Physical Chemistry of Solids
Concepts and formulae III
K.D. Becker
unter Mitwirkung von K. Talk
(WS 2007/08)
III. Defect chemistry: Lattice defects and defect equilibria
1. Macroscopic evidence of point defects
a) Heat capacity
b) Thermal expansion
2. Microscopic models of point defects
a) Element crystals
b) Ionic crystals
3. Thermodynamics of point defects
a) Statistical treatment, element crystals
b) The chemical potential of vacancies
c) Statistical treatment, ionic crystals
d) Disorder parameters
e) Chemical thermodynamics
4. Adjustment and control of point defect concentrations
a) Doping
b) Equilibration with external phases
c) Electroneutrality of crystals
d) Notation, structure elements and rules
5. Examples
6. Abstract
7. Literature
Learning objectives:
-
point defects, types of disorder
lattice defects as quasi-chemical species
point defect thermodynamics
calculation of defect concentrations
1
III. Defect chemistry: lattice defects and defect equilibria
III.1 Macroscopic evidence of point defects
a) Heat capacity
Many solids show an anomalous increase in heat capacity close their melting points ("extra
contribution"):
Extra contribution
due to defects
Tm
Description:
c p (T ) = c p0 (T ) + Δc p (T )
T
ΔH(T ) = ∫ Δc p dT = x def ⋅ h def
0
Ansatz:
x def = Const ⋅ exp( −hdef / RT )
Δc p (T ) = ∂ΔH(T ) / ∂T = Const ⋅ (h def / RT 2 ) ⋅ exp(−h def / RT )
Plot:
ln (T 2 Δc p (T )) = −h def / RT + konst
Example: AgBr
Tm
Degree of disorder at the melting point:
x def (Tm ) =
2
∫ Δc
p
(T ) dT
0
H def
With Hdef = (135,6 ± 1,2) kJ/mol = 1,40 eV, Tm = 421,5°C
and
Tm
∫ ΔC
p
(T ) dt = 3,25
0
kJ
one obtains xdef(Tm)= 2,4⋅10−2
mol
(Kubaschewski, Nölting)
b) Thermal expansion L(T) & a(T)
V(T): crystal volume, L(T): length of crystal slab, a(T): lattice constant
L(T)
a(T)
L(T)
a(T)
Tm T
Crystal volume
thus:
Atomvolumen a3
V = (N + NV ) a 3
(Index V: vacancies)
dV = a d(N + NV ) + 3a da(N + NV )
3
2
= a 3 dNV + 3(N + N V ) a 2 da
N + NV a 2
dV
a3
=
dN
+
3
da
V
V
N + NV a 3
( N + NV ) a 3
dV dN V
da
=
+3
V
N
a
dN V dV
da
⎛ dL da ⎞
=
−3
=3⎜
−
⎟
N
V
a
a ⎠
⎝ L
T
Integration:
dN V N V
=
N
N
273 ° C
∫
mit
N V (273°C) = 0
NV
⎛ ΔL Δa ⎞
= xV = 3 ⎜
−
⎟
N
a ⎠
⎝ L
By means of so-called Simmons-Balluffi experiments (simultaneous determination of the
macroscopic length expansion and of the lattice constant of a crystal) absolute values can be
obtained for the concentrations of vacancies in crystals.
3
III.2 Microscopic models of point defects
a) Element crystals
Vacancy disorder:
Interstitial disorder:
b) Ionic crystals
Schottky disorder
vacancy
Na+ ClCl-
Cl-
+
Cl-
-
Na+ Cl
+
Cl-
Na
Cl
Na
Na+
Na+ Cl
Na+ Cl
Na+ Cl
Na+
Na+ Cl-
Na+ Cl
Cl- Na+ Cl
+
Cl- Na Cl-
Na+
Na+ Cl
Na+ Cl
Na+ Cl
Na+ Cl
Na+ Cl-
Ag+ Br-
Ag+ Br-
Br-
Ag+ Br-
Ag+ Br-
Ag+
Ag+ Br+
Br+
Ag
Ag+ Br-
Ag+ Br-
Ag+ Br-
Br-
Ag
Ag+ Br-
Ag+ Br
Ag+
Ag+ Br
+
Ag
+
Br- Ag Br-
Ag+ BrP
P
Ag+ Br
Ag+ BrP
P
P
Br-
Ag+ BrP
P
Na+ Cl
Cl-
Cl-
Na+ Cl
Na+ Cl
Na+ Cl
Na+ Cl-
Frenkel disorder
+
Br- Ag Br-
Na+ Cl-
Cl-
Na+
Ag+
Ag+ BrP
P
4
Na+ Cl-
Na+ Cl
Na+ Cl-
Na+
Na+ Cl-
Na+
Na+
Na+ ClNa+
Na+ Cl
III.3 Thermodynamics of point defects
a) Statistical treatment of vacancy disorder in element crystals
Task: Minimization of Gibbs energy of a disordered crystal
G(T)=G0(T) + ΔGdef(T)
Ansatz:
Here, G0 is the Gibbs energy of the defect-free crystal.
