Cell, unit interplanar spacing

Bragg s equation gives an easy way to understand XRPD. Powder X-ray diffraction data collected on crystalline samples gives information about peak intensities and peak positions. Peak intensities are determined by the contents of unit cells, and peak positions are closely related to the cell constants. Interplanar spacing is a function of Miller indices and cell constants. Therefore, if the cell constants are known for a crystalline compound, peak positions corresponding to Miller indices can be obtained from the Bragg s equation the wavelength. A, is machine-specific. The determination of cell parameters in structure determination of XRPD pattern is a reverse process to find cell constants from peak positions. Here, note that the cell constants for a unit cell are not affected by the contents in the unit cell. The contents in the unit cell have effects on the peak intensities. 

Now envision this lattice of imaginary points in the same space occupied by the crystal. For a small real unit cell, interplanar spacings dhkl are small, and hence the lines from the origin to the reciprocal lattice points are long. Therefore, the reciprocal unit cell is large, and lattice points are widely spaced. On the other hand, if the real unit cell is large, the reciprocal unit cell is small, and reciprocal space is densely populated with reciprocal lattice points. 

The reciprocal lattice is spatially linked to the crystal because of the way the lattice points are defined, so if we rotate the crystal, the reciprocal lattice rotates with it. So now when you think of a crystal, and imagine the many identical unit cells stretching out in all directions (real space), imagine also a lattice of points in reciprocal space, points whose lattice spacing is inversely proportional to the interplanar spacings within the crystal. 

X-ray diffraction photographs were recorded on flat film using a camera with pinhole collination (250 ym diam.) and Ni-filtered Cu Ko radiation. Calcite (characteristic spacing 0.3035 nm) was used as an internal diffraction calibration. Interplanar spacings were measured off the original photographs and the unit cell dimensions were refined, where possible, by a least-squares procedure. 

It may be recalled that an alternative description for a crystal stmcture can be made in terms of sets of lattice planes, which intersect the unit cell axes at ua, VU2, and was-The reciprocals of the coefficients are transformed to the smallest three integers having the same ratios, h, k, and I, which are used to denote the plane (hkl). Of course, the lattice planes may or may not coincide with the layers of atoms. Any such set of planes is completely specihed by the interplanar spacing, dhU7 and the unit vector normal to the set, since the former is given by the projection of, for example, u ui onto n kh that is dhki = u ui- n ki- The reciprocal lattice vector is defined as ... 

The number of Bragg reflections N that can be measured to any particular interplanar spacing, dmjn, depends upon the volume of the unit cell. 

The Powder Diffraction File ° contains data on single-phase X-ray powder diffraction patterns. These are stored as lists of interplanar spacings dhki and relative intensities characteristic of each compound, as shown in Figure 16.5. Unit cell dimensions, space groups and densities are also listed. 

If one knew how atoms were disposed about many families of planes of different spacings and orientations, the contents of the unit cell could, in principle, be mapped in considerable detail by recombination of all the planes in a suitable fashion. The level of detail that resulted (i.e., the resolution) would be a function of the smallest interplanar spacing, or... 

The summation of exponential terms on the right is a Dirac delta function, a discrete function, which is everywhere zero except when the argument is zero or integral. The summation on the left is a continuous function, which determines the value of the entire transform at those nonzero points. Now d ki is normal to the set of planes of a particular family, and d ki I is the interplanar spacing. In order for dhu s = 1, s must be parallel with dhki and have magnitude 1/ Smreciprocal lattice vector. If s h, then there is destructive interference of the waves diffracted by different unit cells, and the resultant wave from the crystal is zero. The elements of the diffraction spectra, the structure factors, for the crystal can therefore be written as... 

From this one can see that the perpendicular distance between the b-c planes on adjacent unit cells along a is the reciprocal of a. We can denote the stack of these b-c planes by their respective intercepts on the abc triplet of real space axes as (a/1, b/0, c/0), or simply (100). Similarly, another stack of planes can be constructed by selection of reciprocal integer intercepts (a//z, b/k, cjl) and denote them with just the denominators hkt). These are the Miller indices for this stack of planes they correspond to a lattice point in reciprocal space coincident with the vector h which has an associated structure factor Fy, as defined in Equation (15). Because of the connection between the real and reciprocal lattices, the vector h is perpendicular to its respective stack of planes and the interplanar spacing is the reciprocal of the magnitude of h this is the (i-spacing for these planes. 

V =unit cell volume, tli is the interplanar spacing of (hikih) and d2 is the interplanar spacing of (hikih) and... 

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