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Lesson 16

Su ested ReadinChapter 3 in Waseda, pp. 67-75

Ch. 6 B.D. Cullity and S.R. Stock, Elements of X-ray Diffraction, 3 rd Edition, Prentice-Hall (2001)

409

. - . . . , , .Ch. 7 Pecharsky and Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of

Materials, 2 nd Edition, Springer (2009)


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10-6

Wavelengthin nm

Frequency incycles per second

Visible Light: ~6000 X-rays : ~0.5 2.5 Electrons : ~0.05

1020

One an strom 10-1

Gamma raysViolet

Indigo

nm

450

400

1015 One micrometer

One nanometer 1

UltravioletX-rays

Visible Light

Blue

Green

Yellow

550500

1010One centimeter

104 Infrared Orange

Red700

650

One kilometer

One meter 109

Broadcast band

Short radio waves 750

EM Radiation105

1014 Long radio waves Self-propagatingwaves with

410

Spectrum of electromagnetic (EM) radiation as afunction of frequency (s -1) and wavelength (nm).

electric and magnetic

components


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Properties of Electromagnetic Waves

y

ec r cField E

particles of energycalled photons .

x

Beam of EMradiation

Pro a ationPhotons interact withelectrons.

z

Magnetic

direction

Thus they can be e

hc E h photon of energy

.

346.626 10h

Planck's constant J s frequency of the wave

411

83.00 10c

speed of light m/s wavelength of radiation


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Scattering of EM Radiation by Crystals For a material to yield a diffraction pattern,

d . This limits us to usin :. Neutrons

X-rays

X-ray scattering (i.e., absorption + re-emission): Elastic (little or no energy loss diffraction peak)

Inelastic ener loss no diffraction eak

412


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Laues Equations When scattering occurs there is a change in the path of the

incident radiation.Diffracted

n

n n

path difference between incident andscattered beams where is an inte er

S

beam

(cos cos ) ( ) x o oa a S hS o

(cos cos ) ( ) y o

z

o

ooc c l S S S oIncident

a[ , , are integers]h k l

Constructive interference occurs when all three equations

beam

413

.


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Braggs Law

A C

d

B

path difference 2 sin B BC d For constructive interference n

2 sind nd

d

4142 sin (or 2 sin )hkl n

d n d


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Braggs Law

A C

d

B

path difference 2 sin B BC d For constructive interference n

2 sind nd

d

4152 sin (or 2 sin )hkl n

d n d


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Braggs Law

ragg s aw can e expresse n vec or orm:hkl *2sino hkl S S d

*d

S o-S o

S -S o

*hkl

hkl d d Incidentbeam

Diffracted

Thus: Trace of (hkl )

reflecting plane* * * *o d ha kb l S S c

eam

This tells us that constructive interference occurs when

S S o coincides with the reciprocal lattice vector of thereflecting planes.

416


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Laues Equations and the Reciprocal Lattice e can represent t e aue equat on grap ca y.

This is similar to Ewalds sphere. * od S S

For diffraction to be

(13)

S . .

law satisfied) S must

end on a reciprocal S (00) (10)

(01)

(11)

2

a ce po n .

Points satisfying this criteria

Reflecting sphere

represent planes that are orientedfor diffraction

Braggs law, which describes diffraction in terms of scalars, isgenerally used for convenience. 417


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Diffractedbeam

*h l

ok d

S S

(13)

S

RADIUS = 2 S o /

oS

(00) (10)

(01) (11)

(10)

INCIDENTBEAM To

satisfyraggs

2

raggslaw

*oS d S sphere

*

2 sin1 2sin

n d

418

hkl d n


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Reciprocal Lattice

from all diffractionvectors (i.e., g ) for acrystal definespossible Braggreflections.

Points that intersect

b*c*the reflecting spherewill satisfy Braggs

law.000

Incidentbeam

a* Changes inwavelength ( )changes the circle

,to diffraction.However, we generallydo not change .