ΔGdef depends on defect concentration:
ΔGdef = ΔHdef − T ΔSdef = ΔHV − T ΔSV
def = V (vacancy)
with ΔH V = N V ⋅ H V
and ΔSV = ΔSkonf + ΔSnkonf
α) Configuration contribution to defect entropy, ΔSV = ΔSkonf
ΔS konf = k ln W
Boltzmann equation
W: Number of discernible arrangements of Nv vacancies and N particles on N + Nv sites (konf ≡
configuration, i.e. geometrical arrangement)
W=
(N + N ) !
V
N !N V !
Hence:
G(N V ) = G 0 + N v H V − TΔS konf
= G0 + N V H V − kT ln
(N + N ) !
V
N !N V !
= G 0 + N V H V − TN V S V + kT ln
NV
N + NV
Stirling approximation:
G(N V ) → Min !
ln N! = N ln N − N
∂G
=0
∂N V
NV
= e −H
N + NV
v
/ kT
= x V = [V]
5
β) Non-configuration contribution to defect entropy, ΔSnkonf
Entropy contribution caused by changes in the vibrational properties of the lattice due to the
formation of defects. Simplest model: only vibrations of atoms/ions neighboring vacancies are
changed. Treatment in the framework of the Einstein model.
The vibrational entropy of the 3N oscillators of the defect-free crystal in the Einstein model:
T
C EV (T )
hν E / kT
dT = 3Nk
− 3Nk ln( 1 − exp( −hν E / kT ))
E
T
exp(
h
ν
/
kT
)
−
1
0
S vib (T ) = ∫
In the high-temperature limit:
S vib (T ) = 3Nk ( 1 − ln
hν E
)
kT
In a crystal with NV vacancies, 3N−Z⋅NV oscillators vibrate with the unchanged frequency νE = ν0
and Z⋅NV oscillators with the changed frequency ν’, Z is the number of nearest neighbours.
Hence
ΔS nkonf = S vib (3N − Z ⋅ N V oscillators with ν 0 , Z ⋅ N V oscillators with ν' ) − S vib (3N oscillators with ν 0 )
= Z NV k ( 1− ln (ν0/ ν’)) = NV SV
Accounting also for his contribution to entropy, the general result fort he defect concentration in
an element crystal reads:
x V = [V] = eS
v
/k
⋅ e −H
v
/ kT
= e −G
v
/ kT
mit
GV = HV − T SV
b) The chemical potential of vacancies
Is it possible to attribute a chemical potential to a vacancy, i.e. to a non-occupied lattice site?
NV
From G(N V ) = G 0 + N V H V − T N V S V + kT ln
N + NV
the chemical potential of vacancies follows as
dG(N V )
= H V − TS V + kT ln x V
dN V
= μ0 + kT ln xV
μ( V ) =
potentials ist he Obviously, a meaningful definition of the chemical potentials of vacancies, i.e.
of vacant lattices sites, is possible! If μV can be defined, vacancies, V, can be considered as
quasi-chemical species. Generally, conventional chemical potential can be attributed to all kinds
of lattice defects ( =ˆ quasi-chemical species). The existence of chemical necessary prerequisit
fort he validity and application of chemical thermodynamics in this field. In particular this means
that mass-action laws can be set up for reactions involving defects.
6
c) Statistical treatment of vacancy disorder in ionic crystals Aa Bb
Schottky disorder
Sublattice
A:
NA particles + NV A vacancies on sublattice A
B:
NB particles + NV B vacancies on sublattice B
B
Aim: Determination of equilibrium concentrations of VA and VB
B
Ansatz: G(N V , N V ) = G 0 + N V ⋅ H V + N V ⋅ H V − T N V S V − T N V S V
A
B
A
A
B
B
− kT ln
A
(N
A
B
)
+ NA !
