Lab 03: Precision X-Ray Diffractometry

Transcription

1 Lab 03: Precision X-Ray Diffractometry Author: Mike Nill Alex Bryant Contents 1 Introduction 2 2 Experimental Procedures 2 3 Results 3 4 Discussion Lattice Parameter Determination Vegard s Law Experimental Error Peaks Shifts Double Peaks Conclusions 9 1

2 Abstract The effect of solid-solution alloying on crystal structure were investigated for the Cu-Ni system. Four polycrystalline Cu-Ni samples of varying composition (100wt% Ni, 50wt% Cu - 50wt% Ni, 75wt% Cu - 15wt% Ni, and 100wt% Cu) were investigated using a standard θ 2θ diffractometer. The lattice parameters were found to be Å, Å, Å, and Å respectively. The lattice parameter of the Cu-Ni system was found to vary linearly with composition. In addition, a novel method of visualizing multiple sources of error was developed to understand precision in the diffraction experiment. 1 Introduction Two elements of similar size, valence, and crystal structure are often miscible enough to form a substitutional solid-solution at any composition. The presence of an atom of a slightly different size distorts the lattice and alters the directions of the diffracted beams (the diffracted beam intensity is also affected, but we are not concerned with that here). This change is lattice parameter can be determined from the measured d-spacings of diffracting planes. The relationship between the change in angular position of the diffracted beam θ and change in the lattice parameter a 0 can be determined by taking the total derivative of Bragg s law[2]. Noting that nλ is a constant, we obtain d(nλ) = d(2d sin θ) = 2(d cos θ θ + sin θ d) d d = a 0 a 0 = cot θ θ (1) Equation (1) indicates that the diffraction pattern is most sensitive to changes at higher angles. This fact aids in the determination of the lattice constant, which can theoretically be used to estimate the composition of the binary alloy. In a random substitutional solution, one can expect that the distorted lattice constant will lie between the lattice constants of the pure elements. If the dependence is linear, the solution is said to obey Vegard s Law[2], which can be stated as follows: a A,B = X A a A + (1 X A )a B (2) Where X A and are the atomic fractions of atoms A and B, respectively. In reality, metallic solid solutions rarely obey Vegard s law precisely. For binary allows that admit total dissolution at any composition, a as a function of composition must equal the pure element lattice constants at either extreme, but may exhibit either a positive or negative curvature in between. In fact, the Cu-Ni system exhibits a noticeable deviation from Vegard s law with a positive curvature[1]. 2 Experimental Procedures Four polycrystalline Cu-Ni samples of varying composition (100 wt% Ni, 50 wt% Cu, 75 wt% Cu, and 100 wt% Cu) were investigated using a Rigaku MiniFlex II bench-top diffractometer. A tube voltage of 30 kv and current of 15 ma were used to generate X-rays from a copper target. A graphite monochromator was used to filter out contributions other than the strong Cu Kα characteristic peak. Each sample was irradiated with a Cu Kα X-ray beam while the diffraction pattern was scanned from 40 to at a scan rate of 10 /min. For the pure copper sample, a second scan was performed from 90 to after the sample had been polished for approximately 5 minutes using 2

3 diamond grit paste. For the 50 wt% Cu sample, a second scan was performed at a scan rate of 2 /min only over the peaks of interest to determine their locations with greater precision. Once the locations of the peaks were determined, the first three peaks were indexed according to the first three allowed reflections observed in FCC crystals. The stability of this method was tested by comparing the calculated lattice parameter given the peak position and assumed hkl indices to tabulated values for copper and nickel[2, p. 507]. The assumed lattice parameter was then used to determine the value of h 2 + k 2 + l 2 for the remaining peaks. These values were then rounded to the nearest integer and indexed according to those combinations allowed in FCC [2, p. 516]. The indexed peaks were then used to determine a trial lattice parameter for each peak via Bragg s law. The lattice constant for each sample was determined by correlating the trial a values against the Nelson-Riley function, calculating a linear regression, and finding the y-intercept (See Section 4.1). 3 Results The X-ray diffraction spectrum for each sample is shown in Figure 1. The expected peaks for pure copper and pure nickel are shown superimposed on their corresponding experimental spectrums. For pure copper and nickel, average peak shifts of and from tabulated values[6] were observed. All observed peaks are tabulated in Table 2. Calculated lattice constants, along with those predicted by Vegard s law, are shown in Table 1. Many of the observed peaks were actually split into two peaks separated by a few tenths of a degree. In all cases, the stronger of the two peaks was used to determine the lattice constant. The measured lattice constants were in close agreement with Vegard s Law (Figure 2). Composition Ni 50Cu-50Ni 75Cu-25Ni Cu Measured a 0 [Å] Predicted a 0 [Å] % Error 0.14% 0.05% 0.003% 0.12% Table 1: Measured lattice constants, and those predicted by Vegard s Law. (Note that for Cu and Ni, the predicted value refers to the tabulated value). 3

