Multi-quantum variable-angle spinning

The following classes are used in simulating multi-quantum variable-angle spinning spectrum correlating the frequencies from the symmetric multiple-quantum transition to the central transition frequencies. The \(p\) and \(d\) pathways for the MQVAS methods are

(28)\[\begin{split}\begin{align} p: &0 \rightarrow M \rightarrow -1 \\ d: &0 \rightarrow 0 \rightarrow 0 \end{align},\end{split}\]

where \(M\) is the multiple-quantum number. The value of \(M\) depends on the spin quantum number, \(I\), and is listed in Table 9.

Affine mapping

The resulting spectrum is sheared and scaled, such that the frequencies along the indirect dimension are given as

(29)\[\langle \Omega\rangle_\text{MQ-VAS} = \frac{1}{1+\kappa}\Omega_{m, -m} + \frac{\kappa}{1+\kappa}\Omega_{1/2, -1/2}.\]

Here, \(\langle \Omega\rangle_\text{MQ-VAS}\) is the average frequency along the indirect dimension, \(\Omega_{m, -m}\) and \(\Omega_{1/2, -1/2}\) are the frequency contributions from the \(|m\rangle \rightarrow |-m\rangle\) symmetric multiple-quantum transition and the central transition, respectively, and \(\kappa\) is the shear factor. The values of the shear factor for various transitions are listed in Table 9.

Table 9 The table lists the multi-quantum transition associated with the spin \(I\), and the corresponding shear factor, \(\kappa\), used in affine mapping of the MQ-VAS methods.

Spin

Symmetric multi-quantum transition

\(M\)

\(\kappa\)

3/2

\(\left(\frac{3}{2} \rightarrow -\frac{3}{2}\right)\)

\(-3\)

21/27

5/2

\(\left(-\frac{3}{2} \rightarrow \frac{3}{2}\right)\)

\(3\)

114/72

5/2

\(\left(\frac{5}{2} \rightarrow -\frac{5}{2}\right)\)

\(-5\)

150/72

7/2

\(\left(-\frac{3}{2} \rightarrow \frac{3}{2}\right)\)

\(3\)

303/135

7/2

\(\left(-\frac{5}{2} \rightarrow \frac{5}{2}\right)\)

\(5\)

165/135

7/2

\(\left(\frac{7}{2} \rightarrow -\frac{7}{2}\right)\)

\(-7\)

483/135

9/2

\(\left(-\frac{3}{2} \rightarrow \frac{3}{2}\right)\)

\(3\)

546/216

9/2

\(\left(-\frac{5}{2} \rightarrow \frac{5}{2}\right)\)

\(5\)

570/216

9/2

\(\left(-\frac{7}{2} \rightarrow \frac{7}{2}\right)\)

\(5\)

84/216

Triple-quantum variable-angle spinning method

class mrsimulator.methods.ThreeQ_VAS(**kwargs)

Simulate a sheared and scaled 3Q 2D variable-angle spinning spectrum.

Parameters

  • channels – A list of isotope symbols over which the method will be applied.

  • spectral_dimensions –

    A list of python dict. Each dict is contains keywords that describe the coordinates along a spectral dimension. The keywords along with its definition are:

    • count:

      An optional integer with the number of points, \(N\), along the dimension. The default value is 1024.

    • spectral_width:

      An optional float with the spectral width, \(\Delta x\), along the dimension in units of Hz. The default is 25 kHz.

    • reference_offset:

      An optional float with the reference offset, \(x_0\) along the dimension in units of Hz. The default value is 0 Hz.

    • origin_offset:

      An optional float with the origin offset (Larmor frequency) along the dimension in units of Hz. The default value is None.

  • magetic_flux_density – An optional float containing the macroscopic magnetic flux density, \(H_0\), of the applied external magnetic field in units of T. The default value is 9.4.

  • rotor_angle – An optional float containing the angle between the sample rotation axis and the applied external magnetic field, \(\theta\), in units of rad. The default value is 0.9553166, i.e. the magic angle.

Note

The attribute rotor_frequency cannot be modified for this method and is set to simulate an infinite speed spectrum.

Returns

A Method instance.