VA
NV ! NA !
B
(N
− kT ln
A
NV + NA
Boundary condition:
A
N V + NB
=
B
g(N V ,N V ) =
or:
A
B
a
b
(
)
+ NB !
VB
N V !NB !
B
to ensure the conservation of lattice structure in AaBb
B
)
(
)
1
1
N V + N A − N V + NB = 0
a
b
A
B
Minimization of G = G ( NV A , NV B ) under boundary conditions
Task:
Lagrangian method:
Minimization of F(N V ,N V ) = G(N V ,N V ) + λ g(N V ,N V ) , where λ is the so-called
A
B
A
B
A
A
Langrangian parameter
∂F
λ
= H V − T S V + kT ln [VA ] + = 0
a
∂N V
A
[ VA ] =
A
NV
A
NA + NV
A
A
and
∂F
λ
= H V − T S V + kT ln [VB ] − = 0
b
∂N V
B
[ VB ] =
B
B
After elimination of λ:
aH V + bH V − T(a S V + b S V ) + kTa ln [VA ] + kT b ln [VB ] = 0
14
4244
3
1442443
A
B
A
HS
B
SS
one arrives at the general result for Schottky disorder:
[V A ]a [VB ]b
= e S s / k e − H s / kT = e −GS / kT = K S
Hence, for a crystal of type AB
[V A ][VB ] = e S / k e − H
s
s
/ kT
= e − GS / kT = K F
holds and the vacancy concentration of a pure crystal is given by
[VA] = [VB] =exp(SS/2k) exp(−HS/2kT) = = K S
B
7
NV
B
NB + N VB
For the Frenkel equilibrium one obtains analogously:
[V A ][AI ] = e S
F
/k
e − H F / kT = e − GF / kT
and for the pure crystal:
[VA]=[VI]=exp(SF/2k) exp(−HF/2kT) = K F
d) Disorder parameters
Metals
Tm/K
Au
xV(Tm)
1333
HV/eV
7,2 · 10
-4
0,87
-4
0,49
Ag
1234
1,7 · 10
Cu
1353
2,0 · 10-4
1,03
Al
983
9,0 · 10-4
0,73
Ionic crystals
Hf/eV
Sf/k
Disorder type
NaCl
2,44
9,8
Schottky
KCl
2,51
8,99
"
AgBr
1,13
6,55
Frenkel, Kation
AgCl
1,45
9,41
SrF2
1,74
CaF2
2,71
"
"
Frenkel, Anion
5,53
Intrinsic disorder in AgBr
HF = 102 kJ mol −1 =ˆ 1,06 eV
S F = 6,8 R = 52,4 JK −1mol −1
8
"
"
T/°C
T/K
KF
[AgI] = [VAg]
-200
73
5,6 · 10-71
7,5 · 10-36
-100
173
1,1 · 10-28
1,1 · 10-14
0
273
2,4 · 10-17
4,9 · 10-9
25
298
1,0 · 10-15
3,2 · 10-8
100
373
4,2 · 10-12
2,1 · 10-6
200
473
4,5 · 10-9
6,7 · 10-5
300
573
4,2 · 10-7
6,5 · 10-4
400
673
1,0 · 10-5
3,2 · 10-3
Tm=422
695
1,8 · 10-5
4,3 · 10-3
e) Chemical thermodynamics
Schottky disorder in AX
Species: VA, VX, AA, XX
dG = ∑ μ i dn i
= μ( VA )dn V + μ( VX )dn V + μ( A A )dn A + μ( X X )dn X
A
X
A
X
-
Conservation of ions:
dn A = 0 and dn X = 0
-
Conservation of lattice structure:
dn V = dn V = dn V
A
X
A
X
dG = (μ( VA ) + μ( V X )) dn V
G → Min ⇔ dG = 0:
dG = (μ( VA ) + μ( VX )) dn V = 0
d.h. μ( VA ) + μ( VX ) = 0
Obviously, the respective reaction, due to the equilibrium condition
thermodynamics, is
0 ⇌ VA + VX
and hence
μ 0V + kT ln x V + μ oV + kT ln x V = 0
A
A
X
X
from which the already known result follows:
[V ][V ] = exp(− (μ
A
X
0
VA
)
+ μ 0V ) / kT = exp( −G S / kT ) = K S
X
9
∑ν
i
μ i = 0 of chemical
Frenkel disorder in AX
Species: AI, VA, AA, XX
dG = μ( A I )dn A + μ( VA )dn V + μ( A A )dn A + μ( X X )dn X
I
A
A
X
Conservation of ions: dn A + dn A = 0 und dn X = 0
I
A
X
dn A = dn V
Reaction:
I
A
dG = (μ( A I ) + μ( VA ) − μ( A A )) dn A I = 0
d.h. μ( A I ) + μ( VA ) − μ( A A ) = 0
and the respective reaction is:
⇄ A I + VA A A
Equilibrium constant of the Frenkel reaction
K F = [ A I ] [ VA ] = e −G
F
/ RT
III.4 Adjustment and control of defect concentrations
a) Doping
Incorporation of heterovalent foreign ions
Example: Doping of AgBr with Cd2+
AgBr
CdBr 2 ⎯⎯
⎯→ Cd 2Ag+ + 2BrBr− + VAg
to conserve crystal structure!
