling, and the Darwin term) and found E (1) K α = m e c (cid:18) α ( Z − + 15128 α ( Z − (cid:19) . (2)By using a relativistic Bohr-Sommerfeld approximation, we will find model fits to our datathat improve upon the perturbation methods used by Soltis et al . This can help studentsunderstand the intricacies behind combining quantum mechanical theories with relativistictheories. III. RELATIVISTIC BOHR-SOMMERFELD APPROXIMATION
To better understand these relativistic corrections, we utilize the Bohr-Sommerfeld ap-proximation. The Bohr-Sommerfeld quantization condition is a semiclassical rule that saysin any closed orbit in a quantum system (cid:82) orbit pdx = 2 πn (cid:126) for an integer n , where p is thesystem momentum, x the coordinate, and (cid:126) = h/ π is the reduced Planck’s constant. Ifmomentum can be thought of as the derivative of a quantum wavefunction’s phase, thenthis condition says that closed orbits are standing waves where the phase is the same at thebeginning and end, as shown in Fig. 1. For circular orbits, the condition requires that an-gular momentum be L = n (cid:126) . This approximation can be combined with classical mechanics3IG. 1: The n = 2, 3, 4, and 5 orbitals in the Bohr-Sommerfeld picture. The sinusoidalradial wiggles do not represent variations in orbital radius, but rather the phase of thewave.to derive Bohr’s atomic theory.In the relativistic case, we still have L = n (cid:126) . However, L is now γm e rv instead of m e rv ,where m e is the mass of the electron, v is the velocity, r is the radius of the orbit, and theshorthand γ ≡ / (cid:112) − v /c indicates the Lorentz gamma factor. The result is E n = m e c (cid:115) − (cid:18) α ( Z − n (cid:19) , (3)where E n is the total energy of the system’s n th energy eigenstate, including the electronmass-energy. The x-ray energy is then E BS K α = E − E = m e c (cid:16)(cid:112) − α ( Z − / − (cid:112) − α ( Z − (cid:17) . (4)Bohr-Sommerfeld calculations are not exact — they are approximations to more accuratewave mechanics calculations. Remarkably, however, they do match the result of the exactwave mechanical calculation in the case of circular orbits. In the nonrelativistic case, thecorrect wave equation to use would be the Schrodinger’s equation. In the relativistic case,the correct one would be Dirac’s equation. However, in both cases, the Bohr-Sommerfeld4pproximations are much more accessible for student understanding than solutions to thefull wave equations. IV. APPARATUS AND PROCEDURE
To test Moseley’s law, we measure the energy of K α radiation for a variety of elements. Todo this, we expose selected elemental samples to radiation from high-energy sources, inducingthe emission of characteristic x rays. The x rays are then measured by a high resolutionenergy detector, generating a counting signal that is recorded as a spectral histogram by amultichannel analyzer (MCA), as in Fig. 2. Once the system’s energy sensitivity is calibrated,we use the MCA’s output to determine the energy of the x rays which hit the detector byidentifying MCA histogram channels with distinct peaks in the counting rate. A. Detector and calibration
Measurements are performed a Canberra model BE2020 broad energy solid state x-ray detector system. The detector itself is a crystal of p-n doped germanium, biased at − Co,
Ba, and
Cs,5 ermanium detectorSource (
Am or
Ba)MCAPre-amplifierAmplifierDetector bias supply Target
FIG. 2: Block diagram of the apparatus and signal chain. The detector contains ap-ndoped, reverse-biased, nitrogen-cooled germanium crystal that generates pulses of electricalcharge proportional to the energy deposited by each incident x ray. These pulses areintegrated to a voltage by the pre-amplifier, amplified, and counted in amplitude bins bythe MCA.each with several micro-Curies of activity; see Fig. 3. These isotopes were chosen becausethey emitted sharp gamma radiation peaks across the range from near 10 keV to just over100 keV, the same range as the x-ray energies to be measured. We collected data onthese calibration spectra until we saw clean peaks, typically after a few minutes. We thencalibrated the MCA channels to those peaks via a linear fit using χ minimization. The exactenergy per channel depended on amplifier gain and detector bias settings that could varybetween experimental runs, so calibration was repeated each time. Typical values rangedbetween 0.05–0.10 keV per channel. 6IG. 3: An example MCA calibration spectrum of an isotope ( Co in this case) withseveral well-known gamma ray emission energies. The data shown is a histogram of x-raycounts versus MCA channel number after five minutes of sampling.
