` is expressed as :ref:`scattering vector magnitude` (:math:`|Q|`) or :ref:`scattering vector` (:math:`\vec{Q}`) Some analyses may require additional information such as the estimation of experimental uncertainties, the wavelength and type of the radiation probe, or the instrumental resolution. These should be provided, where possible. Note that, for example, reduced SAS data does not *require* a number representing the distance from sample to detector as this common instrument-specific term has already been factored into the data reduction process. .. index:: ! Q .. _Q: Definition of :math:`Q` ========================= :math:`Q` may be represented either as the magnitude of the scattering vector, :math:`|Q|` or by the three-dimensional scattering vector :math:`\vec{Q}`. When we write :math:`Q`, we may refer to either or both of :math:`|Q|` or :math:`\vec{Q}`, depending on the context. .. _Q scalar: Q vector magnitude: :math:`|Q|=(4 \pi / \lambda) \sin(\theta)` ------------------------------------------------------------------ where :math:`\lambda` is the wavelength of the radiation, and :math:`2\theta` is the angle through which the detected radiation has been scattered. This is a one-dimensional reduction of the general case below. .. _Q geometry: .. figure:: graphics/Q-geometry.jpg :alt: Q geometry :height: 200 px definition of :math:`|Q|` geometry for small-angle scattering .. _Q vector: Q vector: :math:`\vec{Q}=\vec{k'}-\vec{k}` ---------------------------------------------- where :math:`\vec{k}` is the wave vector of the incident radiation and :math:`\vec{k'}` is the wave vector of the scattered radiation. Here, :math:`\vec{k}` is a vector of magnitude :math:`2\pi/\lambda` that points along the trajectory of the radiation. .. _Q vector geometry: .. figure:: graphics/q-vector.png :alt: Q vector geometry :height: 400 px definition of :math:`\vec{Q}` geometry for small-angle scattering [#]_ .. [#] A hearty nod for this graphic is given to the guide: **neutron scattering: A Primer**, by Roger Pynn (LANSCE), published in the Summer 1990 edition of *Los Alamos Science*. .. index:: ! I .. _I: Definition of Intensity: :math:`I` ========================================== The intensity may be represented in one of these forms: **absolute units**: :math:`d\Sigma/d\Omega(Q)` differential cross-section per unit volume per unit solid angle (typical units: 1/cm/sr) **absolute units**: :math:`d\sigma/d\Omega(Q)` differential cross-section per unit atom per unit solid angle (typical units: cm^2) **arbitrary units**: :math:`I(Q)` usually a ratio of two detectors but units are meaningless (typical units: a.u.) This presents a few problems for analysis software to sort out when reading the data. Fortunately, it is possible to analyze the *units* to determine which type of intensity is being reported and make choices at the time the file is read. But this is an area for consideration and possible improvement. One problem arises with software that automatically converts data into some canonical units used by that software. The software should not convert units between these different types of intensity indiscriminately. .. index:: I(Q) A second problem is that when arbitrary units are used, then the set of possible analytical results is restricted. With such units, no meaningful volume fraction or number density can be determined directly from :math:`I(Q)`. In some cases, it is possible to apply a factor to convert the arbitrary units to an absolute scale. This should be considered as a possibility of the analysis process. .. index:: ! coordinate axes .. _coordinate axes: Coordinate Axes =========================== The canSAS standard assumes a right-hand rule coordinate system, [#]_, [#]_, [#]_ consistent with a variety of software packages and data formats. [#]_ :z: :math:`z` is along the trajectory of the incident radiation (positive value in the direction towards the detector) :x: :math:`x` is orthogonal to :math:`z` in the horizontal plane (positive values increase to the right when viewed towards the incoming radiation) :y: :math:`y` is orthogonal to :math:`z` and :math:`x` in the vertical plane (positive values increase upwards) .. [#] http://en.wikipedia.org/wiki/Coordinate_system#Cartesian_coordinate_system .. [#] http://en.wikipedia.org/wiki/Polar_coordinate_system .. [#] http://en.wikipedia.org/wiki/Spherical_coordinate_system .. [#] http://www.nexusformat.org/Coordinate_Systems .. _Coordinate Axes geometry: .. figure:: graphics/translation-orientation-geometry-2.png :alt: Coordinate Axes :height: 400 px definition of the coordinate axes for translations and rotations .. index:: ! orientation roll pitch yaw .. _orientation: Orientation ----------- Orientation angles [#]_ in the canSAS standard describe single-axis rotations. Rotations about multiple axes require more information. The rotations are described in terms of roll, pitch, and yaw [#]_ : :**roll**: a rotation about the :math:`z` axis, where zero rotation is along the positive :math:`x` axis :**pitch**: a rotation about the :math:`x` axis, where zero rotation is along the positive :math:`z` axis :**yaw**: a rotation about the :math:`y` axis, where zero rotation is along the positive :math:`z` axis .. [#] http://en.wikipedia.org/wiki/Orientation_%28geometry%29 .. [#] http://en.wikipedia.org/wiki/Tait-Bryan_angles#Aircraft_attitudes