For every Cd2+ ion taken up by the crystal, one vacancy is generated on cation sites. For
sufficiently high doping levels (see below) the vacancy concentration is determined by the
dopant.
b) Equilibration with external phases
Example:
Gas-solid equlibrium between oxid AO and O2(g)
1
AO
O 2 (g) ←⎯
⎯→ O 02− + VA + 2h +
2
conservation of structure electroneutrality
10
With the incorporation of oxygen as oxygen(2-) ion, one vacancy in the cation sublattice and two
holes are created. Application of the mass-action law yields the following relation between these
defect concentrations and the (external) oxygen partial pressure:
[ VA ] ⋅ [h] 2 ∝ p O−1 / 2
2
c) Electroneutrality (EN) of crystals
EN of Coulomb charges of ionic crystal:
∑q
= 0 = e∑ z G
G
G
G
all sites G,
regular & interstitial,
occupied (qG≠0) or unoccupied (qG=0)
Effective charge on defects ε:
⎛
⎞
⎛
⎞
= 0 = ∑ (qreG − qidG ) = e∑ (z reG − z idG ) = e∑ ε def
⎜ ∑ qG ⎟ − ⎜ ∑ qG ⎟
G
G
G
⎝ G
⎠ real ⎝ G
⎠ ideal
123
εdef
EN of effective charges on defects
∑ε
def
=
G
∑ε
def
def
N def = 0 = ∑ ε def [def ]
def
defect
species
ε def = z real − z ideal
Effective charges
Consequence
≡ ε
all regularly occupied sites (mit
Symbolics
'
x
negative
neutral
•
positive
ε = 0 ) disappear from the EN!
Example: A 32+ B 32−
A xA
BBx
A B5⋅
VA'''
VB••
A I•••
3-3
(-2)-(-2)
3-(-2)
0-3
0-(-2)
3-0
ε
0
0
5
-3
2
3
symbol
x
x
•••••
'''
••
•••
ε=zreal-zid
11
If one "subtracts" the ideal structure (middle column) from the real structure (left column),
defects result as relevant (quasi-) particles (according to Maier)
d) Notation, structure elements, and rules
Kröger-Vink notation
Structure elements are completely characterised by specifiing
i) species (type of defect, component)
ii) lattice site
iii) effective charge
12
. cha rge
Structure element: S Gε ←←effect
site
species
Examples:
ii)
(A
VA''' , VB•• , A I••• , A AX
i)
3+
2
B 32− )
(AgBr )
'
, VBr• , AgI•
VAg
Rules for reactions involving structure elements
Equations involving reactions with structure elements S Gε (quasi-chemical species)
have to fulfill the following conditions:
1) conservation of particles (atoms/ions)
2) EN of effective charges:
∑ ε [def ] = 0
i
i
i
3) conservation of crystal structure
(conservation of ratio of regular cation-to-anion sites)
III.5 Examples
a) Doped AgBr
Incorporation of Cd2+-Ionen into AgBr:
'
AgBr
CdBr2 ⎯⎯
⎯→ Cd •Ag + VAg
+ 2 BrBrX
[ ] [ ] [
'
Elektroneutrality: VAg
= AgI• + Cd•Ag
]
[ ][ ]
'
Frenkel equilibrium: K F = VAg
AgI⋅
If x0 = [CdBr2], the following relations hold
[V ] = [VK ] + x
'
Ag
[V ]
'
Ag
F
'
Ag
2
0
[ ]
'
− x 0 VAg
− KF = 0
13
[V ] = x2
'
Ag
+ ( x 02 / 4) + K F
0
[V ]
[Ag ]
'
Ag
log [ ]
•
I
log[Cd´]
~ KF
Incorporation of S2−-Ionen into AgBr:
X
'
AgBr
Ag 2 S ⎯⎯
⎯→ Ag Ag
+ Ag I• + S Br
[S ] + [V ] = [Ag ]
'
Br
'
Ag
I
und
[V ][Ag ] = K
'
Ag
•
I
F
x0 = [Ag2S]
K
[Ag ] = [Ag
]+ x
•
I
F
'
I
[Ag ] = X2
•
I
0
0
+ ( x 02 / 4) + K F
[Ag ]
•
I
log [ ]
[V ]
'
Ag
~
KF
log[S´]
14
b) ZrO2 - CaO
Phase diagramm (Stubican et al.)
