Basic XRD Student Notes PDF

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QuantumPhysics StudentNotes

SKILLS G AINED

A SSUMED K NOWLEDGE

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Ionising radiation safety

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PHYS 1121/1131

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Radiation counters

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PHYS 1221/1231

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Error analysis

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1 st year math

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Data visualisation and presentation.

1 Experimental aim In this experiment we will measure the distance between atoms in simple single crystals using the diffraction of X-rays off the crystal planes formed by ordered atom arrangement. In this experiment you are required to do the following: 1. Complete the tasks outlined in Section 3 to fulfill the aim outlined in each section. 2. During each sub-experiment, setup the relevant experimental parameters and justify why you chose these settings. 3. Collect enough data to analyse the data and answer the questions. 4. Fit the data, analyse the errors and their sources. Justify your results. 5. Produce a report based on your experiment. Note: Before starting the lab you will need to complete and be ready to show pre-work At the end of your experiment you will use your prework answers together with your dataset and analysis to produce and submit an experimental report.

2 Background The discovery of X-rays by Röntgen in 1895 revolutionized medicine and provided a source of electromagnetic radiation with wavelengths much shorter than those of visible light. The discovery won Röntgen the Nobel Prize in 1901. It was soon discovered that X-rays can interact with objects of atomic dimensions and thus provide a powerful technique for the analysis of crystal structures. William Henry Bragg and William Lawrence Bragg were awarded the Nobel Prize in Physics in 1915 for the first experiments on x-ray diffraction as a method for determining crystal structure. X-ray diffraction has since played an important role in biology. Rosalind Franklin’s X-ray diffraction studies of DNA were vital in determining DNA’s double helix structure, research that won Crick, Watson and Wilkins the Nobel Prize in Physiology/Medicine in 1962.1 X-ray diffraction now plays a vital role in determining the structure of everything from simple crystals to complex biomolecules in fields from materials science to biology. X-ray diffraction has evolved significantly from the bench-top set-up you will use. It is now commonly 1

Many, myself included, would say somewhat unfairly, see: http://www.pbs.org/wgbh/nova/tech/rosalind-franklin-legacy.html

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performed using large Synchrotron facilities around the world (ESRF in France, HASYLAB in Germany, APS in USA, SPring-8 in Japan and the Australian Synchrotron in Melbourne) which provide high energy and high brightness X-rays for maximal resolution.

2.1 Theoretical background 2.1.1

X-ray Emission

A simple schematic of a X-ray tube appears in Figure 1. A low voltage power supply heats a filament, which ejects electrons by thermionic emission (electrons use thermal energy to escape the metal. The electrons are then accelerated by an electric field established by placing a potential difference V between the filament, which serves as the cathode, and a copper target, which serves as the anode. The potential difference V is large, of order 2050 kV accelerating the electrons to high velocity. When these electrons hit the copper target, they are rapidly decelerated. Deceleration of a charged particle produces electromagnetic waves, in this case X-rays.

Figure 1: Sketch of X-ray tube in operation.

The highest possible frequency of emitted radiation occurs when all the electron’s kinetic energy is converted into a single photon. In this case the photon frequency  is:    where e is the electron charge, h is Planck’s constant and V is the accelerating potential difference. However some of the electron’s kinetic energy is usually lost and consequently the emitted photons will have a lower than maximum frequency. The X-rays produced by this mechanism are called ‘Bremsstrahlung’ (German for ‘braking radiation’). The Bremsstrahlung spectrum is a smooth background curve depicted in Figure 2.

Figure 2: Idealized X-ray spectrum for Copper featuring a smooth background due to bremsstrahlung radiation and a pair of intense peaks K and K  due to the Auger effect.

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X-rays are also produced by another mechanism known as the Auger effect, discovered by Lise Meitner, but named after Pierre Auger who independently discovered it a year later (for those who can read German: doi: 10.1007/BF01326962). The Auger effect is a two-step process. First, an inner shell electron is knocked out of the atom, either by a highly energetic incoming electron (as here) or a high energy photon. Second, an outer-shell electron ‘drops down’ to fill the inner shell vacancy, dumping its excess energy as an X-ray. As you’ll know from your quantum mechanics lectures, the transition energy is a very precise amount and it will always be converted into a single photon with exactly that amount of energy. Hence the very sharp peaks in the emission spectrum in Fig. 2. As Fig. 3 indicates, there are two processes that commonly occur here: one has the inner shell vacancy filled from the L orbitals, and another where it is filled from the M orbitals. This gives the two lines you see: K  and K . Note that the emission wavelength for these is specific to the composition of the target. If you use molybdenum instead of copper, for example, the bremsstrahlung peak will stay roughly the same, but the ‘Auger peaks’ will shift in wavelength.

Figure 3: Atomic electron transitions responsible for the Auger emission peaks K and K in Fig. 2.

Figure 2 shows these two emission peaks K and K added to the Bremsstrahlung spectrum. The intensities of these two emission lines are several orders of magnitude greater than those of the Bremsstrahlung, consequently the figure is not to scale. 2.1.2

X-Ray Diffraction by Crystals

X-rays can be diffracted by an appropriate grating just like visible light. Single crystals can be used for this purpose here because the interatomic distance in crystals is comparable to the X-ray wavelength.