Example

>>> method = ThreeQ_VAS(
...     channels=["87Rb"],
...     magnetic_flux_density=7,  # in T
...     spectral_dimensions=[
...         {
...             "count": 256,
...             "spectral_width": 4e3,  # in Hz
...             "reference_offset": -5e3,  # in Hz
...             "label": "Isotropic dimension",
...         },
...         {
...             "count": 512,
...             "spectral_width": 1e4,  # in Hz
...             "reference_offset": -4e3,  # in Hz
...             "label": "MAS dimension",
...         },
...     ],
... )
>>> sys = SpinSystem(sites=[Site(isotope='87Rb')])
>>> method.get_transition_pathways(sys)
[TransitionPathway(|-1.5⟩⟨1.5|, |-0.5⟩⟨0.5|)]

Five-quantum variable-angle spinning method

class mrsimulator.methods.FiveQ_VAS(**kwargs)

Simulate a sheared and scaled 5Q variable-angle spinning spectrum.

Parameters

  • channels – A list of isotope symbols over which the method will be applied.

  • spectral_dimensions –

    A list of python dict. Each dict is contains keywords that describe the coordinates along a spectral dimension. The keywords along with its definition are:

    • count:

      An optional integer with the number of points, \(N\), along the dimension. The default value is 1024.

    • spectral_width:

      An optional float with the spectral width, \(\Delta x\), along the dimension in units of Hz. The default is 25 kHz.

    • reference_offset:

      An optional float with the reference offset, \(x_0\) along the dimension in units of Hz. The default value is 0 Hz.

    • origin_offset:

      An optional float with the origin offset (Larmor frequency) along the dimension in units of Hz. The default value is None.

  • magetic_flux_density – An optional float containing the macroscopic magnetic flux density, \(H_0\), of the applied external magnetic field in units of T. The default value is 9.4.

  • rotor_angle – An optional float containing the angle between the sample rotation axis and the applied external magnetic field, \(\theta\), in units of rad. The default value is 0.9553166, i.e. the magic angle.

Note

The attribute rotor_frequency cannot be modified for this method and is set to simulate an infinite speed spectrum.

Returns

A Method instance.

Example

>>> method = FiveQ_VAS(
...     channels=["17O"],
...     magnetic_flux_density=9.4,  # in T
...     spectral_dimensions=[
...         {
...             "count": 256,
...             "spectral_width": 4e3,  # in Hz
...             "reference_offset": -5e3,  # in Hz
...             "label": "Isotropic dimension",
...         },
...         {
...             "count": 512,
...             "spectral_width": 1e4,  # in Hz
...             "reference_offset": -4e3,  # in Hz
...             "label": "MAS dimension",
...         },
...     ],
... )
>>> sys = SpinSystem(sites=[Site(isotope='17O')])
>>> method.get_transition_pathways(sys)
[TransitionPathway(|-2.5⟩⟨2.5|, |-0.5⟩⟨0.5|)]

Seven-quantum variable-angle spinning method

class mrsimulator.methods.SevenQ_VAS(**kwargs)

Simulate a sheared and scaled 7Q variable-angle spinning spectrum.

Parameters

  • channels – A list of isotope symbols over which the method will be applied.

  • spectral_dimensions –

    A list of python dict. Each dict is contains keywords that describe the coordinates along a spectral dimension. The keywords along with its definition are:

    • count:

      An optional integer with the number of points, \(N\), along the dimension. The default value is 1024.

    • spectral_width:

      An optional float with the spectral width, \(\Delta x\), along the dimension in units of Hz. The default is 25 kHz.

    • reference_offset:

      An optional float with the reference offset, \(x_0\) along the dimension in units of Hz. The default value is 0 Hz.

    • origin_offset:

      An optional float with the origin offset (Larmor frequency) along the dimension in units of Hz. The default value is None.

  • magetic_flux_density – An optional float containing the macroscopic magnetic flux density, \(H_0\), of the applied external magnetic field in units of T. The default value is 9.4.

  • rotor_angle – An optional float containing the angle between the sample rotation axis and the applied external magnetic field, \(\theta\), in units of rad. The default value is 0.9553166, i.e. the magic angle.

Note

The attribute rotor_frequency cannot be modified for this method and is set to simulate an infinite speed spectrum.

Returns

A Method instance.

Example

>>> method = SevenQ_VAS(
...     channels=["51V"],
...     magnetic_flux_density=9.4,  # in T
...     spectral_dimensions=[
...         {
...             "count": 256,
...             "spectral_width": 4e3,  # in Hz
...             "reference_offset": -5e3,  # in Hz
...             "label": "Isotropic dimension",
...         },
...         {
...             "count": 512,
...             "spectral_width": 1e4,  # in Hz
...             "reference_offset": -4e3,  # in Hz
...             "label": "MAS dimension",
...         },
...     ],
... )
>>> sys = SpinSystem(sites=[Site(isotope='51V')])
>>> method.get_transition_pathways(sys)
[TransitionPathway(|-3.5⟩⟨3.5|, |-0.5⟩⟨0.5|)]