Transition Frequency Components¶

The NMR spectral simulation in MRSimulator is based on Symmetry Pathways in Solid-State NMR by Grandinetti et al. [1]

Introduction to NMR frequency components¶

The nuclear magnetic resonance (NMR) frequency, \(\Omega(\Theta, i, j)\), for the \(\left|i\right> \rightarrow \left|j\right>\) transition, where \(\left|i\right>\) and \(\left|j\right>\) are the eigenstates of the stationary-state semi-classical Hamiltonian, can be written as a sum of frequency components,

()¶\[\Omega(\Theta, i, j) = \sum_k \Omega_k (\Theta, i, j),\]

where \(\Theta\) is the sample’s lattice spatial orientation described with the Euler angles \(\Theta = \left(\alpha, \beta, \gamma\right)\), and \(\Omega_k\) is the frequency component from the \(k^\text{th}\) interaction of the stationary-state semi-classical Hamiltonian.

Each frequency component, \(\Omega_k (\Theta, i, j)\), is written as the product,

()¶\[\Omega_k(\Theta, i, j) = \omega_k ~ \Xi_L^{(k)}(\Theta) ~ \xi_\ell^{(k)}(i, j),\]

where \(\omega_k\) is the size of the \(k^\text{th}\) frequency component, and \(\Xi_L^{(k)}(\Theta)\) and \(\xi_\ell^{(k)}(i, j)\) are the sample’s spatial orientation and quantized NMR transition functions corresponding to the \(L^\text{th}\) rank spatial and the \(\ell^\text{th}\) rank spin irreducible spherical tensors, respectively.

The spatial orientation function, \(\Xi_L^{(k)}(\Theta)\), in Eq. (), is defined in the laboratory frame, where the \(z\)-axis is the direction of the external magnetic field. This function is the spatial contribution to the observed frequency component arising from the rotation of the \(L^\text{th}\)-rank irreducible tensor, \(\varrho_{L,n}^{(k)}\), from the principal axis system to the lab frame via Wigner rotation which follows,

()¶\[\Xi_L^{(k)}(\Theta) = \sum_{n_0=-L}^L D^L_{n_0,0}(\Theta_0) \sum_{n_1=-L}^L D^L_{n_1,n_0}(\Theta_1) ~ ... ~ \sum_{n_i=-L}^L D^L_{n_i,n}(\Theta_i) ~~ \varrho_{L,n}^{(k)}.\]

Here, the term \(D^L_{n_i,n_j}(\Theta)\) is the Wigner rotation matrix element, generically denoted as,

()¶\[D^L_{n_i,n_j}(\Theta) = e^{-i n_i \alpha} d_{n_i, n_j}^L(\beta) e^{-i n_j \gamma},\]

where \(d_{n_i, n_j}^L(\beta)\) is Wigner small \(d\) element.

In the case of the single interaction Hamiltonian, that is, in the absence of cross-terms, MRSimulator further defines the product of the size of the \(k^\text{th}\) frequency component, \(\omega_k\), and the \(L^\text{th}\)-rank irreducible tensor components, \(\varrho_{L,n}^{(k)}\), in the principal axis system of the interaction tensor, \(\boldsymbol{\rho}^{(\lambda)}\), as the frequency scaled spatial spherical tensor (fsSST) component,

()¶\[\varsigma_{L,n}^{(k)} = \omega_k \varrho_{L,n}^{(k)},\]

of rank \(L\), also defined in the principal axis system of the interaction tensor. Using Eqs. () and (), we re-express Eq. () as

()¶\[\Omega_k(\Theta, i, j) = \sum_{n_0=-L}^L D^L_{n_0,0}(\Theta_0) \sum_{n_1=-L}^L D^L_{n_1,n_0}(\Theta_1) ~ ... ~ \sum_{n_i=-L}^L D^L_{n_i,n}(\Theta_i) ~~ \varpi_{\ell, L, n}^{(k)},\]

where

()¶\[\varpi_{\ell, L, n}^{(k)} = \varsigma_{L,n}^{(k)}~~\xi_\ell^{(k)}(i, j)\]

is the frequency tensor component (FT) of rank \(L\), defined in the principal axis system of the interaction tensor and corresponds to the \(\left|i\right> \rightarrow \left|j\right>\) spin transition.