B. X ray sources and targets
We used two high-energy radiation sources,
Am (10 mCi) and
Ba (7 µ Ci), to generatex rays from pure samples of various elements. The
Am source is part of a variable energyx-ray source assembly. It emits alpha particles with energies near 5.48 MeV and gammarays near 59.5 keV as it decays to
Np. The alpha and gamma radiation bombard one ofsix metals in a rotatable wheel, causing the metals to fluoresce characteristic x rays whichexit the assembly as a beam (see Fig. 4). The experimenter can rotate the wheel to selectdifferent metals, generating x rays from copper, rubidium, molybdenum, silver, barium, andterbium.In addition to this variable x-ray source, we also used a
Ba source that emits x raysat around 80 keV (as well as gamma rays with a few hundred keV each) to bombardprepared metal targets of high purity. We used this technique to measure tantalum, tungsten,platinum, gold, and lead — all of which have x-ray lines below 80 keV. In this setup, thegold and platinum targets are foils held together by kapton tape. (We also bombarded apure ball of kapton tape with radiation from the Ba source in order to ensure that thetape did not affect the measured peaks.)The final sample was uranium. The uranium sample came not from a pure elementalsample as above, but rather from a red-orange glazed Fiesta brand ceramic dinnerware7 a) Front view of the Amersham AMC.2084x-ray source. (b) Side view of the Amersham AMC.2084 x-raysource.
FIG. 4: The Amersham AMC.2084 10 mCi
Am variable energy x-ray source containssix different metals on a rotatable wheel, each of which has its own characteristic x-rayspectrum. Radiation from the source, in the form of 5.48 MeV alpha particles and59.5 keV gamma rays, bombards the metal samples, causing them to fluoresce x rays whichexit the assembly towards a detector.plate. The bright red-orange glaze (branded “Fiesta red”) on these ceramics produced inthe years 1936–1943 contains natural uranium oxide, while those produced in the years 1959–1972 contain the oxide of isotopically depleted uranium. Whether the present sample isof depleted or natural uranium has not been determined. Regardless, the sample’s ownradioactivity is enough to induce x-ray fluorescence.The sample was left in the detector for three days in order to determine the peak towithin one channel. This is an important data point because of its high atomic number( Z =92), which gives a larger deviation from nonrelativistic theories. It is worth noting thatthere are several additional sources of error in the case of the uranium sample. These willbe discussed in the next section.Note the wide range of atomic numbers and the substantial gap between the secondheaviest element, lead, and the heaviest element, uranium. The relativistic model predictsthat the innermost electrons in copper are moving with velocity 0 . c ( γ = 1 . . c ( γ = 1 . K α x-ray energies for each element tested. Each peak energy has astatistical uncertainty of about 0.06 keV. Atomic number Element Peak (keV)29 Cu 8.137 Rb 13.4542 Mo 17.4647 Ag 22.1456 Ba 32.265 Tb 44.473 Ta 57.574 W 59.378 Pt 66.779 Au 68.682 Pb 74.992 U 98.26
V. DATA AND ANALYSIS
Once the system was calibrated, we used it to measure x rays produced by the methodsdescribed above. Table I shows the data collected for each element. Each point has astatistical error of about 0.06 keV due to the resolution of the peaks on the MCA.In addition to statistical errors mentioned above, the uranium plate introduces severalsources of possible error. First, the plate contains an oxide rather than elemental uranium.Since the experiment is mostly concerned with inner shell phenomena, we need to know ifthe valence-shell bonding would matter. Since the valence electrons are several times furtheraway then the nucleus and have far less effective charge, we estimate that they cannot inducean error of more than one percent.Another error may be introduced by the spectrum’s many peaks, as seen in Fig. 5a. Theplate contains uranium, a host of decay products, and many other fluorescing materials. Itis possible that there was another peak overlapping with that of the uranium. However, the9 a) The many peaks shown on the MCA fromthe uranium-containing Fiesta ceramic plate. (b) A close up of the peak used to determine theuranium K α energy. FIG. 5peak, shown in Fig. 5b, is narrow, with a full width at half maximum of about 2.3 keV.The global maximum of counts falls between two adjacent channels, but the counts for thesechannels are higher than the counts for either of the next lower or higher energy channelswith Poisson-based probability of 1 . × − and 2 . × − respectively. Because of this,we believe the error introduced by possible overlap is at most one channel. This lineshape-model-agnostic analysis takes advantage of the high number of counts in the integration toavoid additional errors which could be introduced by a theory