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Modern Physics Lab 359

X-ray Scattering on Binary Alloys

Alistair Armstrong-Brown

January 7, 2005

Contents1 Introduction 1

2 Equipment 2

2.1 Start-up Procedure . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Shutdown Procedure . . . . . . . . . . . . . . . . . . . . . . . 32.3 Mounting the Sample . . . . . . . . . . . . . . . . . . . . . . 3

3 Software 4

3.1 Before You Start . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 How to Measure . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2.1 Edit DQL . . . . . . . . . . . . . . . . . . . . . . . . . 53.2.2 Job Measurement . . . . . . . . . . . . . . . . . . . . . 63.2.3 Status Display . . . . . . . . . . . . . . . . . . . . . . 73.2.4 Diffrac Files Exchange . . . . . . . . . . . . . . . . . . 7

1 Introduction

This manual aims to initiate you in the use of the diffractometer. Care mustalways be taken when using X-rays as they can be harmful. The diffractome-ter is a powerful yet sensitive tool which does all of the hard work for you.Everything is controlled by the computer linked to the diffractometer. There

are many stages to go though before you can start to make a measurement,but once the process has been setup there is very little else to be done froman experimental point of view. It is imperative that you are shown how touse the diffractometer by someone who has used it before, but all of thesetup instructions are listed here in the manual.

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seconds. Invariably this causes the green LED to start blinking. If this is

indeed the case go to the keyboard and press:

shift,

X-Ray,

1,

enter.

Finally press the On button which is located just under the generatorkey.

The X-ray diffractometer is now ready to go, it will go to its default

settings of 20kV and 5mA. This is the voltage by which the electrons areaccelerated onto the target, and the flow of these electrons makes up thecurrent. Both of these can be adjusted for a particular measurement. Careshould be taken to always return these settings to their start-up values beforethe diffractometer is turned off. All the diffractometer needs now is a sampleand instructions from the software.

2.2 Shutdown Procedure

Similarly to start-up the shutdown proceeds as follows:

lower current to 5mA,

lower voltage to 20kV,

Press Off,

Turn generator key to 0,

Turn off green Mains button,

Turn off pump at circuit breaker (leave H.V. and electronics on),

close valves in next room. Inflow then outflow.

2.3 Mounting the Sample

The binary alloy samples have been mounted on aluminium plates. Theseensure that the top of the sample will be at the right height to encounterthe X-ray beam. You, however, are responsible for positioning the samplemount in the x-y plane. There are three spring loaded bolts to hold the

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sample mount in place. Simply release these springs by pulling on the bolt,

and put your sample inside. Try to get the sample in the very center of theexposed region. It is always best to get someone else to show you an idealsample position. Once you have positioned the sample, release the bolt,allowing the pins to hold the sample. Next, shut the front door. You arenow ready to give instructions to the diffractometer via the software.

3 Software

3.1 Before You Start

First obtain a login name and password from the administrator: Saverio Bi-unno. This will allow you to access the network and the computer controllingthe diffractometer.

Next, adjust the environment so that all the data you take will be savedin a desirable location. This can be done by going to the start button onthe desktop and going to:

Start Programs SiemensDIFFRACplusD5000#1 environment

From this point on the program SiemensDIFFRACplus D5000#1 willbe abbreviated to Siemens. In this environment window there are threedirectories. They are asking where the data is to be sent. The default di-rectories need to be changed to the following ones which have already been

created. All of your data will be stored here after each run and can be sentover the network or stored on a floppy disc for later analysis.

in DFNT$SYSTEM enter C : \Diffplus\in SYSTEM1 : DFNT$FIL 1 enter C : \DIFFDAT1\User\Data\in SYSTEM1 : DFNT$FIL 2 enter C : \DIFFDAT1\User\Data\

When erasing the default entries the program has the irritating habit ofinserting a space before your first letter, make sure this is removed otherwisethe program will not be able to store your data. Be sure to add the finalbackslash after the path as well.

