Scattering from imperfect crystals (see Cowley Sect. 7.1)Scattering from imperfect crystals (see Cowley Sect. 7.1)

Two types

average lattice exists (point defects,dislocations, thermal vibrations)

no average lattice (stacking faults, twinning)

Two types

average lattice exists (point defects,dislocations, thermal vibrations)

no average lattice (stacking faults, twinning)

Scattering from imperfect crystals (see Cowley Sect. 7.1)Scattering from imperfect crystals (see Cowley Sect. 7.1)

Two types

average lattice exists (point defects,dislocations, thermal vibrations)

no average lattice (stacking faults, twinning)

Two types

average lattice exists (point defects,dislocations, thermal vibrations)

no average lattice (stacking faults, twinning)

Scattering from imperfect crystals (see Cowley Sect. 7.1)Scattering from imperfect crystals (see Cowley Sect. 7.1)

Two types

can't put in all atoms; consider average of atoms surrounding particular set of N atoms

Two types

can't put in all atoms; consider average of atoms surrounding particular set of N atoms

Scattering from imperfect crystalsScattering from imperfect crystals

Two types

can't put in all atoms; consider average of atoms surrounding particular set of N atoms

for a monatomic solid

Two types

can't put in all atoms; consider average of atoms surrounding particular set of N atoms

for a monatomic solid

Random vacancies (see Cowley Sect. 7.3)Random vacancies (see Cowley Sect. 7.3)

Suppose n random vacancies in monatomic solid w/ N atom sites

Consider vectors ri - rj

Suppose n random vacancies in monatomic solid w/ N atom sites

Consider vectors ri - rj

Random vacancies (see Cowley Sect. 7.3)Random vacancies (see Cowley Sect. 7.3)

Suppose n random vacancies in monatomic solid w/ N atom sites

Consider vectors ri - rj

Suppose n random vacancies in monatomic solid w/ N atom sites

Consider vectors ri - rj

Random vacanciesRandom vacancies

Rearranging:Rearranging:

Random vacanciesRandom vacancies

Rearranging:

scattering power of ordered structure - no defects, reduced f

Rearranging:

scattering power of ordered structure - no defects, reduced f

Random vacanciesRandom vacancies

Rearranging:

scattering power of ordered structure - no defects, reduced f conts distrib of scatt power

- decreases w/ u- approx proportional to n

Rearranging:

scattering power of ordered structure - no defects, reduced f conts distrib of scatt power

- decreases w/ u- approx proportional to n

Random vacanciesRandom vacancies

Rearranging:

scattering power of ordered structure - no defects, reduced f conts distrib of scatt power

- decreases w/ u- approx proportional to n

Rearranging:

scattering power of ordered structure - no defects, reduced f conts distrib of scatt power

- decreases w/ u- approx proportional to n

Random vacanciesRandom vacancies

Now consider PattersonSuppose n random vacancies in monatomic solid w/ N

electron density for deviation from ordered structure ordered structure

Now consider PattersonSuppose n random vacancies in monatomic solid w/ N

electron density for deviation from ordered structure ordered structure

Random vacanciesRandom vacancies

Now consider PattersonSuppose n random vacancies in monatomic solid w/ N

electron density for deviation from ordered structure ordered structure

Now consider PattersonSuppose n random vacancies in monatomic solid w/ N

electron density for deviation from ordered structure ordered structure

Random vacanciesRandom vacancies

Vacancy clustersVacancy clusters

Use Patterson approach againUse Patterson approach again

InterstitialsInterstitials

Assume n small interstitials w/ negligible scattering power

Average structure is

Assume n small interstitials w/ negligible scattering power

Average structure is

InterstitialsInterstitials

Assume n small interstitials w/ negligible scattering powerAssume n small interstitials w/ negligible scattering power

InterstitialsInterstitials

Assume n small interstitials w/ negligible scattering powerAssume n small interstitials w/ negligible scattering power

Thermal vibrationsThermal vibrations

Einstein: monatomic, independent, harmonic, 1-D

Spread electron density by Gaussian

Einstein: monatomic, independent, harmonic, 1-D

Spread electron density by Gaussian

Thermal vibrationsThermal vibrations

Except for origin peak, all Patterson peaks spread by

Except for origin peak, all Patterson peaks spread by

Thermal vibrationsThermal vibrations

Except for origin peak, all Patterson peaks spread by

Intensity is

Except for origin peak, all Patterson peaks spread by

Intensity is

Thermal vibrationsThermal vibrations

Except for origin peak, all Patterson peaks spread by

Intensity is

Except for origin peak, all Patterson peaks spread by

Intensity is

No average latticeNo average lattice

Except for origin peak, all Patterson peaks spread by

Except for origin peak, all Patterson peaks spread by

No average latticeNo average lattice

Except for origin peak, all Patterson peaks spread by

d(z) is a set of fcns

(r) considered periodic in x,y

Except for origin peak, all Patterson peaks spread by

d(z) is a set of fcns

(r) considered periodic in x,y

No average latticeNo average lattice

Except for origin peak, all Patterson peaks spread by

d(z) is a set of fcns

(r) considered periodic in x,y

reciprocal lattice

Except for origin peak, all Patterson peaks spread by

d(z) is a set of fcns

(r) considered periodic in x,y

reciprocal lattice

No average latticeNo average lattice

Use Gaussian distrib w/ mean = c

Use Gaussian distrib w/ mean = c

No average latticeNo average lattice

Use Gaussian distrib w/ mean = c

Use Gaussian distrib w/ mean = c

No average latticeNo average lattice

No average latticeNo average lattice

No average latticeNo average lattice

No average latticeNo average lattice