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Physical Metallurgy PrinciplesPhysical Metallurgy Principles

Chapter 2.ANALYTICAL METHODS

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The Bragg LawThe Bragg Law

Fig. 2.1 An X-ray beam is reflected with constructive interference when the angle of incidence equals the angle of reflection

Fig. 2.2 The Bragg law

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The Bragg LawThe Bragg Law

Ex) When the {110} Planes of BCC have a seperation of 0.1181nm.

ⅰ) λ=0.1541nm Cu target X-ray (order)

n=1 →

n=2 → Impossible

ⅱ) λ=0.02090nm Tu target X-ray

n=1 Θ=5.07˚, n=2 Θ=10.19˚, n=3 Θ=15.39˚, n=4 Θ=20.73˚

n=5 Θ=26.27˚, n=6 Θ=32.07˚, n=7 Θ=38.27˚, n=8 Θ=45.06˚,

n=9 Θ=52.78˚, n=10 Θ=62.23˚, n=11 Θ=76.74˚

7.40)1181.0(2

1541.0sin)2

(sin 11

dn

1181.021541.02sin 1

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The Bragg LawThe Bragg Law

Fig. 2.3 Four of the eleven angles at which Bragg reflections occur using a crystal with an interplanar spacing of 0.1181 nm and X-rays of wavelength 0.02090 nm

Fig. 2.4 X-ray refelections from planes not parallel to the surface of specimen

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Laue TechniquesLaue Techniques

Fig. 2.5 laue back-reflection camera.

Fig. 2.6 Laue back-reflection photographs.(A) Photograph with X-ray beam perpendicular to the basal plane(0001). (B) Photograph with X-ray beam perpendicular to a prism plane (11 0). Dashed lines on the photograph are drawn to show that the back-reflection spots lie on hyperbolas.

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Laue TechniquesLaue Techniques

Fig. 2.7 (A) Laue back-reflection photographs record the refelections from planes nearly perpendicular to the incident X-ray beam, (B) Laue transmission photographs record the refelctions from planes nearly parallel to the incident X-ray beam.

Fig. 2.8 Asterism in a Laue back-reflection photograph. The reflections from distorted- or curved-crystal planes form elongated elongated spots.

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The Debye-Scherrer or Powder MethodThe Debye-Scherrer or Powder Method

Fig. 2.12 Schematic representation of the Debye or powder camera. Specimen is assumed to be simple cubic. Not all reflections are shown.

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The Debye-Scherrer or Powder MethodThe Debye-Scherrer or Powder Method

Fig. Diffraction by Powder Method

Fig. Debye-Scherrer Camera

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The X-Ray DiffractometerThe X-Ray Diffractometer

Fig. 2.14 X-ray diffractometer.

Fig. 2.15 The X-ray diffractometer records on a chart the reflectedintensity as a function of Braggangle. Each intensity peakcorresponds to a crystallographicplane in a reflecting position.

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The X-Ray DiffractometerThe X-Ray Diffractometer

222sin2 lkhad

222

2

2

2

sin4 lkha

Cubic Hexagonal

hkl

hk

Simple FCC BCC diamond

1

100 1 10

2 110 110 2

3 111 111 111 3 11

4 200 200 200 4 20

5 210 5

6 211 211 6

7 7 21

8 220 220 220 220 8

9 300,221 9 30

10 310 310 10

222 lkh BCC : Diffract when

is even

FCC : Diffract when unmixed

h, k, l

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The Transmission Electron MicroscopeThe Transmission Electron Microscope

Fig. Design of a TEM(schematic).

a) Ray path for bright field imaging (three-stage magnification).

b) Ray path after Boersch for selected area diffraction(SAD).

The excitation of the intermediate lens is weaker in b) than in a); thus, the primary diffraction pattern is imaged by the intermediate lens instead of the first-stage sample image.

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The Transmission Electron MicroscopeThe Transmission Electron Microscope

Fig. Helical path of electrons through the field. This is due to the summation of the forces : (1) The potential difference between the cathode and anode which tends to propel the electron in a straight line down the column, and (2) The magnetic field which tends to cause the electron to a circular path at right angles to the electron optical axis.

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The Transmission Electron MicroscopeThe Transmission Electron Microscope

Fig. 2.16 Schematic drawing of a transmission electron microscope.

Fig. 2.17 Images can be formed in the transmissionelectron microscope corresponding to the direct beam or to a diffracted beam .(Images from more than one diffracted beam arealso possible.)

specimen – rotation, tilting

Bright field image, Dark field image->aperture at I ; Fig. 2.18Bright field Image of Dislocation ; Fig 4.9

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The Transmission Electron MicroscopeThe Transmission Electron Microscope

De Brogliemvh

Let us assume V = 2*108 m/s, potential = 100kV

By the above equation, wavelength is 4*10-3 nm

If d = 0.2nm,

∴ 0.01

∴ This means that when a beam of electrons is passed through a thin

layer of crystalline material, only those planes nearly parallel to the beam

can be expected to contribute to the resulting diffraction pattern.

sin2d sin

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Interactions Between The Electrons in an Electron

Beam and a Metallic Specimen

Interactions Between The Electrons in an Electron

Beam and a Metallic Specimen

① Elastic scatteringThe path or trajectory of the moving electron is changed, but its energyor velocity is not altered significantly. (TEM)

2

incident electron beam

scattered beam

The change in path direction isnormally less than about 5° for asingle collision.

If the direction change is more than 90°and if the electron exits the specimen, it is said to have been elasticallybackscattered.

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Interactions Between The Electrons in an Electron

Beam and a Metallic Specimen

Interactions Between The Electrons in an Electron

Beam and a Metallic Specimen

②Inelastic scatteringWhen the moving electron loses some of its kinetic energy as a result of an interaction with the specimen.

(a) excitation of phonons; in heating the specimens

(b) plasmon excitation ; oscillation of free electron gas

(c) ionization of inner shells; emission of characteristic X-ray and Auger electron

(d) excitation of conduction electrons leading to secondary electron ; beam electrons can cause loosely bound conduction band electronsto be ejected from a specimen’s surface, < 50eV

(e) generation of continuous X-ray

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The Scanning Electron MicroscopeThe Scanning Electron Microscope

Fig. 2.22 Aschematic drawing of ascanning electron microscope(SEM).

Fig. 2. 23 The way the electron beam is moved across the specimen surface in an SEM.

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The Scanning Electron MicroscopeThe Scanning Electron Microscope

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The Depth of FocusThe Depth of Focus

MmPES 100

WDR

R = 100 m

WD = 10mm

=0.01rad

rrD

2

Images in effective focus

Probe(beam) diameter = 5nm

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The Characteristic X-raysThe Characteristic X-rays

Fig. 2.31. Illustration of how Kα radiation and Auger electrons are formed. The ejection of anelectron from the K shell is the first step. (B) If an atom from the L shell falls into the hole inthe K shell, it is possible for a Kα photon to be released. (C) Alternatively, the drop of an Lshell electron into the K shell hole may result in the ejection of an Auger electron from the Lshell.

for X-rayEK – EL3 =

∴ =

for MoEK = 19.9995 KeVEL3 = 2.6251 KeV

∴ = 0.07093nm

hch

1

K 3K

1.2398

LEE

1

K

for Auger electron

Eke = EK - 2EL -

= work function of detector

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ExamplesExamples

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ExamplesExamples