3.2 How to Measure

There are four steps you have to take in order to take data, which involvefour separate programs. It is advisable that you place shortcuts to these

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four programs on the desktop in the following order for quick access. All of

the programs are found in Start Programs Siemens.

1. Edit DQL,

2. Job Measurement,

3. Status Display,

4. Diffrac Files Exchange.

The first program, Edit DQL, is the recipe that the diffracometer followswhen it takes data. The second, Job Measurement, takes the role of thechef and controls the process. The Status Display does just what it says.

It is just a window to check that everything is progressing as planned. Thefourth program converts your data into a more universally accepted format.

3.2.1 Edit DQL

Once you open this program it will just display a blank screen. Go to Fileand chose New. It will next ask what kind of format to use so pick Qualita-tive Extended. This gives you control over all aspects of the diffractometer.There is space for a comment at the start of the DQL, fill this in at you ownleisure, it can also be left blank.

Scan Type: Choose Locked Coupled so that the sample moves at the

same time as the detector.Scan Mode: Choose Step Scan so that the data will be put into bins of

finite width.User Task File: Leave this blank.

The next section is called Default Scan Parameters and deals with thesettings of the sweep, here you can adjust all the parameters you like. Noticethat at the end of this section is the calculated sweep time. Play aroundwith the settings until you have a sweep that balances the amount of datayou need with the amount of time you have available.

Start: The initial angle of the sweep in degrees. The detector cannot bein line with the X-ray beam or it will be damaged by the high intensity sostay above 5 degrees. Note that this is 2.

Stop: The final sweep angle. The detector must not crash into the X-raytube, so be sure to choose angles below 140 degrees.

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Step Size: The width of the bins. You can go down to 0.02 degrees.

Time per Step: You will need to spend a few seconds at each step toallow the detector to collect counts.

Delay time: Enter zero.Scintillation: Set it to One.

You can also choose the generator settings here and control the voltagethrough which the electrons are accelerated, and the current they produce.A setting of 40kV and 30mA is high enough for good readings on these sam-ples. If you would like to change these speak to the technician to check thatyou will not be overloading anything.

Now you have completed the recipe, press ok. You must save it underC : \DIFFDAT1\User\DQLs.

3.2.2 Job Measurement

Be careful as there are two types of program with similar names. JOBMeasurement is the correct one and JOB Measurement is not. Once youopen the right program a window will open with a table large enough tostore several jobs at once. As you only want to do one job per sample it isrecommended that your job only consist of one run. You can then start anew job for the next sample.

In the first cell of the column called Raw Filename, enter the name you

want your file to have.Next, click on the first cell in the column named, Parameter File. You

will then need to browse for the DQL you have just saved. The programwill access this recipe and use it for the scan it is about to perform.

Beneath the table are listed the entries you have just made above for theDQL and RAW data file. Check also that the Default Directory has thecorrect pathway that you should have entered in the environment window.See section (3.1).

You must now save the job before you start, save it in

C : \DIFFDAT1\User\job.

The default name of the job is untitled which may not be a valid filenameso use save as and choose an appropriate job name. Sometimes the savingprocess deletes the raw filename you have just entered and it will need tobe entered again.

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Now press the button labelled Execute Job. You should be prompted

to enter the sample at this point. You will then hear the machine rotatingand aligning itself to the settings you have chosen for the sweep.

A job can be aborted at any time in this program under the Job icon,then Abort Job. You can choose between aborting the job immediately orat the end of your sweep.

3.2.3 Status Display

This is a window which gives the data on screen as it comes in. If you hangaround (and if the sample has been aligned correctly) you will slowly seethe peaks appear before your very eyes. . .

3.2.4 Diffrac Files Exchange

Once the sweep is finished a box will pop-up announcing that the run isover. The Job Measurement window will also display the Job Finishedicon. The next thing to do is to adjust the format in which your data canbe transported. The software for the diffractometer will output a file that isa .RAW extension. The Diffrac Files Exchange program converts the .RAWinto a .UXD file, which is can be read by Origin or other data analysisprograms. It consists of columns of numbers in ASCII format. You need totell the conversion program how many columns you want the data to be putinto, otherwise it will use the default setting of ten columns.

