Diffractometer¶

This class provides the transformation from the setting angles of a six circle diffractometer into reciprocal space. There for we use mainly four frames in reciprocal Space: $\theta$ -frame, $\phi$ -frame, $U$ -frame and $(H,K,L)$ -frame.

Lab Frame¶

We define the lab frame $\Sigma$ like in the paper about the four circle diffraction setup (Busing1967), i.e. the $Z_4$ -axis up, the $Y_4$ -axis along the X-ray beam and the $X_4$ -axis forms a right handed orthonganl system with the others. The figure shows the setup with all six angles and a diffrent lab frame from the paper Lohmeier1993 ( $X_6=Z_4$ , $Y_6=Y_4$ , $Z_6=-X_4$ ).

$\bf{a}_4 = V_{46} \bf{a}_6 = \begin{pmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \bf{a}_6$

Rotations and Setting Angles¶

The rotation by $\mu$ along the $+Z_4$ -axis leads to $\Sigma'$ followed by the angles familiar from the four circle setup, i.e. rotation by $\theta$ along the $+X_4'$ -axis leads to $\Sigma''$ ( $\theta$ -frame, $\Sigma^\theta$ ), rotation by $\chi$ along the $+Y_4''$ -axis leads to $\Sigma'''$ , and rotation by $\phi$ along the $+X_4'''$ -axis leads to $\Sigma''''$ ( $\phi$ -frame, $\Sigma^\phi$ ) The detector rotations start in $\Sigma'$ with rotation by $\delta$ along the $+X_4'$ -axis into $\Sigma^*$ , and rotation by $\gamma$ along the $+Z_4^*$ -axis into $\Sigma^{**}$ . Note that in the origanal paper $\mu$ is called $\alpha$ . After this introduction we will drop most times the indices at the axes and refer to the four circle lab frame.

Rotations and frames for six circle geometry
Angle	Axis₄	Axis₆	New Frame
$\mu$	$+Z_4$	$+X_6$	$\Sigma'$
$\theta$	$+X_4'$	$-Z_6'$	$\Sigma''$ ( $\Sigma^\theta$ )
$\chi$	$+Y_4''$	$+Y_6''$	$\Sigma'''$
$\phi$	$+X_4'''$	$-Z_6'''$	$\Sigma''''$ ( $\Sigma^\phi$ )
$\delta$	$+X_4'$	$-Z_6'$	$\Sigma^*$
$\gamma$	$+Z_4^*$	$+X_6^*$	$\Sigma^{**}$

Rotation matrcies¶

Rotation of a vector $\bf{a}$ along the $Z$ -axis by the angle $\alpha$ is discribed by the rotaion matrix $R_Z(\alpha)$

$R_Z(\alpha) &= \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) & 0 \\ \sin(\alpha) & \cos(\alpha) & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ \bf{a}' &= R_Z(\alpha) \bf{a}$

The back transformation is the transposed $R_Z(\alpha)^\mathrm{T}$ or $R_Z(-\alpha)$ , the back rotation. If the coordinate system is rotated by the angle $\alpha$ , $R_Z(-\alpha)$ is the transformation. For completness the other two rotation matrices along the principle axes.

$R_X(\alpha) &= \begin{pmatrix} 1 & -\sin(\alpha) & 0 \\ 0 & \cos(\alpha) & -\sin(\alpha) \\ 0 & \sin(\alpha) & \cos(\alpha) \end{pmatrix} \\ R_Y(\alpha) &= \begin{pmatrix} \cos(\alpha) & 0 & \sin(\alpha) \\ 0 & 1 & 0 \\ -\sin(\alpha) & 0 & \cos(\alpha) \end{pmatrix}$

Calculation of $Q$ ¶

The initial and final (scattered) wavevectors are $\bf{k}_\mathrm{i}$ , $\bf{k}_\mathrm{f}$ and have the same length $k$ (elastic scattering). For all setting angels set to 0, $\bf{k}_\mathrm{i}$ and $\bf{k}_\mathrm{f}$ lay along the $Y$ -axis.

