Module direction_cosine

Functions

calc_spheroid_di(data, diff_data)
Function for calculating the direction cosine dz.

calc_spheroid_ddi(data, diff_data)
Function for calculating the partial derivatives of the direction cosine dz.

calc_spheroid_d2di(data, diff_data)
Function for calculating the second partial derivatives of the direction cosine dz.

calc_ellipsoid_di(data, diff_data)
Function for calculating the direction cosines dx, dy, and dz.

calc_ellipsoid_ddi(data, diff_data)
Function for calculating the partial derivatives of the direction cosines dx, dy, and dz.

calc_ellipsoid_d2di(data, diff_data)
Function for calculating the second partial derivatives of the direction cosines dx, dy, dz.

Variables

__package__ = 'lib.diffusion'

Imports: cos, sin, dot

Function Details

calc_spheroid_di(data, diff_data)

Function for calculating the direction cosine dz.

dz is the dot product between the unit bond vector and the unit vector along Dpar and is given by:

   dz = XH . Dpar.

The unit Dpar vector is:

            | sin(theta) * cos(phi) |
   Dpar  =  | sin(theta) * sin(phi) |
            |      cos(theta)       |

calc_spheroid_ddi(data, diff_data)

Function for calculating the partial derivatives of the direction cosine dz.

The theta partial derivative of the unit Dpar vector is:

   dDpar      | cos(theta) * cos(phi) |
   ------  =  | cos(theta) * sin(phi) |
   dtheta     |      -sin(theta)      |

The phi partial derivative of the unit Dpar vector is:

   dDpar     | -sin(theta) * sin(phi) |
   -----  =  |  sin(theta) * cos(phi) |
   dphi      |           0            |

O is the orientational parameter set {theta, phi}

calc_spheroid_d2di(data, diff_data)

Function for calculating the second partial derivatives of the direction cosine dz.

The theta-theta second partial derivative of the unit Dpar vector is:

   d2Dpar      | -sin(theta) * cos(phi) |
   -------  =  | -sin(theta) * sin(phi) |
   dtheta2     |      -cos(theta)       |

The theta-phi second partial derivative of the unit Dpar vector is:

     d2Dpar        | -cos(theta) * sin(phi) |
   -----------  =  |  cos(theta) * cos(phi) |
   dtheta.dphi     |           0            |

The phi-phi second partial derivative of the unit Dpar vector is:

   dDpar     | -sin(theta) * cos(phi) |
   -----  =  | -sin(theta) * sin(phi) |
   dphi2     |           0            |

O is the orientational parameter set {theta, phi}

calc_ellipsoid_di(data, diff_data)

Function for calculating the direction cosines dx, dy, and dz.

Direction cosines

dx is the dot product between the unit bond vector and the unit vector along Dx. The equation is:

   dx = XH . Dx

dy is the dot product between the unit bond vector and the unit vector along Dy. The equation is:

   dy = XH . Dy

dz is the dot product between the unit bond vector and the unit vector along Dz. The equation is:

   dz = XH . Dz

Unit vectors

The unit Dx vector is:

          | -sin(alpha) * sin(gamma) + cos(alpha) * cos(beta) * cos(gamma) |
   Dx  =  | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
          |                    cos(alpha) * sin(beta)                      |

The unit Dy vector is:

          | cos(alpha) * sin(gamma) + sin(alpha) * cos(beta) * cos(gamma) |
   Dy  =  | cos(alpha) * cos(gamma) - sin(alpha) * cos(beta) * sin(gamma) |
          |                   sin(alpha) * sin(beta)                      |

The unit Dz vector is:

          | -sin(beta) * cos(gamma) |
   Dz  =  |  sin(beta) * sin(gamma) |
          |        cos(beta)        |

calc_ellipsoid_ddi(data, diff_data)

Function for calculating the partial derivatives of the direction cosines dx, dy, and dz.

Dx gradient

The alpha partial derivative of the unit Dx vector is:

    dDx       | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
   ------  =  | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
   dalpha     |                   -sin(alpha) * sin(beta)                      |

The beta partial derivative of the unit Dx vector is:

    dDx      | -cos(alpha) * sin(beta) * cos(gamma) |
   -----  =  |  cos(alpha) * sin(beta) * sin(gamma) |
   dbeta     |       cos(alpha) * cos(beta)         |

The gamma partial derivative of the unit Dx vector is:

    dDx       | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
   ------  =  |  sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
   dgamma     |                             0                                  |

Dy gradient

The alpha partial derivative of the unit Dy vector is:

    dDy       | -sin(alpha) * sin(gamma) + cos(alpha) * cos(beta) * cos(gamma) |
   ------  =  | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
   dalpha     |                    cos(alpha) * sin(beta)                      |

The beta partial derivative of the unit Dy vector is:

    dDy      | -sin(alpha) * sin(beta) * cos(gamma) |
   -----  =  |  sin(alpha) * sin(beta) * sin(gamma) |
   dbeta     |       sin(alpha) * cos(beta)         |

