Smithells Metals Reference Book Part 2 ppsx

Ir
77
Pt
78
Au
79
Hg
80
TI
81
Pb
82
Bi
83
Po
84
At
85
Rn
86
Fr
87
Ra
88
Ac
89
Th
90
Pa
91

U
92
63.40
46.M
65.68
47.82
67.84
49.48
69.90
51.07
71.77
52.52
73.74
54.07
76.60
56.26
79.42
58.44
81.40
60.05
82.38
60.91
83.36
61.77
85.43
63.47
86.25
64.30
88.04
65.81

90.63
67.91
93.18
69.94
71.32
52.10
73.93
54.02
76.t9
55.87
78.46
57.62
80.49
59.23
82.63
60.93
85.78
63.37
88.86
65.77
90.97
67.53
91.97
68.44
92.98
69.35
95.17
91.20
95.95
72.04

97.82
73.66
88.60(i1)
89.92(1,)
78.17(L)
75.95(L’)
80.33
58.98
83.09
61.12
85.67
63.16
88.15
65.11
90.37
66.87
92.67
68.75
96.13
71.44
99.50
74.1
1
76.00
76.96
77.92
79.91
80.75
82.48(L)
72.19(i1,)

74.20(1,,)
87.38(L)
101.7
102.7
103.7
106.0
106.7
97.75(i1)
84.95(L)
115.6
119.2
122.5
125.7
128.4
131.2
135.7
103.0
139.9
106.5
108.8(L)
122.5(1,)
109.8(L)
110.8(L)
113.1(L)
86.32
89.24
91.96
94.51
96.90
99.31

122.1(1,)
91.18(In)
95.03(1,,)
97.50(1,)
102.1(1,)
100.3(In)
75.07(l,,)
I03.2(1,,)
77.4(111)
105.9(i,,)
79.70(1,,)
168.4
131.7
178.5
140.0
188.3
148.1
198.8
156.8
206.7
163.6
215.4
171.1
226.8
180.0
237.7
190.3
249.2
200.2
249.0

201.1
258.8
210.1
266.6
217.8
273.6
224.7
278.6
230.6
290.1
240.2
294.9
246.7
201.9
155.3
213.7
164.8
224.9
173.9
237.1
183.8
245.7
191.2
255.6
199.6
268.3
210.3
280.3
220.6
293.1

231.6
291.9
231.7
302.2
241.1
309.9
248.9
314.6
255.8
320.5
261.1
332.8
272.2
336.7
275.2
-
*Reproduced
by
permission
from
the
International Tables for X-ray Crystallagrapby.
242.9
188.3
256.3
199.4
269.2
210.0
283.2
221.6

292.5
229.9
303.1
239.2
317.4
251.5
330.3
263.1
344.4
275.5
341.5
274.7
351.8
284.8
358.7
292.7
364.3
299.5
366.1
304.0
316.0
320.3
*
4
9
378.6
E
25
380.7
5

e
292.8
229.1
308.2
242.0
322.7
254.3
338.6
267.7
348.2
276.8
359.5
287.2
375.0
301.0
388.5
313.7
403.4
327.4
397.8
325.1
407.2
335.5
412.0
342.7
415.4
348.8
353.7
279.6
370.9

294.5
386.9
308.6
404.8
324.0
414.0
333.6
425.6
344.7
441.5
359.9
454.6
373.4
469.6
388.1
459.8
383.4
466.7
393.1
467.3
398.6
427.3
342.1
451.9
359.0
463.2
374.8
482.7
392.3
490.3

401
.I
500.5
413.0
516.2
4292
527.3
442.5
540.7
457.5
524.3
448.6
526.0
4562
515.6
419.0
736.1
623.5
535.0
437.6
552.3
454.7
572.6
474.0
576.1
481.9
583.1
492.2
596.0
508.1

602.0
519.5
611.0
533.2
749.6
641.7
758.7
656.8
773.1
675.9
1
W
v,
4-36
4.4
X-ray
results
4.4.1
Metal
Working
X-ray
analysis
of
metallic
materids
Table
4.18
GLIDE
ELEMENTS
AND

FRACTURE
PLANES
OF
METAL CRYSTALS
LOW
Eleuared Most closely
temperatures temperatures packed
Glide Glide Glide Glide Lattice Lattice Fracture
Structure Metal plone direction plane directian plane direction
plane
&Cu-Zn
(110)
a-Fe-Si,
(110)
5%
Si
(112)(?)
1
Hexagonal,
Mg
close
zn
packed
Cd
Zn-Cd
ZnSn
Tetragonal
PSn
(110)
(white)

(100)
(101)
(121)
hedral
Sb
(111)
Bi
(1 11)
Rhombo-
As
-
Hg
(100)
and
Approximately 450°C
l(111)
l[lOIl
-
(100)
[loll
2(100) 2[1Oo]
-
3(110) 3[112]
-
-
-
1

-
ClOi]

-
[Ili]
-
11117
-
[lli]
-
c1m
-
[iii]
(110)
[iii]
-
Ciii]
(123)
[iii]
-
-
-
[lli]
(110)
[llfl
- -
-
[lli]
-
l(101)
l[lll]
2(100) 2[100] (001)
z}

3(111) 3[110]
I
-
-
-
I
-
(123) [111]
-
- -
- -
-
-
-
-
-
-
-
- - -
-
[Ool]
Approximately
150°C
1(100) 1[001]
-
[Ool]
(110) [TI11 2(11O) 2[111]
-
- -
-

3(101) 3[100]
-
-
4[101]
-
ClOi]
-
-
complex
Hexagonal Te
(ioio)
~11~01
-
-
(lOT1)
-
(ioio)
4.4.2
Crystal
Structure
Crystal structural
data
for
free
elements are given in Table
4.25.
The coordination number, that
is
the number
of

nearest neighbours
in
contrast
with
an atom, is listed in column
4
and the distances
in column
5.
In
complex structures, such
as
a
Mn
where the coordination
is
not exact, no symbol
is
used
and the range
of
distances between
near
neighbours is given.
X-ray
results
4-37
TaMe
4.19
PRINCIPAL

TWINNING
ELEMENTS FOR
METALS
-
Twinning Second
Crystal
Twinning direerion, undisrorted
Direction
structure
plane,
Kl
VI
K2
v2
Shear
From
C.
S.
Barrctt and
T.
B.
Magplski,
‘Structurr
of
Metals’.’
A
co-ordination symbol
x
in column
4

indicates that each atom has
x
equidistant nearest
neighbours, at a distance from it (in kX-units) specified in column
5.
The symbol
x,
y
indicates that
a given atom has
x
equidistant nearest neighbours, and
y
equidistant neighbours lying a small
distance further away. These distances are given in
column
5.
In complex structures, such
as
z-Mn,
where the co-ordination is not exact, no symbol
is
used,
and the range of distances between near
neighbours is given in column
5.
The Goldschmidt atomic radii given in column
6
are the radii appropriate to 12-fold co-
ordination In the case of the f.c.c. and c.p.h. metals the radius given is one-half

of
the measured
interatomic distance, or of the mean
c?f
:he two distances for the hexagonal packing. In the case of
the b.c.c. metals, where the measured interatomic distances are for 8-fold co-ordination, a
numerical correction has been applied. In some cases, where the pure element crystallizes in a
structure having a low degree of co-ordination, or where the co-ordination is not exact, it
is
possible to find some compound or solid solution in which the element exists in 12-fold co-
ordination, and hence to calculate its appropriate radius. In a few cases no correction for co-
ordination has been attempted, and here the figures, given in parentheses, are one-half
of
the
smallest interatomic distances. It should
be
emphasized that the Goldschmidt radii must not be
regarded as constants subject only to correction for co-ordination and applicable to all alloy
systems: they may vary with the solvent or with the degree
of
ionization, and they depend to some
extent on the filling of the Brillouin zones.
Ionic radii vary largely with the valency, and to
a
smaller extent with co-ordination. The values
given in column
8 are appropriate to &fold co-ordination, and have been derived either by direct
measurement or by methods similar to those outlined for the atomic radii.
All
are based,

ultimately, on the value of
l.32A
obtained for
Oz+
ions by Wasastjerna,28 using refractivity
measurements. Ionic radii are also dected by the charge on neighbouring ions: thus in CaF, the
fluorine ion is
3%
smaller than in KF, where the metal ion carries a smaller charge. It is not
possible to give
a
simple correction factor, applicable to all ions: the effect is specific and is
especially marked in structures
of
low co-ordination. Figures in arbitrary units indicating the
power of one
ion to bring about distortion in a neighbour (its ‘polarizing power’), and indicating
the susceptibility of
an
ion
to
such distortion (its ‘polarizability’) are given in columns
9
and
10,
respectively.
The crystal structures
of
alloys and compounds are listed in Chapter
6,

Table
6.1.
Other sources
of data are references
7
and PearsonZ3 which is particularly valuable
as
the variation of lattice
parameters with composition
as
well
as
structure is given. Structures are generally referred to
standard
types
which are listed in Pearson
and
in Table
62
in Chapter
6.
Further information on
pure crystallography
can
be
obtained from International Tables For X-ray Cry~tallography.~
Table
4.20
ROLLING
TEXTURES

IN
METALS
AND ALLOYS
2
-
00
Texture Texture
2
3
Metal
or
alloy
I
2
3
Y
Metal or alloy
I
-
Facecentred
cubic
cu
cu
CU'
Cu 70Yo-211 30%
Cu
70Yo-Zn
30%*
Cu+12
at.

