DRAFT DICTIONARY CBF/imgCIF Extensions Dictionary Draft version 1.5 for comment

Index

# Category AXIS

Name:
'axis'

Description:

```    Data items in the AXIS category record the information required
to describe the various goniometer, detector, source and other
axes needed to specify a data collection or the axes defining the
coordinate system of an image.

The location of each axis is specified by two vectors: the axis
itself, given by a  unit vector in the direction of the axis, and
an offset to the base of the unit vector.

The vectors defining an axis are referenced to an appropriate
coordinate system.  The axis vector, itself, is a dimensionless
unit vector.  Where meaningful, the offset vector is given in
millimetres.  In coordinate systems not measured in metres,
the offset is not specified and is taken as zero.

The available coordinate systems are:

The imgCIF standard laboratory coordinate system
The direct lattice (fractional atomic coordinates)
The orthogonal Cartesian coordinate system (real space)
The reciprocal lattice
An abstract orthogonal Cartesian coordinate frame

For consistency in this discussion, we call the three coordinate
system axes X, Y and Z.  This is appropriate for the imgCIF
standard laboratory coordinate system, and last two Cartesian
coordinate systems, but for the direct lattice, X corresponds
to a, Y to b and Z to c, while for the reciprocal lattice,
X corresponds to a*, Y to b* and Z to c*.

For purposes of visualization, all the coordinate systems are
taken as right-handed, i.e., using the convention that the extended
thumb of a right hand could point along the first (X) axis, the
straightened pointer finger could point along the second (Y) axis
and the middle finger folded inward could point along the third (Z)
axis.

THE IMGCIF STANDARD LABORATORY COORDINATE SYSTEM

The imgCIF standard laboratory coordinate system is a right-handed
orthogonal coordinate similar to the MOSFLM coordinate system,
but imgCIF puts Z along the X-ray beam, rather than putting X along the
X-ray beam as in MOSFLM.

The vectors for the imgCIF standard laboratory coordinate system
form a right-handed Cartesian coordinate system with its origin
in the sample or specimen.  The origin of the axis system should,
if possible, be defined in terms of mechanically stable axes to be
be both in the sample and in the beam.  If the sample goniometer or other
sample positioner has two axes the intersection of which defines a
unique point at which the sample should be mounted to be bathed
by the beam, that will be the origin of the axis system.  If no such
point is defined, then the midpoint of the line of intersection
between the sample and the center of the beam will define the origin.
For this definition the sample positioning system will be set at
its initial reference position for the experiment.

| Y (to complete right-handed system)
|
|
|
|
|
|________________X
/       principal goniometer axis
/
/
/
/
/Z (to source)

Axis 1 (X): The X-axis is aligned to the mechanical axis pointing from
the sample or specimen along the  principal axis of the goniometer or
sample positioning system if the sample positioning system has an axis
that intersects the origin and which form an angle of more than 22.5
degrees with the beam axis.

Axis 2 (Y): The Y-axis completes an orthogonal right-handed system
defined by the X-axis and the Z-axis (see below).

Axis 3 (Z): The Z-axis is derived from the source axis which goes from
the sample to the source.  The Z-axis is the component of the source axis
in the direction of the source orthogonal to the X-axis in the plane
defined by the X-axis and the source axis.

If the conditions for the X-axis can be met, the coordinate system
will be based on the goniometer or other sample positioning system
and the beam and not on the orientation of the detector, gravity etc.
The vectors necessary to specify all other axes are given by sets of
three components in the order (X, Y, Z).
If the axis involved is a rotation axis, it is right-handed, i.e. as
one views the object to be rotated from the origin (the tail) of the
unit vector, the rotation is clockwise.  If a translation axis is
specified, the direction of the unit vector specifies the sense of
positive translation.

Note:  This choice of coordinate system is similar to but significantly
different from the choice in MOSFLM (Leslie & Powell, 2004).  In MOSFLM,
X is along the X-ray beam (the CBF/imgCIF Z axis) and Z is along the
rotation axis.

In some experimental techniques, there is no goniometer or the principal
axis of the goniometer is at a small acute angle with respect to
the source axis.  In such cases, other reference axes are needed
to define a useful coordinate system.  The order of priority in
defining directions in such cases is to use the detector, then
gravity, then north.

If the X-axis cannot be defined as above, then the
direction (not the origin) of the X-axis should be parallel to the axis
of the primary detector element corresponding to the most rapidly
varying dimension of that detector element's data array, with its
positive sense corresponding to increasing values of the index for
that dimension.  If the detector is such that such a direction cannot
be defined (as with a point detector) or that direction forms an
angle of less than 22.5 degrees with respect to the source axis, then
the X-axis should be chosen so that if the Y-axis is chosen
in the direction of gravity, and the Z-axis is chosen to be along
the source axis, a right-handed orthogonal coordinate system is chosen.
In the case of a vertical source axis, as a last resort, the
X-axis should be chosen to point North.

