Introduction Solid State 2021 Assignment 1694786473 PDF

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this is a solid state assignment where you need to need to find the patterns in the metals. you have been assigned two metals....

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CHEM2001/3 INTRODUCTION TO SOLID STATE CHEMISTRY ASSIGNMENT DUE: MAY 28, 2021 INSTRUCTIONS: PLEASE READ THE OUTLINE, COMPLETE THE ASSIGNMENT AND ANSWER ALL OF THE QUESTIONS THAT FOLLOW

PART 1: INDEXING AND UNIT CELL IDENTIFICAATION OF METALS AND SALTS: THE CALCULATION OF METALLIC RADII FROM POWDER X-RAY DIFFRACTION PATTERNS

X-RAY DIFFRACTION OF POWDER SAMPLES THEORY: X-RAY DIFFRACTION OF POWDER SAMPLES Crystals can be seen as three-dimensional, repeating patterns of atoms. The smallest repeating unit is known as a unit-cell, and may be arbitrarily chosen:

Figure 1 Three-dimensional point lattice showing three different choices of unit cell.

The dimensions of the unit cell are determined by the symmetry contained within the crystal, and are accordingly assigned to one of seven crystal systems and one of fourteen Bravais lattices. This is very well

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illustrated by Figure 16.6 on page 770 of Laidler and Maiser. The three unit cells shown in Figure 1 are thus primitive triclinic, while the third is a B-centred orthorhombic cell. The latter cell is clearly the best choice as the angles are all 90, thus simplifying calculations. The points in a crystal lattice lie in planes that are identified by their (Miller) index, hkl, which consists of three positive or negative integers. Please consult your notes for further examples and explanations of Miller planes. The indices are assigned by first expressing the distance between two of the set of planes cutting a unit cell edge as a fraction of that unit cell length, and then calculating the inverse. Therefore, the indices of the sets of planes shown below in Figure 2 are (123), (312), (103), (330) and (030). Note: If a plane is parallel to a unit cell edge the distance between two of the planes in that direction will be infinite, with the inverse (and hence the index in that direction) being equal to 0.

Figure 2 Sets of planes in a unit cell, illustrating the assignment of indices. 2

If the crystal is irradiated with X-rays, the X-rays will only diffract from a set of planes at a specific angle of incidence of the radiation, namely the Bragg angle,



 d d

The diffraction condition is described by Bragg's law: n = 2d sin

[EQUATION 1]

where λ is the wavelength of the radiation, d is the interplanar spacing and n is an integer, the order of the reflection usually taken as 1. Thus, if the wavelength and Bragg angle for a specific set of hkl planes are known it is possible to calculate the distance, d, between two of those planes. In a crystalline sample the crystallites are randomly distributed, which means that with the large number of very small crystallites in the sample some will be orientated such that diffraction will occur when the angle of incidence equals a Bragg angle. For every small crystal that is correctly orientated a small spot of diffracted X-rays will be observed, so with a large number of crystallites a cone of diffraction will be obtained:

Figure 3 The same set of planes in two crystallites with different orientation around the direction of the incident beam forming the cone of diffraction 3

Figure 3 also shows that the reflection occurs at an angle of 2Ɵ when the incident beam is at an angle of Ɵ with respect to the sample. In the Debye-Scherrer method (also known as the Debye-Hull-Scherrer method) a length of X-ray sensitive photographic film is placed inside a Debye-Scherrer camera. Since the camera is equipped with a collimator to focus the X-rays and a beam-stop to "catch" the undiffracted beam (only a very small portion of the beam is actually diffracted) two holes are punched in the film to accommodate these. For an example of the setup of a Debye-Scherrer camera see Figure 16.16 on p 781 in Laidler and Maiser. The radius is chosen such that each millimetre of film is equivalent to a Bragg angle of either 1 or 0.5 (depending on the camera). However, shrinkage of the film during developing is possible leading to an error in the conversion from length to degrees, thus a correction should be made. These detectors have since been replaced with silicon or germanium doped with lithium (Si(Li) or Ge(Li)) semiconductor detectors. These detectors convert X-ray photons to electron-hole pairs in the semiconductor and are collected to detect the X-rays. This approach makes the detectors more sensitive and capable of acquiring data much faster than film-based detectors. These detectors are usually coupled to X-ray diffractometers which encompass a radiation proof enclosure for the X-ray source, generator, sample, goniometer and all other components (Figure 4).

