Xray Emission and Absorption
When an electron beam of energy around
20 keV strikes a metal target, two di®erent
processes produce xrays. In one process, the
deceleration of beam electrons from collisions
with the target produces a broad continuum
of radiation called bremsstrahlung (braking ra-
diation) having a short wavelength limit that
arises because the energy of the photon hc=¸
can be no larger than the kinetic energy of the
electron. In the other process, beam electrons
knock atomic electrons in the target out of in-
ner shells. When electrons from higher shells
fall into the vacant inner shells, a series of dis-
crete xrays lines characteristic of the target
material are emitted.
In our machine, which has a copper target,
only two emission lines are of appreciable in-
tensity. Copper K® xrays (¸ = 0:1542 nm)
are produced when an n = 2 electron makes
a transition to a vacancy in the n = 1 shell.
A weaker K¯ xray with a shorter wavelength
(¸ = 0:1392 nm) occurs when the vacancy is
¯lled by an n = 3 electron.
The reverse process of xray absorption by
an atom also occurs if the xray has either an
energy exactly equal to the energy di®erence
between an energy level occupied by an atomic
electron and a vacant upper energy level, or an
energy su±cient to eject the atomic electron
(ionization). For the xray energies and metals
considered in this experiment, the ionization
of a K-shell electron is the dominant mech-
anism when the xray energy is high enough,
which leads to the absorption of the initial
xray photon and the ejection of an electron|a
XDA 1
XDA 2 Advanced Physics Laboratory
sin d
2
d
Figure 1: The ray re°ected from the second plane
must travel an extra distance 2d sin µ.
process known as the photoelectric e®ect). If
an xray does not have enough energy to cause
a transition or to ionize an atom, the only
available energy loss mechanism is Compton
scattering from free electrons.
The Xray Di®ractometer
Thus, the spectrum of xrays from an xray
tube consists of the discrete lines superim-
posed on the bremsstrahlung continuum. This
spectrum can be analyzed in much the same
way that a visible spectrum is analyzed using
a grating. Because xrays have much smaller
wavelengths than visible light, the grating
spacing must be much smaller. A single crys-
tal with its regularly spaced, parallel planes
of atoms is often used as a grating for xray
spectroscopy.
The incident xray wave is re°ected spec-
ularly (mirror-like) as it leaves the crystal
planes, but most of the wave energy continues
through to subsequent planes where additional
re°ected waves are produced. Then, as shown
in the ray diagram of Fig. 1 where the plane
spacing is denoted d, the path length di®er-
ence for waves re°ected from successive planes
is 2d sin µ. Note that the scattering angle (the
angle between the original and outgoing rays)
is 2µ.
Constructive interference of the re°ected
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Figure 2: The xray di®raction apparatus.
waves occurs when this distance is an integral
of the wavelength. The Bragg condition for
the angles of the di®raction peaks is thus:
n¸ = 2d sin µ (1)
where n is an integer called the order of di®rac-
tion. Note also that the lattice planes, i.e., the
crystal, must be properly oriented for the re-
°ection to occur. This aspect of xray di®rac-
tion is sometimes used to orient single crystals
and determine crystal axes.
Our apparatus is shown schematically in
Fig. 2. The xrays from the tube are collimated
to a ¯ne beam (thick line) and re°ect from the
crystal placed on the sample table. The de-
tector, a Geiger-M?uller (GM) tube, is placed
behind collimating slits on the detector arm
which can be placed at various scattering an-
gles 2µ. In order to obey the Bragg condition,
the crystal must rotate to an angle µ when the
detector is at an angle 2µ. This µ : 2µ relation-
ship is maintained by gears under the sample
table.
Assuming d is known (from tables of crystal
spacings), the wavelength ¸ of xrays detected
at a scattering angle 2µ can be obtained from
May 18, 2009
Xray Di®raction and Absorption XDA 3
Eq. 1.
Powder Di®raction
An ideal crystal is an in¯nite, 3-dimensional,
periodic array of identical structural units.
The periodic array is called the lattice. Each
point in this array is called a lattice point.
The structural unit|a grouping of atoms or
molecules|attached to each lattice point is
called the basis and is identical in composi-
tion, arrangement, and orientation. A crystal
thus consists of a basis of atoms at each lattice
point. This can be expressed:
lattice + basis = crystal (2)
The general theoretical treatment for the
determination of the di®racted xrays|their
angles and intensities|was ¯rst derived by
Laue. Starting with Huygens principal, it in-
volves constructions such as the Fourier trans-
form of the crystal s electron distribution and
develops the concept of the reciprocal lattice.
The interested student is encouraged to ex-
plore the Laue treatment. (See Introduction to
Solid State Physics, by Charles Kittel.) How-
ever, we will only explore crystals that can be
described with a cubic lattice for which a sim-
pler treatment is su±cient.
There are three types of cubic lattices:
the simple cubic (sc), the body-centered cu-
bic (bcc) and the face-centered cubic (fcc).
The simple cubic has lattice points equally
spaced on a three dimensional Cartesian grid
as shown by the dots in Fig. 3. As a viewing
aid the lattice points are connected by lines
showing the cubic nature of the lattice.
The body-centered cubic lattice has lattice
points at the same positions as those of the
sc lattice and additional lattice points at the
center of each unit cube de¯ned by the grid.
The face-centered cubic lattice also has lattice
points at the same positions as those of the
y ( y
y (
(
Figure 3: The simple cubic lattice points (dots)
and connecting lines showing the cubic structure.
sc lattice but it has additional lattice points
at the center of each unit square (cube face)
de¯ned by the grid.
A primitive unit cell is a volume contain-
ing a single lattice point that when suitably
arrayed at each lattice point completely ¯lls
all space. The number of atoms in a primi-
tive unit cell is thus equal to the number of
atoms in the basis. (There are many ways of
choosing a primitive unit cell.) The primitive
unit cell for the sc lattice is most conveniently
taken as a cube of side a0. a0 is called the
lattice spacing or lattice constant.
The bcc and fcc primitive unit cells (rhom-
bohedra) will not be used. Instead, bcc and
fcc crystals will be treated using the sc lat-
tice and sc unit cell (which would then not be
pr