Peak Position.pdf

TEQIP WORKSHOP ON HIGH RESOLUTION X-RAY AND ELECTRON DIFFRACTION, FEB 01, 2016, IIT-K.

Introduction to x-ray diffraction Peak Positions and Intensities

Rajesh Prasad

Department of Applied Mechanics Indian Institute of Technology New Delhi 110016 [email protected]

Question 1: Why crystal have regular external shapes? Postulate of Kepler, Hooke et al.: Because they have regular arrangement of “building blocks” (“atoms” in modern language)

1895 Wilhelm Röntgen Discovered xrays First Nobel Prize in physics: 1901

First x-ray picture November 1895

Hand of Roentgen’s wife

Question2: Are x-rays waves or particles? X stands for the unknown

Laue’s Postulate If crystals are periodic arrangement of atoms And

If x-rays are waves Then

Crystals should act as a 3D diffraction grating for x-rays

Two GREAT results from a single experiment: 1.  X-rays are waves 2. Crystals are periodic arrangement of atoms

One of the greatest scientific discoveries of twentieth century

X-Ray Diffraction

=

Crystal Structure

=

Peak Positions

+

Peak Intensities

Lattice

+

Motif: Atom Positions

X-Ray Diffraction

Sample Incident Beam

Transmitted Beam

Diffracted Beam

X-Ray Diffraction Sample

Incident Beam

≡ Bragg Reflection

Transmitted Beam

Braggs Law (Part 1): For every diffracted beam there exists a set of crystal lattice planes such that the diffracted beam appears to be specularly reflected from this set of planes.

X-Ray Diffraction Braggs’ recipe for Nobel prize? Call the diffraction a reflection!!! “The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them”. W.L. Bragg

A father-son team that shared a Nobel Prize

Nobel Prize (1915)

William Lawrence Bragg (1890–1971)

William Henry Bragg (1862–1942

X-Ray Diffraction Braggs Law (Part 1): the diffracted beam appears to be specularly reflected from a set of crystal lattice planes.

r

i θ

θ plane

Specular reflection: Angle of incidence =Angle of reflection (both measured from the plane and not from the normal)

The incident beam, the reflected beam and the plane normal lie in one plane

X-Ray Diffraction r

i θ

θ

dhkl Bragg’s law (Part 2):

nλ = 2d hkl sin θ

r

i θ

θ θ θ

P

R

dhkl

Q Path Difference =PQ+QR= 2d hkl sin θ

i

r θ

θ P

R Q

Path Difference =PQ+QR= 2d hkl sin θ Constructive inteference

nλ = 2d hkl sin θ Bragg’s law

Two equivalent ways of stating Bragg’s Law

1st Form

nλ = 2d hkl sin θ d hkl ⇒λ =2 sin θ n d nh,nk ,nl

d hkl = = 2 2 2 n (nh ) + (nk ) + (nl ) a

⇒ λ = 2d nh nk nl sin θ

2nd Form

Two equivalent ways of stating Bragg’s Law

nλ = 2d hkl sin θ nth order reflection from (hkl) plane

⇒ λ = 2d nh nk nl sin θ

≡

1st order reflection from (nh nk nl) plane

e.g. a 2nd order reflection from (111) plane can be described as 1st order reflection from (222) plane

X-rays Characteristic Radiation, Kα Target Mo Cu Co Fe Cr

Wavelength, Å 0.71 1.54 1.79 1.94 2.29

Experimental Diffraction Settings Laue method

Variable λ

Fixed θ

Rotating crystal method Fixed λ

Variable θ

Powder method

Variable θ

Fixed λ

Powder Method λ  is fixed (Kα radiation) θ  is variable –

millions of powder particles randomly oriented in space rotation of a crystal about all possible axes

Diffracted beam 2 Incident beam sample

i

θ

plane

t

θ θ

Strong intensity Zero Diffracted intensity beam 1 X-ray detector 2θ1 Transmitted beam Intensity

Powder diffractometer geometry

r 2θ1

2θ2

2θ

Crystal monochromator

detector X-ray tube

X-ray powder diffractometer

The diffraction pattern of austenite Austenite = fcc Fe

Bcc crystal

x λ λ/2

d100 = a

z 100 reflection= rays reflected from adjacent (100) planes spaced at d100 have a path difference λ

y No 100 reflection for bcc No bcc reflection for h+k+l=odd

Extinction Rules Bravais Lattice

Allowed Reflections

SC

All

BCC

(h + k + l) even

FCC

h, k and l all odd, or h, k and l all even

DC

h, k and l are all odd Or if all are even then (h + k + l) divisible by 4

X-Ray Diffraction

=

Crystal Structure

=

Peak Positions

+

Peak Intensities

Lattice

+

Motif: Atom Positions

Intensity of Powder Diffraction Peaks 1. Scattering by an electron 2. Scattering by an atom (Atomic scattering factor) 3. Scattering by a unit cell (Structure factor) 4. Polarization factor 5. Multiplicity factor 6. Lorentz factor 7. Absorption factor 8. Temperature factor B.D. Cullity, Elements of X-Ray Diffraction, 2nd. Edn., Addison Wesley, 1978, Ch. 4.

