periodic function as a series of
trigonometric functions. Thus,
f(x) = a 0 + (a 1 cosx + b 1 sinx) +
(a 2 cos2x + b 2 sin2x)+...,where a 0 , a 1 , b 1 , b 2 , etc., are con-
stants, called Fourier coefÜcients. The
series wasÜrst formulated by Joseph
Fourier and is used in *Fourier analy-
sis.
Fourier transformAn integral
transform of the type:
F(y) =∫∞- ∞f(x)e
- xydy.
The inverse is:
f(x) = (1/2π)∫∞- ∞F(y)e
ixydy.Fourier transform techniques are
used in obtaining information from
spectra, especially in NHR and in-
frared spectroscopy (see fourier-
transform infrared).
Fourier-transform infrared (FT-IR)
Infrared spectroscopy in which com-
puters are part of the spectroscopic
apparatus and use *Fourier trans-
forms to enable the curve of inten-
sity against wave number to be
plotted with very high sensitivity.
This has allowed spectra to be ob-
tained in the far infrared region; pre-
viously it was difÜcult to attain
spectra in this region as the resolu-
tion was obscured by the signal-to-
noise ratio being too high to resolve
the vibrational and/or rotational
spectra of small molecules in their
gas phase. FT-IR has been used in re-
search on the atmosphere. Another
application of this technique is the
detection of impurities in samples of
condensed matter.
four-level laser A laser in which
four energy levels are involved. The
disadvantage of a three-level laser is
that it is difÜcult to attain population
inversion because many molecules
have to be raised from their ground
state to an excited state by pumping.
In a four-level laser, the laser transi-
tionÜnishes in an initially unoccu-
pied state F, having started in a state
I, which is not the ground state. As
the state F is initially unoccupied,
any population in I constitutes popu-
lation inversion. Thus laser action is
possible if I is sufÜciently metastable.
If transitions from F to the ground
state G are rapid, population inver-
sion is maintained since this lowers
the population in F caused by the
transition in the laser action.fractalA curve or surface generated
by a process involving successive sub-
division. For example, a snowÛake
curvecan be produced by starting
with an equilateral triangle and di-
viding each side into three segments.
The middle segments are then re-
placed by two equal segments, which
would form the sides of a smaller
equilateral triangle. This gives a 12-
sided star-shapedÜgure. The next
stage is to subdivide each of the sides
of thisÜgure in the same way, and so
on. The result is a developingÜgure
that resembles a snowÛake. In the
limit, thisÜgure has ‘fractional di-
mension’ – i.e. a dimension between
that of a line (1) and a surface (2); the
dimension of the snowÛake curve is
1.26. The study of this type of ‘self-
similar’Ügure is used in certain
branches of chemistry – for example,
crystal growth. Fractals are also im-
portant in *chaos theory and in com-
puter graphics.fractionSee fractional distilla-
tion.
fractional crystallizationA
method of separating a mixture of
soluble solids by dissolving them in a
suitable hot solvent and then lower-
ing the temperature slowly. The least
soluble component will crystallize
outÜrst, leaving the other compo-
nents in solution. By controlling the
temperature, it is sometimes possible
to remove each component in turn.233 fractional crystallization
f