2900
Liquid
CaZrO3+Liq.
2500
Cub ss + Liq.
2250 ± 20°
2100
Cub ss
Cub ss + CaZrO3
1700
1300
Tet ss
Tet ss
+
Cub ss
1310 ± 40°
1140 ± 40°
Tet ss + CaZr4O9
900
CaZr4O9 + CaZrO3
Mon ss
CaZr4O9
500
0
10
ZrO2
Cub ss + CaZr4O9
20
30
40
50
CaZrO3
Mol %
CaO
Strategies of CaO incorporation
i) cation sublattice intact
ZrO
CaO ⎯⎯
⎯→ Ca 'Zr' + O OX + VO•• r
or
ii) anion sublattice intact
2
ZrO
2CaO ⎯⎯
⎯→ 2O OX + Ca 'Zr' + CaI••
2
The experimental data shown in the Figure demonstrate that the incorporation of CaO into ZrO2 is
accompanied by the formation of oxygen vacancies.
c) Magnetite, Fe3O4
15
c) Magnetite, Fe3−δO4
Phase diagram Fe-O
1400
δ-Fe
fl. Oxid
1600
1300
1400
800
1 -Δ O
1000
1200
Fe
[T · °C -1 ]
3
1200
γ-Fe
Fe3-δO 4
1100
600
Temperature (°C)
1000
400
Fe2O 3-ε
FeO
10
20
30
40
60
50
80
90
900
Fe2O3
Weight %
α-Fe
-20
FeO·Fe2O3
-15
-10
-5
log a O 2
The spinel structure
AB2O4
Cation sites of
tetrahedral
a0
octahedral
coordination
oxygen ions
a0
Disorder
a0
16
0
Oxygen incorporation
⇄
2 O 2 ( g)
⇄
2+
8 h ⋅ + 8 FeFe
2+
2 O 2 (g) + 8 FeFe
4 O Ox + 3 VFeε + 8 h ⋅
3+
8 Fe Fe
3+
+ 4 O Ox
⇄ 3 VFeε + 8 Fe Fe
For small defect concentrations, the application of mass-action law yields
[V ]
K=
3
Fe
2
O2
a
and for the oxygen partial pressure dependence of vacancy concentration
[V ] = K
Fe
a O2 / 3
V
2
Because of the prevailing Frenkel disorder of iron ions in magnetite
[ ][ ]
K F = VFe FeI
one obtains for the concentration of iron interstitials
[Fe ] = K a
I
I
−2 / 3
O2
Magnetite is a non-stoichiometric compound, which can accommodate iron excess as well as
iron deficit: Fe3−δO4. The stochiometric parameter δ is directly determined by the defect
concentrations:
[Fe]
tot
= 3 − δ = 3 − [VFe ] + [Fe I ]
and hence:
δ = [VFe ] − [Fe I ]
with the following predicted dependence on oxygen partial pressure
δ = K V a O2 / 3 − K I a O−2 / 3
2
2
17
Experimental determination of δ by (thermo)-gravimetrie.
