1cos

Interplanar Spacing in a Simple Cubic

Interplanar Spacing in a Simple Cubic

Lattice

Lattice

1

35 35

Diffraction in TEM

Diffraction in TEM

ƒ ƒ

Reinforced conditionsReinforced conditions nλnλ = 2d sin θ= 2d sin θ

ƒ ƒ

Miller index notation is used for Miller index notation is used for defining

defining crystalographiccrystalographic planes and planes and directions, e.g., for simple cubic directions, e.g., for simple cubic crystal

crystal

ƒ ƒ

By convention for e-By convention for e-diffraction usediffraction use

ƒ ƒ

^11stst order n=1 and, e.g., 2order n=1 and, e.g., 2^ndnd order order use multiple Miller indices.

use multiple Miller indices.

ƒ ƒ

^22ndnd order (n = 2) from a plane (131) order (n = 2) from a plane (131) can be called the 1

can be called the 1^stst order order diffraction from a plane (262).

diffraction from a plane (262).

Bragg

Bragg’’s law can be then s law can be then λλ = 2d sin θ= 2d sin θ

Significance of

Significance of θθ angle at eangle at e-diffraction-diffraction

ƒ ƒ

For 100 keV^{For 100} keV, , λλ = 0.0037 nm; for Al = 0.0037 nm; for Al crystal d = 0.4nm

crystal d = 0.4nm…… sinsinθθ = = 0.0046

0.0046……θθ= 0.26.5= 0.26.5^oo. Hence λ. Hence λ = 2d θ= 2d θ or λor λ/d = 2 /d = 2 θθ

2 2

2 k l

h d

_hkl

a

+

= +

E-beam strongly diffract only from planes of atoms almost being parallel to e-beam.

Diffraction Planes in TEM

Diffraction Planes in TEM

Back focal plane of objective Objective

Back focal plane of post specimen lens

37 37

Formation of Diffraction Patterns in TEM

Formation of Diffraction Patterns in TEM

ƒ ƒ For small diffraction angle For small diffraction angle

ƒ ƒ Combining with eq. Combining with eq. λ λ / / d = 2 d = 2 θ θ

ƒ ƒ

^L

^L

λ

λ - - camera constant camera constant

ƒ ƒ

^L

^{L – –} camera length is not physical distance camera length is not physical distance but notional adjusted by operator.

but notional adjusted by operator.

ƒ ƒ ^{Distance Distance} g g of diffraction spot is inversely of diffraction spot

is inversely

proportional

proportional

to to d d spacing spacing

θ

= 2 L g

λ λ

L d

g d or

L

g = =

Crystal

with spacing d

Diffraction Pattern and Reciprocal Lattice Diffraction Pattern and Reciprocal Lattice

g = g ∝

α

39 39

Ewald

Ewald Sphere Construction

Sphere Construction

Formal demonstration

Formal demonstration between between the reciprocal lattice and

the reciprocal lattice and diffraction pattern

diffraction pattern

a) a)

Diffraction crystal

Diffraction crystal

–

–

presented

presented by its reciprocal lattice

by its reciprocal lattice

b) b)

E

E

-

-

beam is presented by a

beam is presented by a vector

vector 1/λ

1/

λ, parallel to the beam

, parallel to the beam direction, and terminating at the

direction, and terminating at the

origin of the reciprocal lattice

origin of the reciprocal lattice

c) c) Sphere of radius 1/ Sphere of radius 1/ λ is drawn λ is drawn about A.

about A.

Ewald

Ewald sphere passes through a sphere passes through a reciprocal lattice point, a

reciprocal lattice point, a distance 1/d from the origin.

distance 1/d from the origin.

From geometry From geometry

θ λ λ

θ λ 2 sin

2 /

2 /

sin 1 or d

d

d = =

=

Diffraction occurs when the Ewald sphere touches a

reciprocal lattice point

ƒ

Material structure can be inferred from analyzing numerous specimen tilts.

ƒ

A SAD Pattern from a single crystal consists of a regular array of diffraction spots.

ƒ

Miller indices can be found by a ratio technique. For a cubic crystal

ƒ

^{Ratio of}

g

squares gives a ratio of the

Σ

of the squares of the plane’s indices.

ƒ

Spacing of the plane producing diffraction spots can be found from

g

_hkl

=λL/d

_hkl

ƒ

Angle between any two diffracting vectors is identical to the angle between corresponding planes.

