F hkl
ð Þ ¼ f Na :
exp2pi h:0 þ k:0 þ l:0
ð
Þ þ exp2pi h:
1
2 þ k:
1
2 þ l:0
À
Á
þ exp2pi h:0 þ k:
1
2 þ l:
1
2
À
Á þ exp2pi h:
1
2 þ k:0 þ l:
1
2
À
Á
"
#
þ f Cl :
exp2pi h:
1
2 þ k:
1
2 þ l:
1
2
À
Á þ exp2pi h:0 þ k:0 þ l:
1
2
À
Á
+ exp2pi h:0 þ k:
1
2 þ l:0
À
Á þ exp2pi h:
1
2 þ k:0 þ l:0
À
Á
"
#
¼ f Na : 1 þ exppi h þ k
ð
Þþexppi k þ l
ð
Þþexppi l þ h
ð
Þ
½
þ exppi h þ k þ l
ð
Þ f Cl : 1 þ exp À pi h þ k
ð
Þþexp À pi k þ l
ð
Þþexp À pi l þ h
ð
Þ
½
Since expðnpiÞ ¼ exp ÀðnpiÞ, therefore the above structure factor expression
reduces to
F hkl
ð Þ ¼ f Na þ f Cl :exppi h þ k þ l
ð
Þ
ð
Þ : 1 þ exppi h þ k
ð
Þþexppi k þ l
ð
Þþexppi l þ h
ð
Þ
½
The terms within the bracket [ ] are identical as in fcc, hence the same criteria
will apply. That is when h, k, l are unmixed indices then the sum (h + k), (k + l)
and (l + h) are all even integers and the structure factor of fcc will be
F hkl
ð Þ ¼ f 1 þ 1 þ 1 þ 1
½
¼4f and I / F(hkl)
j
j
2 ¼ 16f
2
Further, if indices hkl are mixed, it can be seen that
F hkl
ð Þ ¼ 0; and I / F(hkl)
j
j
2 ¼ 0
Now considering the complete equation for unmixed hkl indices, we can write
F hkl
ð Þ ¼ 4 f Na þ f Cl :exppi h þ k þ l
ð
Þ
f
g
¼ 4ðf Na þ f Cl Þ; if h þ k þ l
ð
Þ is an even integer;
¼ 4ðf Na À f Cl Þ; if h þ k þ l
ð
Þ is an odd integer
Similarly,
F hkl
ð Þ
j
j
2 ¼ f Na þ f Cl :cosp h þ k þ l
ð
Þ
f
g
2 þ f Cl :sinp h þ k þ l
ð
Þ
f
g
2
¼ 16 f Na þ f Cl
ð
Þ
2 ; if h þ k þ l
ð
Þ is an even integer;
¼ 16 f Na À f Cl
ð
Þ
2 ; if h þ k þ l
ð
Þ is an odd integer
The NaCl structure shows a similar behavior as fcc unit cell. Also, the sodium
ions are exactly in phase or exactly out of phase with the chlorine ions. However,
when they are out of phase they do not exactly cancel because the atomic scattering
factor of chlorine ion is much higher than that of sodium ion. Therefore, the
intensities of reflections from some unmixed indices of planes [when (h + k + l) is
an even integer] are increased and some others [when (h + k + l) is an odd integer]
are decreased.
8.2 Determination of Phase Angle, Amplitude …
319
ð Þ ¼ f Na :
exp2pi h:0 þ k:0 þ l:0
ð
Þ þ exp2pi h:
1
2 þ k:
1
2 þ l:0
À
Á
þ exp2pi h:0 þ k:
1
2 þ l:
1
2
À
Á þ exp2pi h:
1
2 þ k:0 þ l:
1
2
À
Á
"
#
þ f Cl :
exp2pi h:
1
2 þ k:
1
2 þ l:
1
2
À
Á þ exp2pi h:0 þ k:0 þ l:
1
2
À
Á
+ exp2pi h:0 þ k:
1
2 þ l:0
À
Á þ exp2pi h:
1
2 þ k:0 þ l:0
À
Á
"
#
¼ f Na : 1 þ exppi h þ k
ð
Þþexppi k þ l
ð
Þþexppi l þ h
ð
Þ
½
þ exppi h þ k þ l
ð
Þ f Cl : 1 þ exp À pi h þ k
ð
Þþexp À pi k þ l
ð
Þþexp À pi l þ h
ð
Þ
½
Since expðnpiÞ ¼ exp ÀðnpiÞ, therefore the above structure factor expression
reduces to
F hkl
ð Þ ¼ f Na þ f Cl :exppi h þ k þ l
ð
Þ
ð
Þ : 1 þ exppi h þ k
ð
Þþexppi k þ l
ð
Þþexppi l þ h
ð
Þ
½
The terms within the bracket [ ] are identical as in fcc, hence the same criteria
will apply. That is when h, k, l are unmixed indices then the sum (h + k), (k + l)
and (l + h) are all even integers and the structure factor of fcc will be
F hkl
ð Þ ¼ f 1 þ 1 þ 1 þ 1
½
¼4f and I / F(hkl)
j
j
2 ¼ 16f
2
Further, if indices hkl are mixed, it can be seen that
F hkl
ð Þ ¼ 0; and I / F(hkl)
j
j
2 ¼ 0
Now considering the complete equation for unmixed hkl indices, we can write
F hkl
ð Þ ¼ 4 f Na þ f Cl :exppi h þ k þ l
ð
Þ
f
g
¼ 4ðf Na þ f Cl Þ; if h þ k þ l
ð
Þ is an even integer;
¼ 4ðf Na À f Cl Þ; if h þ k þ l
ð
Þ is an odd integer
Similarly,
F hkl
ð Þ
j
j
2 ¼ f Na þ f Cl :cosp h þ k þ l
ð
Þ
f
g
2 þ f Cl :sinp h þ k þ l
ð
Þ
f
g
2
¼ 16 f Na þ f Cl
ð
Þ
2 ; if h þ k þ l
ð
Þ is an even integer;
¼ 16 f Na À f Cl
ð
Þ
2 ; if h þ k þ l
ð
Þ is an odd integer
The NaCl structure shows a similar behavior as fcc unit cell. Also, the sodium
ions are exactly in phase or exactly out of phase with the chlorine ions. However,
when they are out of phase they do not exactly cancel because the atomic scattering
factor of chlorine ion is much higher than that of sodium ion. Therefore, the
intensities of reflections from some unmixed indices of planes [when (h + k + l) is
an even integer] are increased and some others [when (h + k + l) is an odd integer]
are decreased.
8.2 Determination of Phase Angle, Amplitude …
319