All rotations are given in degrees and all translations are given in mm.

Axes may be dependent on one another.  The X-axis is the only goniometer
axis the direction of which is strictly connected to the hardware.  All
other axes are specified by the positions they would assume when the
axes upon which they depend are at their zero points.

When specifying detector axes, the axis is given to the beam centre.
The location of the beam centre on the detector should be given in the
DIFFRN_DETECTOR category in distortion-corrected millimetres from
the (0,0) corner of the detector.

It should be noted that many different origins arise in the definition
of an experiment.  In particular, as noted above, it is necessary to
specify the location of the beam centre on the detector in terms
of the origin of the detector, which is, of course, not coincident
with the centre of the sample.

The unit cell, reciprocal cell and crystallographic orthogonal
Cartesian coordinate system are defined by the CELL and the matrices
in the ATOM_SITES category.

THE DIRECT LATTICE (FRACTIONAL COORDINATES)

The direct lattice coordinate system is a system of fractional
coordinates aligned to the crystal, rather than to the laboratory.
This is a natural coordinate system for maps and atomic coordinates.
It is the simplest coordinate system in which to apply symmetry.
The axes are determined by the cell edges, and are not necessarily
othogonal.  This coordinate system is not uniquely defined and
depends on the cell parameters in the CELL category and the
settings chosen to index the crystal.

Molecules in a crystal studied by X-ray diffracraction are organized
into a repeating regular array of unit cells.  Each unit cell is defined
by three vectors, a, b and c.  To quote from Drenth,

"The choice of the unit cell is not unique and therefore, guidelines
have been established for selecting the standard basis vectors and
the origin.  They are based on symmetry and metric considerations:

"(1)  The axial system should be right handed.
(2)  The basis vectors should coincide as much as possible with
directions of highest symmetry."
(3)  The cell taken should be the smallest one that satisfies
condition (2)
(4)  Of all the lattice vectors, none is shorter than a.
(5)  Of those not directed along a, none is shorter than b.
(6)  Of those not lying in the ab plane, none is shorter than c.
(7)  The three angles between the basis vectors a, b and c are
either all acute (<90\%) or all obtuse (>=90\%)."

These rules do not produce a unique result that is stable under
the assumption of experimental errors, and the the resulting cell
may not be primitive.

In this coordinate system, the vector (.5, .5, .5) is in the middle
of the given unit cell.

Grid coordinates are an important variation on fractional coordinates
used when working with maps.  In imgCIF, the conversion from
fractional to grid coordinates is implicit in the array indexing
specified by _array_structure_list.dimension.  Note that this
implicit grid-coordinate scheme is 1-based, not zero-based, i.e.
the origin of the cell for axes along the cell edges with no
specified _array_structure_list_axis.displacement will have
grid coordinates of (1,1,1), i.e. array indices of (1,1,1).

THE ORTHOGONAL CARTESIAN COORDINATE SYSTEM (REAL SPACE)

The orthogonal Cartesian coordinate system is a transformation of
the direct lattice to the actual physical coordinates of atoms in
space.  It is similar to the laboratory coordinate system, but
is anchored to and moves with the crystal, rather than being
schored to the laboratory.  The transformation from fractional
to orthogonal cartesian coordinates is given by the
_atom_sites.Cartn_transf_matrix[i][j]  and
_atom_sites.Cartn_transf_vector[i]
tags.  A common choice for the matrix of the transformation is
given in the 1992 PDB format document

| a      b cos(\g)   c cos(\b)                            |
| 0      b sin(\g)   c (cos(\a) - cos(\b)cos(\g))/sin(\g) |
| 0      0           V/(a b sin(\g))                      |

This is a convenient coordinate system in which to do fitting
of models to maps and in which to understand the chemistry of
a molecule.

THE RECIPROCAL LATTICE

The reciprocal lattice coordinate system is used for diffraction
intensitities.  It is based on the reciprocal cell, the dual of the cell,
in which reciprocal cell edges are derived from direct cell faces:

a* = bc sin(\a)/V  b* = ac sin(\b)/V  c* = ab sin(\g)/V
cos(\a*) = (cos(\b) cos(\g) - cos(\a))/(sin(\b)  sin(\g))
cos(\b*) = (cos(\a) cos(\g) - cos(\b))/(sin(\a) sin(\g))
cos(\g*) = (cos(\a) cos(\b) - cos(\g))/(sin(\a) sin(\b))
V = abc  SQRT(1 - cos(\a)^2^
- cos(\b)^2^
- cos(\g)^2^
+ 2 cos(\a) cos(\b) cos(\g) )

In this form the dimensions of the reciprocal lattice are in reciprocal
\%Angstroms (\%A^-1^).  A dimensionless form can be obtained by
multiplying by the wavelength.  Reflections are commonly indexed against
this coordinate system as (h, k, l) triples.

References:

Drenth, J., "Introduction to basic crystallography." chapter
2.1 in Rossmann, M. G. and Arnold, E. "Crystallography of
biological macromolecules", Volume F of the IUCr's "International
tables for crystallography", Kluwer, Dordrecht 2001, pp 44 -- 63

Leslie, A. G. W. and Powell, H. (2004). MOSFLM v6.11.
MRC Laboratory of Molecular Biology, Hills Road, Cambridge, England.
http://www.CCP4.ac.uk/dist/X-windows/Mosflm/.

Stout, G. H. and Jensen, L. H., "X-ray structure determination",
2nd ed., Wiley, New York, 1989, 453 pp.

__, "PROTEIN DATA BANK ATOMIC COORDINATE AND BIBLIOGRAPHIC ENTRY
FORMAT DESCRIPTION," Brookhaven National Laboratory, February 1992.

