Lecture7 - Universitetet i oslo · Lecture7: •vacancies, derivationofequilibriumconcentration...

Lecture 7:

• vacancies, derivation of equilibrium concentration

• estimation of vacancy concentration as a function of temperature

• vacancy concentration as a function of pressure

• excess concenrations of defects – studing vacancy

clustering by PAS

M/12/1/2009: Introduction to the course. Crystals structures 2h

W/14/1/2009: Reciprocal lattice 1h

M/19/1/2009: Crystals and x-ray diffraction 2h

W/21/1/2009: Crystals and x-ray diffraction 1h

M/26/1/2009: Crystal binding 2h

W/28/1/2009: Introduction and analysis of elastic strains 1h

M/2/2/2009: Point defects, case study – vacancy 2h

W/4/2/2009: Diffusion – phenomenology and mecahnisms 1h

M/9/2/2009: Diffusion under pressure. Dislocations 2h

11/2/2009:

16/2/2009:

18/2/2009: …

M/23/2/2009: Crystal vibrations 2h

W/25/2/2009: Crystal vibrations 1h

M/2/3/2009: Phonons 2h

W/4/3/2009: Heat capacity 1h

M/9/3/2009: Thermal conductivity 2h

W/11/3/2009: Energy bands in solids 1h

M/16/3/2009: Thermal conductivity 2h

W/19/3/2009: Free electron Fermi gas 1h

M/23/3/2009: Free electron Fermi gas 2h

W/25/3/2009: Repetition 1h

30/3/2009: Mid-term exam

Vacancy: A point defect

Defects Dimensionality Examples

Point 0 Vacancy

Line 1 Dislocation

Surface 2 Free surface,

Grain boundary

There may be vacant sites in a crystal

Surprising Fact

There must be a certain fraction of vacant

sites in a crystal in equilibrium.

Fact

Case study: vacancy

• Crystal in equilibrium

• Minimum Gibbs free energy G at constant T

and P

• A certain concentration of vacancy lowers

the free energy of a crystal

Case study: vacancy

Gibbs free energy G involves two terms:

1. Enthalpy H

2. Entropy S

G = H – T S

=E+PV

=k lnW

T Absolute temperature

E internal energy

P pressure

V volume

k Boltzmann constant

W number of microstates

Case study: vacancy

∆ ∆ ∆ ∆ H = n ∆ ∆ ∆ ∆ ΗΗΗΗf

Vacancy increases H of the crystal due to energy required to break bonds

Case study: vacancy

Configurational entropy due to vacancy

Number of atoms: N

Number of vacacies: n

Total number of sites: N+n

How many distinguished configurations,

so called microstates?

We calculate this explicitly


N ln N! N ln N− N

1 0 −1

10 15.10 13.03

100 363.74 360.51


∆G = ∆H − T∆S

neq

G of a

perfect

crystal

n

∆G ∆H

fHnH ∆=∆

−T∆S]lnln)ln()[( NNnnnNnNkS −−++=∆

Equilibrium concentration of vacancy

0=∂∆∂

= eqnnn

G


∆−=kT

H

N

n feqexp

Measure a material property

which is dependent on neq/N vs T

Find the activation

energy from the slope

neq

N

T

exponential dependence

1/T

N

neqln

1

-QD/k

slope


∆−=kT

H

N

n feqexp

– Copper at 1000 ºC

Hf = 0.9 eV/at ACu = 63.5 g/mol ρ = 8400 kg/m-3

First find N in atoms/m-3

( )( )

( )( )3

3

23

//

Check units

0635.0

840010023.6

m

at

molkg

mkgmolatN

N

A

NN

Cu

A

=→

=

×==

ρ


• Since Units Chk:

328 /1097.7 msitesatN −×=

• Now apply the Arrhenius relation @1000 ºC

( )325

5

28

/1018.2

1273/1062.8

/9.0exp1097.7

exp

mvacN

KKateV

ateV

kT

HNN

v

f

v

×=

−×−

×=

−=

−


∆−=kT

H

N

n feqexp

Al: ∆Ηf=0.70 ev/vacancyNi: ∆Hf=1.74 ev/vacancy

1.78x10-105.59x10−300 Ni

1.12x10−41.45x10−120Al

900 K300 K0 Kn/N


• Neighboring atoms tend to move into the

vacancy, which creates a tensile stress field

• The stress/strain field is nearly spherical

and short-range.ao

Equilibrium concentration of vacancy – pressure dependence

∆Gf=Ef+PVf - TSf

kTVkTEkSkTGeq

Vffff eeeeC//// σ−∆− ==

Hf=Ef+PVf

Vf = Ω + relaxation volume

fH∆

Equilibrium concentration of vacancy – pressure dependence

How big the pressure should be to make

a measurable effect on vacancy concentration?