Once you have opened the program click on File then UXD Format.A window will appear with all of the settings for the file exchange.

Raw Data Format: Check Angle+IntensityColumn Width: Check 10.Items per line: 1.Peak List Format: Check Angle+IntensityField Separator: Normal

Also check the Skip Header Information so that all of you files do notstart with 30 lines of jargon.

Now you can open the file you have just measured. It is found in C :

\DIFFDAT1\User\Data. Once the file appears just press the UXDbutton on the toolbar and you will then be prompted to give the name andpath of the new file in ASCII format. You now have your data and the realwork can begin.

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X-Ray Scattering on Binary Alloys

Alistair Armstrong-Brown

February 16, 2004

Contents

1 Introduction 1

2 Setup 2

3 Binary Alloys and Lattice Constants 2

3.1 Results for CuNi . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Results for PbSn . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Calculating Intensities 8

4.1 Beyond the Structure Factor . . . . . . . . . . . . . . . . . . 84.2 Bad Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 Tin in Lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Peak Broadening 15

5.1 K1 and K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1 Introduction

This report aims to cover all of the basic elements of the X-ray diffractionexperiment for the 359 Modern Physics Lab, including both the experimentaland data analysis aspects. Students should be encouraged to go beyond thesuggestions in the manual as the experiment is designed to be open ended.

It will be seen, however, that because of the user friendly diffractometer,more emphasis must be put on the data analysis side of the investigation inorder to do a thorough and in depth report.

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2 Setup

The experiment was performed on the diffractometer. Information on theequipment can be found in the manual. The diffractometer emits a beamof X-rays with a wavelength of 0.15418 nm by bombarding a copper targetwith high energy electrons.

The samples used were composed of small grains of filed down poly-crystal, which were then compressed into a manageable size. In this waythe orientation of all crystalline axes among every grain of powder is moreclosely randomised than if a single poly-crystalline sample was used. Itturns out that in making a binary alloy there exist correlations between

domains of different axis orientation. This will manifest itself as peaks ofunexpected intensity. The powder diffraction method described in the man-ual explains how the pattern will be equally spread over of cone of angle2B from the incident beam, where B is the Bragg angle. This patternwill only be uniformly distributed, however, if the grains of the crystal areoriented completely at random (see section 4.2).

3 Binary Alloys and Lattice Constants

The manual explains possible ways in which a sample can include differentelements into its lattice. If we assume for the moment that we are only deal-ing with interstitial substitution (an assumption which we carry through tothe end of the report), then introducing a larger element B into a latticeof A will at first just slightly increase the lattice parameter above that ofelement A. If the two lattices have the same structure (as is the case forCu and Ni), then the lattice constant will increase smoothly with increasingforeign element density until the element B dominates the original elementand finally you end up with pure element B.

Figure 1 shows the data collected for the alloy of Copper and Nickel.Five samples were prepared containing steadily increasing concentrations ofNickel in Copper from 0 to 100%. The position of the peaks can be ob-tained by fitting them to a Gaussian. In the following calculations the erroron the peak position is given by 1/8 the width of the peak at half maximum(FWHM) just like the Rayleigh Criterion for resolving peaks. This perhapsunusual choice of error is due to the fact the the fitting error will be givenby the quality of the entire fit with respect to peak area, background counts,

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peak position and peak height.

2 Theta

Counts

0% Cu 100% Ni

25% Cu 75% Ni

50% Cu 50% Ni

75% Cu 25% Ni

40 50 60 70 80 90

0

500

1000

1500

100% Cu 0% Ni

Figure 1: Peaks from X-ray diffraction on CuNi Alloy. Angle is twice theBragg angle B. All traces have same scale as shown for pure Cu. =

0.15418nm. Traces have been offset.