%
AI
Cufl.5
at. %AI
Cu+3
at.
%
Au
Cu +29.6
at.
%
Ni
Cu+49
at.
%
Ni
Ni
Au
Au+
10
at.
%
Cu
A1
AI
A1+2
at.
%
Cu
Al+

1.25
at.
%
Si
A1+0.7 at.
%
Mg
Ag
Pb+2
wt
%
Sb
Body-centred
cubic
a-Fe
a-Fe
Mo
W
V
Fe+4.16 wt
%
Si
Body-centred
cubic
(continued)
Fe+35
wt
YO
Co
Fe+35 wt

%
Ni
,&Brass
Hexagonal close-packed
Be,
5
=
1.5847
a
Ti,
5
=
1.5873
a
Zr,
5
=
1.5893
a
Mg,
5
=
1.6235
/I-Co,
5
=
1.623
a
a
Zn,

E=
1.8563
Cd,
5
=
1.8859
a
a
(11 1)/C1121
(1 12)K110]
Mg+AI
(<4%
by
wt)
Mg+2% Mn
(1 W[lfOl (1 11)/C1121
Scatter
increases
with
Si
content Mg+0.4%
Co
Rhombohedral
a-U
6
2.
F
(OOOI)
tilted approx.
2MO"

round
RD
out
of
rolling plane;
;
[IOiO]
parallel
RD
n
(0001)/[11m
(OOO1)
tilted
30-40'
round
RD
out
of
rolling plane;
[IOiO]
parallel
RD
(OOO1)
parallel rolling plane
(OOOI)
parallel rolling plane
(OOO1)
tilted 2O"round transversedirection
out
ofrollingplane

(W1
)/C~OfOJ
(Oool)
tilted
-
15"
out
of
rolling plane around
transverse direction
-
*
Straight-reverse
rolling
treatment.
From
A.
Taylor
'X-ray
Metallography',
John
Wiley
and
Sons
Inc.
X-ray
results
4-39
‘Fable
421

FIBRE
TEXTURES
OF
DRAWN AND EXTRUDED
WIRE*
Metal
Facecentred cubic
Al, Cu
Ni,
Pd,
Ag, Au, Pb, Cu+O.47%Ag,
Cu+0.45% Sb, CU+ l.O%As,
Cu+0.009%Bi
Cu-Zn
(<2.35%Zn)
Cu-Ni
(<32%Ni), Cu-A1 (<2.16%A1),
Cu-AI (>4.4%Al),
Cu-Zn
(r4.8%Zn)
a-Brass,
a-Bronze,
Ni+20% Cr, Ni-Fe,
austenite,
18/8
and
12/12 Cr-Ni
steel
Body-centred cubic
a-Fe,

8-Brass
Ma,
a-Fe,
W,
V,
Nb,
Ta
Hexagonal close-packed
klg,
2
=
1.6235
a
Zn,
=
1.8563
Ti,
5
=
1.5873
a
a
ParalIel ta
drawing
direction Parallel to
extrusion
1 2
direction
__-__
-

C~lOl
Cllll
[loo1
[llO], [1!3]
and
[llO]
Cllll
w11
ClOOl
[lll]and[100]
CoO01ll
[IlZO]
[ioio]
[Oool]
approx.
72“
to drawing
direction
[ioio]
[lOil]
Zr,
=
1.5893
W@lll
a
Se.
F=l.i31 [llZOJ
a
*After
E.

Schmid.
From
A.
Taylor,
‘X-ray Metallography’.
Table
4.22
TEXTURES
IN
ELECTRODEPOSITS*
Metal Fibre textures
A
u
Fe
co
Cr
Sn
Cd
Bi
*From
C.
S.
Bamtt,
Structure
of
Metals’,
McGraw-Hill,
New
YoIk,
1943.

640
X-ray analysis
of
metallic
materials
Table
4.23
TEXTURES
IN
EVAPORATED AND SPUTTERED
FILMS'
Metal
deposited Texture Technique
Face
centred cubic
Ag
[ill];
[IOo];
[IlO]
Evaporated
AI
[Ill];
[lOO];
[llO] Evaporated
Au
[110]:[111] Evaporated
Pd, Cu, Ni Cllll Evaporated
Body centred cubic
Fe
c1111 Evaporated

Mo
c1101 Evaporated
Hexagonal Cd, Zn
c@-w
Evaporated
Rhombohedral Bi
[Ill]:
[llO]
Evaporated
Pt c1@31; c1111 sputtered
~
'From
C.
S.
Barrett,
'Structure
of
Meials',
McGraw-Hill,
New
York,
19432
Table
434
TEXTURES
OF
CAST
METALS'
Structure
Metal Normal

to
cold
surface
Body centred cubic
Face
centred cubic
Hexagonal close packed7
Rhombohedral
Tetragonal
Fe-Si
(4.3%
Si)
8-Brass
Au
Pb
a-Brass
Cd(c/a=
1.885)
Zn(c/a=
1.856)
Mg(c/a=
1.624)
Bi
8%
Columnar
grains,
[OOl>(iOO)
11
to
surface

Chilled surface, [Ool]
Columnar grains, [Wl];
(100)
I[
to
surface
Chilled surface,
[Ool]
Columnar grains, [l00]; (205)
11
to
surface
and (001)
37'
from it
*
From
C.
S.
Banctt,
'Structure
of
Metals',
McGmw-Hill,
New
York,
1943.
t
Three
indias

system;
equivalent
idees
in
four
idices
systems
are
as
follows:
(oOl)=(aool)=basal
plane;
[l~]-~ZTTO]=diegonal
axis
of
type
I=close
packed
TOW
of
atoms
in
basal
plaac; [la01
normal
to
surface=(lZO)
parallel
to
surface.

The density of a material is calculated from crystallographic data with the relation
nA
p
=-
'
VN
where
n
is the number of atoms contained in
the
unit
cell
of
volume
V,
A
is
Avogadro's number
and
A
is the mean atomic weight of the atoms.
A
is computed from the atomic percentages
pi,
pz,
etc, of the elements forming the alloy and their atomic weights
A,,
A,,
etc, using the formula
X-ray

results
4-41
Table
4.25
ATOMIC
AND
IONIC
RADII
I
2
3
6
7
4
5
As
elmt
CO-
ordina- Inter-
tion atomic
No.
distances
6, 6
-
8
9
10
In ionic crystals
Gold-
Schmidt Polarizing Polariza-

ionic radii power bility
1.54 0.62
-
-
I
-
0.78
1.64
0.075
0.34
17.30 0.028
02
-
0.014
Gold-
Schmidt
at. radii
0.46
1.57
1.13
0.97
0.77
0.71
0.60
1.60
1.92
1.60
1.43
[1.17]
-

-
-
c1.091
State
of
ionization
Type
of
Symbol structure
H
c.p.h
He
-
Li
b.c.c.
Be
c,p.h.
B
-
Atomic
number
I
H
-
8 3.03
6, 6 2.22; 2.28
Li
+
Be'+
B3+

4 1.54
3 1.42
-
-
6
7
8
9
10
I1
12
13
14
15
e+
Ns+
0'-
F-
-
-
10.2
-
0.142
-
-
1.32
1.15 3.1
1.33
0.57 0.99
-

-
-
N
cub.
0
orthorh.
F
Ne
f.c.c.
Na
b.c.c.
Mg
c.p.h.
AI
f.c.c.
Si
d.
P
orthorh.
-
-
-
12 3.20
8 3.71
6, 6 3.19; 3.20
12 286
4 2.35
3 218
Na+
Mg2+

~13
+
{
$;
PS
+
0.98
1.04
0.21
0.78 3.29 0.12
0.57 9.23 0.065
1.98
0.39 26.30 0.043
0.3-0.4
-
I
1.74
0.66 7.25
1.81 0.30
3.05
1.33 0.57 0.85
1.06
1.78 0.57
-
-
0.34
51.90
-
-
-

-
{
c1-
S
f.c.
orthorh.
CI
orthorh.
A
f.c.c.
-
2.12
1 2.14
12 3.84
8 4.62
12 3.93
6.6 3.98; 3.99
I2 3.20
6, 6 3.23; 3.30
6, 6 291; 2.95
c1.041
c:.on
1.92
2.38
1.97
2.00
1.60
1.64
1.47
16

