the muppet’s guide to: the structure and dynamics of solids phase diagrams
TRANSCRIPT
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The Muppet’s Guide to:The Structure and Dynamics of Solids
Phase Diagrams
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Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.
-independent variables: T and Co (P = 1 atm is almost always used).
• PhaseDiagramfor Cu-Niat P=1 atm.
• 2 phases: L (liquid)
a (FCC solid solution)
• 3 phase fields: LL + aa
wt% Ni20 40 60 80 10001000
1100
1200
1300
1400
1500
1600T(°C)
L (liquid)
a (FCC solid solution)
L + aliquidus
solid
us
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.
-independent variables: T and Co (P = 1 atm is almost always used).
• PhaseDiagramfor Cu-Niat P=1 atm.
wt% Ni20 40 60 80 10001000
1100
1200
1300
1400
1500
1600T(°C)
L (liquid)
a (FCC solid solution)
L + aliquidus
solid
us
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Liquidus:Separates the liquid from the mixed L+ aphase
Solidus:Separates the mixed L+ a phase from the solid solution
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wt% Ni20 40 60 80 10001000
1100
1200
1300
1400
1500
1600T(°C)
L (liquid)
a (FCC solid solution)
L + a
liquidus
solid
us
Cu-Niphase
diagram
Number and types of phases• Rule 1: If we know T and Co, then we know: - the number and types of phases present.
• Examples:
A(1100°C, 60): 1 phase: a
B(1250°C, 35): 2 phases: L + a
B (
1250
°C,3
5) A(1100°C,60)
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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wt% Ni20
1200
1300
T(°C)
L (liquid)
a(solid)L + a
liquidus
solidus
30 40 50
L + a
Cu-Ni system
Composition of phases• Rule 2: If we know T and Co, then we know: --the composition of each phase.
• Examples:TA
A
35Co
32CL
At TA = 1320°C:
Only Liquid (L) CL = Co ( = 35 wt% Ni)
At TB = 1250°C:
Both a and L CL = C liquidus ( = 32 wt% Ni here)
Ca = C solidus ( = 43 wt% Ni here)
At TD = 1190°C:
Only Solid ( a) Ca = Co ( = 35 wt% Ni)
Co = 35 wt% Ni
BTB
DTD
tie line
4Ca3
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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wt% Ni20
1200
1300
30 40 50110 0
L (liquid)
a (solid)
L + a
L + a
T(°C)
A
35Co
L: 35wt%Ni
Cu-Nisystem
• Phase diagram: Cu-Ni system.
• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.
• Consider Co = 35 wt%Ni.
Cooling a Cu-Ni Binary - Composition
4635
4332
a: 43 wt% Ni
L: 32 wt% Ni
L: 24 wt% Ni
a: 36 wt% Ni
Ba: 46 wt% NiL: 35 wt% Ni
C
D
E
24 36
Figure adapted from Callister, Materials science and engineering, 7 th Ed. USE LEVER RULE
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• Tie line – connects the phases in equilibrium with each other - essentially an isotherm
The Lever Rule – Weight %
How much of each phase? Think of it as a lever
ML M
R S
RMSM L
L
L
LL
LL CC
CC
SR
RW
CC
CC
SR
S
MM
MW
00
wt% Ni
20
1200
1300
T(°C)
L (liquid)
a(solid)L + a
liquidus
solidus
30 40 50
L + aB
T B
tie line
CoC L Ca
SR
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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• Rule 3: If we know T and Co, then we know: --the amount of each phase (given in wt%).• Examples:
At T A : Only Liquid (L) W L = 100 wt%, W a = 0
At T D : Only Solid ( a) W L = 0, W a = 100 wt%
C o = 35 wt% Ni
Weight fractions of phases – ‘lever rule’
wt% Ni20
1200
1300
T(°C)
L (liquid)
a(solid)L + a
liquidus
solidus
30 40 50
L + a
Cu-Ni system
TA A
35C o
32C L
BT B
DT D
tie line
4Ca3
R S
= 27 wt%
43 3573 %
43 32wt
At T B : Both a and L
WL= S
R + S
Wa= R
R + S
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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wt% Ni20
120 0
130 0
30 40 50110 0
L (liquid)
a (solid)
L + a
L + a
T(°C)
A
35C o
L: 35wt%Ni
Cu-Nisystem
• Phase diagram: Cu-Ni system.• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.• Consider Co = 35 wt%Ni.
