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doi:10.1039/9781788013291-FP012

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List of Symbols

Doi:

https://doi.org/10.1039/9781788013291-FP012

Published:

03 Aug 2018
Special Collection: 2018 ebook collection

Series: New Developments in NMR

Page range:

P012 - P015

1: identity matrix
γ_I: nuclear magnetogyric ratio
γ_S: electron magnetogyric ratio
δ: chemical shift
δ^cs: contact shift
δ^dia: diamagnetic chemical shift
δ^hs: hyperfine shift
δ^pcs: pseudocontact shift
Δ $_{t}^{2}$: mean squared fluctuation of the ZFS
Δν^dfs: dynamic frequency shift
Δν^dia: diamagnetic residual dipolar coupling
Δν^rdc: paramagnetic residual dipolar coupling
Δν $_{0}^{EPR}$: overall breadth of the EPR spectrum
Δν^hom: homogeneous linewidth of an EPR spin packet
Δσ^dip: dipolar shielding anisotropy
Δχ: magnetic susceptibility anisotropy tensor
Δχ_ax: axial magnetic susceptibility anisotropy
Δχ_rh: rhombic magnetic susceptibility anisotropy
Δω_p: paramagnetic chemical shift of solvent ligand nuclei
ΔΕ_ZFS: energy shift due to zero-field splitting
ε: DNP enhancement factor
ζ: electron spin tilting angle due to electron–electron coupling in the ZQ space
η_α/β: branching angle
η_I(J): nuclear Overhauser enhancement
μ₀: permeability of the vacuum
μ_B: electron Bohr magneton
μ_I: nuclear magnetic moment
μ_S: electron magnetic moment
〈μ_S〉: electron induced magnetic moment
ξ: Overhauser effect coupling factor
ρ: electron spin density at the nucleus
ρ_I/ρ_S: longitudinal relaxation rate of spin I/S
σ^DIP: Dipolar shielding tensor
σ^FC: Fermi contact shielding tensor
σ^hs: hyperfine shielding tensor
σ_I(J): cross-relaxation rate
σ_IS: electron–nuclear cross-relaxation rate
τ_1e: longitudinal electron relaxation time
τ_2e: transverse electron relaxation time
τ_c: correlation time
τ_D: diffusional correlation time
τ_e: electron relaxation time
τ_l: correlation time for the local motions
τ_M: exchange time
τ_r: reorientation correlation time
τ_v: correlation time for electron relaxation
χ: magnetic susceptibility per molecule
χ_iso: isotropic part of the magnetic susceptibility
χ_q: quadrupolar coupling constant
ω_I: nuclear Larmor frequency times 2π
ω_S: electron Larmor frequency times 2π
ω_1S: electron Rabi frequency times 2π
ω_µw: microwave frequency times 2π
ω_ZFS: zero-field splitting shift in units of frequency times 2π
Ω_S: EPR frequency offset in the µw rotating frame
A: hyperfine coupling operator
A^DIP: dipole–dipole operator
A^FC: Fermi-contact operator/coupling constant
A: secular hyperfine coupling constant
B₀: external magnetic field in tesla (magnetic induction)
B₁: oscillating μw field in tesla (magnetic induction)
B: pseudo-secular hyperfine coupling constant
d: distance of closest approach
D: axial zero-field splitting constant
D: ZFS tensor
D^ee: electron–electron coupling tensor
d^ee: electron–electron dipolar coupling constant
D $_{d}^{ee}$: diagonal e–e coupling element
D $_{0}^{ee}$: off-diagonal e–e coupling element in the ZQ space
D_M/D_L: diffusion coefficients of the paramagnetic molecule/ligand molecule
E: rhombic zero-field splitting constant
f: Overhauser effect leakage factor
f_M: mole fraction of ligand nuclei in bound positions
g: g tensor
g: g factor
g_e: free-electron g factor
ħ: Planck's constant divided by 2π
H_hc: hyperfine coupling Hamiltonian
H_SE: solid effect Hamiltonian