17
18
19
20
-
K'
Ca'
+
sc3
+
Ti''
Ti3
+
Cr'
+
Cr6
+
K
b.c.c.
21
0.83
0.76
0.69
0.64
0.65
0.61
-0.4
0.64
0.3 -0.4
0.91

0.70
0.52
22
Ti
c.p.h.
23
V
b.c.c.
8 2.63
8 2.49
6, 6 271; 2.72
-
2.24
-
2.96
-
236
-
2.68
8, 4 258; 2.67
8 248
12 2.52
6,
6 2.49; 2.51
12 2.51
6, 6 2.49; 2.49
12 249
12 255
6, 6 2.66; 2.91
-

2.43
-
2.79
4 244
3,
3
2.51; 3.15
2, 4 2.32; 3.46
1
2.38
12 3.94
1.36
1.28
1.36
c1.181

1.37
1.28
1.26
1.25
1.26
1.25
125
128
1.37
1.35
1.39
c1.251
[l.l2]
W6l

[1.19]
1.97
b.cc.
(a)
Cr
{c.p.h.
(8)
cub.
(a)
Mn
(
(0)
f.c.t.
(y)
24
25
0.87
- -
0.67
0.82
0.65
0.78
-
-
0.96
0.83 2.90
-
0.62 7.80
-
0.44

20.66
-
0.69
-0.4
-
1.91
0.55
6.4
0.3-0.4
-
-
1.96 026 4.17
- -
-
-
- -
-
-
-
-
-
26
27
28
29
N)
31
32
33
34

35
36
co
.z$&h(l;;'
Ni
{f.c.c.
(J?)
c.p.h.
(a)
c11
r.c.c.
Ga
orthorh.
Ge
d.
As
r.
Se
hex.
Br
otthorh.
k
f.c.c
Zn
c.p.h.
642
Table
4.25
ATOMIC
AND

IONIC
RADII-continued
X-ray
analysis
of
metallic materials
1
2
3
4
5
6
7
8
9
10
As
ekment
In
ionic crystals
CO-
ordina- Inter-
Gold-
Gold-
Atomic Typeof rion atomic schmidt State
of
schmidt polarking polariza-
number
Symbol
structure

No.
distances
at.
radii ionization ionic radii
power
bility
37
Rb
38
Sr
39
Y
40
Zr
41
Nb
42
Mo
43
Tc
44
Ru
45
Rh
46
Pd
47
Ag
48
Cd

49
In
50
Sn
51
Sb
52
Te
53
I
54
xe
55
cs
b.c.c.
f.c.c.
c.p.h.
{
2:;
bc.c
b.c.c.
c.p.k
fJ2.C.
f.c.c.
f.c.c.
f.c.t.
-
c.p.h.
cira.
r.

hex.
orthorh
f.c.c.
bee.
4.87
4.30
3.59;
3.66
3.16; 3.22
3.12
285
2.72
2.64, 2.70
2.68
2.75
2.88
2.97; 3.29
3.24; 3.37
2.80
3.02; 3.18
290,
3.36
286;
3.46
270
4.36
5.24
-
251
2.15

1.81
1.60
1.61
1.47
1.40
1.34
1.34
1.37
1.44
1.52
1.57
1.58
-
-
1.61
C1.431
C1.361
2.18
270
1.49
1.27
1.06
0.87
0.69
0.69
0.68
0.65
0.65
0.68
0.65

0.50
1.13
1.03
0.92
2.15
0.74
0.90
2.11
0.89
2.20
0.94
1.65
-
-
0.45
1.24
2.67
5.28
-
10.50
-
-
-
-
-
-
-
0.78
1.88
3.54

7.30
-
-
0.45
0.21
-
-
-
0.37
4.5
X-ray
fluorescence
X-ray fluorescence occurs after an electron has
been
ejected from
a
shell surrounding the nucleus
of an atom. The X-radiation is characteristic
of
the atom from which the electron has been ejected,
and hence provides a means
of
identifying the atomic species. The ejection of an electron may
be
induced by irradiating the sample with photons
(X
or y-rays) electrons, protons, charged particles
or, indeed, any radiation capable
of
creating vacancies in the inner shells of the atoms of interest in

the sample. The relative merits of each technique are given in Table
4.26.
A
further comparison of
X-ray or radio-isotope
sources
for X-ray fluorescent spectroscopy is given
in
Table
4.27.
Details
of suitable available isotope sources are given in Table
4.28.
Analysis
of
fluorescent X-rays is achieved by wavelength dispersion using crystal analyser (or
several in
a
multichannel instrument),
or
by energy dispersion with solid-state detectors.
Wavelength dispersion offers more accurate quantitative analysis, especially for the detection of
small concentrations of elements where X-ray spectra from several elements overlap. Energy
dispersion is preferred when rapid or quantitative analysis is required of an unknown sample.
Examples of the detection limits for X-ray excited samples are given in Tables
4.29
and
4.30,
and for ion excited samples in Table
4.31.

Accuracy levels for elemental analysis are typically:
for X-ray excitation
for
electron and ion excitation
better than
1%.
1-2%.
These values can be improved with very carefully calibrated standards, but are frequently much
worse, especially when the specimen
surface
is rough. Unlike X-ray diffraction, powdered samples
are the most difficult sample form to analyse.
X-ray
fluorescence
4-43
Table
4.25
ATOMIC
AND
IONIC
RADII-continued
I
2
3
4
5
6 7
8
9
10

As
element
In
ionic crystals
CO-
ordina- Inter-
Gold-
Gold-
Atomic Type
of
tion atomic schmidt Stage
of
schmidt
Polmizing
polariza-
number
Symbol
structure
No.
distances at. radii ionization ionic radii power bility
56
57
58
59
60
61
62
63
64
65

66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
a2
83
84
85
86
87
88
89
90
91
92
93
94
95

96
Ba
La
Ct:
Pr
Nd
-
Sm
Eu
Gd
Tb
DY
Ho
Er
TITI
Yb
Lu
Hf
Ta
W
Re
OS
Ir
Pt
Au
Hg
T1
Pb
Bi
Po

At
Rn
Fr
Ra
Ac
Th
Pa
U
NP
Pu
Am
Cm
b.c.c.
c.p.h.
f.c.c.
c.p.h.
[hex. f.c.c.
f.c.c.
hex.
-
-
b.c.c.
c.p.h.
c.p.h.
cp.h.
c.p.h.
c.p.h.
c.p.h.
f.c.c.
c.p.h.

c.p.k
b.c.c.
C.D.4.
c.p.h
f.c.c.
f.c.c.
f.c.c.
r.
f.c.c.
r.
monocl.
-
-
-
-
-
f.c.c.
-
orthorh.
-
-
-
c
4.34
3.72; 3.75
3.75
3.63; 3.65
3.63
3.63; 3.66
3.64

3.62; 3.65
-
-
3.96
3.55; 3.62
3.51; 3.59
3.M; 3.58
3.48; 3.56
3.46; 3.53
3.45; 3.52
3.87
3.44; 3.51
3.13;
3.20
2.85
274
12;
2,
4 2.82; 2.52
282
2.73; 276
2.67; 2.73
2.71
2.77
288
3.00
3.40; 3.45
3.36
3.49
3.1 1; 3.47

2.81
-
-
-
-
-
3.60
-
2.76
-
I
-
-
2.24
1.87
1.87
1.82
1.82
1.83
1.82
1.82
-
-
204
1.80
1.77
1.77
1.76
1.75
1.74

1.93
1.73
1.59
1.47
1.41
1.41
1.38
1.35
1.35
1.38
1.44
1.55
1.71
1.73
1.75
1.82
c1.41
-
-
-
-
-
1.80
-
C1.381
-
-
-
-
1.43

1.22
1.18
1.02
1.16
1.00
1.15
-
1.13
1.13
1.1
1
1.09
0.89
1.07
1.05
1.04
1.04
1
.oo
0.99
0.84
0.68
0.68
0.65
-
0.67
0.66
0.52
0.55
1.37

1.12
1.49
1.06
2.15
1.32
0.84
1.20
-
-
-
-
1.52
1.10
-
-
1.05
-
-
-
-
4-44
X-ray
analysis
of
metallic
materials
.___
Table
4.26
COMPARISON

OF
X-RAY
FL.UORESCENCE TECHNIQUES
Exciting
radiation
Adtiantages
Limitations
-
________
Electrons
High-intensity energy-regulated
sources
easily Specimen must
be
in vacuum with source
produced Signal-to-background ratio relatively poor
Can
be
focused
into submicron
spot
si7x
Low cost
Good light element detection
Positive Better signal-to-background ratio than
ions electrons Expensive equipment
Specimens must
be
in vacuum with source
Can

be focused
Very sensitive to low concentrations
Convenient
-
specimen need not be in vacuum
Wavelength can
be
chosen
for
maximum semi-
tivity
for
element
of
interest
Photons
X-rays Widely
used
or
vys
Cannot
be
focused
Not
as
sensitive to small samples
as
positive
ion excited methods
Light elements(