Cooling a Cu-Ni Binary – wt. %
46344332
a: 27 wt%
L: 73 wt%
L: 8 wt%
a: 92 wt%
Ba: 8 wt% L: 92 wt%
C
D
E
24 36
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Equilibrium cooling
• Multiple freezing sites– Polycrystalline materials– Not the same as a single crystal
• The compositions that freeze are a function of the temperature
• At equilibrium, the ‘first to freeze’ composition must adjust on further cooling by solid state diffusion
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Diffusion is not a flow
Our models of diffusion are based on a random walk approach and not a net flow
http://mathworld.wolfram.com/images/eps-gif/RandomWalk2D_1200.gif
Concept behind mean free path in scattering phenomena - conductivity
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Diffusion in 1 Dimension
• Fick’s First Law
dCJ D T
dx
J = flux – amount of material per unit area per unit timeC = concentration
Diffusion coefficient which we expect is a function of the temperature, T
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Diffusion cont….• Requires the solution of the continuity equation:
The change in concentration as a function of time in a volume is balanced by how much material flows in per time unit minus how much flows out – the change in flux, J:
• giving Fick’s second law (with D being constant):
2
2
C C CD D T
t x x x
0C Jt x
dC
J D Tdx
BUT
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Solution of Ficks’ Laws
C
x
CCo
t = 0
t = t
For a semi-infinite sample the solution to Ficks’ Law gives an error function distribution whose width increases with time
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Consider slabs of Cu and Ni.
Interface region will be a mixed alloy (solid solution)
Interface region will grow as a function of time
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wt% Ni20
120 0
130 0
30 40 50110 0
L (liquid)
a (solid)
L + a
L + a
T(°C)
A
35C o
L: 35wt%Ni
Cu-Nisystem
Co = 35 wt%Ni.
Slow Cooling in a Cu-Ni Binary
a: 43 wt% Ni
L: 32 wt% Ni
L: 24 wt% Ni
a: 36 wt% Ni
Ba: 46 wt% NiL: 35 wt% Ni
C
D
E
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Enough time is allowed at each temperature change for atomic diffusion to occur. – Thermodynamic ground state
Each phase is homogeneous
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Non – equilibrium
cooling α
L
α + L
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Reduces the melting
temperature
No-longer in the thermodynamic
ground state
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• Ca changes as we solidify.• Cu-Ni case:
• Fast rate of cooling: Cored structure
• Slow rate of cooling: Equilibrium structure
First a to solidify has Ca = 46 wt% Ni.
Last a to solidify has Ca = 35 wt% Ni.
