Parameters and Variables

Parameters

In LSQ a parameter is something which it is sensible to try to adjust. It represents some physical thing like ``the x coordinate of the 3rd atom'', ``an overall temperature factor'' or ``the scale factor for all observations coded number 2''.

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.

Basic Variables

It is often the case that parameters are related, either inherently by the symmetry of the problem, or by a strict constraint applied by the user. If two variables are related, they must not both be refined by LSQ; the process expects the shifts to be independent. Every constraint or relation reduces by 1 the number of variables to be refined.

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 $n_p$ parameters, $n_f$ of which are fixed and $n_v$ varied,

$$n_p = n_f + n_v$$

and if, of those $n_v$, $n_b$ are basic and $n_r$ are redundant,

$$n_v = n_b + n_r$$

Strict Constraints

The existence of $n_r$ redundant variables implies that there are $n_r$ strict constraints on the problem. Derivatives are calculated (and shifts are required) for all variables, not just the basics.

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 $(x,x,x)$ will give rise to one basic variable, $x$, with two redundant variables $y$ and $z$ 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:

$$a_1\Delta p_1 = a_2\Delta p_2$$

that is, constant $a_1$ times the shift in parameter 1 = constant $a_2$ times the shift in parameter 2.

Type 2 constraints involve a linear combination of parameters, with constant coefficients, thus:

$$a_1\Delta p_1 + a_2\Delta p_2 + a_3\Delta p3 + \ldots \mbox{etc}\ = 0$$

Note that the way these expressions are written involves differentiating the original constraint. For example, if a type 2 constraint is needed to fix the sum of three parameters to be 3, the constraint

$$p_1 + p_2 + p_3 = 3\quad\mbox{becomes}\quad \Delta p_1 + \Delta p_2 + \Delta p_3 = 0$$

Other more complicated types of constraint could be added if needed.

Parameter Names

The need to fix, vary or relate parameters implies the need to call these parameters by name. The CCSL LSQ facilities allow each problem to be set up individually. Names for the parameters of a problem are given in its main program; the user needs to know what these names are in order to refer to the parameters in his Crystal Data.

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 ).

Families of Parameters

Structure parameters such as those introduced in the previous section, being a commonly occurring group in LSQ, are said in CCSL to belong to family 2 of the parameters. Family 1 contains all other parameters in the simple LSQ applications we are considering here. Other families are used in PR, and in other LSQ programs.

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.