The space group diagram is shown in Figure 1 where we can see that it is a primitive cell with 4 mirror lines within the unit cell in 2-D (2 horizontal and 2 vertical). There is 1 unique mirror plane. Also, there are 4 double glide planes (dashed/dotted lines: 2 horizontal and 2 vertical) by which there are a total of 2 unique glide planes. In a double glide plane, there is vertical and horizontal translation and a periodic repeat of the basis to produce a choral pair. In addition, it has 2 fold rotation points (aid) at the corners, centre and at the middle along the sides.

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The 2 fold rotation points are also centers of inversion. The second structure being studied is the tetragonal retile structure that is commonly found in metal dioxides and idealizes. With analysis of the tetragonal retile, there are 2 possibilities that the basis atoms (octahedron’s) can be arranged in 3-D space. One of the APP/MN occurs with the origin at mom and mom at In this case, there are 2 unique mirror lines that runs diagonally across the plane cell.

The dashed lines represents a glide line in the unit cell where there is reflection and translation of objects.

Root-inversion twofold screw axis with centre of yammerer is found at the centre and corners of the unit cell. Fourfold screw axis with inversion centers are observed to be at the middle of the cell parameters. In the other possible plane group of APP/MN with origin at mom and mom at the glide lines run diagonally across the plane cell. The fourfold screw axis with inversion centers are found in the middle and corners while the twofold screw axis with inversion centers are positioned along the middle of the cell parameters.

The temperature transition from the orthorhombic phase to the tetragonal phase occurs in the effective presence of changes in lattice parameter ND the tilting of octahedral. At higher temperatures, it is postulated that the atoms tend to occupy more space. The tilting of octahedral about twofold axis in the octahedral phase would effect in elongation of the a axis and the reduction of the b axis to form tetragonal phase. Figure 1: Space symmetry operators in APP/MN and Penn.

Paragraph 2 In both polymorph, the connections are the same, the central Ca has a coordination number of 6, the octahedron consists of 4 short bonds and 2 long bonds, all Ca-CLC bonds. However, in tetragonal structure, the long bonds are anger compared to the shorter bonds which is more similar to a tetragonal stress field as predicted by Kahn Teller effects. In Cacao and Caber, phase transformation from orthorhombic to tetragonal occurs at lower temperature. But for Crack, the phase transformation temperature occurs probably above its melting temperature, hence no tetragonal Crack is possible.

The higher the temperature, according to Kahn Teller effects, the long bonds will elongate so as to achieve stability (tetragonal considered as long octahedron’s is more stable than short octahedron’s) and the octahedral tilt as a result, when it reaches zero, t would have zero orthorhombic strain. At the same time, octahedral tilt of zero leads to the tetragonal structure. However for Crack, the high coupling constant K forbids it to reach tetragonal structure before melting. The first ATOMS drawing (Figure 2(a)) is the tetragonal retile structure, and the second the orthorhombic calcium chloride (Figure 2(b).

The main similarity would be the octahedral in the middle and the 900 unit cells. The main differences would be the lattice parameters. For tetragonal, a=b, and for orthorhombic, a is slightly larger than b, although it might not be obvious from the drawing. One can also notice with figures that the octahedral in retile structure is more of a octahedral that is having very similar apical and equatorial bond lengths and the octahedron in tetragonal is more of a tetragonal, that is having a significantly longer apical bond that equatorial bond.

The orthorhombic structure has lattice parameters all not equals to each other. However, as temperature changes, lattice parameters also change such that a=b, and that forms our tetragonal structure. In orthorhombic Cacao structure, in terms of angstroms, a=6. 45 A, b=6. 25 A, c=4. 18 ?, while a=b=6. ?, c=4. 21 ? in tetragonal structure. The c increased from orthorhombic to tetragonal, but a decrease with b increasing to meet and form tetragonal. The point where bond length difference is the biggest is just below the phase transition.

The apical (longer) bond increases and equatorial (shorter) bond decreases such that the difference increase to a maximum where the tetragonal structure is formed, that is also where the orthorhombic strain is the lowest as the octahedral is more of a tetragonal shape hence more stable according to Kahn Teller effect. Octahedral tilting from 5. Degrees for orthorhombic Cacao and 8. 5 degrees for orthorhombic Caber to zero in tetragonal Cacao and Caber. Figure 2: (a) On right tetragonal APP/MN Tie structure, and (b) on left, orthorhombic Penn Cacao structure.

Paragraph 3 With increasing temperature, lattice parameter ‘a’ and ‘b’ of Cacao and Caber converges to a single value while ‘c’ increases steadily. Phase transformation takes place as the cell structure changes from orthorhombic to tetragonal. The octahedral shape, tilting orientation and distortion must also vary with temperature. We can hence find the quadratic solution to approximate the imperative dependence by using the general solution w(T) = a(T)2 + b(T) + c. Exact values of a & b can be approximated using system of equation method. Using plotted values of Cacao for reference: Substituting (0,6. ) c=6. 2 Substituting (240,0) + b(240) +6. 2 Substituting (150,4) + b(150) + 6. 2 Solving (1) & (2) simultaneously, we obtain the values a = -1. EYE-04 & b = -0. 004 By ignoring coefficient ‘b’ attached to the T term with power 1 , we obtain the new quadratic solution to be w(T) = -1. EYE-24TH + 6. 2 within an allowable range of [0 < T < 2400C l. The approximation of temperature dependence of CaC12 w a 1. 24E-04 T2 +6. 2 The approximation of temperature dependence of CaBr2 is roughly similar to that of CaC12 as the plotted graph curvature is similar with different temperature ranges (Figure 3).

The relationship between bond length of Cacao and temperature is noticeably minute. Theoretically, bond length will increase with temperature as increased thermal vibrations results in the mean equilibrium interaction distance to widen. A small increase of less than 0. 02 ? can be observed within a range of temperatures from 100-ICC. Lattice parameters of Cacao, however, changes to a large extent from 100-ICC with a’ decreasing from 6. 42 to 6. 38 , ‘b’ increasing from 6. 29 to 6. 38 A. If the changes in lattice parameters is purely influenced by changing bond lengths, ‘a’ & ‘b’ should be increasing simultaneously.

This, therefore, suggests a process of phase transition for both Cacao and Caber. Cacao transform at ICC and Caber transform at ICC. Based on Equation (2) & (3), ‘a’/b’ can be expressed as / However, Crack does not follow this temperature dependence. The apical, equatorial bond lengths, & lattice parameters all increases steadily with temperature. This could indicate that the increase in lattice parameters s only due to increased bond lengths caused by greater thermal vibrations at higher temperatures.