Graef M. Introduction to conventional transmission electron microscopy

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678 List of symbols

(q) Drift envelope function

(p. 617)

Fermi energy level (p. 180)

ijk

Normalized permutation

symbol (p. 19)

Total energy of a system

(p. 84)

F Electric ﬁeld applied to

ﬁlament (p. 180)

f Focal length (p. 151)

f physical observable (p. 81)

f (E, T ) Fermi–Dirac distribution

(p. 180)

f ( f ) Defocus spread function

(p. 608)

(s) Atomic scattering factor for

electrons (p. 111)

(s) Atomic scattering factor for

x-rays (p. 112)

Asymptotic image focal

length (p. 157)

Eigenvalue for eigenstate

|

 (p. 82)

Asymptotic object focal

length (p. 157)

g(E) Normalized energy

distribution for electron

emission (p. 190)

G(i, j ) Gain reference image

intensity (p. 224)

∗

Reciprocal 4-index metric

tensor (p. 27)

∗

Reciprocal metric tensor

(p. 14)

4-index metric tensor

(hexagonal) (p. 25)

Direct metric tensor (p. 6)

Normal component of

reciprocal lattice vector

(p. 325)

, G

Three-fold astigmatism

(Cartesian) (p. 629)

max

Maximum spatial

frequency (p. 441)

H Hamiltonian (classical)

(p. 84)

h Planck’s constant (p. 83)

Matrix elements of

beam–crystal interaction

(p. 124)

(q) Diffuse intensity (p. 572)

Dark ﬁeld intensity (p. 350)

Bright ﬁeld intensity

(p. 350)

Intensity of a diffractogram

(p. 627)

Schottky current density

(p. 183)

Thermal ﬁeld emission

current density (p. 184)

Thermionic current density

(p. 183)

k Wave number (p. 85)

Boltzmann constant

(p. 127)

General wave number

(p. 94)

Wave number corrected for

refraction (p. 305)

Normal component of

wave vector (p. 325)

, K

Coma coefﬁcients

(Cartesian) (p. 629)

L Camera length (p. 206)

L Column width (p. 466)

, L

Dimensions of sample

(p. 347)

Column width (p. 347)

Bloch wave matrix for

ALCHEMI-like

computations (p. 415)

M Transverse magniﬁcation

of round lens (p. 158)

m Electron mass (relativistic)

(p. 90)

Electron rest mass (p. 83)

Angular magniﬁcation of

round lens (p. 159)

n Diffraction order (in Bragg

equation) (p. 97)

n Electron-optical refractive

index (p. 95)

Linear electron density

(p. 195)

, N

Number of pixels in image

(p. 347)

p Momentum (classical or

relativistic) (p. 83)

Q Normalized spatial

frequency (p. 594)

( j )

Imaginary part of Bloch

wave vector k

( j )

(p. 328)

List of symbols 679

Nyquist critical frequency

(p. 222)

Frequency corresponding

to point resolution (p. 601)

Complex combination of

extinction distance and

absorption length (p. 309)

R 180

◦

symmetry operator of

a diffraction disk (p. 336)

, r

General solutions of

paraxial ray equation

(p. 156)

s Signal-to-noise ratio

(p. 614)

S(z) Amplitude of the diffracted

beam (p. 353)

s(q) Source spread function

(p. 608)

Amplitude of scattered

beam (p. 311)

Structure matrix for

ALCHEMI-like

computations (p. 415)

Excitation error/deviation

parameter (p. 121)

T Absolute temperature

(p. 127)

T Kinetic energy (p. 83)

T (z) Amplitude of the

transmitted beam (p. 353)

T (r) Point spread function

(p. 223)

t, t



Time variable (p. 90)

Transmission

cross-coefﬁcient (p. 607)



(r) Scaled electrostatic lattice

potential (imaginary part)

(p. 305)

U (r) Scaled electrostatic lattice

potential (real part)

(p. 305)

U (r) Scaled electrostatic lattice

potential (p. 95)

, U

Normal and tangential

energy component of

emitted electron (p. 181)

, u

Complex coordinates in the

object, lens, and image

planes (p. 168)

V Potential energy (p. 84)

v Velocity (p. 83)

V (z) Barrier potential (p. 180)

V (r) Electrostatic lattice

potential (p. 95)

(r) Electrostatic atom potential

(p. 112)

(r) Finite crystal lattice

potential (p. 119)

Mean inner potential

(p. 304)

(r) Complex electrostatic

lattice potential (p. 125)

Projected potential (p. 317)

Potential of sheared

structure (p. 565)

Block potential (p. 565)

Fourier coefﬁcients of

electrostatic lattice

potential (p. 116)

W Work function (p. 180)

w Dimensionless excitation

error (p. 350)

W (r)

Optical potential (p. 125)

Z Atomic

number (p. 112)

Crystal thickness (p. 312)

Crystal thickness along

beam direction (p. 466)

, z

Asymptotic object and

image focal points (p. 156)

