experimental physics - extrapolation of the s constant. in this experiment, an undergraduate...

Mr. Benjamen P. Reed (110108461) [email protected]

Experimental Physics: Extrapolation of the Bremsstrahlung continuum to determine Planck’s constant

Mr. Benjamen P. Reed (110108461)

[email protected]

Aberystwyth University

May 5, 2013

Abstract

Crystals of Sodium Chloride and Lithium Fluoride were irradiated with Bremsstrahlung x-rays and the subsequent scattering pattern was recorded to plot the Bremsstrahlung continuum curves with respect to scattering angle. The continuums were then extrapolated using Gaussian and linear regression techniques, to determine the minimum wavelength of the incident radiation and hence calculate the value of Planck’s constant. The Sodium Chloride results provided a Planck’s constant of (7.87±0.10)×10-34 Js and the Lithium Fluoride results gave a value of (9.10±0.08)×10-

34 Js. These are precise but inaccurate values, given that the accepted value of Planck’s constant is 6.626×10-34 Js. Possible sources of error were also discussed as well as the advantages of using an automated experiment in conjunction with LabVIEW.

Mr. Benjamen P. Reed (110108461) [email protected] 1

Table of Contents

Abstract…………………………………………………………………...Title Page

I. Introduction & Theory…………………………………………………..2 – 3

II. Experimental Procedure………………………………………………...3 – 5

III. Data Analysis…………………………………………………………....5 – 9

IV. Discussion………………………………………………………………….10

V. Conclusions………………………………………………………………...11

Acknowledgements……………………………………………………………..…11

References……………………………………………………………………...…12

A Error Analysis……………………………………………………………...13


I. Introduction & Theory

In the laboratory, X-rays are usually generated by means of an X-ray tube. Such a device will heat a filament to produce free electrons, which are then accelerated through vacuum conditions by a uniform electric field with a potential difference in the range of 20 – 60kV. These electrons will accelerate toward and collide with a metal anode (e.g. Copper), in the process creating and emitting X-rays of a continuous spectrum. This process is immensely inefficient, with as much as 99% of the electrons kinetic energy being dissipated as thermal energy in the metal target [1]. In this situation, X-ray radiation is emitted from the anode in two ways: the sudden retardation or acceleration of the incoming electrons, and the de-excitation of electrons in the metal target’s atoms. In the latter case, if an incoming electron has enough kinetic energy, and collides with another electron in lowest energy level of one of the metal’s atoms, the bound electron is given enough energy to free itself from that atom. Electrons in the higher energy levels (i.e. shells) will then drop down to fill the vacancy, emitting a photon with energy equal to the difference of potential energy between the two levels. As electron energy levels are discrete values, the energies of the emitted photons will also have discrete energies [1, 2]. X-rays emitted in this fashion are referred to as ‘Characteristic X-rays’. The most probable characteristic x-rays are the Kα and Kβ energies, which occur when electrons drop down into the lowest energy level, also known as the K-shell, hence the name. In the former case, incoming electrons are deflected and accelerated, thus reducing their kinetic energy. Due to the conservation of energy, the kinetic energy lost is emitted as a photon, usually in the energy range of X-rays (figure 1). The possible range of energies is dependant on the kinetic energy of the electrons, the amount they are accelerated and the angle they are deflected. Due to the continuous nature of these variables, the X-rays emitted this way can have a continuum of energies. This method of emission is called ‘Bremsstrahlung’, which is German for ‘Braking radiation’, so named due to the reduction of the electron’s kinetic energies. The Bremsstrahlung continuum refers to the relative intensity of the radiation against either the wavelength or energy of the radiation, or angle of Bragg’s deflection (which is proportional to the wavelength). Figure 2 demonstrates the ‘whale back’ shape of the Bremsstrahlung continuum. The intensities of the photons trail off, as their wavelengths increase and the whale back will intercept the x-axis at a minimum wavelength corresponding the maximum amount of kinetic energy lost [3]. This Emax can be used to determine the physical constant, Planck’s constant, which has an accepted value of 6.626×10-34 Joule-seconds [4]. A basic method of ascertaining this constant is by the use of x-ray crystallography; Bremsstrahlung X-rays are emitted towards transparent crystals

Figure 1 - Graphical representation of Bremsstrahlung emission. The deceleration of the incoming electron

causes a photon of x-ray radiation to be emitted, with energy equal to the kinetic energy lost.

