A mathematical construction is given for arbitrarily many distinct crystal structures all of which would give the same diffraction pattern. A. L. Patterson's concept of homometric sets is analyzed, and examples are given in one, two and three dimensions.Let al, a2,a3 be linearly independent vectors in threedimensional space. Let A be the three-by-three matrix whose columns are the aj. The vectors aj determine a lattice of points An = nlat + n2a2 + n3a3(1)where the nj are integers. The basic cell of the lattice is the set of points x = ~al + ~za2 + ~3a3 with 0 < #j < 1 .(2)The reciprocal lattice has the matrixIts columns bj satisfyThe reciprocal lattice consists of the points Bh, where the hj are integers (called Miller indices).Let the atoms in a crystal be located at rt,...,rN in the basic cell (2) and at all congruent points rj + An. By X-ray analysis, one tries to find the positions rj.
The solid-state phase transitions of the KNO 3 -NH 4 NO 3 solid solutions have been determined by high temperature X-ray diffractometry, and lattice parameter calculation has also been performed. Ammonium nitrate ͑AN͒ is of great use for gas generators of automobile air bag systems. The X-ray diffraction results showed the single ͑AN͒ phase III from 5% to 20% KNO 3 in NH 4 NO 3 and up to 373 K, which is the important temperature range for the air bag gas generator applications. The X-ray diffraction patterns of the low temperature KNO 3 phase ͑KN II͒ are from 92% to 100% KNO 3 composition range and up to 393 K temperature. The high temperature KNO 3 phase ͑KN I͒ showed very broad composition range from 20% up to 100% KNO 3 at various temperature ranges. The lattice parameters of the NH 4 NO 3 -rich ͑AN III͒ and KNO 3 -rich ͑KN II and KN I͒ solid solutions have been calculated at different temperature range. The volumes of AN III phase decrease from 0.3201͑4͒ to 0.3166͑1͒ nm 3 at room temperature and from 0.3250͑6͒ to 0.3215͑3͒ nm 3 at 373 K as the compositions increase from 5% to 20% KNO 3 . The lattice constants of the hexagonal KN I phase show that there is no significant change in a direction when the temperature increases. Details of X-ray results, lattice expansions, and equations during heating are presented. © 2005 International Centre for Diffraction Data.
The solid-state phase transitions in ammonium nitrate (AN)-potassium nitrate (KN) system, and the equilibrium AN-KN phase diagram have been determined by using differential scanning calorimetry and high-temperature in situ x-ray diffractometry. Sample preparation was performed in a special "dry room" with very low humidity. A single phase region (AN III) with no phase transitions to 373 K was observed in the composition range 5 to 20% KN; this is critical for use in air bag gas generators. The high-temperature KN phase (KN I) has a wide range of stability from 20 to 100 wt.% KN. There are one eutectic, two eutectoid, three peritectoid, and one congruent transformations in this phase diagram. Two new nonstoichiometric phases were found at lower temperatures in the mid-composition range between the AN and KN terminal solid solutions. Details of the phase equilibria are presented.
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