176 5.3 X-Ray Tools
of a periodic arrangement of magnets transverse to the beam, with adjacent magnets on
each side of the beam arranged with alternating pole geometries (Figure 5.3a) and having a
period length parallel to the beam axis of usually a few tens of millimeters, which generate a
B-field amplitude of ~1 T. This causes a wiggle on the electron beam to generate a sinusoidal
electron path around the main beam axis such that the high curvature at the peak sinus
oidal amplitudes results in the release of synchrotron radiation generated toward the forward
beam axis direction, which is highly coherent. However, unlike a visible light laser, there are
no equivalent mirror for x-rays, which could be used to generate a resonant cavity (i.e., to
reflect the synchrotron radiation back along the undulator thereby amplifying the x-ray laser
output) so instead an extended undulator length is used up to a few meters.
The wavelength range of an XFEL currently is ~10−10 to 10−9 m, with an average brightness
~100 times greater than the most advanced synchrotron sources. However, since the magnets
in the undulator have a well-defined periodicity, the laser output is pulsed, with a pulse dur
ation of ~10−13 s, compared to an equivalent pulse duration of ~10−11 s for a synchrotron
source, and so the peak brightness of the XFEL can be several orders of magnitude greater.
This ultrashort pulse duration is having a significant impact into conventional x-ray crys
tallography for determining the structure of biomolecules in reducing the sample radiation
damage dramatically—the rapid pulse x-ray beam results in diffraction before destruction.
One such application is in X-ray pump–probe experiments. Here, ultrashort optical laser
pulses are directed onto crystal to generate transient states of matter, which can subsequently
be probed by hard x-rays. The fast pump rate of the XFEL (pulse duration of a few tens of
femtoseconds) enables time-resolved investigation, that is, more than one shot to be made
on the same crystal to monitor rapid structural dynamics. Also, as discussed in the following,
there is significant benefit from having a coherent x-ray source in obtaining direct phase
information from x-ray scattering atoms in a biomolecule sample.
5.3.2 X-RAY DIFFRACTION BY CRYSTALS
A 3D crystal is composed of a regular, periodic arrangement of several thousand individual
molecules. When a beam of x-ray photons propagates through such a crystal, the beam is
diffracted due to interference between backscattered x-rays from the different crystal layers. The
scattering effect is due primarily to Thompson elastic scattering, which results from the inter
action of an x-ray photon with a free outer shell valence electron, unlike electron scattering,
which is from atomic nuclei, and is also influenced mainly by the electron orbital density. The
angle of an emergent diffracted x-ray beam is inversely related to the length of separation within
the periodic structures involved in the scattering, in exactly the same way that was discussed for
electron diffraction, modeled by Bragg’s law discussed previously for electron diffraction.
The smallest repeating structure in a crystal is called the “unit cell,” and for the simplest
crystal shape, which is that of an ideal cubic crystal, the unit cell can be characterized by a
crystal lattice parameter a0 and the interplanar spacings, dhkl of planes, which are labeled by
Miller indices (h, k, l):
(5.10)
d
a
h
k
l
hkl =
+
+
0
2
2
2
Similar relations for dhkl exist for each different-shaped unit cells in a crystal (e.g., ortho
rhombic, tetragonal, hexagonal). As an example, for the cubic unit cell, the diffractive inten
sity maxima generated at angle θhkl satisfies
(5.11)
θ
λ
hkl
h
k
l
a
=
+
+
−
sin 1
2
2
2
0
2
Broadly, there are three practical methods for observing clear diffraction peaks from
crystalline samples. Some samples may consist of heterogeneous crystals, that is, are poly
crystalline, and the Debye–Scherrer method uses a monochromatic source of x-rays, which