The gamma partial derivative of the unit Dy vector is:

    dDy       |  cos(alpha) * cos(gamma) - sin(alpha) * cos(beta) * sin(gamma) |
   ------  =  | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
   dgamma     |                             0                                  |

Dz gradient

The alpha partial derivative of the unit Dz vector is:

    dDz       | 0 |
   ------  =  | 0 |
   dalpha     | 0 |

The beta partial derivative of the unit Dz vector is:

    dDz      | -cos(beta) * cos(gamma) |
   -----  =  |  cos(beta) * sin(gamma) |
   dbeta     |        -sin(beta)       |

The gamma partial derivative of the unit Dz vector is:

    dDz       | sin(beta) * sin(gamma) |
   ------  =  | sin(beta) * cos(gamma) |
   dgamma     |           0            |

calc_ellipsoid_d2di(data, diff_data)

Function for calculating the second partial derivatives of the direction cosines dx, dy, dz.

Dx Hessian

The alpha-alpha second partial derivative of the unit Dx vector is:

    d2Dx       | sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
   -------  =  | sin(alpha) * cos(gamma) + cos(alpha) * cos(beta) * sin(gamma) |
   dalpha2     |                  -cos(alpha) * sin(beta)                      |

The alpha-beta second partial derivative of the unit Dx vector is:

       d2Dx         |  sin(alpha) * sin(beta) * cos(gamma) |
   ------------  =  | -sin(alpha) * sin(beta) * sin(gamma) |
   dalpha.dbeta     |      -sin(alpha) * cos(beta)         |

The alpha-gamma second partial derivative of the unit Dx vector is:

       d2Dx          | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
   -------------  =  |  cos(alpha) * sin(gamma) + sin(alpha) * cos(beta) * cos(gamma) |
   dalpha.dgamma     |                             0                                  |

The beta-beta second partial derivative of the unit Dx vector is:

    d2Dx      | -cos(alpha) * cos(beta) * cos(gamma) |
   ------  =  |  cos(alpha) * cos(beta) * sin(gamma) |
   dbeta2     |      -cos(alpha) * sin(beta)         |

The beta-gamma second partial derivative of the unit Dx vector is:

       d2Dx         | cos(alpha) * sin(beta) * sin(gamma) |
   ------------  =  | cos(alpha) * sin(beta) * cos(gamma) |
   dbeta.dgamma     |                 0                   |

The gamma-gamma second partial derivative of the unit Dx vector is:

    d2Dx       | sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
   -------  =  | sin(alpha) * cos(gamma) + cos(alpha) * cos(beta) * sin(gamma) |
   dgamma2     |                            0                                  |

Dy Hessian

The alpha-alpha second partial derivative of the unit Dy vector is:

    d2Dy       | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
   -------  =  | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
   dalpha2     |                   -sin(alpha) * sin(beta)                      |

The alpha-beta second partial derivative of the unit Dy vector is:

       d2Dy         | -cos(alpha) * sin(beta) * cos(gamma) |
   ------------  =  |  cos(alpha) * sin(beta) * sin(gamma) |
   dalpha.dbeta     |       cos(alpha) * cos(beta)         |

The alpha-gamma second partial derivative of the unit Dy vector is:

       d2Dy          | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
   -------------  =  |  sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
   dalpha.dgamma     |                             0                                  |

The beta-beta second partial derivative of the unit Dy vector is:

    d2Dy      | -sin(alpha) * cos(beta) * cos(gamma) |
   ------  =  |  sin(alpha) * cos(beta) * sin(gamma) |
   dbeta2     |      -sin(alpha) * sin(beta)         |

The beta-gamma second partial derivative of the unit Dy vector is:

       d2Dy         | sin(alpha) * sin(beta) * sin(gamma) |
   ------------  =  | sin(alpha) * sin(beta) * cos(gamma) |
   dbeta.dgamma     |                 0                   |

The gamma-gamma second partial derivative of the unit Dy vector is:

    d2Dy       | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
   -------  =  | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
   dgamma2     |                             0                                  |

Dz Hessian

The alpha-alpha second partial derivative of the unit Dz vector is:

    d2Dz       | 0 |
   -------  =  | 0 |
   dalpha2     | 0 |

The alpha-beta second partial derivative of the unit Dz vector is:

       d2Dz         | 0 |
   ------------  =  | 0 |
   dalpha.dbeta     | 0 |

The alpha-gamma second partial derivative of the unit Dz vector is:

        d2Dz         | 0 |
   -------------  =  | 0 |
   dalpha.dgamma     | 0 |

The beta-beta second partial derivative of the unit Dz vector is:

    d2Dz      |  sin(beta) * cos(gamma) |
   ------  =  | -sin(beta) * sin(gamma) |
   dbeta2     |        -cos(beta)       |

The beta-gamma second partial derivative of the unit Dz vector is:

       d2Dz         | cos(beta) * sin(gamma) |
   ------------  =  | cos(beta) * cos(gamma) |
   dbeta.dgamma     |           0            |

The gamma-gamma second partial derivative of the unit Dz vector is:

    d2Dz       |  sin(beta) * cos(gamma) |
   -------  =  | -sin(beta) * sin(gamma) |
   dgamma2     |            0            |