chapter. 2 physical methods for characterizing...

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Chapter. 2 Physical Methods for Characterizing SolidsChapter. 2 Physical Methods for Characterizing Solids

2.1 Introduction

▪ Single Crystal X-ray diffraction: precise atomic positions, bond angles, bond lengths

long range ordered str.

: weakness → less suited on the str. positions of defects

→ difficult to grow single crystal

▪ powder X-ray diffraction - most commonly used

- phase purity analysis & assessing

- str. determination

Solid-State Chemistry 2011 Spring T.-S.Y

2. 2 X2. 2 X--ray Diffractionray Diffraction

2.2.1 Generation of X-rays

▪ Discovery: 1895, German Physist, William Rötgen → 1901, Noble Prize in Physics


Wilhelm Conrad Röntgen (1845-1923)

A famous photo of his wife’s hand appeared in Nature and Science

“Eine Neue Art von Strahlen” (“On a New Kind of Rays”)

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▪ Nature of X-ray: X-rays are electromagnetic waves (EW)

: Every time a charge accelerate or decelerates, EWs are generated.



© 2003 by Kluwer Academic Publisher

▪ white X-rays (origin):cathode (-) anode (+)

accelerating voltage

e-


Ek = 1/2mv2 ≡ EE = eV

(kinetic and electro static E)

Eph = hcv = hc/λ

(E of a photon)

accelerating voltage

hc/λ = eV → λ = hc/eV

m = e- mass


λ = (12.4 x 10-7)m/V

v = e- velocitye = e- charge (1.6 x 10-19 C)V = accelerating voltageh = Planck’s constant (6.625 x 10-34 J/sec) c = velocity of light in vacuum (2.998 x 108 m/sec)ν = frequencyλ = wavelength (λ = ν-1)

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▪ an angström: 1Å = 10-10 m

▪ λ = (12.4 x 10-7)m/v establishes the short wavelength limit (SWL)

- e.g.) V = 1000V; λSWL = 12400/1000 = 12.4 Å


V = 10000V; λSWL = 12400/1000 = 1.24 Å

(L)

(M)


▪ X-ray generation: ‘white-radiation’ + Kα, Kβ

: Kα, Kβ’s wavelength – characteristic of the anode metal (e.g. Cu, Mo)

: bombarding → knock-out e- @ K shell (n=1) → vacancy → filled by descending

e- from L shell (n=2): Kα or from M shell (n=3): Kβ

(K)© 2003 by Kluwer Academic Publisher


▪ X-ray generation: Kα, Kβ’s wavelength – characteristic of the anode metal (e.g. Cu, Mo)



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▪ monochromatic radiation (single wavelength or a very narrow range of wavelengths) is required.

Kα, is used, but Kβ is not.

screened by a filter made of thin metal foil of the element adjacent (Z-1)

e.g.) Cu : uses Ni

Mo : uses Nb

by graphite (single crystal)



▪ sealed X-ray tube:



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▪ X-ray generation:


© 2003 by Kluwer Academic Publisher © 2003 by Kluwer Academic Publisher


2.2.2 Diffraction of X-rays

▪ crystalline solids: regular arrays of atoms, ions or molecules w/ interatomic spacing of the order

of 100 pm.

f d ff h l h f h d l h h f hfor diffraction, the wavelengths of the incident light ≈ the spacing of the

grating

▪ X-ray diffraction str. determination: W.H Bragg, W.L. Bragg (father and son)

str. of NaCl, KCl, ZnS, CaF2, CaCO3

: X ray diffraction acts like “reflection” from the planes of : X-ray diffraction acts like reflection from the planes of

atoms within the crystal at specific orientation

reflection only occurs when the constructive

interference conditions are fulfilled!!