Figure 4: Bragg diffraction in a crystal lattice with diffraction angle and Bragg planes indicated

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Figure 4 shows three adjacent planes of a crystal irradiated with a monochromatic X-ray beam with wavelength  and incident angle . The vertices of the unit cells of the crystal are shown as dots and the paths of two neighbouring rays are traced. The path difference between the two emerging rays is 2y. From Figure 4 we can observe the following relation: 2  2 

(1)

If this path difference 2y is equal to an integer number of wavelengths , the emerging rays will reinforce each other (constructive interference) and strong X-ray reflection will occur. Thus the condition for this type of reflection is: 2   

(2)

where n = 1,2,3,… is called the ‘order of diffraction’. This equation is known as Bragg’s Law. X-ray diffraction is a very powerful tool for both crystal analysis and for the measurement of X-ray wavelengths. If the incident beam is a continuous wavelength spectrum such as in Figure 5 then for each value of  there is a value of  that satisfies Bragg’s Law. Therefore a plot of the intensity of the reflected beam as a function of  should look similar to that shown in Figure 5. (think about this for a few moments – the argument is a little subtle and vital to interpreting your data). This spectrum contains the first (n = 1) and the second (n = 2) order diffraction patterns, with a K and K diffraction peak present for each order of diffraction. Pay close attention to the order of the peaks here.

Figure 5: Idealized angular-dependence of X-ray intensity for X-ray diffraction from a single crystal with X-rays generated by a Cu target.

3 Prework The following prework should be completed and marked before starting the lab experiment.

3.1 Theoretical prework Question 1: If electrons in an X-ray tube are accelerated through a potential difference V = 20 kV, calculate the shortest possible wavelength appearing in the bremsstrahlung radiation. Question 2: A cubic crystal reflects the K  radiation of copper (  = 1.38 × 10 10 m) at an angle of  = 18 ° in the first order (n = 1). Calculate the corresponding inter-planar spacing d for the crystal. Question 3: The crystal structure of NaCl is face-centred cubic (FCC) as shown in the diagram below. The unit cell is a cube with side of length a. (pro tip: if you don’t know what a unit cell is, google it ).

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The chlorine atoms occupy the centre of each face and the eight corners of the cube. For clarity the other three faces of the unit cell are not shown. The spacing d measured in this experiment is the separation between the upper face and that passing through the centre of the cube; thus d = a/2. (a) Show that each unit cell of the crystal contains four ‘molecules’ of NaCl. You will need to consider how many chlorine and sodium atoms are in each unit cell, and how they are shared between neighbours. For example each chlorine atom located at a corner of a unit cell is shared with 7 neighbouring cells. Consequently each chlorine atom at a corner makes a contribution of 1/8 of an atom to each unit cell. (b) Prove that the density of the NaCl crystal is given by  = M/(2NAd3), where M is the molar mass and NA is Avogadro’s number.

3.2 Experimental plan You will be expected to create your own experimental plan for this experiment. How much data you take, how you take it, how you graph, analyse and present it, and how you draw and justify your conclusions is for you to decide. You will be partially assessed on the quality of your plan and your ability to explain and justify it. Sections 4 and 5 may be helpful to consider when putting together your plan for this experiment.

4 Experimental hints and tips 4.1 Conducting the experiment Follow the setup instructions in the operating manual before starting the experiment, as outlined here: 

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Using the left hand side of the diffractometer, vary 2 from an angle of 14° up to 80° in steps of 2°. Measure the intensity of the reflected beam, in terms of counts per minute, as a function of incidence angle . In the vicinity of the K and K peaks (first and second orders) take the readings in steps of 0.5°.

Question: Why is it important to take more readings around these peaks? *Hint: Consider errors - your X-ray photons should obey Poisson statistics and the error for n counts would be expected to be within ± n1/2.

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4.2 Optional Extra part to experiment Only do this part if you have sufficient time. Now replace the NaCl crystal with KCl and concentrate only the angular positions of the K and K peaks in the first and in the second order range. (Remember that they will occur at different angles for KCl than NaCl). Note that you do not need to measure the whole spectrum as you did for NaCl.

5 Analysis hints and tips At the very least, you should plan to: 

Plot the X-ray intensity (counts/min) versus  showing both the bremsstrahlung and Auger line spectra diffractions for NaCl.

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Use the first and second orders of the K and K lines to calculate the spacing of the atoms in NaCl, with error estimates.

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Remember that for K  = 0.154 nm and for K  = 0.138 nm.

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Extrapolate your bremsstrahlung curve at the low angle end to estimate the cutoff angle. From this you can calculate the shortest X-ray wavelength emitted from the source and compare your result with that calculated in the Pre-work Question 1.

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From your measured spacing for the NaCl crystal calculate Avogadro’s number. The density of NaCl is 2.16 × 103 kg m-3 and its molar mass is 58.46 × 10-3 kg mol-1.

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If you have the data, calculate the interatomic spacing of your KCl crystal and estimate the error.

6 Writing your report, hints and tips You are expected to plot the results you have collected on graph paper by hand; you do not need a computer to do the graphing or analysis for this experiment. Carefully consider how you will scale your axis to ensure all relevant features can be seen. *Hint: choose a scale which allows you to see both peaks and the shape of the background. Consider: 

How does your curve compare to the ideal intensity spectrum for NaCl? Are there any extra features?

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What are the sources of errors you obtained from the experiment?

7 References and useful reading 7.1 Background references If you’re curious (you should be as a physics undergraduate ), here’s some information on how synchrotrons work: http://www.synchrotron.org.au/about-us/our-facilities/accelerator-physics/how-doesthe-australian-synchrotron-work http://www.clt.uwa.edu.au/__data/assets/pdf_file/0007/2301676/chapter02_3.pdf

7.2 Prework references X-ray characteristics: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xrayc.html

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7.3 Other links for interested students Quantum numbers and shell filling: http://hyperphysics.phy-astr.gsu.edu/hbase/pertab/perfill.html http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html X-rays transitions and shells: http://epmalab.uoregon.edu/pdfs/X_natur3%20_Chap%203_.pdf

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