gigablochs.bloch.construct_B_field

gigablochs.bloch.construct_B_field#

gigablochs.bloch.construct_B_field(rf_am, G=0, position=0, *, off_resonance=0, B1_sensitivity=1, rf_fm=0, time_axis=0, space_axis=-1)[source]#

Construct the magnetic field vector from the RF and gradient waveforms.

Parameters:

  • rf_am (ndarray) – RF amplitude waveform in Tesla.

  • G (ndarray, optional) – Gradient waveform in Tesla/m. The length of the time axis should match rf_am. Default is 0.

  • position (ndarray, optional) – Spatial position vector in meters. Shape should match G. Default is 0.

  • off_resonance (float, optional) – Off-resonance frequency in Hz. Default is 0.

  • B1_sensitivity (float or ndarray, optional) – B1 sensitivity factor. Default is 1.

  • rf_fm (ndarray, optional) – RF frequency waveform in Hz. Default is 0.

  • time_axis (int, optional) – Axis along the input arrays which represents the time axis. Default is 0.

  • space_axis (int, optional) – Axis along the field array which represents the 2D or 3D spatial field vector. Also used for the dot product between gradient and position vectors. Default is -1.

Returns:

B_eff – Magnetic field vector in Tesla. Shape is extended from rf_am by \(G \cdot position\), off_resonance, B1_sensitivity, and coordinates for the spatial axis. Explicitly, when space_axis=-1, the shape of the magnetic field vector is (time, …, dB0, dB1, 3), where time is len(rf_am), dB0 = len(off_resonance), dB1 = len(B1_sensitivity), 3 is the spatial axis (x, y, z), and represents any extra dimensions in rf_am (for example to parametrize bandwidth), followed by any extra dimensions in dot(G, position, space_axis) (for example to parametrize spin motion).

Return type:

ndarray

Notes

As this function uses numpy-style broadcasting, the input arrays should have unique shapes, for example, parametrized dimensions like len(off_resonance) should NOT match len(rf_am), the number of time steps.

The magnetic field vector is computed as:

\[ \begin{align}\begin{aligned}B_x = \Delta B_1 \Re{RF_{AM}}\\B_y = \Delta B_1 \Im{RF_{AM}}\\B_z = \vec{G} \cdot \vec{r} + \frac{RF_{FM}}{\gammabar} + \frac{\Delta f}{\gammabar}\end{aligned}\end{align} \]

where \(B_x\), \(B_y\), and \(B_z\) are the magnetic field components, \(\Delta B_1\) is the unitless B1 sensitivity factor, \(RF_{AM}\) is the RF amplitude modulation waveform in Tesla, \(\vec{G}\) is the gradient waveform in Tesla/m, \(\vec{r}\) is the spin’s spatial position waveform in meters, \(RF_{FM}\) is the RF frequency modulation waveform in Hz, \(\gammabar\) is the reduced gyromagnetic ratio in Hz/T, and \(\Delta f\) is the off-resonance frequency in Hz.

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