<
Mg difficult)
-____
Table
4.27
COMPARISON
OF
X-RAY
AND
RADIOACTIVE
SOURCES
FOR
X-RAY
FLUORESCENT
SPEmosCoPY
Rauiutwn
soiww
Ad
can raps Limitations
X-rays Controllable high-intensity source which
can
be
switched
off
when not required
lntensity can
be
102-104
times that
of

radio-
isotope
source
-___

___
-
-
Bulky, expensive equipment
-
requires
high voltage generator
Radio Cheaper, portable, and smaller
than
X-ray Permanent radioactive
hazard
isotopes systems
Low
intensity
means
long exposure times
and/or larger samples
Relatively
small
number
of
available
iso-
topes
(see

Table
4.28)
Can be built into
process
plant
for
local
on-
stream analysis
Table
4.B
RADIO-ISOTOPE
SOURCES FOR
X-RAY
FLUORESCENCE
Nuclide
Ha[f-life
Emission
energies
keV
5SFe
2.7
years
5.9
losCd
453
days
22.1,
87.7
241Am

458
years
12.17.
60
"Co
270
days
6.4,132,144
238Pu
86.4
years
12-17
1251
6oday~
21
Snnee:
Jaklevic and Goulding, in
ref.
27,
p.
33.
4.6
Radiation screening
In using X-rays, radioisotopes or accelerator-based sources
of
radiation, exposure to individuals
must
be
controlled to
be

as
low
as is reasonably practicable but in any event
to
be
less
than the
values indicated in Table
4.32.
In most countries, persons processing, using, selling or transporting
radioactive materials must
be
licensed by a national authority. In England the authority is the
Department
of
the Environment, in Wales it is the Welsh Office, and in Scotland it
is
the Scottish
Development Department.
2
500
W
Wavelength dispersion Cr
tube
100
s
analysis time
Table
4.30
3a

DETECTION LIMITS
FOR
BULK
SAMPLES
Experimental conditions: Wavelength dispersion;
All
measurements
in
p.p.m.
iron and steel sample
W
target,
Fe
and
Ni
base
alloys
Wtarget
Element
2
240
W
10
min
analysis
time
2025W 100s
Si 170
4
P

35
S
8
Ti
1.0
V
1.9
Cr
4.0
1
MI2
1.4
5
Ni
5.4
Ca
8.5 12
As
6.8
Zr
4.6
Mo
4.5 22
Sn
3.9
Emgy dispersive
Ag
tube
10
min

analysis time
Ag
filter
1.8
W
~
Agf
W8lter
22
W
Source:
J.
V.
Giltrich,
in
rd.
2l,
p.
408.
Table
A31
EXAMPLES
OF
DETECTION
LIMITS
FOR
ION
EXCITATION
~~ ~~
Energy Detection limit

Sample
mount Ions
MeV
Current or charge Time Detection limit criterion
-
1
mgcm-2
tt
50
1
nA
VYNS
10-2opgcm-2
p*
5
5
PC
carbon
or
nitrocellulose
a5
5
IrC
40pgcm-’
p+
1.5
5
PA
carbon
400

s
Cu
1.9
x
10-’2g
P/B=O.I
Sn
3.2 x
g
Pb
5.5
x
10-I2g
100-
K
1
x
lo-’
g cm-’
3a
Bgd
200s
cu
2 x
10-9gcrn-2
Br
1
x
10-qgcm-’
AU

5x
10-9~~m-2
100-
K
1
x
g
em-’
3a
Bgd
200s
Br
lO~lO-~gcm-’
Au
20x
10-ggcm-’
30
min
Ca
0.3
x
lo-’*
g
cu
1
x
10-12 g
Ba
20x
g

Pb 10
x
10-’’~
100 counts
above
Bgd
4-46
Table
431
EXAMPLES
OF
DETECTION LIMITS
FOR
ION
EXCITATION-continued
X-ray analysis
of
metallic materials
Detection
limit
Energy
Sample
mount
ions
MeV
Cwent
or
charge Time Defection limit eritm'on
4
pm

P+ 1
10
pc
500
s
Ca
7
x
g
cmW2
3a
Bgd
Mylar
Zn
18
x
10-9gcm-2
Zr
300x
IOu9
g
cm-2
Pb
90
x
g
0.3
10
pc
500

s
Ca
3
x
g
cm-2
3s
Bgd
Zn 5
x
10-9gcm-2
Pb
23
x
g
zr
30
%io-9
g
a-2
Source:
I.
V.
GiUrich,
m
ref.
27,
p.
406.
441

Definitioas
Exposure
Exposure is a measure of the intensity of ionizing radiation multiplied by time. It is measured in
coulombs per kilogram
(C
kg-
')
in SI units. An exposure
of
1
C
kg- implies the production of a
stated number
of
ion pairs per unit mass
of
air:
Absorbed
dose
Absorbed dose is a measure of the energy absorbed per unit mass in a stated material when exposed
to ionizing radiation under stated conditions.
It
is
measured in grays (Gy) in SI units.
1
Gy
=
1
J
kg

-
'
.
Dose equivalent
An absorbed dose has a biological effect which depends upon the type of ionizing radiation and
on the end-effect under consideration. The relationship
between
the absorbed dose
(A)
and dose
equivalent
(0)
is determined by the quality factor
(Q)
(originally described
as
relative biological
effectiveness)
in
the equation
D=QxA
Dose
equivalent is measured in sieverts (Sv) in
SI
units.
1
Sv
=
1
J

kg-
'.
Q is approximately
1
for
X-rays,
prays
and electrons; it is in the range of approximately
1-11
for neutrons, and can
be
as
high as
20
for
tl
particles,
Dose
may be received from external radiation or, following
an
intake of radioactive material,
from internal radiation. Dose received from internal radiation is called committed dose. The terms
dose equivalent and committed dose equivalent refer to dose received by individual organs or
tissues,
etc, from external and internal radiation respectively. The corresponding terms for dose
received by the whole body are effective
dose
equivalent and committed effective dose equivalent
respectively. The dose limits given in Table
4.32

refer to the
sum
of external and internal dose for
the part of the body concerned (unless stated otherwise).
External dose to the whole body
is
normally taken in practice to be the penetrating component
of the dose measured by a film badge or thermoluminescent dosemeter worn
on
or near the chest
or
trunk of the body, plus any neutron dose measured by a neutron badge.
Table
4.32
CURRENT
STATUTORY
DOSE
LIMITS
(Ionizing
Radiations
Regulations
1985)
Organ,
tissue,
or
part
of
body
Whole body
50

mSv
Individualorganortissue (otherthanthelensofthe
500
mSv
eye),
body
extremity
or
skin
Lens
of
the eye l5OmSv
Annual limit
for
occupationally exposed employees
aged 18 or over (lOmSv=lrem)
Radiation screening
4-47
Actiuity
The
SI
unit of activity is the becquerel
(Bq),
equal to one nuclear transformation
per
second.
In the
UK,
in addition to the annual dose limits, the Health and Safety Executive requires that
an investigation

be
camed out forthwith when any adult employee receives a whole
body
dose
greater than three-tenths of the dose limit, i.e. lSmSv, in a calendar year. Notification
to
the
Executive is required if
an
adult employee receives a whole body dose greater
than
three-fifths of
the dose limit, Le. 30mSv, in any calendar quarter.
Adult
fede
occupationally
exposed
persons
As
in Table 4.32 and above, but abdomen dose, from external radiation, in any three consecutive
months not to exceed 13 mSv; after declaration ofpregnancy, abdomen dose, from external radiation,
for remainder of pregnancy not
to
exceed 10mSv.
Young
persons
No
person under
16
years

of
age should be employed
in
work which may involve exposure to
ionizing radiation. Persons who are over
16
but under
18
years of age
are
limited to 0.3
times
the
maximum doses
set
out
in
Table 4.32.
Other
persons
All other persons are limited to one tenth of the annual maximum permissible doses set out in
Table 4.32.
Table
4%
THICKNESS
OF
CONCRETE REQUIRED
TO
REDUCE BROAD BEAM PULSATING OR CONSTANT
POTENTIAL X-RAYS