Frequency-scaled spatial spherical tensor (fsSST) components in PAS, \(\mathbf{\varsigma}_{L,n}^{(k)}\)¶

Single nucleus scaled spatial orientation tensor components¶

Nuclear shielding interaction¶

The nuclear shielding tensor, \(\boldsymbol{\rho}^{(\sigma)}\), is a second-rank reducible tensor, which can be decomposed into a sum of the zeroth-rank isotropic, first-rank anti-symmetric, and second-rank traceless symmetric irreducible spherical tensors. In the principal axis system, the zeroth-rank, \(\rho_{0,0}^{(\sigma)}\) and the second-rank, \(\rho_{2,n}^{(\sigma)}\), irreducible tensors follow,

()¶\[\begin{array}{c c c c} \rho_{0,0}^{(\sigma)} = -\sqrt{3} \sigma_\text{iso}, & \rho_{2,0}^{(\sigma)} = \sqrt{\frac{3}{2}} \zeta_\sigma, & \rho_{2,\pm1}^{(\sigma)} = 0, & \rho_{2,\pm2}^{(\sigma)} = - \frac{1}{2}\eta_\sigma \zeta_\sigma, \end{array}\]

where \(\sigma_\text{iso}, \zeta_\sigma\), and \(\eta_\sigma\) are the isotropic nuclear shielding, shielding anisotropy, and shielding asymmetry of the site, respectively. The shielding anisotropy and asymmetry are defined using Haeberlen notation.

First-order perturbation

The size of the frequency component, \(\omega_k\), from the first-order perturbation expansion of Nuclear shielding Hamiltonian is \(-\omega_0=\gamma B_0\), where \(\omega_0\) is the Larmor angular frequency of the nucleus, and \(\gamma\), \(B_0\) are the gyromagnetic ratio of the nucleus and the macroscopic magnetic flux density of the applied external magnetic field, respectively. The relation between \(\varrho_{L,n}^{(\sigma)}\) and \(\rho_{L,n}^{(\sigma)}\) follows,

()¶\[\begin{split}\varrho_{0,0}^{(\sigma)} &= -\frac{1}{\sqrt{3}} \rho_{0,0}^{(\sigma)} \\ \varrho_{2,n}^{(\sigma)} &=\sqrt{\frac{2}{3}} \rho_{2,n}^{(\sigma)}\end{split}\]

A list of scaled spatial orientation tensors in the principal axis system of the nuclear shielding tensor, \(\mathbf{\varsigma}_{L,n}^{(k)}\) from Eq. (), of rank L resulting from the Mth order perturbation expansion of the Nuclear shielding Hamiltonian is presented.¶
Order, \(M\)	Rank, \(L\)	\(\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}\)
1	0	\(\varsigma_{0,0}^{(\sigma)} = -\omega_0\sigma_\text{iso}\)
1	2	\(\varsigma_{2,0}^{(\sigma)} = -\omega_0 \zeta_\sigma\), \(\varsigma_{2,\pm1}^{(\sigma)} = 0\), \(\varsigma_{2,\pm2}^{(\sigma)} = \frac{1}{\sqrt{6}} \omega_0\eta_\sigma \zeta_\sigma\)

Electric quadrupole interaction¶

The electric field gradient (efg) tensor, \(\boldsymbol{\rho}^{(q)}\), is also a second-rank tensor; however, unlike the nuclear shielding tensor, the efg tensor is always a symmetric second-rank irreducible tensor. In the principal axis system, this tensor is given as,

()¶\[\begin{array}{c c c} \rho_{2,0}^{(q)} = \sqrt{\frac{3}{2}} \zeta_q, & \rho_{2,\pm1}^{(q)} = 0, & \rho_{2,\pm2}^{(q)} = - \frac{1}{2}\eta_q \zeta_q, \end{array}\]

where \(\zeta_q\), and \(\eta_q\) are the efg tensor anisotropy and asymmetry of the site, respectively. The efg anisotropy and asymmetry values are defined using the Haeberlen convention.

First-order perturbation

The size of the frequency component from the first-order perturbation expansion of Electric quadrupole Hamiltonian is \(\omega_k = \omega_q\), where \(\omega_q = \frac{6\pi C_q}{2I(2I-1)}\) is the quadrupole splitting angular frequency. Here, \(C_q\) is the quadrupole coupling constant, and \(I\) is the spin quantum number of the quadrupole nucleus. The relation between \(\varrho_{L,n}^{(q)}\) and \(\rho_{L,n}^{(q)}\) follows,

()¶\[\varrho_{2,n}^{(q)} = \frac{1}{3\zeta_q} \rho_{2,n}^{(q)}.\]

Second-order perturbation

The size of the frequency component from the second-order perturbation expansion of Electric quadrupole Hamiltonian is \(\omega_k = \frac{\omega_q^2}{\omega_0}\), where \(\omega_0\) is the Larmor angular frequency of the quadrupole nucleus. The relation between \(\varrho_{L,n}^{(qq)}\) and \(\rho_{L,n}^{(q)}\) follows,

()¶\[\varrho_{L,n}^{(qq)} = \frac{1}{9\zeta_q^2} \sum_{m=-2}^2 \left<L~n~|~2~2~m~n-m\right> \rho_{2,m}^{(q)}~\rho_{2,n-m}^{(q)},\]

where \(\left<L~M~|~l_1~l_2~m_1~m_2\right>\) is the Clebsch Gordan coefficient.