It is known that both Copper and Nickel have a face centered cubicstructure (fcc). Before the lattice constant can be extracted from the data,we need to identify to which Bragg planes the peaks correspond. To do thisit is necessary to calculate the structure factor (1) which takes the structure

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of the basis cell into account:

Fhkl =Nn=1

fne2i(hun+kvn+lwn), (1)

where N is the total number of atoms in the unit cell, fn is the atomicfactor (tabulated in terms of sin

in appendix 10 of reference [1]) h,k and l

are the Miller indices and un, vn, wn are the positions of the atoms within theunit cell. For an fcc lattice there are four atoms located at the origin (0, 0, 0)aand the face centers (1/2, 1/2, 0)a, (1/2, 0, 1/2)a and (0, 1/2, 1/2)a. Puttingthese into equation (1) shows that not all of the peaks will be present,some of them, like (100), will have an intensity of zero due to destructive

interference with other atoms in the basis. In fact it is possible to inspectequation (1) and deduce rules governing which peaks will be absent. For thefcc structure it turns out that

Fhkl =

0 if h,k,l have mixed parity4fn if h,k,l have same parity.

(2)

This puts some fairly strict limits on which peaks will be present. BraggsLaw (3) then can be applied to the remaining peaks to calculate their posi-tion

n = d sin, (3)

where d is the distance between scattering planes and is determined bythe Miller indices and the lattice constant a

1

d2=

h2 + k2 + l2

a2(4)

for a cubic lattice. Of course in order to identify the peaks a knowl-edge of the lattice constant is required, when in fact it was extraction ofthe lattice constant from the peaks which motivated the peak identification.Thankfully from figure 1 we can deduce that the lattice constant does notshift much from pure Copper (red trace) to pure Nickel (black trace) as the

peak positions do not move much. Therefore it is possible to use either tab-ulated lattice constant to identify the peaks using equation (3). Three peaksare visible in figure 1 corresponding to (111), (200) and (220) in increasingvalues of .

Having identified the peaks we take our known value for wavelengthto calculate the lattice constant as a function of concentration. For now

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we assume that the diffractometer has no offset in . This issue will be

addressed in section 5.1.

3.1 Results for CuNi

A Gaussian was fit to all three peaks in figure 1 for every concentration of thealloy. This gave a value for the peak position and the FWHM. We can thenuse equation (3) in reverse to give a value for the lattice constant. In facteach peak will give a different (but hopefully similar) value for the latticeconstant. All the data and a fit for lattice constant against concentration isshown in figure 2. The fit is a second order polynomial with the coefficientsdisplayed in table 1.

0 20 40 60 80 100

3.52

3.54

3.56

3.58

3.60

3.62

Percentage of Copper

LatticeConstant[Angstroms]

Lattice Constant CuNi Alloy

(100)

(200)(220)

Fit

Figure 2: Extracted values of the Lattice constant a using a Gaussian fit forpeaks (100), (200) and (220). Assumed no offset. Error from 1/8 FWHM.

It can be seen that the lattice constant does indeed change smoothlywith increasing concentration, but that it is not quite a linear relationship.The error in table 1 is generated by the fitting program. Another pointto mention is that closer inspection of the points in figure 2 show that the(100) in black squares are always lowest and the (220) in green triangles

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a = b + cx + dx2

b 3.52258 +/ 7.24942 104c 7.57064 104 +/ 4.09167 105

d 1.4605 106 +/ 3.98489 107

Table 1: Polynomial fit of lattice constant to peak position using BraggsLaw in figure 1. x is percentage of Copper.

always yield higher values of the lattice constant, even though they areconsistent within the error bars. This may be due to an offset in the zeroangle measured by the diffractometer and can be corrected if there are more

peaks present, as in the case of Lead or Tin.Because the lattice constant varies smoothly and we do not see any extra

peaks in figure 1, we can conclude that we have seen a binary alloy changingfrom an fcc pure Copper lattice to an fcc pure Nickel lattice without anyother phase in between.