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▪ Bragg‘s Law: relationship among, 1) the diffraction angle (Bragg angle)

2) wavelength

3) interplanar spacing


∴ nλ = 2dhklsinθhkl


2. 3 Powder Diffraction2. 3 Powder Diffraction

2.3.1 Powder Diffraction Patterns

▪ crystallites: a finely ground crystalline powder

: randomly oriented to one another in powder samples

diff i h h lli i d h l f lfill h B - diffraction occurs when the crystallite are oriented at the correct angle to fulfill the Bragg

calculation

- 2θ: angle b/w incident & diffracted beam

- reflections lie on the surface of core whose semi-apex angles are equal to the deflection angle

2θ (Fig. 2.4(a))


▪ Debye-Scherrer methods: a strip of film wrapped around the inside of a X-ray camera(Fig.2.4(b))

w/ a hole)

: a sample is rotated for more reflections into the diffracting condition

© by Taylor & Francis Group, LLC

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2. 3 Powder Diffraction2. 3 Powder Diffraction

cont’d

▪ Debye-Scherrer methods: cores were recorded as arcs on the film

using radius of the camera

distance along the film from the center

2θ, dhkl can be calculated.

▪ recent method: automated diffractometer (Fig.2.5(a))

using CCD detector or scintillation

angle and intensities are recorded (Fig. 2.5(b))


▪ ‘indexing the reflection’: assigning the corrected hkl index to each reflection

detecting which planes are responsible for each reflection

▪ possible for simple compound w/ a high symmetry, but extremely difficult for larger & less

symmetrical system

2.3.12.3.1--2. Reciprocal Lattice2. Reciprocal Lattice

▪ Introduced by P. Ewald in 1921

▪ Let a, b, c be the elementary translations (vectors) of a lattice (i.e. a directional lattice)

▪ A second lattice, reciprocal to the first one, is defined by introducing the translations a*, b*,

c*, which satisfy the following conditions:

a = 1/V(b x c)

b = 1/V(a x c)

a*·a = 1

b*·b = 1


b 1/V(a x c)

c = 1/V(a x b)

b b = 1

c*·c = 1


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2.3.12.3.1--3.3. Properties of Reciprocal LatticeProperties of Reciprocal Lattice

▪ the family of planes (hkl) w/ interplanar distance d becomes a vector in reciprocal space

▪ d* = 1/d, and it is perpendicular to the corresponding (hkl) planes

▪ in an orthogonal lattice,,

a* = 1/a and it is perpendicular to bcb* = 1/b and it is perpendicular to acc* = 1/c and it is perpendicular to ab



2.3.12.3.1--4.4. Reciprocal Lattice and Braggs’ LawReciprocal Lattice and Braggs’ Law



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2.3.12.3.1--5.5. Reciprocal lattice and Ewald’s SphereReciprocal lattice and Ewald’s Sphere



2.3.12.3.1--6.6. Stationary vs. Rotating Crystal TechniqueStationary vs. Rotating Crystal Technique



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2.3.12.3.1--6.6. Rotating Crystal TechniqueRotating Crystal Technique



2.3.12.3.1--7.7. Origin of Powder Diffraction PatternOrigin of Powder Diffraction Pattern



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2.3.12.3.1--8.8. Powder Diffraction Pattern of CuPowder Diffraction Pattern of Cu



2.3.12.3.1--9.9. Powder Diffraction Pattern of LaBPowder Diffraction Pattern of LaB66




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2. 3. 2 Absences due to lattice centering2. 3. 2 Absences due to lattice centering

▪ For a primitive cubic system: reflection w/ the smallest Bragg angle → the largest dhkl spacing

e.g.) 100 plane → largest separation → reflection

010, 001 at the same position (∵ a = b = c)

dhkl = a/(√h2 + k2 + l2)

w/ Bragg equation,

λ = 2asinθhkl/(√h2 + k2 + l2)


all integer #

h2 + k2 + l2 ↑ → 2θ ↑

2. 3. 2 Absences due to lattice centering2. 3. 2 Absences due to lattice centering

▪ Table 2.1

▪ body-centered, face-centered cubic: different line from the primitive system

∵ centering → destructive interference → extra missing reflection

known as systematic absences


▪ 200 plane in the F-centered cubic unit cell

cell dimension: a → spacing: a/2

Fig.2.6© by Taylor & Francis Group, LLC

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