4 (a) Copper Line 2θ B d [Å] hkl a [Å] (c) 75%Cu-25%Ni Line 2θ B d hkl a (b) 50%Cu-50%Ni Line 2θ B d hkl a (d) Nickel Line 2θ B d hkl a Table 2: All observed peaks are tabulated for each sample. Rows without a line number are double peaks (See section 4.5). 4

5 Diffraction Spectrum: 100 wt% Cu Sample PDF wt% Cu - 50 wt% Ni Intensity [arb. units] wt% Cu - 25 wt% Ni wt% Ni Detector angle 2θ Figure 1: Diffraction spectrums for all four samples are shown. dashed lines for the pure elements. Tabulated peaks are shown as 5

6 Effects of Composition on Lattice Parameter Experimental Data Vegard s Law 3.6 a0 [Å] wt% - Cu Figure 2: A plot of lattice parameter vs. composition reveals a clear linear dependence. 4 Discussion 4.1 Lattice Parameter Determination For each sample, the indexed peaks were used to determine a trial lattice parameter for each peak. Noting that Equation (1) predicts greater experimental precision with higher angle, these higher-angle peaks should be given greater weight in determining the lattice constant. One way of accomplishing this is by reducing the data; correlating calculated a constants against a function which predicts expected precision. In a thorough examination of error, Nelson and Riley[5] determined that the relation a a ( cos θ 2 = K sin θ ) cos θ2 + θ accurately predicts the accuracy of a given measurement. The lattice constant is determined by plotting the calculated a values vs. the Nelson-Riley function (Figure 3), constructing a linear regression, and determining the y-intercept (where the Nelson-Riley function, and thus a/a goes to zero). 4.2 Vegard s Law Vegard s law predicts that the lattice constant of a binary solid solution will be a weighted average of that of the its two constituents (Equation (2)). In this investigation a small negative deviation from Vegard s law was observed. Barret also reports a negative deviation[1]. De Graef and McHenry also note close agreement with Vegard s Law for 75Cu-25Ni[3, p. 369]. Vegard s law can also be used to confirm the composition of binary alloys. Treating Vegard s law as a fact, one can determine the composition by solving Equation (2) for X A yielding 6

7 3.632 Calculated Lattice Constants vs. NR Function Lattice Constant a Copper Linear Regression Nelson-Riley Function cos θ2 sin θ + cos θ θ Figure 3: Determination of the lattice constant by reduction against the Nelson-Riley function X A = a 0 a B a A a B Using experimentally derived lattice constants for the nominal 50%-Cu alloy and nominal 75%-Cu alloy, compositions of 51.41% and 75.2% are obtained. This demonstrates that Vegard s law is a useful measure against which experimental data can be compared. 4.3 Experimental Error There are two types of error present in experiments. The first is random error, whose sources include electrical noise, background radiation, unwanted reflections, and counter anomalies. Random error is unavoidable, and can only be mitigated by noting its effects carefully. Random errors decrease the precision, or repeatability, of the measurement. The second type of error is systematic error, which results in a constant shift of the measured data. Sources of systematic error include presence of impurities, misalignment of the sample or detector, non-linearity of the X-ray counter response, preferred grain orientation of the sample, and thermal effects. Systematic error usually cannot be totally eliminated, but can be avoided through meticulous experimental procedure. Systematic error reduces the accuracy, or vicinity to the correct value of the measurement. Both sources of error crop up the determination of the lattice constant. Analysis by Nelson and Riley showed that systematic error in the measured lattice constant should decrease linearly with N R(θ) (The Nelson-Riley function). Therefore, a steeper slope would indicate the presence of large 7