, z

Asymptotic object and

image nodal points (p. 159)

, z

Asymptotic object and

image principal points

(p. 157)

Foil thickness in units of

extinction distance

(p. 350)

χ(q) Aberration function

(p. 594)

δ Lattice misﬁt parameter

(p. 479)

T Kinetic energy operator

(p. 84)

V Potential energy operator

(p. 84)

I Average image intensity

(p. 223)

Dynamical or Bethe

potential (p. 436)

V Scaled complex

electrostatic lattice

potential (p. 316)

680 List of symbols

r Vector in normal

coordinates (p. 41)

A Bloch wave structure

matrix (p. 325)

Slice operator (p. 440)

A Magnetic vector potential

(p. 146)

a, b, c Bravais lattice basis vectors

(p. 2)

A, B, C, I Lattice centering vectors

(p. 3)

∗

, b

∗

, c

∗

Reciprocal basis vectors

(p. 11)

∗

Reciprocal hexagonal basis

vectors (4-index) (p. 27)

∗

Reciprocal basis vectors

(p. 11)

Bravais lattice basis vectors

(p. 2)

B Magnetic ﬁeld vector

(p. 144)

b Burgers vector (p. 481)

⊥

In-plane magnetic

induction (p. 291)

C Center of Ewald sphere

(p. 98)

( j )

Column vector of Bloch

wave coefﬁcients

(p. 325)

E Electric ﬁeld vector

(p. 146)

∗

Reciprocal Cartesian basis

vectors (p. 19)

Cartesian basis vectors

(p. 19)

, e

Basis vectors in cylindrical

coordinates (p. 147)

F Foil normal (p. 122)

F Force vector (p. 90)

Lorentz force (p. 291)

G Shortest reciprocal lattice

vector of the FOLZ

(p. 213)

g Reciprocal lattice vector

(p. 11)

hkl

Normal to the plane (hkl)

(p. 13)

J Electron ﬂux (p. 125)

k, k



,... Wave vectors (p. 98)

( j )

Wave vector of ( j)th Bloch

wave (p. 322)

Wave vector corrected for

refraction (p. 305)

Tangential component of

incident wave vector

(p. 334)

2D wave vector (p. 316)

Tangential component of

diffracted wave vector

(p. 334)

Screw axis (p. 37)

n Unit normal (p. 13)

O Origin of reciprocal space

(p. 98)

p Momentum vector (p. 83)

p, q General position vectors

(p. 5)

q Position vector in

reciprocal space (p. 101)

(k)

Star of q (p. 46)

r Position vector (p. 5)

R(r) Displacement ﬁeld

(p. 461)

(k)

Orbit of r (p. 46)

, R

Rhombohedral centering

vectors (p. 54)

S Column vector of diffracted

amplitudes (p. 312)

S Scattering vector (p. 373)

s Scattering vector (p. 111)

T Pitch vector of a screw axis

(p. 37)

t Translation vector (p. 2)

u Line direction (dislocation)

(p. 481)

u(t) Instantaneous atomic

vibration amplitude

(p. 126)

U(z) Paraxial ray vector

(p. 157)

(t) Instantaneous displacement

of atom i (p. 374)

v Velocity vector (pp. 90,

146)

A Crystal transfer matrix

(p. 311)

C Matrix of Bloch wave

eigenvectors (p. 325)

E(z) Diagonal matrix with

thickness dependent part of

S (p. 327)

G General group (p. 42)

List of symbols 681

Matrix used for thickness

integrated intensity

computations (p. 415)

Lens phase factor (p. 164)

P Phase matrix for planar

defect (p. 500)

Fresnel propagator

(p. 163)

R, R



,... Signal functions (p. 107)

S Scattering matrix (p. 312)

T Bravais lattice (p. 2)

T Diagonal part of A (p. 314)

T Instrument point spread

function (p. 107)

T Lens transfer matrix

(p. 158)

∗

Reciprocal lattice (p. 12)

V Off-diagonal part of A

(p. 314)

Planck’s constant divided

by 2π (p. 83)

H Hamiltonian operator

(p. 84)

Crystal Hamiltonian

(p. 123)

int

Interaction Hamiltonian

(p. 123)

p Momentum operator

(p. 83)

f Quantum mechanical

operator (p. 82)

{a, c} Hexagonal lattice

parameters (p. 54)

c Velocity of light (p. 90)

DQE Detective quantum

efﬁciency (p. 223)

var(I ) Variance of image intensity

(p. 223)

Greek letters

α Angle between k + g and

foil normal (p. 122)

α, β, γ Angles between Bravais

basis vectors (p. 3)

( j )

Bloch wave excitation

amplitude (p. 322)

∗

,β

∗

,γ

∗

Angles between reciprocal

basis vectors (p. 14)

0, j

Warren–Cowley

short-range order

parameters (p. 572)

General coordinate

transformation matrix

(p. 48)

Difference between Bloch

wave absorption

coefﬁcients (p. 415)