BY
THE TRANSMISSION FACTOR GIVEN
Transmission factor
1/10 1/100 1/10oO 1/1oOOo l/l00oao l/lo0ooao
~~~
Tube
Total
potential finration
kV
mmAl
Conerete thickness
cm
50
io
100
125
150
m
250
300
300
400
1
.o
1 2
1.5 2 4.5
2.0
3.5 8.5
3.0
4.0 10.5

3
.O
5.5 12.5
3
.O
6.5
15
3.0
7.5
17
3.0 8.0 18
3.001
13 22
3.001 14
24
3
8
14
17
19.5
23.5
26
28
31.5
34
4.5 6
12 15.5
19.5 25
23.5 30
26.5 33.5

32 40.5
35 44.5
38 48
41
50.5
44
54
7
19.5
30.5
36.5
41
49
54
58
60
64
T&k
434
BY THE TRANSMISSION FACTOR GIVEN
THICKNESS
OF
LEAD
REQUIRED
TO
REDUCE BROAD BEAM PULSATING
POTENTIAL
X-RAYS
-
~ ~

Transmission factor
1/10 1/100 1/1oO0 1/1oOOo 1/1ooOOo
l/loO0oO0
Tube TotaI
potential fillration
Lead
thickness
kV
mm
AI
cm
50
1.8
0.01
0.02
0.04
0.065
0.085
0.11
125 3.2
0.03
0.085
0.165 0.26
0.36
0.46
75 3.8
0.015
0.05
0.10
0.155

0.215 0.27
100
4.0
0.023
0.075
0.15
0.24
0.330 0.417
150 3.2
0.035
010
0.185 0.28
0.38
0.48
200 3.3
0.055
0.13
0.25
0.39
0.53
0.68
250
3.0Cu
0.08
0.24
0.46
0.72
1.02 1.32
300 4.0Cu 0.17
0.46

0.88
1.32
1.80
2.30
4-48
X-ray analysis
of
metallic materials
FURTHER READING
‘Recommendations of the International Commission on Radiological Protection’, ICRP Publication
9, Pergamon, Oxford, 1966.
‘Protection against Ionising Radiation from External
Sources’,
ICRP Publication
15,
Pergamon,
Oxford, 1970.
‘Data for Protection against Ionising Radiation from External Sources’, ICRP Publication 21,
Pergamon, Oxford, 1973.
‘Recommendations of the International Commission
on
Radiological Protection’, ICRP Publication
26, Pergamon, Oxford, 1977.
‘Limits for Intakes of Radionuclides by Workers’, ICRP Publication
30,
Pergamon, Oxford 1979.
‘Basic Radiation Protection Criteria’, NCRP Report No. 39, 1971.
‘Radiation Protection Design Guidelines for
0.1
to

100
MeV Particle Accelerator Facilities’, NCRP
Report
No.
51, 1977.
Concrete and Lead Screening
for X-rays, ‘Handbook of Radiological Protection’, National
Radiation Protection Board, 1971.
Ionising Radiation Regulations 1985 (ISBN01 10573331).
Approved Code
of
Practice ‘Protection of Persons against Ionising Radiations arising from any
Work Activity, Part
2,
Section 9’ (ISBN0118838385). Available from
HMSO
Publication Centre,
PO Box 276, London SW8
5DT.
‘A Guide to Radiation Protection in the
Use
of X-ray
Optics
Equipment’, Science Reviews Ltd,
28
High Ash Drive, Leeds LS17 8RA.
References
1.
A.
Taylor, ‘X-ray Metallography’, John Wiley and

Sons
Inc., 1961.
2. C.
S.
Barrett and
T.
B. Massalski, ‘Structure
of
Metals Crystallographic Methods, Principles and Data’,
Pergamon
Press,
Oxford,
1980
(new edition
of
C.
S.
Barrett, Structure
of
Metals’, McGraw-Hill,
New
York,
1943).
3.
H.
P.
Klug and L. E. Alexander, ‘X-ray Diffraction Procedures
for
Polycrystalline and Amorphous Materials’,
John Wiley and

Sons,
1974.
4. ‘Advances in X-ray Analysis’, Plenum
Press,
New York.
5.
‘International Tables
for
X-ray Crystallography’, Vol.
III,
p79,
Kynoch
Press, 1962.
6.
H.
F.
Gobel, Advances
in X-ray Analysis,
Vol. 22, ~255,1978,
Plenum
Press,
New York;
Advances
in
X-ray
7. Joint Committee
on
Powder Diffraction Standard-International Centre for
Diffraction
Data, 1601 Park

Analysis,
Vol. 25, ~~273,315, 1981, Plenum Press, New York.
Lane, Swarthmore, Pennsylvania 1981.
8.
K.
J. Pickard, R.
F.
Walker and N. G. West,
Ann.
Occup.
Hyg.,
Vol. 29, No. 2, pp149-167, 1985.
9.
F.
H.
Chung,
J.
Appl. Cryst.,
1973.
10.
F.
H. Chung,
Advances in X-ray Analysis,
Vol. 17, p106, Plenum Press, New York, 1973.
11. H. P.
Klug
and L.
E.
Alexander, ‘X-ray Diffraction Procedures’, 2nd Edition, pp531-565, John Wiley and
12.

J.
Durnin and K.
A.
Ridal,
Journal
of
the Iron and
Steel
Institute,
January 1988,
p60.
13. D. Kirk,
Strain,
Vol. No. 2, ~75,1970.
14.
A.
Taylor, ‘X-ray Metallography’, John Wiley and
sons
Inc., ~~788426,1961.
15. B. D. Cullity,
Aduunces in X-ray Analysis,
Vol. 20, Plenum Press, p259, 1976.
16.
M.
R.
James and J.
B.
Cohen,
Advances in X-ray Analysis,
Vol. 22, p241, 1978.

17.
D.
Lonsdale, P. Doig
and
P.
E.
J. Flewitt.
18. A.
J.
C. Wilson, ‘Mathematical Theory
of
X-ray Powder Diffractometry’, Philips
Technical
Library, 1963.
19.
A.
R.
Stokes, ‘Line Profiles’ X-ray Diffraction by Polycrystalline Materials, H.
S.
Peiser,
H.
P.
Rooksby, A.
20.
E.
F.
Sturcken and W.
G.
Duke, ‘AEC Research and Development
Report

DP-607‘, El. du Pont
de
Nemours
21. H. J. Bunge, ‘Texture Analysis in Mateiak Science Mathematical Methods’, Buttemofis, 1982.
22. ‘International Tables for X-ray Crystallography’, Vols
I,
Il
and
111,
Kynoch Press, 1962.
23.
W.
P.
Peanon,
‘A
Handbook
of
Lattice Spacings and Structure
of
Metals and Alloys’, Pergamon Press,
24. D. K. Smith and C.
S.
Barrett,
Advances in X-ray Analysis,
Vol. 22, pl, 1978, Plenum Press, New York.
25. C. H. White, The Nimonic alloys’, W. Betteridge and
J.
Heslop (Eds), p63, 2nd ed., 1974, Edward Arnold
26.
I.

S.
Brammar and M.
A.
P. Dewey, ‘Specimen Preparation
for
Electron Metallography’, Blackwell Scientific
27.
H.
K. Herglotz and
L.
S.
Birks
(eds),
‘X-ray Spectrometry’, Marcel Dekker, New York, 1978.
28. J.
A.
Wasastjerna,
Comenf.
Phys. Math.,
Helsinaf, 1923,1,38.
Sons,
1974.
J. C. Wilson.
(Eds)
The
Institute
of
Physics, London, 1955.
and
Co.

Savannah River Laboratory, 1961.
New York.
(Publishers) Ltd.
Publications, Oxford, 1966,
5
Crystallography
5.1
The
structure
of
crystals
5.1.1
Translation
groups
Metals and alloys, like most solid matter, are aggregates of crystals; they are built up of units,
consisting of small groups
of
atoms regularly and indefinitely repeated throughout the body by
parallel translations. If the co-ordinates of the atoms within such a group are given, then three
independent translations represented by vectors
a,b,c,
which are not all parallel to the same
plane, suffice to specify the position
of
any other atom in the crystal. Let the vector from an
arbitrary origin to an atom
be
r
=
xu

+
yb
+
IC
then atoms
of
the same kind will
be
found at all points
rn
=
(n,
+
x)a
+
(n2
+
y)b
+
(n,
+
z)c
where
n,,
n2,
n3
may
be
any positive or negative integers. Such a succession of regularly arranged
points in space constitutes

a
space
lattice.
The lattice
may
be regarded either
as
a system
of
translations relating identical points in a
structure, or as a system
of
points arranged
in
parallel and equidistant nets, each net consisting of
series of parallel and equidistant .rows in which the points are spaced at equal distances. The
points of such
an
array
can
be arbitrarily arranged
in
an
infinite number of ways in parallel
equidistant linear rows
or
planar nets; they can, in other words, be referred to
an
infinite number
of

systems of three primitive vectors, but investigation
has
shown that any structure possessing the
symmetry observed in crystals
can
be
referred to
one
of
14
lattices, defined by its primitive vectors
and by the character
of
its unit cell, the latter being the parallelepiped formed by the three
translations selected as units. In general, unit translations
are
selected
so
as to give the simplest
cell having edges as short as possible, but there are several cases in which
a
more complex cell is
chosen
so
as
to display the symmetry
of
the lattice,
or
its relation

to
other lattices, to greater
advantage.
The system of three vectors
a,