H_CE: cross effect Hamiltonian
H_Δ: electron–electron ZQ Hamiltonian
H_Σ: electron–electron DQ Hamiltonian
H_ZFS: zero-field splitting Hamiltonian
I: nuclear spin
〈I_z〉: expectation value of I_z
〈I_z〉₀: expectation value of I_z in thermal equilibrium
¹J: J-coupling
J^ee: electron–electron exchange interaction constant
J(ω,τ): spectral density function
k: Boltzmann constant
k: rate matrix
K: cross effect transition moment
K_D: dissociation constant
k_ex: exchange rate
K_M: Michaelis constant
k_off: dissociation rate constant
k_on: association rate constant
M: magnetization
m_I: nuclear magnetic spin quantum number
m_S: electron magnetic spin quantum number
n: orientation vector in the unit sphere
N_A: Avogadro's constant
N_α/β: population number of spin ground/excited state for I=½
P: probability tensor
p_A: weighting factor for A
q: hydration number
r: vector connecting the positions of the nuclear and electron spins
r₁: longitudinal relaxivity
R: relaxation matrix
R₁/R₂: longitudinal/transverse nuclear relaxation rate
R_1ρ: nuclear longitudinal relaxation rate in the rotating frame
R $_{1}^{dia}$ /R $_{2}^{dia}$: diamagnetic longitudinal/transverse nuclear relaxation rate
R_1M/R_2M: paramagnetic enhancement of the nuclear longitudinal/transverse relaxation rate
R $_{1 M}^{Curie}$ /R $_{2 M}^{Curie}$: longitudinal/transverse Curie spin relaxation rate
R $_{1 M}^{dip}$ /R $_{2 M}^{dip}$: longitudinal/transverse dipole–dipole relaxation rate
R $_{1 M}^{FC}$ /R $_{2 M}^{FC}$: longitudinal/transverse Fermi contact relaxation rate
R_1p/R_2p: paramagnetic enhancement of the longitudinal/transverse relaxation rate of ligand nuclei
R $_{1p}^{OS}$ /R $_{2p}^{OS}$: paramagnetic outer-sphere longitudinal/transverse relaxation rate of ligand nuclei
R_ex: exchange contribution to relaxation
s: EPR saturation factor
s: null average part of the electron spin
S: electron spin
S $_{Acc}^{c}$: solvent accessibility
S_C: thermal average of the electron spin
S_LS: Lipari–Szabo parameter
〈SS〉: effective electron spin dyadic
〈S_z〉: expectation value of S_z
S/N: signal-to-noise ratio
s.t.: subject to
t: time
T: temperature
T_1M: longitudinal nuclear relaxation time due to the electron–nucleus interaction
T_1q: quadrupolar longitudinal relaxation time
T_B: time constant for exponential build-up of polarization
Tr: trace
v: reaction rate
V_max: maximal rate achievable at saturating concentration of the substrate
w₀: electron–nuclear ZQ transition probability induced by fluctuation of hyperfine coupling
w₁: nuclear SQ transition probability induced by fluctuation of hyperfine coupling
w₂: electron–nuclear DQ transition probability induced by fluctuation of hyperfine coupling
w°: nuclear SQ transition probability not induced by hyperfine coupling
w_ij: transition probabilities
W_max: largest weight

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References

New Developments in NMR

Paramagnetism in Experimental Biomolecular NMR

Edited by

DOI:

https://doi.org/10.1039/9781788013291

Hardback ISBN:

978-1-78801-086-3

PDF ISBN:

978-1-78801-329-1

EPUB ISBN:

978-1-78801-496-0

Special Collection: 2018 ebook collection

Series: New Developments in NMR

No. of Pages:

316

Published online:

03 Aug 2018

Published in print:

07 Aug 2018

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Paramagnetism in Experimental Biomolecular NMR

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