ABSTRACT
The condition j cos'j 1 implies n = f1; 0; 1; 2; 3g: By into the former expression we obtain the solutions listed
TABLE 1.1 Allowed symmetry axes in periodic crystals.
n ' AXIS -1 2-fold 0 2/3 3-fold 1 /2 4-fold 2 /6 6-fold 3 0 identity
The simplest illustration of processes which are simultaneously periodic in space and time can be found in wave phenomena. For instance, sinusoidal waves of the form (r; t) = 0 sin(k:r!t), where k = 2= is the wave number and measures the wavelength, often occur in waves propagating in gases, liquids or solids as well as in electromagnetic waves propagating in vacuum. Their characteristic wave function describes a periodic pattern in space if we …x the time variable (i.e., t t0). Alternatively, if we …x the space variable (i.e., r r0), it describes a harmonic motion in time at every point of space, where the quantity k:r0 measures the relative dephasing between the oscillations of two points separated by a distance r0. The double periodicity (in space and time) of wave motion can be traced back to the very structure of the corresponding wave equation, which reads
r2 + 1 c2 @2
@t2 = 0; (1.5)
where c = !=k is the phase velocity of the wave. The …rst (second) term in Eq.(1.5) describes the periodicity in space (time) of the propagating wave, while its phase velocity couples its spatial pattern to its propagation rhythm. A key feature of sinusoidal waves, signi…cantly contributing to pervade pe-
riodic thinking in scienti…c thought, is that any non-sinusoidal, periodic wave can be represented as a collection of sinusoidal ones (with di¤erent frequencies) blended together in a weighted sum of the form [cf. Eq.(1.2)]
f(t) = a0 2 +
[am cos(!mt) + bm sin(!mt)] ; (1.6)
where !m = 2m=T; and
am = 2
T
Z f(t) cos(!mt)dt; bm =
T
Z f(t) sin(!mt)dt; (1.7)
in
Fourier coe¢ cients, after the French mathematician Joseph who introduced this procedure in 1822. Closely related to one can consider the so-called Fourier transform, which into a continuous spectrum of its frequency components expression
F (!) =
f(t)ei!tdt: (1.8)
analogous expressions hold for periodic functions in replacing the corresponding variable in Eqs.(1.6)-(1.8). In this way, a Fourier transform can be envisioned as a linear transformation relating two di¤erent mathematical domains: that corresponding to usual time or space variables (which come closer to our everyday experience), and that corresponding to the related frequency or reciprocal space spectrum, which encloses a more abstract view of the underlying order in the considered phenomenon. Remarkably enough, there exist processes in Nature able to Fourier-transform material structures in a natural way, namely, di¤raction of electromagnetic (x-ray) or matter quantum waves (electrons, neutrons) by atomic scatters in condensed phases. The resulting di¤raction spectra exhibit regular arrangements of bright spots (the so-called Bragg peaks) disclosing the abstract information encoded within Fourier space to our eyes. In this way, the workings of Nature translate wave motion into geometrical patterns engraved in reciprocal space through the orchestrated interaction of matter and energy in condensed matter.