419

000


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Reciprocal Lattice

orientation of theincident beam relativeto the crystal changesthe orientation of thereciprocal lattice,reflecting sphere, and

.

Chan e will eventuall

000

Incidentbeam

yield a condition wherediffraction is possible.

We mentioned this afew lectures ago.

420


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X-ray Diffractometer X-ray source is generally

fixed.Focal circle

Rotate sample andDetector Source

e ec or o a us .

Rotates onfocal circle

Fixed

our Bruker D8 and PhilipsMPD, the source and

2

Sample

e ec or move w e esample remainsstationary.

421

-


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Diffraction Directions

combining Braggs law with the interplanar spacinge uations for a cr stals.

Cubic:

2

2 2 2 22

2 sin

sin

d

h k l

2 2 2

2 21 h k l

d a

This equation predicts, for a particular incident anda particular cubic crystal of unit cell size a , all of the

422

possible Bragg angles for the diffracting planes ( hkl )


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Diffraction Directions contd Example:

planes in a cubic crystal?

Solution:2

2 2 2 2111 2 2sin 1 1 14 4a a

h k l

423


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Diffraction Directions What about other systems?

Tetragonal:

2 2 2 22

2 2 2

2 sinsin

4

d h k l

a a c

2 2 21 h k l

d a c

22 2 1For 111 , sin

Solution: must know c and a . Will be one eak.

a a c

424 What about {110}? Ive intentionally given you a family here


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Sphere

Cone Debye

Ring

2*hkl d

S

In powder diffraction you

Beam S o

S

generate an in inite num er orandomly oriented, but identical,reciprocal lattice vectors. Detector

placed on the surface of Ewaldssphere.

The roduce owder diffraction

* 1* hkl hkl

R g d d

425

cones at different Bragg angles

(see the next slide). Adapted from Vitalij K. Pecharsky and Peter Y. Zavalij, Fundamentals

of Powder Diffraction and Structural Characterization of Materials, Kluwer Academic Publishers (2005)


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Ewaldssphere

* 1hkl

hkl

d d

Incident SoBeam

Fi ure 8.2

426

Adapted from Vitalij K. Pecharsky and Peter Y. Zavalij, Fundamentals of Powder Diffractionand Structural Characterization of Materials, 2 nd Edition, Kluwer Academic Publishers

(2009), p. 154.


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Diffraction vector cone Diffraction cone

Ewald sphere

/S

hkl

X-ray2

* 1hkl

hkl

d d

2 /oS

/S

hkl

430Diffraction cone and diffraction vector cone illustrated on the Ewald sphere. Adapted from B.B. He, Two-Dimensional X-ray Diffraction, Wiley (2009), p. 20.


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(a) (b)

2 1

2 2

Increasing 2

2 1

2 22 3

(c) (d)

431

X-ray diffraction patterns: (a) from single crystal, (b) Laue diffraction pattern from single crystal, (c)diffraction cones from polycrystalline solid, and (d) diffraction frame from a polycrystalline solid. (a)and (c) from B.B. He,Two-Dimensional X-ray Diffraction , Wiley (2009), p. 22. (d) from PhotonicScience (http://photonic-science.co.uk ).


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Incidentelectrons

Ewald sphere

Figure 2.11 Formation of a single crystal diffraction pattern in transmission electron microscopy.The short wavelength of electrons makes the Ewald sphere flat. Thus, the array of reciprocal lattice

432

po n s n a rec proca p ane ouc es e sp ere sur ace an genera es a rac on pa ern on escreen. Adapted from Y. Leng , Materials Characterization, Wiley (2008), p. 55.


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Linedetector

In a linear diffraction pattern,the detector scans through anarc that intersects each Debye

cone at a sin le oint.

Figure 8.4This gives the appearance of a

discrete diffraction peak.

433

Adapted from Vitalij K. Pecharsky and Peter Y. Zavalij, Fundamentals of PowderDiffraction and Structural Characterization of Materials, 2 nd Edition, Kluwer

Academic Publishers (2009), p. 156.

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