Figure 2 - The 'whale-back' curve, characteristic of the Bremsstrahlung continuum.


with lattice atomic arrangements, and the scattering of the radiation is observed to determine to angles, wavelengths or energies at which certain events occur. The wavelength of the incident X-rays dictates the scattering angle and so in theory, individual photons can be isolated. This is particularly useful for ascertaining the minimum wavelength of the incident radiation and hence, Planck’s constant.

In this experiment, an undergraduate laboratory X-ray scattering unit known as the Tel-X-ometer 580 was used to find minimum wavelength and hence calculate a value for Planck’s constant. Due to the angular limitations of this apparatus, the subsequent Bremsstrahlung continuum achieved had to be extrapolated to the x-axis intercept, hence the name of this report. Details of the experimental procedure will be described and the extrapolation techniques will be outlined. The practical undertaking was the collaborative effort of several authors, whom have been acknowledged at the end of this report.

II. Experimental Procedure

For this experiment, a Tel-X-ometer 580 X-ray scattering unit was used to find the minimum wavelength of the incident radiation. Developed by Tel-Atomic Inc. (formerly Tetron Ltd.), it is a useful tool in undergraduate laboratories for various X-ray studies, from creating Röntgen’s observations, to emission and absorption, to x-ray crystallography. The unit used in this experiment belongs to the Institute of Mathematics and Physical Sciences at Aberystwyth University, and dates back to about 1974. Much of the experimental procedure was taken from the Tel-X-ometer’s instruction manual that came with the unit. It outlined the general operating procedures of the unit as well as the theory and experimental methodology of well over a dozen investigations. The experiment conducted by the author and their collaborators is described in full in section D15 of this manual [5]. This experimental procedure was initially followed to the letter, but it was found to be inaccurate and imprecise. A scalar was used in conjunction with a stopwatch timer, whose precision was determined by human reaction time, about 100ms. At extremely high-count rates, for example 1000 counts-second-1, this leads to a minimum precision of 100 counts-second-1. Even at low count rates, the precision of the count readings will rarely fall below 10%, which is a large error to say the least. Experiments were conducted manually with the scalar, and the resulting Bremsstrahlung continuums were not well defined and contained anomalies. It was decided that automating the experiment and removing human error from the equation, was the best approach and so some time was invested in creating a LabVIEW virtual instrument that would run the experiment semi-automatically. Figure 3 illustrates the basic set up of the automated experiment.

MiniLab 1008

3 12 60 120 Run Stop

AutoScan Control

30kV

20kV

2θ

Range counts

GM Tube supply

To LabVIEW AutoScan driver unit

G-M Tube

Ratemeter

AutoScan control Crystal post X-ray tube with

Cu anode

X-ray tube voltage

Carriage arm Slave plate

Collimating slits

Figure 3 - Basic schematic of the automated experiment


The VI was designed around the chart output of a ratemeter, which was measured using a MiniLab 1008 pod. The ratemeter would pick up a voltage from a Geiger–Müller tube and display a counts-per-second reading on an analogue needle chart. The voltage used to move the needle ranged from 0 – 1V and could be measured by connecting a multimeter or MiniLab into the ratemeter chart output. This voltage reading was then sent to a computer, loaded into the LabVIEW VI and converted into a count-rate reading.

Another source of human error was the adjustment of the Tel-X-ometer’s carriage arm. The minimum increment that could be used was 1 arc minute, but using this increment would have taken a considerable amount of time. The human percentage error of the timing would have gotten worse in this time frame as fatigue increased and concentration dwindled. Hence, the smallest increment used was 0.5o. To reduce this increment but maintain a high level of precision, an AutoScan driver was attached to the carriage arm. This would move the carriage arm around the Tel-X-ometer in 0.013o pulses exactly, increasing the precision of the experiment and reducing any possible error. The AutoScan driver unit was connected to a control box that allowed differing pulse rates to be selected (3, 12, 60 and 120 pulses per second), as well as the stop/start controls. The control box was also connected to the MiniLab pod, which was measuring the pulses from the AutoScan driver. This information was then sent to the LabVIEW VI, and thus gave users of the apparatus the number of pulses that had allotted since the AutoScan unit was set to ‘Run’. The VI was also programmed to start taking count values from the ratemeter as soon as ‘Run’ was selected on the AutoScan control. The ratemeter would on average take 3 count readings per pulse (when running at 3pps). Both the pulses and count readings were saved automatically into a comma separated variable file upon conclusion of an experimental run.