Figure 4 X-ray powder diffractometer setup inside its radiation safe housing For further information visit: https://www.youtube.com/watch?v=ZYzKd2qMn1o

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UNIT CELLS Bragg's law can be used to obtain the lattice spacing of a particular cubic system. The interplanar

spacing, d, for a set of hkl planes is related to the unit-cell lengths, and for a cubic cell with all sides equal and all angles being 90 this relationship is: 𝑑=

𝑎 √ℎ2 +𝑘 2 +𝑙 2

[EQUATION 2]

where a is the unit cell length. Clearly, the unit cell length can be determined by combining equations [1] and [2] to obtain:

𝑠𝑖𝑛2 𝜃 =

𝜆2 (ℎ 2 4𝑎2

+ 𝑘 2 + 𝑙2 )

[EQUATION 3]

Thus, for each hkl reflection the unit cell length, a, may be calculated using the Bragg angle. As a result of symmetry some hkl reflections are absent in face (F) or body (I) centred lattices (also known as FCC and BCC), which are present in primitive cells. This is known as a systematic absence or extinction when the structure factor is zero. The structure factor is a mathematical function describing the amplitude and phase of a wave diffracted from crystal lattice planes characterised by Miller indices h, k, l due either to the centring of the lattice or to the presence of additional symmetry elements. In simpler terms, the structure factor describes how the atomic arrangement influences the intensity of a scattered X-ray beam. It tells us which reflections (i.e. peaks, hkl) to expect in a diffraction pattern from a given crystal-structure with atoms located at position u, v and w within the crystal. Equation 2 includes the sum (h2 + k2 + l2) where h, k, l are integers. The sum (h2 + k2 + l2) include instances that do not produce all possible numbers (integers). For example, no combination of integers can be found where the sum of the squares is equal to 7 or 15, and these values are therefore "forbidden". Therefore, all cubic cells are characterised by the absence of the 7th and 15th lines. Table 2 includes a list of all of the (h2 + k2 + l2) values calculated for the FCC, BCC and PC crystal systems.

MY SCOPE For an interactive X-ray diffractometer showing the basic operation and sample preparation please visit the virtual X-ray diffractometer at: https://myscope.training/legacy/xrd/practice/virtualxrd/. This is a very useful visualization tool. Please good make use of it. 5

Table 1 (h2 + k2 + l2) values for cubic unit cells

(h2 + k2 + l2)

hkl P

Peak No.

100 110 111 200 210 211

1 2 3 4 5 6

1 2 3 4 5 6

220 300, 221 310 311 222 320 321

8 9 10 11 12 13 14

7 8 9 10 11 12 13

I

Peak No.

F

SYST. ABSENCE SYST. ABSENCE 3 4 2 4 SYST. ABSENCE SYST. ABSENCE 6 3 SYST. ABSENCE 2 2 2 7 IS FORBIDDEN: (h + k + l ) ≠ 7 8 4 8 SYST. ABSENCE SYST. ABSENCE 10 5 SYST. ABSENCE SYST. ABSENCE 11 12 6 12 SYST. ABSENCE SYST. ABSENCE 14 7 SYST. ABSENCE 2 2 2 15 IS FORBIDDEN: (h + k + l ) ≠ 15 16 8 16

Peak No.

SYST. ABSENCE 2 1

1 2

3

4 5

400 16 14 6 etc P = PRIMITIVE CUBIC; I = BODY CENTRED; F = FACE CENTRED; SYST ABSENCE – SYSTEMATIC ABSENCE – PLEASE SEE LECTURE NOTES SET 4 HOW TO INDEX AN X-RAY POWDER DIFFRACTION PATTERN To solve a powder X-ray diffraction pattern (PXRD), students have to set up a table and solve Braggs law for a cubic crystal system. This is done by splitting Braggs law into separate mathematical terms: 𝑠𝑖𝑛2 𝜃 =