3.2 Results for PbSn

Figure 5 shows the data collected for a different binary alloy consisting ofTin and Lead. In this alloy things are not a simple as for CuNi. The latticestructure for pure Lead is fcc, the same as Copper and Nickel. This time,

however, pure Tin has a tetragonal structure. We expect that intermediateconcentrations will show peaks of both an fcc Lead and a tetragonal Tin.It can be seen from a phase diagram of the alloy that they do not dissolvewithin each other very well (see page 1848 of reference [2]). Therefore differ-ent domains within the sample, even within each grain in the powder, willbe either predominantly fcc Lead or predominantly tetragonal Tin. In factLead can take a small percentage of tin inside it, thereby shifting the latticeconstant. This is investigated in section 4.3.

There is the possibility that a new, entirely different phase is encoun-tered at intermediate concentrations. In this case new peaks, not belonging

to either of the pure sample diffraction patterns should be seen.

In fact we do not observe any new peaks in figure 5 at intermediate con-centrations and conclude that we get two domains of fcc Lead and tetragonalTin with varying concentration. Every peak in the alloy samples can be seento derive from one of the pure traces of either Lead in black or Tin in red.

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2 Theta

Counts

0% Sn 100% Pb

25% Sn 75% Pb

50% Sn 50% Pb

75% Sn 25% Pb

0 10 20 30 40 50 60 70 80 90

0

500

1000

1500

100% Sn 0% Pb

Figure 3: Lead Tin Alloys Diffraction Pattern.

Also very little shifting of the peaks occur, this is another clue that they do

not dissolve within each other very well. If they did we would expect thatthe larger atom would squeeze apart the lattice as it pushed its way inside,increasing the lattice parameter and shifting the peak positions.

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4 Calculating Intensities

4.1 Beyond the Structure Factor

Calculation of the structure factor for complicated basis can be a lengthyand tedious process. This then gives a value for the intensity of the radiationscattered from the sample immediately after interacting with the atoms inthe sample. It will not, however, represent the measured intensity of thepeaks in any experimental setup. Chapter four of reference [1], which alsoappears in the manual, thoroughly explains all of the factors which need tobe taken into account. These factors are:

1. polarisation factor,

2. structure factor F,

3. multiplicity factor p,

4. Lorentz factor,

5. absorption factor,

6. temperature factor.

The structure factor has already been mentioned (1) and evaluated for anfcc lattice in equation (2). The polarisation factor accounts for the fact that

we are using unpolarised radiation. The multiplicity describes how some-times many different planes will contribute to each peak. For example the(111) peak seen in figure 1 will contain a contribution from the (-1,1,1) andin fact all eight combinations of positive and negative ones. The multiplicityfactor is tabulated for all peaks in appendix 11 of reference [1].

The Lorentz factor accounts for all of the geometric factors involvedin taking a measurement. One example is how the diffractometer sweepsin an arc of a circle around the sample. As mentioned above the X-raysare diffracted into a cone of angle 2B from the incident beam. Thereforea circle will intercept a different fraction of this cone depending on thescattering angle. The ratio of cosine and sines in equation (5) accounts for

the polarisation factor which is the numerator and the Lorentz factor whichis the denominator.

The absorption factor has not been included in the subsequent calcula-tions for the following reason: it is directly proportional to the volume ofthe sample irradiated. For small scattering angles the beam does not pen-etrate the sample very far but the area irradiated is large. Conversely, for

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Element B [m2]

Pb 2.101020Cu 5.301020

Ni 4.251021

Table 2: Element coefficients for temperature factor.

large scattering angles, there is deeper penetration but the irradiation areais smaller. The volume of the sample irradiated remains constant and forcalculations of the relative intensity between peaks this term can be omitted.

The final factor covers the thermal vibrations of the atoms in the lattice.

The atoms undergo an average displacement of u from their mean position.Two peaks at different scattering angles will correspond to different planespacings, d (4). It is the relative displacement u/d which determines theeffect on the scattering intensities. In this way an angular dependence creepsinto the temperature factor. This is the exponential term in equation (5).