b,
e,
is described by their lengths
a,
b,
c
and by the angles between
them:
(bc)
=
tl,
(ca)
=
/3,
(ab)
=
y.
The face of the unit cell which is parallel to the plane of the
(a)
and
(b)
axes, and which therefore intersects the
(c)
axis at distance
c

from the origin is termed the
c
face. Similarly, the face parallel to the
b-c
axial plane is the
a
face, and that para!lel to the
a-e
axial plane the
b
face.
The simplest cell, having points only at its corners, is termed ‘primitive’ and is given the symbol
r
(Schoenflies)
or
P
(Hermann). Other cells, termed ‘face centred‘, have points at the comers and at
the centres of two
or
more of their faces, and
are
given symbols indicating the faces carrying these
additional points. Thus
A,
B,
C,
F
represent centring
on
the

a,
b,
c
and all faces respectively.
Finally there is the ‘body centred‘ cell, having points at its corners and one additional point at the
intersection
of
the body diagonals. This
is
given the
symbol
I
(Hermann). Centred cells are
indicated by Schoenflies by dashes. The
14
lattices are listed in Table
5.1.
5.1.2
Symmetry elements
Symmetry elements may be classified as axes, planes, and centres.
A
body has an axis of symmetry
when rotation through a definite angle about some line through it (the axis
of
rotation) causes it
to
5-1
5-2
Crystallography
Table

5.1
THE
FOURTEEN
MTTICES
S~mbOlS
Sysrem
Axes Angles
Unit
cells Schoenflies
Hermann
Triclinic
a#b#c
Monoclinic
a#b#c
Orthorhombic
a#b#c
Tetragonal
a=b#c
Cubic
a=b=c
Rhombohedral
a=b=c
Hexagonal
a=b#c
a
=
j3
=
y
#

900
a
=
p
=
90"
y
=
120"
Primitive
I
Primitive
2
c
face centred
I
Primitive
2
Face centred
3
All face centred
4
Body centred
I
Primitive
2
Body
centred
I
Primitive

2
All
face
centred
3
Body
centred
Primitive
Primitive
*
By choosing diflerrnt
a
and
b
axes the centred monoclinic
cell
will
be
seen to
be
equivalent to a primitive cell in which
t
By suitable change
of
axes it is possible to convert
the
orthorhombic
C
and
F

cells into primitive and body centred cells
f
This is
a
special form
of
the alternative setting described in the preceding note
for
the orthorhombic
C
cell.
It
is therefore given
a
=
b
#
e,
K
=
fl
=
SO',
y
#
a.
Even
if
these axes are selected, the symbol
C

is
retained for this cell.
respectively having
a
E
b
#
c,
c(
=
fl
-
W',
y
obtuse. The symbols for
thcse
alternative settings remain
C
and
F.
the
symbol
C.
assume its original aspect. Crystals have been observed to have axes of
2-,
3-,
4-
and 6-fold
reflection, involving coincidence after rotation through
180",

120",
90" and
60"
respectively. If
a
plane
can
be passed through a body such that every point on one side of the plane stands in
mirror-image relationship to a corresponding point on the other side, the plane is said to be
a
reflecting plane or plane of symmetry.
A
point within a body is a centre of symmetry or centre of
inversion
if
a line drawn from any point of the body to the centre and extended to an equal
distance beyond it encounters a corresponding point. Other symmetry operations are:
Rotary reflection,
involving rotation through a definite angle, combined with reflection in a
plane normal to the axis.
Screw axes
of
rotation,
combining rotation about an
n-fold
axis with a translation of a
specified length in the direction of the axis.
Glide planes,
combining reflection with a translation parallel to the plane of the mirror.
In the case of screw axes, the amount of the shift must be a rational fraction of the translation

along the same axis, the denominator of the fraction being the multiplicity
of
the rotation. Thus
for a &fold axis, the shift may
be
1/6, 2/6,3/6
.
of
the translation. In the case of the glide plane,
the shift must be one half of some translation in that plane. Thus it may be a/2 or
b/2
parallel
to
the
a
and
b
axes, or of hag the face diagonal in the direction parallel to that diagonal. If the
cell
is
centred on that particular face, the shift may be one half
of
the distance to the centre, is. of the
face diagonal.
513
The
point
group
The point group may be defined as a group of symmetry elements distributed about a point in
space, and may be conveniently visualized as

an
assembly of points generated by the operation of
the symmetry elements in question upon a single point having co-ordinates
xyz
referred to
specified axes, the symmetry elements passing through the origin. Thus a symmetry plane, passing
through the origin and containing the
x
and
y
axes,
will
generate, from the point
xyz,
an
equivalent point of which the co-ordinates are
xyZ.
These two points serve to characterize the
point group.
The 32-point groups define all the ways in which axes, planes and centres
of
symmetry can be
distributed
so
as to intersect in a point in space, and correspond to the 32 classes of morphological
crystallography.
The Hermann-Mauguin system of point- and space-group notation
5-3
5.1d
Thespmeegroop

The
space
group may
be
defined as an extended network of symmetry elements distributed about
the points of
a
space lattice, and may be visualized
as
an assembly of points generated by the
operation of symmetry elements on a series of points situated identically in each cell of the lattice.
Whereas in the point group the repeated operation of any symmetry element must ultimately
bring each point back to its original position, in the space group an operation need only bring the
point to an analogous position in the same or in another cell of the lattice. Thus in the space
group the more complex symmetry operation of screw axes and glide planes are possible,
combining translation with reflection and rotation.
Point groups, placed at the points of space lattices belonging to the same system of symmetry,
give rise to the simplest of the 230 space groups. The remainder are generated by replacing the
simple planes and axes of the point group by glide planes and screw axes.
5.2
The
Schoenflies
system
of
point- and space-group notation
The symmetry elements chosen by Schdies are axes of n-fold rotation, reflection planes and
centres of inversion. The symbols assigned to the various point groups are as follows:
C,,
=
groups having a single n-fold axis.

C:
=
C;
=
D,
=
V
=
0
=
groups having a single vertical n-fold axis, together with a horizontal reflection
plane
groups having a single vertical n-fold axis, together with
n
vertical reflection planes
groups having
an
n-fold axis and
n
two-fold axes at right angles to it
a symbol frequently used
as
an
alternative to
Dz
the two cubic groups which possess the maximum possible number of rotation
axes, namely four 4-fold axes parallel to the cube edges, four 3-fold axes parallel to
the cube diagonals and six 2-fold axes parallel to the face diagonals
T
=

the three remaining groups of the cubic system
S,
=
groups having an n-fold axis
of
rotary reflection
The suffix
i
signifies a centre
of
inversion.
The suffix
s
signifies a single plane of symmetry.
The suflix
d
signifies a diagonal reflection plane, bisecting the angle between two horizontal
axes.
The symbols for the space groups are simple modifications of those for the point groups: the
index and subscript of the point group are combined to give the subscript of the space group
symbol, and an index is added representing the order in which Schoenflies deduced the symmetry