$\bf{k}_i(0) = \bf{k}_f(0) = \begin{pmatrix} 0 \\ k \\ 0 \end{pmatrix}$

To get $\bf{k}_\mathrm{i}''$ , i.e. in $\Sigma_\theta$ , we have to apply the rotation matrix along $Z$ by the angle $-\mu$ and the rotation matrix along $X'$ by the angle $-\theta$ , because $\bf{k}_\mathrm{i} = k_i(0)$ for all angles. To get $\bf{k}_\mathrm{f}''$ we have to considered that $\bf{k}_\mathrm{f}^** = k_f(0)$ for all angles, becuase the measured scattered X-rays hit directly the detector. In a similar sence we need rotations by $+\gamma$ along $Z^*$ and by $-\theta + \delta$ along $X'$ .

$\bf{k}_i'' &= R_X(-\theta) R_Z(-\mu) \bf{k}_i(0) \\ \bf{k}_f'' &= R_X(-\theta + \delta) R_Z(\gamma) \bf{k}_f(0)$

Now we get $\bf{Q}^\theta$ in $\Sigma^\theta$ .

$\bf{Q}^\theta = \bf{Q}'' = \bf{k}_f'' - \bf{k}_i''$

If we take the rotations by $\chi$ and $\phi$ into account we come to $\Sigma^\phi$ .

$\bf{Q}^\phi = R_Z(-\phi) R_Y(-\chi) \bf{Q}^\theta$

The last step is transforming from $\Sigma^\phi_4$ to $\Sigma^\phi_6$ and applying the inverse orientation matrix $UB^{-1}$ from e.g. Spec SIXC to get the $HKL$ -values.

$\bf{Q}^{HKL} = UB^{-1} V_{46}^\mathrm{T} \bf{Q}^\phi$

Diffractometer Class¶

class pyspec.diffractometer.Diffractometer(mode='sixc')¶

Diffractometer class

This class provides various functions to perform calculations to and from the sample and instrument frames

used lab frame: Z up, Y along X-ray beam, X = Y x Z; sample rotations: mu : along +Z -> S’ (mu-frame), theta : along +X’ -> S’’ (theta-frame), chi : along +Y’’ -> S’‘’ (chi-frame), phi : along +X’‘’ -> S’‘’’ (phi-frame); detector rotations: mu : along +Z -> S’ (mu-frame), delta : along +X’ -> S* (delta-frame), gamma : along +Z* -> S** (gamma-frame)

getQCart()¶

Calculate (Qx, Qy, Qz) set in cartesian reciprocal space from (Qx, Qy, Qz) set in theta-frame

still under construction

getQHKL()¶

Calc HKL values from (Qx, Qy, Qz) set in theta-frame with UB-matrix

QHKL = UB^-1 sixcToFourc^T QPhi

getQPhi()¶

Calculate (Qx, Qy, Qz) set in phi-frame from (Qx, Qy, Qz) set in theta-frame

QPhi = rotZ(-phi) rotY(-chi) QTheta

getQTheta()¶: Return transformed coordinates

setAllAngles(angles, mode='deg')¶: Sets angles for calculation. Angles are expected in spec ‘sixc’ order: Delta Theta Chi Phi Mu Gamma

setAngles(delta=None, theta=None, chi=None, phi=None, mu=None, gamma=None, mode='deg')¶: Set the angles for calculation

setEnergy(energy)¶: Set the energy (in eV) for calculations

setLambda(waveLen)¶: Set the wavelength (in Angstroms) for calculations

setUbMatrix(UBmat)¶: Sets the UB matrix

Diffractometer¶

Lab Frame¶

Rotations and Setting Angles¶

Rotation matrcies¶

Calculation of $Q$ ¶

Diffractometer Class¶

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Calculation of $Q$ ¶