All of the factors can be brought together to give the (as yet un-normalised)relative intensity, I, as a function of the scattering angle, ,

I= |F|2 p

1 + cos2 2

sin2 cos

e

2B sin2

2 (5)

where B has been given by Debye as

B =6h2T

mk2D

(x) +

x

4

. (6)

Here T is the sample temperature at measurement in Kelvin, m is themass of the atom in question, k is Boltzmanns constant and D is theDebye temperature. The function (x) and the Debye temperature can befound in appendix 13 of reference [1], where x is D

T. Table 2 gives the B

values calculated for the elements in question.Putting in values for all of these factors is best done on a spread sheet,

but results for the first few peaks of Lead are shown in table 3. The peaks

are normalised to the highest intensity peak, in this case (111).Data taken to higher scattering angles for Lead is shown in figure 4.

The next thing to do is to compare whether the results calculated in table3 match with those recorded. One of the parameters in the fitting processis the area of the Gaussian and should be directly proportional to the totalnumber of counts within the peak. This can be compared to the calculated

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h k l sin 2 sin

fPb |F|2 p Lorz+Pol Temp. I Rel. I

1 1 1 15.6 0.27 31.3 0.17 70 78400 8 24.7 0.88 13 106 10.02 0 0 18.1 0.31 36.3 0.20 67.5 72900 6 17.9 0.84 6.5 106 4.842 2 0 26.1 0.44 52.3 0.29 61 59536 12 7.89 0.71 4 106 2.941 1 3 31.1 0.52 62.2 0.33 57 51984 24 5.33 0.62 4.1 106 3.052 2 2 32.7 0.54 65.3 0.35 56 50176 8 4.79 0.60 1.1 106 0.85

Table 3: Relative Intensities for Pb.

peak position and relative intensity. Tables (4, 5, 6) compare the calculatedrelative intensities and area under the measured peaks for Lead, Copper and

Nickel respectively.

30 40 50 60 70 80 90 100 110 120 130 140

0

6000

12000

Counts

2 Theta

Data

Fit

Figure 4: Data taken up to higher scattering angle for wide slits. Gaussianfit of (111) peak is also shown. Lead.

Inspection of the final two columns of tables (4, 5, 6) show that thecalculation of the peak position is to within 0.2% of the measured positionfor all of the pure samples. The intensity, which involves a much morelengthy procedure, in to within 10% for most peaks. The notable exceptionsare the (333) peak of Lead and the (220) peaks of both Copper and Nickel.

Interestingly both the (244) and (006) fall on the same angle. It is impossibleto fit two differently sized Gaussians on the same position, but we noticethat the sum of the calculated intensities 0.52 and 0.13 yields 0.65, which isclose to the measured intensity of 0.73 for the combined peak. Again within10%.

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Peak Calculated Measured Measured/Calculatedh k l 2 Int. Rel. Int. 2 Area Rel. Area 2 Rel. Intesity

1 1 1 31.30 1.36E7 10.00 31.38 3980 10.00 1.00263 0.999652 0 0 36.30 6.59E6 4.84 36.38 1875 4.71 1.00232 0.972712 2 0 52.26 4.00E6 2.94 52.36 961 2.41 1.00173 0.821551 1 3 62.20 4.15E6 3.05 62.29 1081 2.72 1.00148 0.890932 2 2 65.30 1.15E6 0.85 65.38 336 0.84 1.00127 0.998890 0 4 77.06 4.52E5 0.33 77.15 114 0.29 1.00114 0.861921 3 3 85.50 1.24E6 0.91 85.59 360 0.90 1.00099 0.989442 0 4 88.29 1.12E6 0.83 88.36 244 0.61 1.00081 0.741672 2 4 99.45 7.93E5 0.58 99.51 236 0.59 1.00061 1.01793

3 3 3 108.0 2.27E5 0.17 108.1 261 0.66 1.00064 3.931610 4 4 123.5 3.20E5 0.23 123.6 83 0.21 1.00041 0.886541 3 5 134.2 1.34E6 0.99 134.3 310 0.78 1.00025 0.793222 4 4 138.3 7.12E5 0.52 138.2 292 0.73 0.99952 1.40380 0 6 138.3 1.78E5 0.13 138.2 292 0.73 0.99952 5.61522

Table 4: Comparison of calculated and measured intensities for Pb.