Cored vs Equilibrium Phases
First a to solidify: 46 wt% Ni
Uniform C a:
35 wt% Ni
Last a to solidify: < 35 wt% Ni
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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2 componentshas a special compositionwith a min. melting temperature
Binary-Eutectic Systems – Cu/Ag
• 3 phases regions, L, a and b and 6 phase fields - L, a and , b L+ , a L+ , +b a b
• Limited solubility – mixed phases
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
a phase:
Mostly copper
b phase:
Mostly Silver
Solvus line – the solubility limit
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Min. melting TE
Binary-Eutectic Systems
• Eutectic transitionL(CE) (CE) + (CE)
• TE : No liquid below TE
TE, Eutectic temperature, 779°CCE, eutectic composition, 71.9wt.%
The Eutectic point
Cu-Ag system
L (liquid)
a L + a L +b b
a + b
Co wt% Ag in Cu/Ag alloy20 40 60 80 1000
200
1200T(°C)
400
600
800
1000
CE
TE CaE=8.0 CE=71.9 CbE=91.2
779°C
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Any other composition, Liquid transforms to a mixed L+solid phase
E
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L + aL +b
a + b
200
T(°C)
18.3
C, wt% Sn20 60 80 1000
300
100
L (liquid)
a 183°C 61.9 97.8
b
• For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find... --the phases present: Pb-Sn
system
Pb-Sn (Solder) Eutectic System (1)
a + b--compositions of phases:
CO = 40 wt% Sn
--the relative amount of each phase:
150
40
Co
11
C
99
C
SR
Ca = 11 wt% SnCb = 99 wt% Sn
Wa =C - CO
C - C
= 99 - 4099 - 11
= 5988 = 67 wt%
SR+S =
W =CO - C
C - C=R
R+S
= 2988
= 33 wt%= 40 - 1199 - 11
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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• 2 wt% Sn < Co < 18.3 wt% Sn• Result: Initially liquid → liquid + then alone finally two phases
a poly-crystal fine -phase inclusions
Microstructures in Eutectic Systems: II
Pb-Snsystem
L + a
200
T(°C)
Co , wt% Sn10
18.3
200Co
300
100
L
a
30
a + b
400
(sol. limit at TE)
TE
2(sol. limit at T room)
La
L: Co wt% Sn
ab
a: Co wt% Sn
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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• Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals.
Microstructures in Eutectic Systems: Co=CE
160 m
Micrograph of Pb-Sn eutectic microstructurePb-Sn
systemL
a
200
T(°C)
C, wt% Sn
20 60 80 1000
300
100
L
a b
L+ a
183°C
40
TE
18.3
: 18.3 wt%Sn
97.8
: 97.8 wt% Sn
CE61.9
L: Co wt% Sn
Figures adapted from Callister, Materials science and engineering, 7 th Ed.
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• Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals.
Microstructures in Eutectic Systems: Co=CE
Pb-Snsystem
L
a
200
T(°C)
C, wt% Sn
20 60 80 1000
300
100
L
a b
L+ a
183°C
40
TE
18.3
: 18.3 wt%Sn
97.8
: 97.8 wt% Sn
CE61.9
L: Co wt% Sn
Figures adapted from Callister, Materials science and engineering, 7 th Ed.
97.8 61.945.2%
97.8 18.3W
61.9 18.354.8%
97.8 18.3W
Pb rich
Sn Rich
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Lamellar Eutectic Structure
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
PbSn
At interface, Pb moves to a-phase and Sn migrates to b- phase
Lamellar form to minimise diffusion distance – expect spatial extent to depend on D and cooling rates.
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• 18.3 wt% Sn < Co < 61.9 wt% Sn• Result: a crystals and a eutectic microstructure
Microstructures IV
18.3 61.9
SR WL = (1- Wa) = 50 wt%
Ca = 18.3 wt% Sn
CL = 61.9 wt% SnS
R + SWa = = 50 wt%
• Just above TE :
Pb-Snsystem
L+b200
T(°C)
Co, wt% Sn
20 60 80 1000
300
100
L
a b
L +a
40
a +b
TE
L: Co wt% Sn LL
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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• 18.3 wt% Sn < Co < 61.9 wt% Sn• Result: a crystals and a eutectic microstructure
Microstructures IV
18.3 61.9
SR
97.8
SR
Primary, a
Eutectic, a Eutectic, b
• Just below TE :Ca = 18.3 wt% SnCb = 97.8 wt% Sn
SR + S
Wa = = 73 wt%
Wb = 27 wt%
Pb-Snsystem
L+b200
T(°C)
Co, wt% Sn
20 60 80 1000
300
100
L
a b
L +a
40
a +b
TE
L: Co wt% Sn LL
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Intermetallic Compounds
Mg2Pb
Note: intermetallic compound forms a line - not an area - because stoichiometry (i.e. composition) is exact.