ƒ

Single Crystal: SAD Pattern Single Crystal: SAD Pattern

( ) ( )

( ) ( )

g h k l

for two difraction vectors g

g

h k l

m n o

hkl

hkl mno

2 2 2 2

2 2

2 2 2

2 2 2

∝ + +

= + +

+ +

41 41

Conventional TED Patterns:

Conventional TED Patterns:

SAD from Different Structures SAD from Different Structures

ƒ Conventional TED pattern

ƒ Amorphous: Series of diffused concentric rings.

ƒ Polycrystalline: Sharp concentric rings.

ƒ Single crystal: Regular spot TED pattern.

Monocrystalline.

ƒ Then structural analysis can be carried out.

ƒ In amorphous matrix, the

positions of atoms are not

normally required.

Experimental Conditions for TED Patterns Experimental Conditions for TED Patterns

ƒ

Specimen area must be flat and horizontal in the

microscope.

ƒ

This area must be at the unique height at which tilting the sample will not cause

moving the image - minimizes any distortion in diffraction pattern produced by e-optics.

ƒ

TED pattern should contain as many complete rings as

possible - for accurate measurement - large SAD aperture.

ƒ

At least 6 innermost rings.

ƒ

Ring diameter should be

measured along one diameter.

43 43

Analysis: Polycrystalline Diffraction Pattern

Analysis: Polycrystalline Diffraction Pattern

ƒ Measure radii of the rings g

₁

, g

₂

, g

₃

, etc.

ƒ Calculate the d spacing d

₁

, d

₂

, d

₃

, etc of planes, give rise to rings g

₁

, g

₂

, g

₃

, etc. from eq. d = Lλ/g

ƒ d spacing obtained identify with cross-referring data to those tabulate in X-ray diffraction files, which list the spacing of thousands of

materials.

ƒ Not all plane diffract – depends on the structure factor, which must be calculated If it is 0 – plane does not diffract. For some crystals the

structure factor rules are simple.

ƒ For face-centered cubic crystal diffract only when Miller indices are unmixed, i.e., odd or even.

ƒ For body centered diffraction - only h+k+l is even.

Analysis: Polycrystalline Diffraction Pattern

Analysis: Polycrystalline Diffraction Pattern

ƒ

g measured from the pattern, we need to match the crystal planes to the rings (index the rings)

ƒ

Since g is inversely proportional to d, the innermost ring has largest d spacing.

ƒ

^(*)

ƒ

Considering all possible values of h,k, and l a possible series of increasing g values would be the sequence 100, 110, 111, 200, 210, 211, 220, 311, etc

ƒ

The structure factor rules tell us that for Au, face centered cubic material h, k, l must be unmixed and therefore rings 100 110, 210 and 211 will not appear and sequence of the ring must be 111, 200, 220, 311, etc

ƒ

Even without knowing the camera constant we can check this by measuring the ratios of the radii g₂/g₁, g₃/g₁etc and comparing them with ratios calculated from equation (*)

) ( h

²

k

²

l

²

a

L d

g = L λ = λ + +

45 45

Energy Dispersive X

Energy Dispersive X - - ray Spectroscopy in TEM ray Spectroscopy in TEM

ƒ

EDS for compositional analysis.

ƒ

X-ray produced specific to chemical element.

ƒ

Number of photons per increment path dt: I_Adt = n_AZ_A dt.

ƒ

For two components A and B of specimen (< 10 nm)

I_A/I_B = n_AZ_A/n_BZ_B

Z_A,B - atomic correction number standard or standardless

ƒ

Intensity measured with a solid state detector.

ƒ

Sensitivity 0.2 atomic %.

ƒ

Spatial resolution ~10 nm

Energy Dispersive X

Energy Dispersive X - - ray Spectroscopy in TEM ray Spectroscopy in TEM

ƒ

Cross-sectional bright field image - an electronic device

ƒ

EDS analysis of ~ 150 nm nanometers thick TiSi₂

metalization layer shows stoichiometric composition.

ƒ

High spatial resolution (<10 nm) - interaction volume is limited by the specimen thickness

ƒ

Peaks are resolved well from the background.

ƒ

47 47

在文檔中 穿透式電子顯微鏡 (頁 34-47)