Peak Calculated Measured Measured/Calculated

h k l 2 Int. Rel. Int. 2 Area Rel. Area 2 Rel. Intesity1 1 1 43.41 351348 10.00 43.38 529 10.00 0.9992 0.999492 0 0 50.57 130779 3.722 50.51 229 4.33 0.9989 1.16242 2 0 74.31 33481 0.95 74.22 121 2.29 0.99876 2.39904

Table 5: Comparison of calculated and measured intensities for Cu.

Peak Calculated Measured Measured/Calculated

h k l 2 Int. Rel. Int. 2 Area Rel. Area 2 Rel. Intesity

1 1 1 44.58 312777 10.00 44.55 520 10.00 0.99925 0.999512 0 0 51.95 125339 4.01 51.9 228 4.31 0.99897 1.075022 2 0 76.55 36337 1.16 76.45 112 2.12 0.9987 1.8217

Table 6: Comparison of calculated and measured intensities for Ni.

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Due to the myriad of terms in equation (5), it is remarkable that the

calculated intensities are this close to the measured ones. A job well done.

4.2 Bad Samples

As alluded to earlier, we now turn to the case of an imperfect sample. Thiswas deliberately prepared as a normal piece of everyday lead. This is a poly-crystal which will contain domains with differently oriented crystalline axes.These domains are not, however, randomly oriented. Figure 5 shows datataken from one of these poly-crystalline samples and compares it to a tracetaken with a powder sample. As the two samples had different monocro-mating slit configurations their absolute intensities are different. They are

therefore plotted on different scales so that the principal peak is the samesize on the figure. In this was we can compare the relative intensities byeye. A more quantitative comparison is also given in table 7.

0 20 40 60 80 100 120 140 160

0

2000

4000

6000

8000

10000

12000

14000

Counts

2 Theta

Powder Sample

0

200

400

600

800

1000

1200

1400

Polycrystaline

Figure 5: Comparison between powder and poly-crystalline samples of Lead.

The final two columns show that the peak positions are again within 0 .2%of their predicted values. The intensities, on the other hand, are sometimes

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Peak Calculated Measured Measured/Calculated

h k l 2 Int. Rel. Int. 2 Area Rel. Area 2 Rel. Intesity1 1 1 31.30 1.36E7 10.00 31.29 212 10.00 0.99975 0.999652 0 0 36.30 6.59E6 4.84 36.26 17 0.80 0.99901 0.165572 2 0 52.27 4.00E6 2.94 52.25 10 0.47 0.99962 0.160491 1 3 62.20 4.15E6 3.05 62.15 38 1.79 0.99923 0.587962 2 2 65.30 1.15E6 0.85 65.27 17 0.80 0.99959 0.94880 0 4 77.06 4.52E5 0.33 77.11 2 0.09 1.00062 0.283881 3 3 85.51 1.24E6 0.91 85.44 7 0.33 0.99924 0.361192 0 4 88.29 1.12E6 0.83 88.26 36 1.70 0.99967 2.054322 2 4 99.45 7.93E5 0.58 99.41 4 0.19 0.99961 0.32393 3 3 108.04 2.27E5 0.17 107.97 9 0.42 0.99935 2.54519

0 4 4 123.52 3.20E5 0.24 123.76 4 0.19 1.00195 0.80211 3 5 134.25 1.33E6 0.98 134.09 10 0.47 0.99884 0.48037

Table 7: Comparison of calculated and measured intensities for poly-crystalline Lead.

only 16% of the calculated value. This shows that there is indeed somecorrelation in the orientations of the domains in a poly-crystalline sample,and in order to measure the correct intensities the samples must be preparedwith care.

4.3 Tin in Lead

As mentioned above, the phase diagram of PbSn shows that Tin is partlysoluble in Lead. Lead is fcc with a lattice parameter of 4.95 Angstroms.Tin is tetragonal with lattice constants of 5.63 and 3.18 Angstroms. Tin isa lighter atom than lead and slightly smaller, so we expect that the samplecontaining 25% Tin will have some domains where some of the lattice pointsin fcc lead will have been replaced by tin. This will act to reduce the latticeparameter in comparison to pure lead.

