Chemistry Reference
In-Depth Information
Figure 4.26. The incident (
S
o
is the unit vector of the incident wave-normal) and
diffracted (“reflected”,
S
is the unit vector of the reflected wave-normal) X-ray beams
on two parallels crystal lattice planes (
hkl
) separated by a distance
d
hkl
. The vector
b
=
S
−
S
0
is normal to the lattice plane (
hkl
) with
|
b
|
=
|
s
−
s
0
|
=2sin
θ
.
relations are expressed in vector notations as below:
a
∗
·
b
=
a
∗
·
c
=
b
∗
·
a
=
b
∗
·
c
=
c
∗
·
a
=
c
∗
·
b
=0
The magnitudes of the reciprocal vectors are the reciprocal of the spacing of
the corresponding planes in the real lattice, i.e.,
a
∗
·
a
=
b
∗
·
b
=
c
∗
·
c
=1.
It means that
|
a
∗
|
is the reciprocal of the spacing of the
a
planes of the
real crystal lattice;
|
b
∗
|
is the reciprocal of the spacing of the
b
planes of
the real crystal lattice; and
|
c
∗
|
is the reciprocal of the spacing of the
c
planes of the real crystal lattice. Thus, as shown in Figure 4.27, if PO
is the vector of length 1
/λ
in the direction of the incident wave-normal
(
S
o
/λ
), and PQ the vector (
S
/λ
) of the diffracted wave in the direction
of a maximum diffraction, OQ must be the vector (
b
/λ
) normal to one of
the lattice planes (
hkl
) with a magnitude of (2
/λ
)sin
θ
. When the Bragg
condition 2d
hkl
sin
θ
=
λ
is fulfilled, (2
/λ
)sin
θ
is equal to 1/
d
hkl
.OQis
therefore in the direction of a vector in the reciprocal lattice, and is equal to
the vector in magnitude (1/
d
hkl
). If the vector PO is drawn so that O lies
at the origin of the reciprocal lattice, Q must lie at another point (
h
,
k
,
l
)
of this lattice. In other words, if a sphere is drawn with P as the center and
1
/λ
as the radius, both the origin O and another point Q of the reciprocal
lattice must lie on the sphere. Otherwise there will be no diffraction.
For certain orientations of the crystal relative to the direction of the inci-
dent X-rays, there will be no reciprocal lattice points lying on the surface of
the diffraction sphere, and therefore no diffraction from the crystal planes.