```
Examples:

 Example 1 - This example shows the axis specification of the axes of a kappa- geometry goniometer [see Stout, G. H. & Jensen, L. H. (1989). X-ray structure determination. A practical guide, 2nd ed. p. 134. New York: Wiley Interscience]. There are three axes specified, and no offsets. The outermost axis, omega, is pointed along the X axis. The next innermost axis, kappa, is at a 50 degree angle to the X axis, pointed away from the source. The innermost axis, phi, aligns with the X axis when omega and phi are at their zero points. If T-omega, T-kappa and T-phi are the transformation matrices derived from the axis settings, the complete transformation would be: X' = (T-omega) (T-kappa) (T-phi) X ``` loop_ _axis.id _axis.type _axis.equipment _axis.depends_on _axis.vector[1] _axis.vector[2] _axis.vector[3] omega rotation goniometer . 1 0 0 kappa rotation goniometer omega -.64279 0 -.76604 phi rotation goniometer kappa 1 0 0 ```

 Example 2 - This example shows the axis specification of the axes of a detector, source and gravity. The order has been changed as a reminder that the ordering of presentation of tokens is not significant. The centre of rotation of the detector has been taken to be 68 millimetres in the direction away from the source. ``` loop_ _axis.id _axis.type _axis.equipment _axis.depends_on _axis.vector[1] _axis.vector[2] _axis.vector[3] _axis.offset[1] _axis.offset[2] _axis.offset[3] source . source . 0 0 1 . . . gravity . gravity . 0 -1 0 . . . tranz translation detector rotz 0 0 1 0 0 -68 twotheta rotation detector . 1 0 0 . . . roty rotation detector twotheta 0 1 0 0 0 -68 rotz rotation detector roty 0 0 1 0 0 -68 ```

 Example 3 - This example show the axis specification of the axes for a map, using fractional coordinates. Each cell edge has been divided into a grid of 50 divisions in the ARRAY_STRUCTURE_LIST_AXIS category. The map is using only the first octant of the grid in the ARRAY_STRUCTURE_LIST category. The fastest changing axis is the gris along A, then along B, and the slowest is along C. The map sampling is being done in the middle of each grid division ``` loop_ _axis.id _axis.system _axis.vector[1] _axis.vector[2] _axis.vector[3] CELL_A_AXIS fractional 1 0 0 CELL_B_AXIS fractional 0 1 0 CELL_C_AXIS fractional 0 0 1 loop_ _array_structure_list.array_id _array_structure_list.index _array_structure_list.dimension _array_structure_list.precedence _array_structure_list.direction _array_structure_list.axis_id MAP 1 25 1 increasing CELL_A_AXIS MAP 1 25 2 increasing CELL_B_AXIS MAP 1 25 3 increasing CELL_C_AXIS loop_ _array_structure_list_axis.axis_id _array_structure_list_axis.fract_displacement _array_structure_list_axis.fract_displacement_increment CELL_A_AXIS 0.01 0.02 CELL_B_AXIS 0.01 0.02 CELL_C_AXIS 0.01 0.02 ```

 Example 4 - This example show the axis specification of the axes for a map, this time as orthogonal \%Angstroms, using the same coordinate system as for the atomic coordinates. The map is sampling every 1.5 \%Angstroms (1.5e-7 millimeters) in a map segment 37.5 \%Angstroms on a side. ``` loop_ _axis.id _axis.system _axis.vector[1] _axis.vector[2] _axis.vector[3] X orthogonal 1 0 0 Y orthogonal 0 1 0 Z orthogonal 0 0 1 loop_ _array_structure_list.array_id _array_structure_list.index _array_structure_list.dimension _array_structure_list.precedence _array_structure_list.direction _array_structure_list.axis_id MAP 1 25 1 increasing X MAP 2 25 2 increasing Y MAP 3 25 3 increasing Z loop_ _array_structure_list_axis.axis_id _array_structure_list_axis.displacement _array_structure_list_axis.displacement_increment X 7.5e-8 1.5e-7 Y 7.5e-8 1.5e-7 Z 7.5e-8 1.5e-7 ```

Category groups:
inclusive_group
axis_group
diffrn_group
Category keys:
_axis.id
_axis.equipment

Mandatory category: no