Because of the increased sweep range of figure 4 it was now possible

to try to correct for the instrumental offset at zero angle. The peaks wereagain all fitted with Gaussians, yielding all of the peak positions for differentvalues ofd. These values, shown in table 8, were then fitted to Braggs law(7) using d and as dependant parameters and a and 0 as the fittingparameters.

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Pure Lead 25% Tin 75% Lead

a 4.9521 +/ 0.0014 4.9485 +/ 0.0007 Offset 0.041 +/ 0.009 0.033 +/ 0.004

Table 9: Change in lattice parameter of Lead with introduction of 25% Tin.Theta offset in degrees.

lattice parameter does indeed drop to a lower value by about 0.07% whichis just outside the error bars. Another thing to note is how small size ofthe offset. Perhaps this is due to too few peaks being used, or a highlyaccurate machine.

5 Peak Broadening

During the fitting process the width of each peak is also generated. We nowlook at how this width varies with scattering angle. Figure 7 shows how thepeaks broaden. The black points represent fitting a single Gaussian to everypeak. In fact most of this broadening is due to peak splitting. The presenceof two peaks is due to two very similar, yet different, wavelengths beingpresent in the beam. This is discussed in the next section. Fitting thesepeaks with double Gaussians has been done for the red and blue points in

figure 7. As can be seen this significantly reduces the measured peak width.

5.1 K1 and K2

The final point to address is the existence of more than one wavelength in theX-ray beam. When the electrons strike the Copper target, low lying boundstates are knocked out of the atoms electron shell which are immediatelyreplaced by 2p electrons from higher up. The 2p electrons however comein two types depending on their spin, and there is a small energy differencebetween them (Fine structure). The X-rays emitted as these 2p electronsfall down the potential well will have a wavelength inversely proportionalto the change in energy. Therefore if we have electrons of two different fill-

ing up the low lying holes, we will have two slightly different wavelengthspresent. These are the K1 and K2 lines. (In fact many more processesare going on during the electron bombardment of the copper target, butthe monochromators will filter out everything else. The K1 and K2 lines,being so close in wavelength, both pass through.)

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20 40 60 80 100 120 140

0.2

0.4

0.6

0.8

1.0

FWHM[degrees]

2 Theta [degrees]

Lead

Single Peak

Ka1

Ka2

Figure 7: FWHM for peaks in Lead.

Peak K1 K2h k l 2 FWHM +/ 2 FWHM +/

2 2 4 99.43 0.39 1.54175 5.56E-4 99.75 0.41 1.5454 5.83E-43 3 3 108.00 0.44 1.54164 5.38E-4 108.48 0.30 1.54632 3.65E-40 4 4 123.45 0.46 1.54163 4.17E-4 124.00 0.32 1.5456 2.87E-4

1 3 5 134.10 0.46 1.54137 3.28E-4 134.78 0.46 1.54522 3.23E-42 4 4 138.10 0.41 1.54137 2.64E-4 138.79 0.40 1.54489 2.54E-4

Table 10: Measurement of the two wavelengths from Lead peaks

The effect of these two wavelengths will manifest itself as a splitting ofthe peaks. This may not be visible at low scattering angle as the peaks willbe so close together, but at higher angles it is quite clear, see figure 8. It ispossible to fit a double Gaussian to the data, and extract a different wave-length from each peak assuming that we now know the lattice parameter

from table 9. This was done for as many peaks as could be resolved seetable 10.

This gives an average for the two wavelengths and an error given by theaverage of 18 of the FWHM. This is summarised in table 11. The two valuesare slightly out from the quoted values found in the manual for the K1 and

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References

[1] B.D. Cullity, S.R. Stock, Elements of X-Ray Diffraction Third Edition,Prentice Hall (2001).

[2] T.B. Massoulsk, Binary Alloy Phas Diagrams, p942 Vol 1 CuNi, p1848Vol 2 PbSn, American Soc. for Metals.

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