of the group. Thus Czrepresents the mth group derived from the point group C;.
Table
5.2
gives the point groups, their symbols and elements of symmetry, the crystal classes
with which they correspond, and the eo-ordinates of equivalent points.
5.3
The Hermann-Mauguin system
of
point- and space-group notation

The symbols used by Hermann and Mauguin to indicate the various symmetry operations are as
follows:
Roration axes: the number
2,
3,
4
or
6
denoting the multiplicity.
Screw axes: the
symbol
denoting the multiplicity of rotation, with a subscript indicating the
magnitude of the shift. The complete set
of
screw axes is
2,;
3,;
32;
4,.
4,,
4,;
6,,
6,,
6,,
6,,
65.
Axes of rotary refection:
2,
3,
T,

6,
the numeral indicating the multiplicity.
Centre of inversion:
i.
Refection plane:
m
Glide
plane: with shift in the a direction: a
with shift in the
b
direction:
b
with shift
of
f
the face diagonal:
n
with shift of
f
the centring translation:
d.
The full space group symbol consists of the translation
(or
1attice)symbol followed by symbols
of
the symmetry elements associated with specified crystallographic directions in a specified order.
The direction associated with a reflection or glide plane is that of its normal: no direction can be
specified
for
a Centre

of
inversion.
5-4
Crystallography
Table
5.2
POINT-GROUP
Crystal
class
Clus.,
Schoenjlies
System
no.
symbol
Schoenjlies
Dana Miers
Triclinic
1
2
Monoclinic
3
4
5
Orthorhombic
6
Tetragonal
Cubic
7
8
9

10
11
12
13
14
15
16
17
18
19
20
Rhombohedral 21
22
23
24
25
Hexagonal 26
27
28
29
30
31
32
Hemihedry
Holohedry
Hemihedry
Hemimorphic hemi-
hedry
Holohedry
Hemimorphic hemi-

Enantiomorphic hemi-
hedry
Holohedry
Tetartohedry
of
2nd
sort
Hemihedry
of
2nd
sort
Tetartohedry
Paramorphic hemi-
hedry
Hemimorphic hemi-
hedry
Enantiomorphic hemi
hedry
Holo hedry
hedry
Tetartohedry
Paramorphic hemi-
hedry
Hemimorphic hemi-
hedry
Enantiomorphic hemi-
hedry
Holohedry
Tetartohedry
Hexagonal tetarto-

hedry
of
2nd
sort
Hemimorphic hemi-
hedry
Enantiomorphic hemi-
hedry
Holohedry
Trigonal paramorphic
hemihedry
Trigonal holohedry
Tetartohedry
Paramorphic hemi-
hedry
Hemimorphic hemi-
hedry
Enantiomorphic hemi-
hedry
Holohedry
Asymmetric
Normal
Clinohedral
Hemimorphic
Normal
Hemimorphic
Sphenoidal
Normal
Tetartohedral
Sphenoidal

Pyramidal hemimor-
phic
Pyramidal
Hemimorphic
Trapezohedral
Normal
Tetartohedral
Pyritohedral
Tetrahedral
Plagihedral
Normal
Not named
Trirhombohedral
Ditrigonal pyramidal
Trapezohedral
Rhombohedral
Not named
Trigonotype
Pyramidal hemimor-
phic
Pyramidal
Hemimorphic
Trapezohedral
Normal
Asymmetric
central
PIanar
Digonal
polar
Digonal equatorial

Didigonal polar
Digonal holoaxial
Didigonal equatorial
Tetragonal alternating
Ditetragonal alternat-
ing
Tetragonal polar
Tetragonal equatorial
Ditetragonal polar
Tetragonal holoaxial
Ditetragonal equa-
torial
Taseral polar
Tesseral central
Ditesseral polar
Tesseral holoaxial
Ditesseral central
Trigonal polar
Hexagonal alternating
Ditrigonal polar
Trigonal holoaxial
Dihexagonal alternat-
ing
Trigonal equatorial
Ditrigonal equatorial
Hexagonal polar
Hexagonal equatorial
Dihmagonal polar
Hexagonal holoaxial
Dihexagonal equa-

torial
The
Hermann-Mauguin system
of
point- and space-group notation
5-5
NOTATION (SCHOENFLIES)
Typical
example Symmetry elements
Co-ordinates
of
equivalent
points
-
None
Copper sulphate Centre of inversion
-
Tartaric acid
Gypsum
Topaz
Sulphur
Barytes
I
Chalcopyrite
Wulfenite
Scheelite
I
-
Zircon
Ullmanite

Pyrites
Blende
Cuprite
Galena
Dioptase
Tourmaline
Quartz
Calcite
I
._
._
Nepheline
Apatite
Greenockite
-
Beryl
Horizontal reflecting plane
2-fold axis
~
.XYZ
xyz,
fp
xyz, xyi
xyz,
jcpz
2-fold axis and horizontal plane
2-fold axis and vertical plane
xyz,
apz,
xyf,

apr
xyz,
apz,
Pyz.
xpz
xyz, Xjf, nyi.
ngz
xyz, xgi, ny5.
iyz,
xy?. xj-z,
xyz,
ip
xyz, gxz,
%jZ,
YE5
4-fold axis, two 24old
axes
and diagonal
xyz, xgi,
nyi,
igz,
yxz, jxZ.
YE,
yiZ,
piz
vertical plane
4-fold axis
xyz, yxz,
npz,
y2z

4-fold axis, horizontal plane
xyz,
gxz,
apz,
yaz,
xyi, yxi,
xyz.
yE
4-fold axis, vertical plane
xyz, jxz,
mjz,
yxz,
iyz, yxz, xp,
yx;
4-fold axis and
four
2-fold
axes
xyz, pxz. agz, yiz, xgz.
paz,
ayz,
yxz
4-fold axis, four 2-fold axes, horizontal
xyz, gxz,
Zgz,
yicz, xpi,
pi,
%y?, gxZ
plane
xyi, jxZ,

npi,
yn5, xyz.
gnz,
xyz, yxz
Three 2-fold axes coincident with cube
:xy. ixp, Eiy,
zxj
Four
3-fold axes coincident with
Three
2-fold axes
Three Mold axes and horizontal plane
4-fold rotary reflection
{
di:znals
xy!, xp:, Ryz.
xpz
Those of
T
plus
zxy, zxy,.
Zip,
ziy
yza,
jif yix, yzx
yxz. pxi.
yxz,
yxz
Those of Tplus
xzy, xfy,

xiy.
Xzj
zyx, ipx, iyx.
zjx
pa,
yaz, pxz. yxl
As
for Tplus six 2-fold axes coincident with
Those
of
Tplus
Ej-,
tzy, xzp, xZy
face diagonals thereby converting the
three original 2-fold axes into 4-folds
As for
0
plus a horizontal plane
3-fold axis
3-fold
axis
and Centre of inversion
3-fold axis and vertical plane
3-fold axis and three 2-fold axes
As for Tplus a diagonal vertical plane
1
ipa,
zyx zgx, iyx
As for
Tplus

a horizontal plane
Those
of
all the
four
preceding classes
xyz, zxy, yzx
referred to rhombohedral axes
xyz, zxy. yzx,
ZgZ,
Ej,
pi3
(rhombohedral axes)
xyz, zxy, yzx, yxr, xzy, zyx
(rhombohedral axes)
xyz, zxy, yzx,
pi,
%j,
5pZ
(rhombohedral axes)
3-fold axis, three 2-fold axes and diagonal
xyz, zxy, yzx,
j-if,
aig,
ip.?
vertical planes
zpz,
znp,
pix,
yxz. xzy, zyx

(rhombohedral
ax4
3-fold axis and horizontal plane
xyz. (y-x)Y.z, Y(x-y)z, xyf (y-x)E, Y(x-y)i
(hexagonal axes)
3-fold axis, three Zfold
axes
and horizontal
xyz,
(
y
-
x)k, J(x
-
y)z, (x
-
y)YZ,
yx%
ir(
y
-
x)i
plane
xyi, (y-x).fi,y(x-y)i, (x-y)yz,yxz, .f(y-x)z
(hexagonal axes)
&fold axes
6-fold
axis
and horizontal plane
xyz, y(y-x)z, Cv-x)az,

apz,
j(x-y)z, (x-ybz
(hexagonal axes)
Those of
C6
plus
xyz, y(y-x)i, (y-x)%Z,
???,
~(x-y)i, (x-y)xZ
(hexagonal axes)
Those
of
C
lus
n(y-x)z,
(J-x)~,
vxz, x(x-
y)z,
(x-y)pz, yxz
(hexagonal axes)
x(x-y)f.
(x
-y)pZ,
jii
(hexagonal axes)
6-fold
axis
six 2-fold axes and horizontal
Those
of

all the four preceding classes
plane
6-fold axis and vertical plane
&fold
axis
and six 2-fold axes Those
of
C6
PIUS
W-X)?,
~-x)YZ;
XYE
6
4-
5-6
Crystallography
The
specified directions are:
Triclinic system:
none
Manoclinic system:
the
b
direction, i.e. the
6
axis.
Orthorhombic system
the
a,
b, e directions, in that order.

the
(I,
6,
and
(a-b)
directions, in that order. The direction
1
c-face.
Tetragonal
Hexagonal
represented by the vector difference
is
one of the diagonals of the
Rhombohedral
Cubic system
the directions
c,
(a
+
b
+
e)
and
(a
-
b),
in
that order, i.e. the
c-axis
the cube

If
a symmetry axis has a symmetry plane normal to it, the two symbols are combined in the form
of a fraction, thus
-,
-,
alternatively written
2/m,
4,
/d.
If
one
of
the specified crystallographic directions has no symmetry element associated with it,
this is indicated by inserting the symbol
1
in the appropriate position in the space group symbol.
The
1
may be omitted without risk
of
misunderstanding
if
it occurs at the end of the space group
symboL Symmetry symbols may also
be
omitted
if
they can be derived from those already
indicated. These abbreviated symbols are termed ‘short’.