A list of scaled spatial orientation tensors in the principal axis system of the efg tensor, \(\mathbf{\varsigma}_{L,n}^{(k)}\) from Eq. (), of rank L resulting from the Mth order perturbation expansion of the Electric Quadrupole Hamiltonian is presented.¶
Order, \(M\)	Rank, \(L\)	\(\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}\)
1	2	\(\varsigma_{2,0}^{(q)} = \frac{1}{\sqrt{6}} \omega_q\), \(\varsigma_{2,\pm1}^{(q)} = 0\), \(\varsigma_{2,\pm2}^{(q)} = -\frac{1}{6} \eta_q \omega_q\)
2	0	\(\varsigma_{0,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{6\sqrt{5}} \left(\frac{\eta_q^2}{3} + 1 \right)\)
2	2	\(\varsigma_{2,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{\sqrt{2}}{6\sqrt{7}} \left(\frac{\eta_q^2}{3} - 1 \right)\), \(\varsigma_{2,\pm1}^{(qq)} = 0\), \(\varsigma_{2,\pm2}^{(qq)} = -\frac{\omega_q^2}{\omega_0} \frac{1}{3\sqrt{21}} \eta_q\)
2	4	\(\varsigma_{4,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{\sqrt{70}} \left(\frac{\eta_q^2}{18} + 1 \right)\), \(\varsigma_{4,\pm1}^{(qq)} = 0\), \(\varsigma_{4,\pm2}^{(qq)} = -\frac{\omega_q^2}{\omega_0} \frac{1}{6\sqrt{7}} \eta_q\), \(\varsigma_{4,\pm3}^{(qq)} = 0\), \(\varsigma_{4,\pm4}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{36} \eta_q^2\)

Coupled nucleus scaled spatial orientation tensor components¶

Weak \(J\)-coupling interaction¶

The \(J\)-coupling tensor, \(\boldsymbol{\rho}^{(J)}\), is a second-rank reducible tensor, which can be decomposed into a sum of the zeroth-rank isotropic, first-rank anti-symmetric, and second-rank traceless symmetric irreducible spherical tensors. In the principal axis system, the zeroth-rank, \(\rho_{0,0}^{(J)}\) and the second-rank, \(\rho_{2,n}^{(J)}\), irreducible tensors follow,

()¶\[\begin{array}{c c c c} \rho_{0,0}^{(J)} = -\sqrt{3} J_\text{iso}, & \rho_{2,0}^{(J)} = \sqrt{\frac{3}{2}} \zeta_J, & \rho_{2,\pm1}^{(J)} = 0, & \rho_{2,\pm2}^{(J)} = - \frac{1}{2}\eta_J \zeta_J, \end{array}\]

where \(J_\text{iso}, \zeta_J\), and \(\eta_J\) are the isotropic \(J\)-coupling, coupling anisotropy and asymmetry parameters, respectively. The \(J\) anisotropy and asymmetry are defined using Haeberlen notation.

First-order perturbation

The size of the frequency component from the first-order perturbation expansion of weak J-coupling Hamiltonian is \(\omega_k = 2\pi\). The relation between \(\varrho_{L,n}^{(J)}\) and \(\rho_{L,n}^{(J)}\) follows,

()¶\[\begin{split}\varrho_{0,0}^{(J)} &= -\frac{1}{\sqrt{3}} \rho_{0,0}^{(J)} \\ \varrho_{2,n}^{(J)} &=\sqrt{\frac{2}{3}} \rho_{2,n}^{(J)}\end{split}\]

A list of scaled spatial orientation tensors in the principal axis system of the J-coupling tensor, \(\mathbf{\varsigma}_{L,n}^{(k)}\) from Eq. (), of rank L resulting from the Mth order perturbation expansion of the J-coupling Hamiltonian is presented.¶
Order, \(M\)	Rank, \(L\)	\(\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}\)
1	0	\(\varsigma_{0,0}^{(J)} = 2\pi J_\text{iso}\)
1	2	\(\varsigma_{2,0}^{(J)} = 2\pi \zeta_J\), \(\varsigma_{2,\pm1}^{(J)} = 0\), \(\varsigma_{2,\pm2}^{(J)} = -\frac{1}{\sqrt{6}} 2\pi\eta_J \zeta_J\)