To reduce the count rate to a level where the range-counts variable on the ratemeter (and LabVIEW VI) could remain constant, collimating slits were used. Collimating slits are designed to provide a narrow parallel beam of x-rays, mainly to reduce the number of photons (which could lead to a saturated G-M tube) and prevent interference within the beam. A 1mm collimating slit was placed into position 4 of an auxiliary carriage arm that was affixed to the X-ray tube, and a 3mm collimating slit was placed at position 14 on the main carriage arm. The 1mm slit’s position was stated in the Tel-X-ometer’s manual, whereas the 3mm slit’s position was chosen because it provided the most stable and usable data.

Two crystals, Sodium Chloride (NaCl) and Lithium Fluoride (LiF), were used in the experiment to provide different Bremsstrahlung continuums, with the hope that they could converge on the same value of Planck’s constant. These were placed into the crystal post, standing up with the short edges parallel to the beam path. The electron acceleration voltage in the X-ray could also be altered from 30kV to 20kV, using the switch on the face of the Tel-X-ometer (see figure 3). Despite this option, only the 30kV value was used because the 20kV setting provided no data apart from the background radiation count.

With the necessary equipment ready, data could now be taken. Before collecting data, the Tel-X-ometer was switched on and left for at least 30 minutes. This allowed the filament inside the X-ray tube to heat up to a point sufficient for electrons to be ‘boiled off’. For each run, the relevant crystal was chosen and placed into the crystal post, and the protective lid of the Tel-X-ometer was closed and centred. If the lid was not centred, microswitches inside the Tel-X-ometer prevented the X-rays being turned on. Next, the carriage arm was returned to its starting angle, which in all the experiments was 12o. A smaller starting angle could not be used because the Tel-X-ometers lid locks at the front of the unit; hence the lock prevented the carriage from reaching 0o. This is why the resulting Bremsstrahlung continuum graphs had to be extrapolated to the minimum wavelength. The G-M tube supply was set to 400V and the range counts setting on the ratemeter was set to 250cps. With the carriage arm in position, the timing switch was set to it’s maximum, and the X-rays were turned on, using the button on the front of the unit. At this point,


the LabVIEW VI was opened and the relevant information was entered in (i.e. file name, path directory, range counts setting etc). With the experiment primed and ready, the AutoScan control was set to 3pps and then ‘Run’ was selected. The carriage arm would now move in 0.013o around the Tel-X-ometer, and the G-M tube would simultaneously take count readings. When the carriage arm reached the final angle (25o for NaCl, 32o for LiF), the VI and AutoScan driver would be terminated, and the data was subsequent saved for data analysis. This process was repeated three times for each crystal.

III. Data Analysis

Data from the experiments was exported by LabVIEW into comma separated variable files and then imported into the data analysis package, SciLab, as element vectors. In SciLab the AutoScan pulse values were converted into degrees using the pulse-to-degree constant stated in the previous section, and by adding each value onto the starting angle of 12o. The count rate values were then plotted against the scattering angle to show the Bremsstrahlung continuum with respect to θ . Figures 4 and 5 show the graphs of count-rate intensity against scattering angle for Sodium Chloride and Lithium Fluoride respectively.

Figure 4 - Graph displaying Sodium Chloride data.


Figure 5 – Graph displaying Lithium Fluoride data

The Lithium Fluoride results were very consistent, whereas the Sodium Chloride results varied in average intensity from run-to-run. The intensity results of the experiments for each crystal were then averaged to produce a single set of data each, representative of all the runs conducted on that crystal. These averaged vectors of intensity were then plotted against their respective angles. Figures 6 and 7 show the average data plots for NaCl and LiF.


Figure 6 - Graph showing all three Sodium Chloride results and the averaged data set.

Figure 7 – Graph showing all three Lithium Fluoride results and the averaged data set.


As discussed in section I, the Bremsstrahlung continuum has an asymmetric ‘whale-back’ shape, which makes it difficult to fit a curve to it for extrapolation. It was reasoned that since the only section of the graphs really needed for the extrapolation, were the angles between the minimum angle and the whale-back peak, a symmetric curve could be used instead, such as a Gaussian distribution. Using SciLab, a Gaussian curve fitting function was created and a curve was plotted that matched the NaCl average data on the left hand side of the whale-back. This revealed an intercept with the x-axis at 9.59o for Sodium Chloride (see figure 8). Due to the 2:1 ratio between the carriage arm and slave plate, the actual minimum angle is half the intercept value, i.e. 4.80o.

Figure 8 - Graph showing the intercept of the Gaussian curve fitted to the Sodium Chloride average data.