As

already explained, the
symmetry
of
a space group
can
be derived from that of
a
point group
by placing the latter at the points of the various lattices appropriate to the crystal system, and by
using glide planes and screw axes as well as reflection planes and rotation axes. Thus the point
group symbol will contain no symbol for the lattice; its symmetry planes will be indicated by
m
and its axes by numbers specifying the multiplicity and without subscripts. Thus the point group
2/m
will
be
associated with the space groups P(2/m); P(Z,/m);
P(2/c);
P(2,lc);
C(2/rn); and C(2/c).
systems:
diagonal and a diagonal of the c-face.
2
41
md
53.1
Notes
on
the
space-group

tables
For a full description of the space groups, reference should
be
made to the
Internationale Tabellen
mr
Bestimmung von Kristallstrukturen.
If
there are
n
symmetry elements associated with any space group, their operation
upon
any
single point having co-ordinates
xyz
will give rise to
a
total of
n
points which may
be
termed
geometrically equivalent. If, however, the co-ordinates
xyz
are such that the point
lies,
say, on an
axis or plane
of
symmetry, then the number

of
equivalent positions will be reduced, while
if
it lies
at the intersection of two elements the number will
be
reduced still further.
A
knowledge of these
so-called special positions is of importance, because experience has shown that they are the
positions which are frequently occupied by the atoms or ions in an actual crystal. In sodium
chloride, for example, the four sodium ions are situated in one set of four equivalent positions,
those having co-ordinates
OOO,
9,
fi,
w,
whilst the four chlorine ions are situated in
another set of
four
points, having co-ordinates
e,
w,
O@,
fioo.
The co-ordinates of all the
special positions for each space group were given by
R.
W.
G.

Wyckoff in
The Analytical
Expression
of
the Results
of
the Theory
of
Space
Groups
(Washington Carnegie Institution,
1930)
and are also listed
in
the
Internationale Tabellen.
The last column of Table
5.3
gives the missing x-ray reflections characteristic
of
each space
group.
If
the unit cell is centred on one or more faces, or is body centred, certain reflections will
be
absent,
because
in directions corresponding to the missing reflections the waves scattered by the
atoms at the face or body centres
will

be exactly out of phase with those scattered by the atoms at
the
cell
corners. In other words, the spacings of certain planes are halved, and odd-order
reflections from these planes are destroyed. Thus with the body-centred lattice all reflections are
absent for which
(h
+
k
+
I)
is odd. Again,
a
glide plane halves the spacings in the direction of
glide, and a 2-fold screw axis halves those along the axis. Consequently, odd order reflections are
missing in these directions. Similarly with a 3-fold axis; the only reflections occurring in the
direction
of
the axis are those for which
(I)
is a multiple of three.
In
Table
5.3,
x-ray reflections
of
the type indicated do not occur unless indices which are
underlined are even, or unless the sum of indices joined together by brackets is even. Thus:
00)
Ok_I

h,kO
hkl
means that reflections will not occur unless
1
is even.
means that reflections will not occur unless
(k
+
I)
is even.
means that reflections will not occur unless both
h
and
k
are even.
means that reflections will not occur unless
(h
+
k
+
I)
is even.
LIJ
The Hermann-Mauguin system
of
point- and space-group notation
5-7
h&J
hhj
means that reflections

will
not occur unless the sums
of
any
two indices are even.
means that reflections will not occur
if
the first two indices are equal, unless the third
index,
l.
is even.
A
subscript
3,4
or
6
means that
the
marked index,
or
the sum
of
the marked indices, must be
a
multiple
of
that number
for
reflections
to

occur.
Thus
O&
hdJ3
means that reflections will not occur unless
(k
+
I)
is a multiple
of
4.
means that reflections will not occur unless
h
+
2k
+
1
is a multiple
of
3.
Table
53
THE
HERMANN-MAUGUIN
SYSTEM
OF
POINT-
AND
SPACE-GROUP
NOTATION

Space group
Hermann-Mauguin Schoen-
Missing
sprctra
flies
Full Shorr
Triclinic
system
Class
I-c,
P1
C;
-
Pi
c:
CIass
1-
C,
-
Monoclinic
system
Class m-C,
Pm
Pc
Cm
cc
class
2-
cz
P2

P2
1
c2
2
m
Class c,,
2
P-
m
2
m
2
P-
2
P-1
C
P-L
2
c-
m
2
c-
Orthorhombic system
Class mm2
(short
mm)-C,,
-
Pmm2 Pmm
c:,
Pmc2, Pmc

c:,.
hOr
Space group
lfermann-Mauguin Schoen-
Missing
spectra
Pies
Full Short
Class
nun2
(short
mm) C,,
continued
Pd
Pmn2,
Pcc2
Peal,
Pcn2
Pba2
Pbn2,
Pnn2
Cmm2
cmc2,
ccc2
Amm2
Ama2
Abm2
Aba2
Fmm2
Fdd2

Imm2
Ima2
Iba2
Pma
Pmn
PCC
Pca
Pcn
Pba
Pbn
Pnn
Cmm
Cmc
ccr
Amm
Ama
Abm
Aba
Fmm
Fdd
Imm
Ima
Iba
CIm
222
(short
22)-
P222
P222,
P212,2

p212121
e222
c222,
F222
1222
12,2121
5-8
Crystallography
Table
5.3
THE HERMANN-MAUGUIN SYSTEM
OF
POINT- AND SPACE-GROUP NOTAlTON-continued
Space group
ffermann-Mmrguin
Schoen-
spectra
flies
Fufl
Short
Space
group
Hermann-Mauguin
Schoen-
Missins
spectra
flies
Cull
Short
Tetragonal

system
class
i-s4
-
P4
s:
14
s:
hdl
Class
4-C4
-
P4
c:
P4,
c:
P4,,P4,
c:,
c::
00/4
14
c:
I&/
141
h!$J
0014
Class
C4h
m
4

m
4
P-
I
P-
c:h
n
c.%
c:h
Ool
42
P-
m
p42
n
c:,
hJO.00~
h&/
4
m
4
I-
c:h
I'
C&,
hkJ,
h_ko
Ool,
Class
&2m

(or,
in
other
orientation,
&m2)-D,&+)
P4h
D:*(V;)
-
The
Hermann-Mauguin
system
of
point-
ana’
spacegroup notation
5-9
Table
53
THE HERMANN-MAUGUIN
SYSTEM
OF
POINT-
AND SPACE-GROUP NOTATION-continued
Class
4mm-C4.
P4mm P4m.
P4,mc
P4mc
P4,cm P4cm
P4cc P4cc

P4bm P4bm
P4,bc P4bc
P4,nm P4nm
P4nc P4nc
1h
14mm
14cm 14cm
14,md I4md
14,cd
14cd
Class
422
(short
42)-
P422 P42
P4422 P422
P4,22 P4,2
P4,22 P4,2
P42,2 P42,
P42212 P422,
P41212 P4121
P43212 P432,
1422 1422
14,22 14,2
422
mmm
Class
(short
4/mmm)-D4h
p ~4/mmm

D:~
-
mmm
Space
group
Hermann-Mauguin
Schoen-
Full
Short
Missing
spectra
pies
Class
-
422
-
-
(short
4/mmm)-D,,,-continued
mmm
hhl
Okl
Ok!,
hhl
Okl
-
Oil,
hhl
OLf
Oil,

hh!
k&O
h@,
hh!
EO,
Okl
-
EO,
OkL
hhl

I&&,
Ob1
EO.
Okf,
hhl
EO.
0,kf
EO,
21.
hh!
h&l
hhl.
Okl
-
hL1,
@O,
hz4
hkl,
hkO.

OE,
w-
Cubic
system
Class
23-T
-
P23 T’
P2,3
7-4
hoo
5-10
Crystallography
Tabk
53
THE
HERMANN-MAUGUIN
SYSTEM OF
POINT-
AND
SPACE-GROUP
NOTATION continued
-
Space
group
Hermann-Mauguin
Full Short
ism
spectra
flies

Class
23-T-continued
F23
T2
I23
T'
12'3
T5
2
m
Class S(short
m3)-Th
Pm3
Pn3
Pa3
2
P-
3
m
2
P-
3
2
P'
3
Fm3
Fd3
2
F;i
3

2-
m
2
I-
3 lm3
la3
123
Class
53m T,
P63m
P43n
F43m
F43C
I43m
lad
Class
432-0
P432 P43
P4,32 P4,3
P4,32 P4,3
P4,32 P4,3
F432 F43
F4,32 F4,3
1432 143
14,32 14,3
T,'
Thz
G6
7;:
Th4

Th5
Th'
Tb
Td4
T,'
TJ
G3
G6
0'
02
O')
06
03
04
05
08
4-2
mm
Class
-
3
-
short m 3m)-Oh
4-2
mm
P-3- Pm3m
0;
Space
group
Hermann-Mauguin Schoen-

Full
Short
flies
Missing
spectra
4-2
Class
-3-(short m3m)-Oh-continued
mm
42
p-1.3-
mn
42
P-1.3-
nm
42
P-3-
nn
42
F-3-
mm
42
F-3-
mc
4,
-
2
dm
4, 2
F-3-

de
42
I-3-
mm
42
12.3
-
ad
F-3-
Pm3n
Pn3m
Pn3n
Fm3m
Fm3e
Fd3m
Fd3c
lm3m
la3d
Rhombohedral system (all
indices
and multiplicities
referred to hexagonal
axes)
Class
3-C3
-
c3
c:
c31,c32
c:,

c;
001
-
R3
c:
hk/3
Class
3-c,,
-
c3
c:i
R3
c:,
%3
Class
3m(to,
indicate orientation distinguish
3ml
and
31m)-Cg
-
C3ml
c
2"
C31m
c:.
-
C3Cl
c:.
hO1

-
C31c
ct.
hkl
R3m
c:,
h&I3
R3c
C&
hz,3
h0'
-