Weak dipolar-coupling interaction¶

The dipolar-coupling tensor, \(\boldsymbol{\rho}^{(d)}\), is a second rank reducible tensor, which can be decomposed as a second-rank traceless symmetric irreducible spherical tensors. In the principal axis system, the second-rank, \(\rho_{2,n}^{(d)}\), irreducible tensors follow,

()¶\[\begin{array}{c c c c} \rho_{2,0}^{(d)} = \sqrt{\frac{3}{2}} \zeta_d, & \rho_{2,\pm1}^{(d)} = 0, & \rho_{2,\pm2}^{(d)} = 0, \end{array}\]

where \(\zeta_d\) is second-rank symmetric dipolar coupling tensor anisotropy given as

()¶\[\zeta_d = \frac{2}{r^3}\]

where \(r\) is the distance between two coupled magnetic dipoles. The dipolar splitting is given as

()¶\[\omega_d = - \frac{\mu_0}{4\pi} \frac{\gamma_1 \gamma_2 \hbar}{r^3} = - \frac{\mu_0}{8\pi} \zeta_d \gamma_1 \gamma_2 \hbar\]

and the dipolar coupling constant, \(D = \frac{\omega_d}{2\pi}\).

First-order perturbation

The size of the frequency component from the first-order perturbation expansion of weak J-coupling Hamiltonian is \(\omega_k = \frac{2\omega_d}{\zeta_d}\). The relation between \(\varrho_{L,n}^{(d)}\) and \(\rho_{L,n}^{(d)}\) follows,

()¶\[\varrho_{2,n}^{(d)} =\sqrt{\frac{2}{3}} \rho_{2,n}^{(d)}\]

A list of scaled spatial orientation tensors in the principal axis system of the dipolar-coupling tensor, \(\mathbf{\varsigma}_{L,n}^{(k)}\) from Eq. (), of rank L resulting from the Mth order perturbation expansion of the dipolar-coupling Hamiltonian is presented.¶
Order, \(M\)	Rank, \(L\)	\(\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}\)
1	2	\(\varsigma_{2,0}^{(d)} = 2\omega_d\), \(\varsigma_{2,\pm1}^{(d)} = 0\), \(\varsigma_{2,\pm2}^{(d)} = 0\)

Spin transition functions, \(\xi_\ell^{(k)}(i,j)\)¶

The spin transition function is typically manipulated via the coupling of the nuclear magnetic dipole moment with the oscillating external magnetic field from the applied radio-frequency pulse. Considering the strength of the external magnetic rf field is orders of magnitude larger than the internal spin-couplings, the manipulation of spin transition functions is described using the orthogonal rotation subgroups.

Single nucleus spin transition functions¶

A list of single nucleus spin transition functions, \(\xi_\ell^{(k)}(i,j)\).¶
\(\xi_\ell^{(k)}(i,j)\)	Rank, \(\ell\)	Value	Description
\(\mathbb{s}(i,j)\)	0	\(0\)	\(\left< j \| \hat{T}_{00} \| j \right> - \left< i \| \hat{T}_{00} \| i \right>\)
\(\mathbb{p}(i,j)\)	1	\(j-i\)	\(\left< j \| \hat{T}_{10} \| j \right> - \left< i \| \hat{T}_{10} \| i \right>\)
\(\mathbb{d}(i,j)\)	2	\(\sqrt{\frac{3}{2}} \left(j^2 - i^2 \right)\)	\(\left< j \| \hat{T}_{20} \| j \right> - \left< i \| \hat{T}_{20} \| i \right>\)
\(\mathbb{f}(i,j)\)	3	\(\frac{1}{\sqrt{10}} [5(j^3 - i^3) + (1 - 3I(I+1))(j-i)]\)	\(\left< j \| \hat{T}_{30} \| j \right> - \left< i \| \hat{T}_{30} \| i \right>\)

Here, \(\hat{T}_{\ell,k}(\bf{I})\) are the irreducible spherical tensor operators of rank \(\ell\), \(k \in [-\ell, \ell]\), for transition \(|i\rangle \rightarrow |j\rangle\). In terms of the tensor product of the Cartesian operators, the above spherical tensors are expressed as follows,

Spherical tensor operator	Representation in Cartesian operators
\(\hat{T}_{0,0}(\bf{I})\)	\(\hat{1}\)
\(\hat{T}_{1,0}(\bf{I})\)	\(\hat{I}_z\)
\(\hat{T}_{2,0}(\bf{I})\)	\(\frac{1}{\sqrt{6}} \left[3\hat{I}^2_z - I(I+1)\hat{1} \right]\)
\(\hat{T}_{3,0}(\bf{I})\)	\(\frac{1}{\sqrt{10}} \left[5\hat{I}^3_z + \left(1 - 3I(I+1)\right)\hat{I}_z\right]\)

where \(I\) is the spin quantum number of the nucleus and \(\hat{\bf{1}}\) is the identity operator.