(r) Defect phase factor

(p. 462)

β Beam brightness (p. 185)

β Velocity in units of velocity

of light (p. 90)

Difference between Bloch

wave eigenvalues (p. 415)

Phase difference between

real and imaginary part of

lattice potential (p. 310)

 Path length difference

(p. 162)

δ Magnetic domain wall

width (p. 296)

δ sampling interval (p. 222)

δ(p − q) Dirac

delta-function

(p. 84)

p Momentum transfer vector

(p. 99)

Kronecker delta (unit

matrix) (p. 9)

 0.978 45 × 10

−6

−1

(p. 93)

 Slice thickness (p. 313)



Permittivity of vacuum

(p. 112)



Strain tensor (p. 482)

( j )

Shorthand notation for

Bethe potential

computation (p. 435)

γ Relativistic correction

factor (p. 90)

( j )

Normal part of Bloch wave

vector k

( j )

with absorption

(p. 328)

( j )

Normal component of

Bloch wave vector

(p. 323)

 Relativistic acceleration

potential (p. 93)



Relativistic acceleration

potential inside crystal

(p. 95)

Diagonal matrix with Bloch

wave eigenvalues (p. 326)

682 List of symbols

λ Wavelength (p. 85)

Compton wavelength

(p. 94)

Relativistic wavelength

inside crystal (p. 95)

Non-relativistic electron

wavelength (p. 87)

≡ λ Relativistic electron

wavelength (p. 92)

sc,a

Spatial coherence width at

condensor aperture plane

(p. 194)

Spatial coherence width

(p. 193)

Temporal coherence length

(p. 193)

 Two-dimensional complex

differential operator

(p. 316)

¯σ

Flynn operator (p. 572)



Refraction corrected

relativistic acceleration

potential (p. 305)

µ Magnetic permeability

(p. 145)

ν Frequency of a wave (p. 85)

ν Poisson’s ratio (p. 482)

 Unit cell volume (p. 11)

ω Angular frequency of a

wave (p. 85)

ω Scalar magnetic potential

(p. 144)

(S) Solid angle (p. 185)



∗

Volume reciprocal unit cell

(p. 21)

FFT twiddle factors

(p. 108)

 Electrostatic potential

(p. 146)

,,... Wave function (p. 81)

(q) Aberrated wave function

(p. 599)

ψ,φ,... General wave functions

(p. 88)

Fourier component of wave

function ψ (p. 306)

ρ Total charge density

(p. 111)

Chromatic information

limit (p. 615)

Detector information limit

(p. 620)

Drift information limit

(p. 617)

Electron charge density

(p. 111)

Nuclear charge density

(p. 111)

Microscope point

resolution (p. 601)

Spatial incoherence

information limit (p. 617)

σ Interaction constant (p. 93)

σ Two-beam parameter

(p. 354)

σ (I ) Standard deviation of

image intensity (p. 223)

Site occupation operator

(p. 571)

Stress tensor (p. 482)

τ Exposure time (p. 228)

θ Angle between two vectors

(p. 7)

θ Bragg angle (p. 96)

Beam divergence angle

(p. 198)

Angle between basis

vectors i and j (p. 6)

Lorentz deﬂection angle

(p. 292)

,θ



Phase of U

and U



(p. 309)



Normal absorption length

(p. 310)



Absorption length (p. 309)

Extinction distance

(p. 309)

Special symbols

(hkl) Miller indices of a plane

(p. 10)

(r,ϕ,z) Cylindrical coordinates

(p. 144)

(x, y, z) = r

Fractional atom

coordinates (p. 5)

(D|t) Seitz symbol for symmetry

operator (p. 40)

(, k

) decomposition of wave

vector incident from below

crystal (p. 334)

List of symbols 683

(⊗, k

) decomposition of wave

vector incident from above

crystal (p. 334)

f ⊗ g Convolution product

(p. 106)

 Crystal eigenstates (p. 123)

|q| Length or norm of vector q

(p. 3)

|

 Eigenstate of quantum

mechanical system (p. 82)

 f  Quantum mechanical

expectation value (p. 81)

uvw Family of directions (p. 45)

| f | Scalar product of functions

(p. 82)



2D Laplacian operator

(p. 316)

1 Identity matrix (p. 312)

p · q Vector dot product (p. 6)

p × q Vector cross product

(p. 18)

−1

[ ] Inverse Fourier transform

operator (p. 104)

F[ ] Direct Fourier transform

operator (p. 104)

W Symmetry matrix in

normal coordinates

(p. 41)

∇ Gradient differential

operator (p. 83)

∇

, Laplacian operator (p. 84)

∇

2D gradient operator

(p. 316)

O General symmetry operator

(p. 39)

,α} Rhombohedral lattice

parameters (p. 54)

{hkl} Family of planes (p. 45)

[uvtw] Miller–Bravais direction

indices (p. 23)

[uvw] Crystallographic

direction

(p. 5)

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