8 systematic errors close to 2θ = 0. Random error manifests itself by deviation from linearity of the data. One quantitative measure of departure from linearity is the coefficient of determination of the linear regression defined as follows: R 2 (ai f i ) 2 = 1 (ai ā) 2 where f i are the fitted values. R 2 gives an indication of how much of the variance of the data is accounted for, or predicted, by the linear regression. R 2 = 1 indicates a perfect fit to the data. Combining these two quantitative measures into a single plot (Figure 4), it is possible to visualize the error in each measurement. The slope of the linear regression is plotted on the ordinate, and the quantity (1 R 2 ) is plotted on the abscissa, so that the positive direction of each axis represents increasing error. The measurement of a 0 for 50 wt% Cu especially suffers from random error. This is because two of the low-angle peaks were shifted significantly and resulted in outlying data points. However, the high correlation of the high-angle data (low systematic error) allows us to justify the final measurement. 4.4 Peaks Shifts A small, constant deviation was observed between the observed peaks and tabulated values. A comparable deviation has been observed for powder diffraction on the same machine in the past so it is very likely that the shift is due to misalignment of the machine. The fact that the shift is relatively constant across a wide range of angles suggests a mechanical source rather than an X-ray optical phenomenon. 4.5 Double Peaks Several of the observed peaks were actually composed of two closely spaced but easily resolvable peaks. As an example, the copper 331 peak is clearly split into two peaks at and (Figure 5). For the purposes of data analysis the stronger of the two peaks was used. This is justified on the basis of (1) the stronger signal and (2) agreement with the systematic shift described in Section 4.4. In all cases, the stronger of the two peaks was offset from the tabulated value by about 0.14, the same as the other peaks. Incomplete filtering of the X-ray beam could account for this shift. The effect of a small shift in X-ray wavelength on diffraction angle can be determined from Bragg s Law: θ B λ = 1 d 4 l2 d 2 Unfortunately, this error increases for higher angles, precisely where higher precision is desired. Noticable presence of the Kα 2 line could cause a shift of up to 0.5. It is also possible that the samples were prepared in such a way as to mechanically induce a preferred grain orientation, or the surface of the sample may have not been perfectly flat. 8

9 2.8 x 10 3 Visualization of Sources of Error Increasing Systematic Error Increasing Random Error (a) 1 R 2 is plotted on the x-axis (increasing random error) while the slope of the linear regression from Figure 4(b) is plotted on the y-axis. No natural scale has been defined for the y-axis. This plot is simply meant to help visualize relative magnitudes of error Calculated Lattice Constants vs. NR Function 50 wt% Cu Linear Regression Lattice Constant a Conclusions Nelson-Riley Function cos θ2 sin θ + cos θ θ (b) Nelson-Riley reduction of 75Cu-25Ni showing significant random error Figure 4: Visualization of sources of error in this diffraction experiment. The limiting geometry of the θ 2θ diffractometer allows for extremely precise measurements of peaks angles not otherwise possible in a film or polychromatic diffraction experiment. It has been shown that sensitivity to changes in the lattice constant increases with higher Bragg angles. This 9

10 25 Double Peak in Copper Spectrum Reflection Intensity [arb. units] Detector Angle 2θ Figure 5: A Close-up of the Cu 311 reflection shows a clear double peak at 137. fact can be used to improve precision of measurements of lattice parameters. Today, software is available to automate such calculations[4]. In a binary substitutional solid solution, the lattice constant as a function of composition can usually be approximated by a straight line connecting the two pure elements. This empirical relation, known as Vegard s law, is useful as a cross-check, but admits numerous exceptions. In diffraction experiments, many sources of error exist which may reduce accuracy or precision. Two types of errors can be differentiated by their effect on the measurement: systematic errors can often be mitigated by careful sample preparation and machine calibration; random errors must be accepted with death and taxes. A novel visual representation of the total error was developed in order to understand propagation of error in diffraction experiments. One such systematic error was apparent in a slight shift of peak positions from nominal values, likely due to mechanical misalignment. Other sources of random error were assessed, although random error was not as prevalent as systematic error, as evidenced by Figure 4. References [1] C.S. Barret, Structure of Materials. McGraw-Hill, New York, 2nd Edition, (1952) [2] B.D. Cullity, Elements of X-Ray Diffraction. Addison-Wesley, Massachusetts, 2nd Edition, (1978) [3] M. De Graef and M. E. McHenry, Structure of Materials. Cambridge, New York, (2007) [4] E.R. Hovestreydt FINAX: A computer program for correcting diffraction angles, reding cell parameters, and calculating powder patterns. J. of Appl. Cryst., Volume 16, Part 6, pg (December 1983) 10