A list of composite single nucleus spin transition functions, \(\xi_\ell^{(k)}(i,j)\). Here, I is the spin quantum number of the nucleus.¶
\(\xi_\ell^{(k)}(i,j)\)	Value
\(\mathbb{c}_0(i,j)\)	\(\frac{4}{\sqrt{125}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(i, j) + \sqrt{\frac{18}{25}} \mathbb{f}(i, j)\)
\(\mathbb{c}_2(i,j)\)	\(\sqrt{\frac{2}{175}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(i, j) - \frac{6}{\sqrt{35}} \mathbb{f}(i, j)\)
\(\mathbb{c}_4(i,j)\)	\(-\sqrt{\frac{18}{875}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(i, j) - \frac{17}{\sqrt{175}} \mathbb{f}(i, j)\)

Weakly coupled nucleus spin transition functions¶

A list of weakly coupled nucleus spin transition functions, \(\xi_\ell^{(k)}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})\).¶
\(\xi_\ell^{(k)}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})\)	Value	Description
\((\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})\)	\(m_{f_I} m_{f_S} - m_{i_I} m_{i_S}\)	\(\left< m_{f_I} m_{f_S} \| \hat{T}_{10}(I) \hat{T}_{10}(S) \| m_{f_I} m_{f_S} \right>\) – \(\left< m_{i_I} m_{i_S} \| \hat{T}_{10}(I) \hat{T}_{10}(S) \| m_{i_I} m_{i_S} \right>\)

Here, \(\hat{T}_{\ell,k}(\bf{I})\) are the irreducible spherical tensor operators of rank \(\ell\), \(k \in [-\ell, \ell]\), for transition \(|m_{i_I} m_{i_S}\rangle \rightarrow |m_{f_I} m_{f_S}\rangle\) in weakly coupled basis.

Frequency tensor components (FT) in PAS, \(\varpi_{\ell,L, n}^{(k)}\)¶

The table presents a list of frequency tensors defined in the principal axis system of the respective interaction tensor from Eq. (), \(\varpi_{\ell,L,n}^{(k)}\), of ranks \(\ell\) and \(L\) resulting from the Mth order perturbation expansion of the interaction Hamiltonian supported in mrsimulator.¶
Interaction	Order, \(M\)	Rank, \(L\)	\(\varpi_{\ell,L,n}^{(k)}\)
Nuclear shielding	1	0	\(\varpi_{1,0,0}^{(\sigma)} = \varsigma_{0,0}^{(\sigma)} ~~ \mathbb{p}(i, j)\)
Nuclear shielding	1	2	\(\varpi_{1,2,n}^{(\sigma)} = \varsigma_{2,n}^{(\sigma)} ~~ \mathbb{p}(i, j)\)
Electric Quadrupole	1	2	\(\varpi_{2,2,n}^{(q)} = \varsigma_{2,n}^{(q)} ~~ \mathbb{d}(i, j)\)
Electric Quadrupole	2	0	\(\varpi_{c_0,0,0}^{(qq)} = \varsigma_{0,0}^{(qq)} ~~ \mathbb{c}_0(i, j)\)
Electric Quadrupole	2	2	\(\varpi_{c_2,2,n}^{(qq)} = \varsigma_{2,n}^{(qq)} ~~ \mathbb{c}_2(i, j)\)
Electric Quadrupole	2	4	\(\varpi_{c_4,4,n}^{(qq)} = \varsigma_{4,n}^{(qq)} ~~ \mathbb{c}_4(i, j)\)
Weak \(J\)-coupling	1	0	\(\varpi_{(1,1),0,0}^{(J)} = \varsigma_{0,0}^{(J)} ~~ (\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})\)
Weak \(J\)-coupling	1	2	\(\varpi_{(1,1),2,n}^{(J)} = \varsigma_{2,n}^{(J)} ~~ (\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})\)
Weak dipolar-coupling	1	2	\(\varpi_{(1,1),2,n}^{(d)} = \varsigma_{2,n}^{(d)} ~~ (\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})\)

References