𝜆2 2 (ℎ + 𝑘 2 + 𝑙 2 ) 4𝑎 2

This equation cab be rewritten as: 𝜆2 𝑠𝑖𝑛2 𝜃 = 2 2 4𝑎 (ℎ + 𝑘 2 + 𝑙 2 ) [EQUATION 4] The wavelength, λ is specific to a particular metal anode. Copper (CuKα1) with a wavelength of λ =

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1.5406 Å and Molybdenum (MoKα1) with a wavelength of λ = 0.70930 Å. The angstrom (Å) is the conventional length scale used for X-ray wavelengths. For a given diffraction pattern do the following: 1. Number the peaks in the diffraction pattern. Stop at 10 if the diffraction pattern has numerous peaks. This is the peak number (Peak No.) listed in Table 1. Since this method is dependent on the 2  value care should be taken to ensure that it is obtained as accurately as possible. Use at least 3 significant figures. 2. For each peak in the PXRD pattern, determine the respective 2-theta (2Ɵ) values. 3. Divide the 2Ɵ value by two to get Ɵ. 4. Calculate sin2 for each Ɵ value. 5. Use the values in Table 1. For each of crystal system namely P, I and F a column of values of the sum (h2 + k2 + l2) are listed. Divide the sin2θ by the (h2 + k2 + l2) value. The peak numbers listed Table 1 corresponds to the integer value (h2 + k2 + l2) that is to be used for the calculation in step 5. Therefore peak 1 will have a different (h2 + k2 + l2) value for each crystal system as will peak 2 etc. This is due to systematic absences. See your lecture notes! 6. Repeat the calculation in step 5 for each crystal system (P, I and F) listed in Table 1. 7. One column of values should produce a set of values that are approximately constant. 8. Take the average of the column of values that is approximately constant. This allows one to identify the cubic crystal system that is represented by the powder diffraction pattern. 9. The wavelength, λ, of the X-ray source is known. Therefore, equation 4 (shown above) can be solved for a, the unit cell length.

ASSIGNMENT Each student has been assigned (link) two different PXRD patterns (link) that were measured on unknown metal samples. For these PXRD patterns use the procedure outlined above to: 1. Identify the cubic crystal systems of the samples represented by the two diffraction patterns assigned to you (please see the list). 2. Calculate the unit cell constants of the samples represented by these diffraction patterns. 3. Using the data obtained in answering Questions 1 and 2, determine the positions of the metal atoms in the unit cell and hence calculate the metallic radius of the metal. PLEASE NOTE: STATE YOUR ANSWER IN ANGSTROM (Å) 4. Using a list of metallic radii and your knowledge of the positions of atoms in the different types of cubic unit cells, determine the probable identity of the unknown metals.

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(Please see: https://environmentalchemistry.com/yogi/periodic/atomicradius.html)

QUESTIONS 5. Peaks with a larger 2θ angle (for a given wavelength, say CuKα1), are found to be more accurate for unit cell determinations. Explain why this is the case. 6. In the theory section it was stated that the crystallites in a powder sample are randomly orientated. This is however not always true. Thinking of the shapes and forms of crystals, what can cause the crystallites not to be randomly orientated? What effect can this have on the powder diffraction pattern? 7. Considering the method set out in the theory section here for indexing the diffraction pattern (leading to Equation 3), explain how the equations would have to change for the case of an orthorhombic unit cell.

QUESTION MARK ALLOCATION FOR QUESTIONS 1 20 MARKS: 10 MARKS PER DIFFRACTION PATTERN SOLVED 2 3 4 5 4 MARKS 6 3 MARKS 7 3 MARKS REFERENCES 1. K.J. Laidler, J.H. Meiser "Physical Chemistry" 3rd Ed. (1999) Houghton Mifflin Company, Boston. 2. P.W. Atkins "Physical Chemistry" 5th Ed. (1994) Oxford University Press, Oxford.

SUGGESTED READING •

Powder Diffraction: Theory and Practice. Editors: RE Dinnebier, SJL Billinge. (2008) [ISBN: 9781847558237]. (Available on Wits library online)

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Explanation of Bragg’s Law: http://skuld.bmsc.washington.edu/~merritt/bc530/bragg/

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X-ray sources: https://www.ruppweb.org/Xray/x-ray_sources.html

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Systematic absences: Please refer to your lecture notes.

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