The Lithium Fluoride average data revealed a similar whale-back curve, but also a strikingly linear section between 16.25o and 18.50o. It was decided, that a linear regression should be used on this section to determine the intercept, if anything for demonstration purposes. In SciLab, a weighted regression was performed and a regression line was plotted through the average LiF data. An intercept of 15.53o was determined using this method (see figure 9). Due to the 2:1 ratio of the carriage arm and slave plate, the actual minimum angle is half the intercept, i.e. 7.77o.


Figure 9 – Graph showing the intercept of the weighted regression line for Lithium Fluoride average data.

From these intercepts, the minimum wavelength of the incident radiation, and hence the value of Planck’s constant can be calculated. The minimum wavelength of the x-rays can be determined by using Bragg’s Law, which is described by the equation…

nλ = 2d sinθ

...where n is the peak order, λ is the wavelength of the incident radiation, d is the atomic lattice spacing of the crystal being used, and θ is the angle of deflection [6]. The value for Planck’s constant is thus calculated by equating the gain in kinetic energy of the X-ray tube’s electrons, to the Planck’s equation of discrete energies in quantized radiation packets. Such that…

Ve = hcλ

…rearranged gives…

h = Veλc

…where V is the electron acceleration voltage, e is the elementary charge, c is the speed of light in a vacuum, and h is Planck’s constant. Using these equations, two values of Planck’s constant were calculated using the values 4.80o and 7.77o from the NaCl and LiF data respectively. For Sodium Chloride, the calculated value of Planck’s constant was (7.87±0.10)×10-34 Js, and for Lithium Fluoride, the calculated value was (9.10±0.08)×10-34 Js.


IV. Discussion

At the conclusion of the experiment, two values of Planck’s constant have been determined. The Sodium Chloride results provided a more accurate value in comparison to the Lithium Fluoride results, despite large discrepancies in the count-rate from run-to-run. Two possible reasons for this were discussed. First, the NaCl crystal itself was in less-than-perfect condition. It was cloudy throughout and it had many marks and scratches on it’s surface. Secondly, the X-ray tube may still have been warming up, so the amount of electrons hitting the Copper anode may have been slightly more each run, thus causing the apparent increase in the count rate between experimental runs. But even with these crystal imperfections and possible lag-time with the X-ray tube, the NaCl crystal did appear to provide the more accurate value for Planck’s constant at (7.87±0.10)×10-34 Js. The Lithium Fluoride crystal, despite being a better quality, gave the more inaccurate value of (9.10±0.08)×10-34 Js. A full error analysis is detailed in appendix A.

Both the errors on the two values of Planck’s constant have turned out to be in the 10-36 Joule-second region, which is very precise given the extremely small value of Planck’s constant. These errors were compared with the results from the manual experimentation, and found to be far more precise. The use of LabVIEW and the AutoScan driver unit, has drastically improved the precision of the entire experiment and it allowed more time to be spent on the data and error analysis.

The calculated values of Planck’s constant from the experimental data are, on average, 23% larger than accepted value. Such a large discrepancy appears to be systematic error propagation, which may have come about from an incorrect constant value. As the speed of light in a vacuum and the elementary charge are accepted and well-known constants, the only other constant used to calculate h, is the acceleration voltage of the X-ray tube’s electrons. Upon doing some extra research, a laboratory report was found written by an undergraduate in 2003 using the same Tel-X-ometer model [8]. In this report, entitled “X-ray Diffraction and X-ray Absorption With an Extension of Measuring Planck’s Constant”, the author stated that his calculated values of Planck’s constant were higher than expected due to reduced electron acceleration voltage. Upon measuring the actual acceleration voltage, the author discovered that it was at least 10kV lower than stated on the Tel-X-ometer. This would have caused a shift in the Bremsstrahlung continuum, which when used with an incorrect value of V, would have given an inaccurate value for Planck’s constant. This also explains the lack of data from the 20kV setting during the manual experiment.

Taking this into account, inquiries were made to IMAPS technicians about measuring the electron acceleration voltage of the Tel-X-ometer. It was learned that measuring the Tel-X-ometer’s internal voltage would require disassembling either the unit or the X-ray tube. Disassembling the unit would have taken a long time given that only the technicians are allowed to undertake such a task, and would have to take time out of their schedules to do so. In regards to the X-ray tube, once the protective glass is removed, a series of safety switches trigger and prevent the Tel-X-ometer from producing X-rays. Reassembling this component requires all the safety switches to be reset and such a process is time-consuming also. It was decided to simply acknowledge this possible systematic error and follow it no further on this occasion.

During the progress of this investigation, improvements were made to increase the precision of the experiment. Despite automating the experiment, the accuracy was subject to a large systematic error in the electron acceleration voltage. In future investigations, it would be worth asking a technician to check the internal X-ray tube voltage to ensure that the correct calculations are made.