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
a phase:
Mostly Mg
b phase:
Mostly Lead
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Eutectoid & Peritectic
Cu-Zn Phase diagram
Eutectoid transition +
Peritectic transition + L
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
mixed liquid and solid to single solid transition
Solid to solid ‘eutectic’ type transition
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Iron-Carbon (Fe-C) Phase Diagram• 2 important points
-Eutectoid (B): g a + Fe3C
-Eutectic (A): L g + Fe3C
Fe3C
(cem
entit
e)
1600
1400
1200
1000
800
600
4000 1 2 3 4 5 6 6.7
L
g(austenite)
g+L
g +Fe3C
a +Fe3C
a+g
L+Fe3C
d
(Fe) Co, wt% C
1148°C
T(°C)
a 727°C = T eutectoid
ASR
4.30Result: Pearlite = alternating layers of
a and Fe3C phases
120 mm
g ggg
R S
0.76
Ceu
tect
oid
B
Fe3C (cementite-hard)a (ferrite-soft)
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Iron-Carbon
http://www.azom.com/work/pAkmxBcSVBfns037Q0LN_files/image003.gif
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The Muppet’s Guide to:The Structure and Dynamics of Solids
The Final Countdown
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CharacterisationOver the course so far we have seen how thermodynamics plays an important role in defining the basic minimum energy structure of a solid.
Small changes in the structure (such as the perovskites) can produce changes in the physical properties of materials
Kinetics and diffusion also play a role and give rise to different meta-stable structures of the same materials – allotropes / polymorphs
Alloys and mixtures undergo multiple phase changes as a function of temperature and composition
BUT how do we characterise samples?
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Probes
Resolution better than the inter-atomic spacings
• Electromagnetic Radiation
• Neutrons
• Electrons
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Probes
Treat all probes as if they were waves:
;
hp k p mv
Wave-number, k:2
k k
Momentum, p:
Photons ‘Massive’ objects
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Xavier the X-ray
hcE
Ex(keV)=1.2398/l(nm)
Speed of Light
Planck’s constant Wavelength
Elastic scattering as Ex>>kBT
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Norbert the Neutron
hmv
222
12 2n
n
hE mvm
En(meV)=0.8178/l2(nm)
De Broglie equation:
mass velocity
Kinetic Energy:
Strong inelastic scattering as En~kBT
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Eric the Electron
• Eric’s rest mass: 9.11 × 10−31 kg.• electric charge: −1.602 × 10−19 C• No substructure – point particle
hmvDe Broglie equation:
mass velocity
Ee depends on accelerating voltage :– Range of Energies from 0 to MeV
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ProbesResolution better than the interatomic spacings
Absorption low – we want a ‘bulk’ probe
• Electrons - Eric
• quite surface sensitive
• Electromagnetic Radiation - Xavier
• Optical – spectroscopy
• X-rays :
• VUV and soft (spectroscopic and surfaces)
• Hard (bulk like)
• Neutrons - Norbert
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Interactions
1. Absorption
2. Refraction/Reflection
3. Scattering Diffraction
EnglebertXavier
Norbert
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Crystals are 2D with planes separated by dhkl. There will only be constructive interference when == - i.e. the reflection
condition.