bias.Fig. 6a shows a linear fit between the atomic number and the square root of the energy,testing Moseley’s original law and the Bohr model of Eq. (1). Our fit suggests an electronmass of 593 ± stat ± sys keV/ c , which is more than seven standard deviations away fromvalues found in other experiments. Visually, the line appears to be a good fit, but the χ value is over 5880 with just 10 degrees of freedom — corresponding to a probability of lessthan 10 − , which we consider negligible — suggesting a bad fit and providing a valuablelearning opportunity.To verify that the theory behind Moseley’s law is inaccurate, we plot the residuals to thislinear fit in Fig. 6b. The clear pattern in the residual supports our belief that a new modelis required.We next test the model described by Soltis et al. , that is, the first-order perturbative10 a) Linear fit suggested by Moseley’s law. (b) Residuals of data to the Moseley’s law fit. FIG. 6: The visually apparent good fit in Fig. 6a is shown to be spurious by the obviouslynonrandom trend in the residuals of Fig. 6b.FIG. 7: Residuals of data to fit suggested by Soltis et al . An obvious nonlinear trend isapparent, suggesting the model does not represent the data well.corrections to the semiclassical Bohr model, as in Eq. (2). We again see a strong trend inthe residuals, shown in Fig. 7, and find a high χ value: 1077 with 10 degrees of freedom,corresponding to a probability value of about 10 − . While this is clearly an improved fitover Moseley’s law, the large χ value provides motivation to search for a still better model.We finally use the fit presented in Eq. 4 to find m e c and test our claim that the fullyrelativistic Bohr-Sommerfeld approximation provides a more accurate model. The residuals11IG. 8: Residuals of data to relativistic Bohr-Sommerfeld approximation fit. No obvioustrend is apparent.from this fit are shown in Fig. 8, which indicates no significant trend. This fit has a χ value of 13.4 and 10 degrees of freedom, consistent with random noise, indicating thatthe fully relativistic theory is the most accurate of the models tested. The probabilityvalue of the relativistic hypothesis is 0.20, so it is not rejected. In addition, we find that m e c = 502 ± VI. CONCLUSIONS
This paper examines a correction to Moseley’s law that accounts for relativistic effects. Itshows that the corrected version better explains measured data than both the nonrelativisticversion and alternative theories tested in the past. The nonrelativistic version of the lawgives an electron mass inconsistent with the other experiments while the relativistic versionagrees within 1.5 standard deviations.All of the samples measured in this experiment are metals. It would be interesting toinclude some heavy elements which are not metals — for example, an iodine tablet. Thiswould allow us to determine whether different categories of elements had different x-raybehavior.This experiment is a valuable teaching opportunity, as it requires experimenters to lookat residual plots to clearly reveal incorrect models. In addition, the theory presented here12s fundamentally both relativistic and quantum. That means that this experiment not onlydemonstrates that a well-known model, Moseley’s law, is inaccurate, but it also tests bothrelativity and quantum mechanics at the same time. The combined theory is accurate to thelimit of the experimental apparatus and allows students to explore the process of discoveringnew explanations as data becomes more accurate.
ACKNOWLEDGMENTS
This experiment was collaborative across both space and time, and we would like to thankall involved both in making the original observation and generating the final results writtenup here. We would especially like to thank the MIT Junior Lab staff for lending their time,expertise, and dinnerware to the experiment. ∗ [email protected] † [email protected] H.G.J. Moseley, “The high-frequency spectra of the elements,” Philos. Mag. Series 6 (156),1024–1034 (1913). H.G.J. Moseley, “The high-frequency spectra of the elements. Part II,” Philos. Mag. Series 6 (160), 703–713 (1914). Niels Bohr, “On the Constitution of Atoms and Molecules,” Philos. Mag. Series 6 (151),1–25 (1913). Tomaz Soltis, Lorcan M. Folan, and Waleed Eltareb, “One hundred years of Moseley’s law: Anundergraduate experiment with relativistic effects,” Am. J. Phys. , 352–358 (2017). C. Patrignani et al. (Particle Data Group), “Review of Particle Physics,” Chin. Phys. C (10),1–1808 (2016). David Kraft, “Relativistic Corrections to the Bohr Model of the Atom,” Am. J. Phys. (10),837–839 (1974). Canberra Industries, Inc.,
Germanium Detectors: User’s Manual (2003). The Radiochemical Centre (Amersham),
Variable energy X-ray source: code AMC.2084 , Datasheet 11196 (June 1975). V.P. Chechev and N.K. Kuzmenko, “133-Ba,” in
Table de radionucleides , Decay Data EvaluationProject (2016). The Hall China Company, “Color history”,