V. Conclusions

An experiment has been conducted with a Tel-X-ometer X-ray studies unit to determine the value of Planck’s constant using extrapolations techniques on Bremsstrahlung continuums. Two values of (7.87±0.10)×10-34 Js and (9.10±0.08)×10-34 Js were ascertained using a Sodium Chloride and Lithium Fluoride crystal respectively.

Despite a high precision, the experimental results were inaccurate due to a possible reduction in the Tel-X-ometer’s X-ray tube voltage since commissioning. This would have shifted the calculated values of Planck’s constant, giving them a higher value than expected. Although this error could not be rectified in the time frame given, it is suggested that this is taken into consideration in future uses of the IMAPS Tel-X-ometers.

Acknowledgements

The author would like to acknowledge and thank the efforts and help provided by their collaborators: Miss. R. E. Cooper, Miss. J. K. Maddocks and Miss. C. J. Barratt. The author would also like to extend their gratitude to Dr. D. Langstaff and Mr. J. McMillan for much needed help with constructing the LabVIEW VI used in this experiment. And finally, the author would like to thank Mr. S. Fearn for their assistance in laboratory.


References

1. Generation of X-rays in an X-ray tube –

http://pd.chem.ucl.ac.uk/pdnn/inst1/xrays.htm (April 31, 2013) http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xtube.html (April 31, 2013)

2. Characteristic and Bremsstrahlung X-rays –

J. Hodak (2008), “X-ray Absorption and Emission”, BSAC Program, http://einstein.sc.mahidol.ac.th/~jose/pdf/Handout_X-ray_Abs_Fluor.pdf (April 31, 2013)

3. Interpretation of the Bremsstrahlung continuum –

Tel-Atomic Inc. (formerly: Teltron Ltd.) (1974), “The Production, Properties and uses of X-rays (Tel-X-ometer instruction manual”, pp.18. http://www.telatomic.com/x-ray/docs/TEL-X-Ometer_Manual.pdf (April 31, 2013)

4. Accepted value of Planck’s constant –

CODATA Value: Planck’s Constant. The NIST Reference on constants, units and uncertainties. US National Institute of Standard and Technology. http://physics.nist.gov/cgi-bin/cuu/Value?h|search_for=Planck%27s+constant (April 31, 2013)

5. Experimental Procedure notes –

Tel-Atomic Inc. (formerly: Teltron Ltd.) (1974), “The Production, Properties and uses of X-rays (Tel-X-ometer instruction manual”, pp.18 -19 http://www.telatomic.com/x-ray/docs/TEL-X-Ometer_Manual.pdf (May 1, 2013)

6. Bragg’s Law equation –

John W. Jewett, Jr. and Raymond A. Serway (2010) “Physics for Scientists and Engineers with Modern Physics, 8th Edition” Cengage Leaning, pp.1188. ISBN-13: 978-1-4390-4875-7. (May 3, 2013)

7. Planck’s relation equation –

John W. Jewett, Jr. and Raymond A. Serway (2010) “Physics for Scientists and Engineers with Modern Physics, 8th Edition” Cengage Leaning, pp.1126. ISBN-13: 978-1-4390-4875-7. (May 3, 2013)

8. Laboratory Report from Mr. B. Evans –

B. Evans (2003), “X-ray Diffraction and X-ray Absorption With an Extension of Measuring Planck’s Constant”, University of Exeter. http://reocities.com/benjaminevans82/Xray_Diffraction.pdf (May 4, 2013)

Figure 1: Graphical representation of Bremsstrahlung radiation - http://www4.nau.edu/microanalysis/Microprobe-SEM/Images/Bremsstrahlung.jpg Figure 2: Bremsstrahlung ‘Whale-back’ continuum – http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xrayc.html


A Error Analysis

The error analysis for this experiment was conducted by using the equations outlined below.

The error in the wavelength, σ lambda is found using…

σλ = 4d 2 cos2θ ×σθ2

…where d is the atomic lattice spacing of the NaCl and LiF crystals, θ is the scattering angle, and σ is the error on the sub-scripted value.

The error on Planck’s constant σh is found using…

σ h =V 2e2

c2×σλ

2

…where V is the X-ray tube voltage, e is the elementary charge and c is the speed of light in a vacuum. The error analysis shown here was performed using methods from I. G. Hughes and T. P. A. Hase’s publication, entitled “Measurements and Uncertainties: A Practical Guide to Modern Error Analysis”.

experimental physics - extrapolation of the s constant. in this experiment, an undergraduate...

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