11 [5] J.B. Nelson and D.P. Riley, Proc. Phys. Soc. (London) 57, 160 (1945) [6] Powder Diffraction File, International Centre for Diffraction Data (ICDD) Appendix A: MATLAB Code Listing 1: LatticeConstant.m % Define Nelson R i l e y f u n c t i o n NR theta ) cos ( theta pi / ). ˆ 2. / sin ( theta pi / ) +... cos ( theta pi / ). ˆ 2. / ( theta pi / ) ; % Pure Cu % Obtain 1 s t order f i t c1 = polyfit (NR(Cu. peaks (Cu. n, 1 ) / 2 ),Cu. a (Cu. n ), 1 ) ; Cu. a0 = c1 ( 2 ) ; % y i n t e r c e p t %R square v a l u e s s e r r = sum( (Cu. a (Cu. n ) ( c1 (2)+ c1 (1) NR(Cu. peaks (Cu. n, 1 ) / 2 ) ) ). ˆ 2 ) ; s s t o t = sum( (Cu. a (Cu. n ) mean(cu. a (Cu. n ) ) ). ˆ 2 ) ; rs Cu = 1 s s e r r / s s t o t ; % Pure Ni Ni. pd = lambda. / (2 sin ( ( Ni. peaks ( :, 1 ) ) / 2 pi / ) ) ; Ni. a = Ni. pd. ( Ni. hkl. ˆ 2 [ 1 ; 1 ; 1 ] ). ˆ 0. 5 ; c2 = polyfit (NR( Ni. peaks ( Ni. n, 1 ) / 2 ), Ni. a ( Ni. n ), 1 ) ; Ni. a0 = c2 ( 2 ) ; %R square v a l u e s s e r r = sum( ( Ni. a ( Ni. n ) ( c2 (2)+ c2 (1) NR( Ni. peaks ( Ni. n, 1 ) / 2 ) ) ). ˆ 2 ) ; s s t o t = sum( ( Ni. a ( Ni. n ) mean( Ni. a ( Ni. n ) ) ). ˆ 2 ) ; r s N i = 1 s s e r r / s s t o t ; % Cu5050. pd = lambda. / (2 sin ( ( Cu5050. peaks ( :, 1 ) ) / 2 pi / ) ) ; Cu5050. a = Cu5050. pd. ( Cu5050. hkl. ˆ 2 [ 1 ; 1 ; 1 ] ). ˆ 0. 5 ; c3 = polyfit (NR( Cu5050. peaks ( Cu5050. n, 1 ) / 2 ), Cu5050. a ( Cu5050. n ), 1 ) ; Cu5050. a0 = c3 ( 2 ) ; %R square v a l u e s s e r r = sum( ( Cu5050. a ( Cu5050. n )... ( c3 (2)+ c3 (1) NR( Cu5050. peaks ( Cu5050. n, 1 ) / 2 ) ) ). ˆ 2 ) ; s s t o t = sum( ( Cu5050. a ( Cu5050. n ) mean( Cu5050. a ( Cu5050. n ) ) ). ˆ 2 ) ; rs Cu5050 = 1 s s e r r / s s t o t ; % Cu75. pd = lambda. / (2 sin ( ( Cu75. peaks ( :, 1 ) ) / 2 pi / ) ) ; Cu75. a = Cu75. pd. ( Cu75. hkl. ˆ 2 [ 1 ; 1 ; 1 ] ). ˆ 0. 5 ; 11

12 c4 = polyfit (NR( Cu75. peaks ( Cu75. n, 1 ) / 2 ), Cu75. a ( Cu75. n ), 1 ) ; Cu75. a0 = c4 ( 2 ) ; %R square v a l u e s s e r r = sum( ( Cu75. a ( Cu75. n )... ( c4 (2)+ c4 (1) NR( Cu75. peaks ( Cu75. n, 1 ) / 2 ) ) ). ˆ 2 ) ; s s t o t = sum( ( Cu75. a ( Cu75. n ) mean( Cu75. a ( Cu75. n ) ) ). ˆ 2 ) ; rs Cu75 = 1 s s e r r / s s t o t ; % c5 = polyfit (NR( Cu10. peaks ( :, 1 ) / 2 ), Cu10. a, 1 ) ; Cu10. a0 = c4 ( 2 ) ; % c6 = polyfit (NR( Cu20. peaks ( :, 1 ) / 2 ), Cu20. a, 1 ) ; Cu20. a0 = c4 ( 2 ) ; % V i s u a l i z e e r r o r r s = 1 [ rs Cu r s N i rs Cu5050 rs Cu75 ] ; s l = [ c1 ( 1 ) c2 ( 1 ) c3 ( 1 ) c4 ( 1 ) ] ; L a t t i c e C o n s t a n t P l o t s clear c1 c2 c3 c4 c5 c6 n = 3 ; Listing 2: IndexLines.m NR theta ) cos ( theta pi / ). ˆ 2. / sin ( theta pi / ) +... cos ( theta pi / ). ˆ 2. / ( theta pi / ) ; A = Cu ; % Place h o l d e r f o r working sample % Assume f i r s t t h r e e l i n e s (FCC) A. hkl = [ ; ; ] ; % Based on the f i r s t 3 peaks, index the r e s t a = A. pd ( 1 : n ). (A. hkl ( 1 : n, : ). ˆ 2 [ 1 ; 1 ; 1 ] ). ˆ 0. 5 ; c = polyfit (NR(A. peaks ( 1 : n, 1 ) / 2 ), a, 1 ) ; a0 = c ( 2 ) ; % Compare aˆ2/dˆ2 to q u a d r a t i c forms r = a0 ˆ2. / A. pd. ˆ 2 12