StepSize Algorithm for BonseHart UltraSmallAngle Scattering Instruments
Pete R. Jemian
2004 July 22
In USAS experiments, it is desired to stepscan the
angle of the analyzer crystals, , in one scan
through the entire angular range of interest, starting at
an initial position of
and taking positions ending with
.
In this case, the step size would be:

(1) 
Unless the angular range is quite small, it is likely
that the of Eq. 1 will be larger
than the angular width of the Bragg reflection optics
and the scan using this step size will fail to sample
the intensity about the angular center, ,
of the rocking curve.
Alternatively, it can be inconvenient to span the entire angular range
with a constant step size
where
is the smallest step size to
use while crossing the rocking curve peak of the Bragg
reflection optics. Often, experimenters will thus divide
the angular range into smaller intervals with different constant
step sizes for each interval.
Ideally, a scan will employ a continuouslyvarying
step size that will take constantsize small steps across
the central part of the Bragg reflection and approximately
proportionately larger steps as the angular distance from
the Bragg reflection increases. One description of such
an equation for the step size algorithm is
that used in the USAXS
instrument.

(2) 
In practice, the minimum step size,
,
is chosen based on the smallest step of the rotary stage.
Also,
.
For the case , equallogspaced steps are taken in the
tails of the rocking curve while constant sized steps are taken
in the vicinity of . As increases, larger steps
are taken in the tails of the rocking curve (thus more points are
packed towards ). Typically, gives good results.
The parameter is adjusted minimizing the error, ,
in arriving at the final step, :

(3) 
so that
.
In summary, is determined from the userspecified terms
of
, ,
,
, and using Eq. 3. Then, steps
are generated as needed according to Eq. 2.
Pete R. Jemian
20040723