fVσ

Compare

kTVkTEkSkTGeq

Vffff eeeeC//// σ−∆− ==

Excess concentration of vacancies

Radiation

Chemical reactions

[ ] [ ][ ] [ ] ( )kTECVVVRDt

VbnnV

n −⋅−=∂∂

− exp4 01π

dissociationgeneration

The amount of singlt vacancies exceed the ”solubility” limit and

vacancies may start cluster

Positron beam

Trapping and annihilation

Potential

diagram

γγγγ

γγγγ

Eγγγγ =511 keV

+ Doppler broadening depending on the amount of electron momentum

10-4

10-3

10-2

10-1

INTE

NS

ITY

(a

rb. units)

3020100ELECTRON MOMENTUM (10-3 m0c)

latticevacancy

W

S

S-parameter characterizes annihilation with low momentum valence

electrons. Increase in S-parameter is naturaly interpreted as an

increase in vacancy concentration

W-paprameter characterizes annihilation with high momentum core

electrons and increase in vacancy concentration results in decrease

of W-parameter

Investigation of vacancies using PAS

Positron beam

Trapping and annihilation

Potential

diagram

γγγγ

γγγγ

Eγγγγ =511 keV

+ Doppler broadening depending on the amount of electron momentum

1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy

Energy (keV)/Mean positron implantation depth (µm)

S p

ara

mete

r

a

30/1.60

bulk reference

Experimental points group around a line in the W-S plane if

there are only two annihilation states vailable in the sample


1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy

Li, as implanted


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference


1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy

Li, as implanted 500oC, 1h


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference


1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy


800oC, 1h


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference


1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy


900oC, 20ms 800

oC, 1h


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference


1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy


900oC, 20ms 800

oC, 1h

1200oC, 20ms


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference


Vacancy cluster formation kinetics and thermal stability

Li interacts and deactivates by

vacancy cluster that consist of at least

3–4 zinc vacancies

Why clusters do not survive 1h anneals at ≥ 800 °C?

1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy


900oC, 20ms 800

oC, 1h

1200oC, 20ms


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference



3–4 VZn and, possibly few VO

[ ] [ ][ ] [ ] ( )kTECVVVRDt

VbnnV

n −⋅−=∂∂

− exp4 01π



1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy


900oC, 20ms 800

oC, 1h

1200oC, 20ms


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference

• clustering take place as long as the vacancy diffusivity (Dv), supersaturation

level ([V]), and clustering reaction radii (R) are high enough

• Eb – dissociation energy – determines the dissociation rate at a given temperature




3–4 VZn and, possibly few VO

1hclusters dissolve

20 ms20 ms20 ms1hclusters survive

14001200900800500temperature (°C)

Eb

= 2.6 ± 0.3 eV

[ ] [ ][ ] [ ] ( )kTECVVVRDt

VbnnV

n −⋅−=∂∂

− exp4 01π


• clustering take place as long as the vacancy diffusivity (Dv), supersaturation

level ([V]), and clustering reaction radii (R) are high enough

• Eb – dissociation energy – determines the dissociation rate at a given temperature


1.00 1.02 1.04 1.06 1.08

0.8

0.9

1.0

b Zn vacancy

W p

ara

mete

r

S parameter

Bulk

1.03

1.08

10/0.3 20/0.8

Zn vacancy


900oC, 20ms 800

oC, 1h

1200oC, 20ms


S p

ara

mete

r

a

30/1.60

Rp(Li)

bulk reference


T.M. Børseth, et al Phys.Rew. B 89, (2006)

Lecture7 - Universitetet i oslo · Lecture7: •vacancies, derivationofequilibriumconcentration...

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Transcript of Lecture7 - Universitetet i oslo · Lecture7: •vacancies, derivationofequilibriumconcentration...