9
1.0
7
-0.5
calculated with: K V=2.3
K I=1.5·10 -6
6
-13
-12
-11
-10
-9
-8
-7
log a O 2
5
4
Fe2+εO 3
-1 4
Fe 1-ΔO
-15
δ·102
-1.0
Fe 3-δO 4
T=1400°C
8
Fe2+εO 3
0
Fe 1-ΔO
δ·102
0.5
Fe 3-dO 4
T=900°C
calculated with: K V=2200
K I=2·10 -13
3
2
1
0
-1
-7
-6
-5
-4
-3
-2
log a O 2
-1
0
The weight change of 1 mol Fe3O4 due to oxygen exchange with an extenal O2 gas phase is
given by:
Δm = m(δ) − m(δ = 0) =
4
δ MO
3
The Figures above show δ as a functin of oxygen activity at different temperatures. The lines
drawn represent a fit to the experimental data according to
δ = K V a O2 / 3 − K I a O−2 / 3
2
2
The equilibrium constants KV and KI obtained from the fits are shown in the Figures. The model
has to be modified for high defect concentrations.
d) Cu2O
1
1
'
O 2 (g) ⇄ O 0x + 2 VCu
+ 2 h• ;
Incorporation of O 2 (g) :
2
2
EN :
[V ] = [h ]
•
'
Cu
K=
[V ]
4
'
Cu
PO
2
[V ] ~ p
'
Cu
1/ 8
O2
18
[V ] [h ] [O ]
K=
2
'
Cu
• 2
PO
2
x
0
6. Abstract
•
Point defects are created in thermodynamic equilibrium (G → Min) und thus constitute
natural elements of crystals.
•
Thermodynamically, defects can be considered as quasi-chemical species, because
(
)
chemical potential can be assigned to them μ V , μ A ,... .
•
A
I
Reactions involving point defects can be treated using mass-action laws (see rules for
structure elements)
•
The intrinsic thermal disorder of crystals, e.g., Schottky disorder (AX)
0 ⇄ VA +VX
K S = [VA ][VX ] = e S
S
/R
e −H
/ RT
/R
e −H
/ RT
S
or Frenkel disorder (AX)
AA ⇄ AI + VA
K F = [VA ] [A I ] = e S
F
F
can be modified and controlled by doping with heterovalent impurities or by equilibration with
external gas phases. The respective defect concentrations and their dependencies on partial
pressure or dopant level can be calculated in the framework point defect thermodynamics.
19
7. Literature
General literature
Physical Chemistry of Solids
Borg, Dines
Wiley
Der feste Zustand
W.J. Moore
Vieweg, Braunschweig 1977
Festkörperchemie
L.Smart, E. Moore
Vieweg, Braunschweig 1998
Solid State Chemistry
A. West
Wiley, New York 1992
Fundamentals of Ceramics
M. Barsoum
McGraw-Hill International Series, New York 1997
Introduction to Ceramics
W.D. Kingery, H.K. Bowen, D.R. Ullmann
Wiley, New York 1976
Einführung in die Festkörperphysik
C. Kittel
Oldenbourg, München 2002
Materials Thermochemistry
O. Kubaschewski, C.B. Alcock, P.J. Spencer
Pergamon, Oxford 1993
Physikalische Metallkunde
P. Haasen
Springer, Berlin 1984
Kristallstruktur und chemische Bindung
A. Weiss, H. Witte
Verlag Chemie1983
Special literature for Chapter III
The Chemistry of imperfect crystals, Vol.2
F.A. Kröger
North-Holland Publ. Comp., Amsterdam 1964
Ionenkristalle, Gitterdefekte und nichtstöchiometrische Verbindungen
N.N. Greenwood
Verlag Chemie, Weinheim 1973
20
Defektchemie: Zusammensetzung, Transport und chemische Reaktion im festen Zustand.
I. Thermodynamik
J. Maier, in : Angew. Chem. 105 (1993) 333, 558
Festkörper – Fehler und Funktion
Prinzipien der Physikalischen Festkörperchemie
J. Maier
Teubner Studienbücher Chemie
B.G. Teubner Stuttgart, Leipzig 2000
Physical Chemistry of Ionic Materials
J. Maier
Wiley 2004
Atomic Transport in Solids
A.R. Allnatt, A.B. Lidiard
Cambridge University Press 1993
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