From Northwestern University Center for Atom-Probe Tomography
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Below are tables of symbols and abbreviations. Please edit them with any new nomenclature that should be common across group documents. All symbols are written in TeX [1]. MediaWiki has multiple methods of rendering this. You may wish to go to you user preferences & change you math rendering mode to "PNG" for the most accurate rendering if you are trying to pick out symbols in MS Word, Writer, or another GUI system.


Abbreviation Description
APFIM atom-probe field-ion microscopy
APT atom-probe tomograph, atom-probe tomography
CNT classical nucleation theory
COM center of mass
CTEM conventional transmission electron microscopy
CVN Charpy V-notch
DSC dispersion-strengthened-cast
FIB focused ion beam
FIM field-ion microscopy
HRC Rockwell hardness C
HREM high-resolution electron microscopy
HSLA high-strength low alloy
HSLC high-strength low carbon
HV Vickers hardness
HVN Vickers microhardness
IDD interprecipitate distance distribution
ISD interface shape distribution
IVAS Imago Visualization and Analysis Software
KV Kuehmann-Voorhees
LEAP® Local-Electrode Atom Probe®
LRO long-range order
LSW Lifshitz-Slyozov-Wagner
NN nearest neighbor
PSD precipitate size distribution
RDF radial-distribution function
RE rare-earth element
SANS Small-angle neutron scattering
SEM scanning-electron microscope, scanning-electron microscopy
SRO short-range order
TEM transmission-electron microscope, transmission-electron microscopy
TM transition metal
UO Umantsev-Olson


Symbol Description
<math>A</math> area
<math>a_{0}</math> lattice constant
<math>B</math> bulk modulus
<math>b</math> Burgers vector
<math>C</math> concentration
<math>C^\prime</math> tetragonal shear modulus
<math>C_{44}</math> trigonal shear modulus
<math>D</math> diffusivity
<math>d</math> grain diameter
<math>d_{max}</math> user-selected maximum-separation distance between solute atoms of interest
<math>E</math> Young's elastic modulus
<math>F</math> Helmholtz free energy
<math>\Delta F_{ch}</math> chemical free energy change on forming a nucleus
<math>\Delta F_{el}</math> elastic free energy change on forming a nucleus
<math>f</math> fraction of ppts interconnected by necks[1]
<math>G</math> Gibbs free energy[2]
<math>G^\alpha_{i,j}</math> Partial derivatives of the molar Gibbs free energy of the <math>\alpha</math> phase
<math>J^{st}</math> stationary-state nucleation rate
<math>K_i^{\alpha^\prime/\alpha}</math> Partitioning ratio of element i between phases <math>\alpha^\prime</math> and <math>\alpha</math> [3]
<math>K_{KV}</math> Coarsening constant for <R(t)> from KV model
<math>k_B</math> Boltzmann's constant
<math>l_g</math> Radius of gyration
<math>M</math> Taylor factor
<math>N_0</math> Total number of possible nucleation sites per unit volume
<math>N_{min}</math> Minimum number of solute atoms in a non-discarded cluster (in envelope method)[4]
<math>N_V</math> Number density of precipitates
<math>n</math> stress exponent
<math>n_{ap}</math> apparent stress exponent
p grain size exponent
<math>p_i</math> magnitude of partitioning of element i between two phases (precipitate over matrix)
<math>Q</math> activation energy
<math>Q_{ap}</math> apparent activation energy
<math>R</math> radius
<math>\langle R \rangle</math> mean radius
<math>\bar{R}</math> mean planar radius[5]
<math>R^*</math> critical radius
<math>R_g</math> universal gas constant
<math>T</math> temperature
<math>t</math> time
<math>t_0</math> initial time[6]
<math>t_c</math> critical time to reach stationary-state coarsening
<math>V</math> Volume
<math>V_a^\alpha</math> Average atomic volume in phase <math>\alpha</math>
<math>V_m^\alpha</math> Average molar volume in phase <math>\alpha</math>
<math>W_R</math> net reversible work required to form a nucleus
<math>W_R^*</math> net reversible work required to form a critical nucleus
<math>\Delta x</math> bin spacing
<math>Z</math> Zeldovich factor


Symbol Description
<math>\beta</math> kinetic coefficient describing the rate of condensation of a single atom on the critical nucleus
<math>\Gamma_i</math> Gibbsian interfacial excess of element i[7]
<math>\gamma</math> interfacial free energy[8]
<math>\delta</math> lattice misfit
<math>\epsilon</math> strain
<math>\dot\epsilon</math> strain rate (often minimum or steady-state)
<math>\eta</math> detection efficiency
<math>\kappa</math> coarsening rate constant for supersaturation
<math>\lambda_{e-e}</math> edge-to-edge interprecipitate distance[9]
<math>\nu</math> Poisson's ratio
<math>\xi</math> Gibbs dividing surface
<math>\rho</math> density[10]
<math>\sigma</math> applied stress (normal); uncertainty; interfacial free energy[8]; conductivity
<math>\sigma_{Or}</math> Orowan stress (normal)
<math>\sigma_{th}</math> threshold stress (normal)
<math>\sigma_{UTS}</math> ultimate tensile strength
<math>\sigma_{YS}</math> tensile yield strength
<math>\tau</math> applied stress (shear)
<math>\tau_{th}</math> threshold stress (shear)
<math>\phi</math> Volume fraction
<math>\phi_{eq}</math> Equilibrium volume fraction

Other Formalism


for ternary alloys:

  • <math>\langle R(t)\rangle^3-\left\langle R\left(t_0\right)\right\rangle^3=K_{KV}\left(t-t_0\right)</math>
  • <math>N_V\left(t\right)^{-1}-N_V\left(t_0\right)^{-1}=4.74\frac{K_{KV}}{\phi_{eq}}\left(t-t_0\right)</math>
  • <math>\Delta C^\alpha_i\left(t\right)=\left\langle C^{\alpha,ff}_i\left(t\right)\right\rangle-C^{\alpha,eq}_i\left(\infty\right)=\kappa^\alpha_{i,KV}\left(t\right)^{-\frac{1}{3}}</math>

Ion Labels

<math>{}^{mass}Element^{charge state}_{number}</math>, such as 18C1+


  • <math>C_i^\alpha</math> concentration of component <math>i</math> in phase <math>\alpha</math>
  • <math>C_i^{\alpha, eq}</math> equilibrium concentration of component <math>i</math> in phase <math>\alpha</math>
  • <math>C_i^{\alpha, ff}</math> far-field concentration of component <math>i</math> in phase <math>\alpha</math>
  • <math>\Delta C_i^\alpha</math> supersaturation of component <math>i</math> in phase <math>\alpha</math>


  1. Some also used it for volume fraction, but this should be discontinued
  2. Some also use it for shear modulus, but this should be discontinued
  3. Source of this? Might change it.
  4. See talk page
  5. Voorhees group uses <math>\left\langle R_{PS} \right\rangle</math> (PS=planar section); see talk
  6. This is NOT the onset of steady-state coarsening or even of quasi-steady-state. It is, rather, some initial time from your experiments (which should be chosen on or after the onset of quasi-stationary-state coarsening).
  7. superscripted, comma-separated phase abbrevs for relative
  8. 8.0 8.1 For the interfacial free energy, Emma used gamma & chantal used sigma. There is a lot of overlap with sigma for quantities (but overlap with gamma for phase identities)
  9. A superscript '2D' or '3D' may be used to describe dimensionality
  10. (subscript element or "th" for theoretical; superscript phase)