a
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Basic Scattering Theory
The number of scattered particles per
second is defined using the standard
expression
I Id
ds 0
Unit solid angle Differentialcross-section
Defined using Fermi’s Golden Rule
INTERACTION POTENTId
dFi Ana L i ll Init a
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Spherical Scattered Wavefield
ScatteringPotential
Incident Wavefield
Different for X-rays, Neutrons and Electrons
2
exp k r r r r
d
dd
i V
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BORN approximation:• Assumes initial wave is also spherical• Scattering potential gives weak interactions
0
2rexp kp rk r rex V i
ddi
d
2
exp q rr r id
dd
V
Scattered intensity is proportional to the Fourier Transform of the scattering potential
q k k0
2
exp k r r r rd
i V dd
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Scattering from CrystalAs a crystal is a periodic repetition of atoms in 3D we can formulate the scattering amplitude from a crystal by expanding the scattering
from a single atom in a Fourier series over the entire crystal
( ) exp q r V
f r i dV
(q) q exp q T rj jT j
A fi
Atomic Structure Factor
Real Lattice Vector: T=ha+kb+lc
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The Structure FactorDescribes the Intensity of the diffracted beams in reciprocal space
exp q r exp u v w 2jj j
i i h k l
hkl are the diffraction planes, uvw are fractional co-ordinates within
the unit cell
If the basis is the same, and has a scattering factor, (f=1), the structure
factors for the hkl reflections can be found hkl
Weight phase
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The Form Factor
Describes the distribution of the diffracted beams in reciprocal space
The summation is over the entire crystal which is a parallelepiped of sides:
1
1
32
2 3
1T 1
2 31 1
q exp q T exp q a
exp q b exp q c
N
n
NN
n n
L i n i
n i n i
1 2 3N a N b N c
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The Form FactorMeasures the translational symmetry of the lattice
The Form Factor has low intensity unless q is a
reciprocal lattice vector associated with a reciprocal
lattice point
1,2,3 1,2,3 1,2,3
sin s sin sq exp s
sin s
i
i
Ni i i i
i ijini i i
N NL i n
s
0
0.5x105
1.0x105
1.5x105
2.0x105
2.5x105
-0.02 -0.01 0 0.01 0.02
Deviation parameter, s1 (radians)
[L(s
1)]
2
N=2,500; FWHM-1.3”
N=500
q d s Deviation from reciprocal lattice point located at d*
Redefine q:
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The Form Factor
0
20
40
60
80
100
-0.6 -0.3 0 0.3 0.6
Deviation parameter, s1 (radians)
[L(s
1)]
2
0
0.5x105
1.0x105
1.5x105
2.0x105
2.5x105
-0.02 -0.01 0 0.01 0.02
Deviation parameter, s1 (radians)
[L(s
1)]
2
The square of the Form Factor in one dimension
N=10 N=500
1,2,3
sin sq i i
ji
NL
s
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Scattering in Reciprocal Space
T
q q exp q r exp q Tj jj
A f i i Peak positions and intensity tell us about the structure:
POSITION OF PEAK
PERIODICITY WITHIN SAMPLE
WIDTH OF PEAK
EXTENT OF PERIODICITY
INTENSITY OF PEAK
POSITION OF ATOMS IN
BASIS
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Powder DiffractionIt is impossible to grow some materials in a single crystal form or
we wish to study materials in a dynamic process.
Powder Techniques
Allows a wider range of materials to be studied under different sample conditions
1. Inductance Furnace 290 – 1500K
2. Closed Cycle Cryostat 10 – 290K
3. High Pressure Up-to 5 million Atmospheres
• Phase changes as a function of Temp and Pressure
• Phase identification
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Search and MatchPowder Diffraction often used to identify phases
Cheap, rapid, non-destructive and only small quantity of sample
Inte
nsi
ty
2 A ngle
JCPDS Powder Diffraction File lists materials (>50,000) in order of their d-
spacings and 6 strongest reflectionsOK for mixtures of up-to 4
components and 1% accuracy
Monochromatic x-rays
Diffractometer
High Dynamic range detector
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Single Crystal Diffraction
2dhkl hklsin Monochromatic radiation so sample needs to moved to the
Bragg condition….
Angular resolution is the Darwin width of analyser crystal (Typically 10-20”)
Detailed Lateral Information obtained
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XMaS Beamline - ESRF
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StrainPeak positions defined by the lattice parameters:
1
1 1, ,
q exp qN
ini a b c
L i n
Strain is an extension or compression of the lattice,
hkl hkld d
Results in a systematic shift of all the peaks
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Ho Thin FilmsXRD measured as a function of temperature
10-4
10-2
100
102
104
20 40 60 80 100
T=294KT=244KT=194KT=94KT=144KT=42KT=10KT=300K
Scattering Angle ()
Inte
nsity
(ar
b. u
nits
)
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Ho Thin FilmsSubstrate and Ho film follow have different behaviour
1
10
100
1000
30 32 34 36
T=294KT=244KT=194KT=144KT=94KT=42KT=10K
Scattering Angle ()
Inte
nsi
ty (
arb
. u
nits
)
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Whole film refinement
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Peak BroadeningDiffraction peaks can also be broadened in qz by:
1. Grain Size 2. Micro-Strains OR Both
The crystal is made up of particulates which all act as perfect but small crystals
, ,
sin sq i i
ii a b c
NL
s
Number of planes sampled is finite
Recall form factor: Scherrer Equation
2
cosSizeBD
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NixMn3-xO4+ (400 Peak)As Grown at 200ºC AFM images (1200 x 1200 nm)
400
0
0.2
0.4
0.6
0.8
1.0
-1.0 -0.5 0 0.5 1.0
900C850C
800C
750C700C
650C
2
Inte
nsi
ty
D
450nm thick films
Annealed at 800ºC
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Peak BroadeningDiffraction peaks can also be broadened in qz by:
1. Grain Size 2. Micro-Strains OR Both
The crystal has a distribution of inter-planar spacings dhkl ±Ddhkl.