Intensity of Powder Diffraction Peaks 1. Scattering by an electron Thompson’s equation

K ! 1+ cos2 2θ $ IP = I0 2 # & (4-2) r " 2 %

P 2θ

O

Location of an electron

Direction of transmitted beam

Ip I0 K R 2θ

= = = = =

Intensity of scattered beam at at P Intensity of incident beam at O constant (7.94 x 10–30m2) distance of P from O scattering angle (angle between transmitted and scattered beam)

Polarization factor

! 1+ cos2 2θ $ # & 2 " % Unpolarization factor!!

http://www-outreach.phy.cam.ac.uk/

Polarization Factor 1

0.5

0

90

2θ

180

Intensity of Powder Diffraction Peaks 2. Scattering by an atom (Atomic scattering factor) Cu

Cullity Fig. 4-5 Cullity Fig. 4-6

f =

amplitude of the wave scattered by an atom amplitude of the wave scattered by one electron

3. Scattering by a unit cell (Structure factor) Phase difference between waves scattered from atom at the origin 000 k th atom at a fractional coordinate uiviwi for waves reflected by (hkl) plane

ϕ k = 2π (huk + kvk + lwk ) Amplitude of this wave

Ak = fk

where fk is the atomic scattering factor of the ith atom In the complex exponential notation this wave is represented by iϕ k

Ak e

Summation of all such scattered waves from the entire unit cell is called the STRUCTURE FACTOR, F n

F = ∑ Ak e

iϕ k

1

n

F = ∑ fn e 2 π i(huk +kvk +lwk ) 1

Intensity

*

I = FF = F

2

Example: Structure factor of a monatomic bcc unit cell Coordinate of atoms in the unit cell = 000; n

F = ∑ f ne

2πi ( hun + kv n + lwn )

1

=fe

2πi ( h 0+ k 0+l 0 )

[

πi ( h + k +l )

= f 1+ e If,

111 222

h + k + l = even h + k + l = odd

+fe

1 1 1 2πi ( h + k +l ) 2 2 2

] then then

F =2f F =0

Intensity of Powder Diffraction Peaks 4. Multiplicity factor {100}cubic = (100), (010), (001), ( 1 00), (0 1 0), (00 1 )

p{100} = 6

{111}cubic = (111), ( 1 11), (1 1 1), (11 1 ) ( 1 1 1 ), (1 1 1 ), ( 1 1 1 ), ( 1 1 1)

p{111} = 8

The ratio of intensities of 100 reflection to the intensity of 111 reflection, other things being equal, is expected to be:

6 3 = 8 4

Intensity of Powder Diffraction Peaks 5. Polarization Factor

1 + cos 2 2θ P= 2 Due to the fact that incident wave is unpolarized

Intensity of Powder Diffraction Peaks 6. Lorentz Factor: (A) reflection at non Bragg angles I max ∝

B ∝

1 sin θ 1 cos θ

Integrated intensity(A) ∝

1 sin 2θ

Intensity of Powder Diffraction Peaks 4. Lorentz Factor: (B) Fraction of properly oriented crystals

ΔN rΔθ 2π r sin(90 − θ B ) = N 4π r 2 =

Δθ cosθ B 2

Figure 4-15 of Cullity

Integrated intensity(B) ∝ cos θ

Intensity of Powder Diffraction Peaks 6. Fraction of diffraction cone recorded

Figure 4-16 of Cullity

Full length of the diffracted line = 2π Rsin 2θ B

1 Integrated intensity(C) ∝ sin 2θ

Intensity of Powder Diffraction Peaks 6. Lorentz Factor: Integrated intensity(A) 1 sin 2θ

cos θ

1 sin 2θ

reflection at non Bragg angles Fraction of properly oriented crystals Fraction of diffraction cone recorded

Lorentz factor

⎛ 1 ⎞ ⎛ 1 ⎞ cosθ L = ⎜ ⎟(cosθ )⎜ ⎟ = 2 sin 2 θ sin 2 θ sin 2θ ⎝ ⎠ ⎝ ⎠

Intensity of Powder Diffraction Peaks 5. Polarization Factor

1 + cos 2 2θ P= 2 6. Lorentz Factor:

L=

cosθ sin 2 2θ

Lorentz-Polarization Factor

1 + cos2 2θ LP = sin 2 θ cosθ

ignoring the numerical constant

Figure 4-17 from Cullity

7. Absorption Factor A Absorption in the specimen Independent of θ:

x

d

x

d

Low θ: High θ: Low penetration depth, d High penetration depth, d Large irradiated area, x Small irradiated area, x Effective irradiated volume constant and independent of θ.

Temperature factor Gist of a Discussion in a Coffee House in Munich before Laue’s famous experiment Atoms are continuously vibrating with random amplitude Thus there cannot be any periodicity of instantaneous positions Therefore no sharp diffraction pattern will be observed. Reference: Kittel, Solid state Physics

Temperature factor 1. Lattice parameter increase -> Peaks shift to lower θ 2. Intensity of the diffraction lines decrease 3. Intensity of background scattering increases Rather surprisingly, there is no significant change in peak width.

Final Intensity expression for Diffractometer

! 1+ cos2 2θ $ −2 M I = F p# 2 &e " sin θ cosθ % 2