A parameter is also something with respect to which the function may be differentiated. Differentiation may be analytic, if it is possible (and sensible) to write down such derivatives algebraically. It may also be numerical in awkward cases, using a simple approximation to the derivative involving function values.
For any particular run of a LSQ refinement program, it is unlikely that the user will want to vary every parameter. The subset of parameters which are actually to be varied (i.e. those for which shifts are required) we call variables .
Parameters are thus either fixed or varied . For a fixed parameter, there is no need for a derivative; but for all parameters to be varied, derivatives will be needed.
The subset of variables for which shifts are actually calculated we call basic variables , or basics for short.
Those variables which are not basic we call redundant .
Thus if a problem has parameters,
of
which are fixed and
varied,
Those constraints which must be imposed because of the crystallographic
symmetry are generated automatically by CCSL, and need not be provided by
the user. For example, the special position will give rise to
one basic variable,
, with two redundant variables
and
related to
it by two constraints, but the user will not need to tell the system
this.
The user may impose additional strict constraints. Two constraint types
are at present available. The simpler is type 1:
Type 2 constraints involve a linear combination of parameters, with
constant coefficients, thus:
Examples of names used in various standard main programs are:
Na2 X Ca B11 SCAL 2 P23 ITF A*
Most names have two parts, which we call the genus and species names; the
genus names above are Na2, Ca, SCAL, P23 , with corresponding
species names X, B11, 2 and ITF . The name A*
is simply a species name.
In LSQ programs which involve structure factors,
those structure parameters which belong to a particular named atom
have the atom name as genus name, and one of:
X Y Z B11 B12 B13 B22 B23 B33 ITF SITE SCAT
as species name.
(SCAT means the scattering factor, when this is represented
by one number and it is sensible to refine it).
MAGLSQ has the additional species names: MU MU1 THET THE1 PHI PHI1 PSI1 PSI2 PSI3 PSI4 .
When LSQ programs refer to parameters in output for shifts, correlations etc, they use these parameter names.
More details about parameter names are to be found in Chapter 3 under L FIX cards. Note the availability of words like XYZ to mean ``all three x,y,z coordinates", CELL to mean ``all six cell parameters", etc. One may also use the word ALL to mean ``all the members of this genus" (as in ALL Na3 for ``all the parameters of atom Na3") or ``all the genera with this species" (as in ALL X , or ALL SITE , or even ALL XYZ ).
Family 1, genus 1 is treated by CCSL as special. It contains the parameters for which the species name by itself is enough, like A* above; another example is an overall temperature factor for a structure, known as TFAC .
Genera 2, 3 etc of family 1 may in general be given any genus name. Within such a genus the species name could be simply an integer, or species could have individual names. This choice is for the writer of the main program. In SFLSQ there are two genera in family 1, the second being named SCAL with species numbered 1,2 etc.
Unless the user is writing or modifying a main program, he need not be concerned with the detailed mechanism for numbering parameters. He only needs to know what names he is allowed to use on his Crystal Data for the programs he runs.