Diffraction over a range, ,Dq of angles
Differentiate Bragg’s Law: 2 2 tanStrain B
Width in radians
Strain Bragg angle
dd
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Peak BroadeningDiffraction peaks can also be broadened in qz by:
1. Grain Size 2. Micro-Strains OR Both
Total Broadening in 2q is sum of Strain and Size:
2 2 tancosTotal B
BD
2 cos 2 sinhkl hkl hklB B D
Rearrange
Williamson-Hall plot
y mx c
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Powder Diffraction
0
100
200
300
400
30 40 50 60
Detector Angle (°)
Inte
nsity
(a
rb.
units
)
Powder of Nickel ManganiteCUBIC Structure
0
0.05
0.10
0.15
0.20
0.25
0 10 20 30
333422
400
222311
220
y=(1.5412/(4*a2))xa=8.348 ± 0.0036
(h2+k2+l2)
sin2
(B)
0.005
0.006
0.007
0.008
0.05 0.10 0.15 0.20 0.25
y=((1.541/d))+(2s)xGrain Size=299 ± 19.5a/a = 0.005 ± 0.001
sin(B)
Wid
th *
cos
(B) Grain size = 30±2nm
Strain Dispersion = 0.005±0.001
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Cubic-Tetragonal Distortions
CUBICTETRAGONAL
a c a c
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High Temperature Powder XRD
25 30 35 40 45 50 55 60 65 702Theta (°)
0
10000
20000
30000
Inte
ns
ity
(c
ou
nts
)
30.8 30.9 31.0 31.1 31.2 31.3 31.4 31.5 31.6 31.72Theta (°)
10000
20000
30000
Inte
ns
ity
(c
ou
nts
)
0.4BiSCO3 - 0.6PbTiO3 (K. Datta)
Tetragonal → Cubic phase transition
Courtesy, D. Walker and K. Datta University of Warwick
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CsCoPO4
Dr. Mark T. Weller, Department of Chemistry, University of Southampton, www.rsc.org/ej/dt/2000/b003800h/
Variable temperature powder X-ray diffraction data show a marked change in the pattern at 170 °C.
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Sn in a Silica Matrix
1. What form of tin
2. Particle size
3. Strain
4. Melting Temperature
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Eutectic’s
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wt% Ni20
120 0
130 0
30 40 50110 0
L (liquid)
a (solid)
L + a
L + a
T(°C)
A
35C o
L: 35wt%Ni
Cu-Nisystem
• Consider Cu/Ni with 35 wt.% Ni
Following Structural Changes
4332
a: 43 wt% Ni
L: 32 wt% Ni
L: 24 wt% Ni
a: 36 wt% Ni
Ba: 46 wt% NiL: 35 wt% Ni
C
D
E
24 36
Figure adapted from Callister, Materials science and engineering, 7 th Ed. USE LEVER RULE
A. Liquid
B. Mixed Phase
C.
D.
E. Solid
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Cored Samples
α
L
α + L
Issues:
Lattice Parameter
Particle Size
Strain Dispersion
2 cos 2 sinhkl hkl hklB B D
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NiCrStructural Changes?
Fcc: hkl are either all odd or all
even.
Bcc: sum of hkl must be even.