The following quantities are calculated by SCALE1. Note that G(j), the reciprocal of the scale factor K(j), is the quantity actually refined.

1. The residual minimized in the least squares refinement is given as

W(ijk) [I(ijk) - G(j)*Y(i)]²,

where i specifies a particular Miller index hkl, j specifies a scale group number and k specifies a particular observation.

W(ijk) =1/σ²I(ijk)

I(ijk) = measured intensity(rel)

G(j) = 1/K(j)

K(j) = scale for group j.

2. The mean intensity for reflection i is given by

summed over j,k.

3. The value used for G(j) is that from the previous cycle as given by

G(j) = ΣW(ijk) I(ijk) Y(i) / ΣW(ijk) Y(i)² summed over i,k.

4. The scaling R factor reported by SCALE1 is given in

Σ W(ijk) [I(ijk)-G(j) Y(i)]²

= ---------------------- summed over i,k.

Σ W(ijk) I(ijk)²

The overall R is obtained by summing all three indices.

Refining G(j), the inverse of the scale, makes this a linear least squares problem. Although this causes the process to require several cycles for convergence, it does make it insensitive to the initial values of the scales. Convergence will occur in relatively few cycles even when 1.0 is used for the initial vaues of all scale factors for a problem in which the final values range from 0.5 to 2.0. Linear independence is assured by the requirement that the mean of the scales stay constant. The G(j) are internal to the program, all data are read and stored in terms of the K(j).

The number of observations that can be processed is limited only by the amount of memory that can be made available to the process.

All of the observations for a reflection are read in (along with any corresponding Friedel-related observations, if merging is being done) and a number of tests are applied to each to see, first, if the reflection is useable and, second, which observations, if any, can contribute to the scaling process. These tests are:

Per Observation

1. Is the group number one of those whose scale factor is being sought?

2. Is the(rel) within the requested range?

Per Reflection

1. Is it inside the specified sinθ/λ limits?

2. Does it have more than one replicate?

3. Does it have replicates from more than one scale group?

If the answers to all of the above questions are yes, then the reflection will contribute some or all of its observations to the normal equations.

Reads multiple observation data from the input archive bdf

Writes new scale factors to the output archive bdf

title PARENT P450 CAD4 DATA (4 SETS)

SCALE1 0 5 1 1 60 59

In this example, a scale will be refined for every scale group whose group number appears in logical record lrexpl: of the input bdf.

Monahan, J.E., Schiffer, M. and Schiffer, J.P. 1967. On the scaling of X-Ray Photographs. Acta Cryst. 22, 322.

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