# User’s Guide¶

Refl1D is a complex piece of software hiding some simple mathematics. The reflectivity of a sample is a simple function of its optical transform matrix \(M\). By slicing the sample in uniform layers, each of which has a transfer matrix \(M_i\), we can estimate the transfer matrix for a depth-varying sample using \(M=\prod M_i\). We can adjust the properties of the individual layers until the measured reflectivity best matches the calculated reflectivty.

The complexity comes from multiple sources:

Determining depth structure from reflectivity is an inverse problem requiring a search through a landscape with multiple minima, whose global minimum is small and often in an unpromising region.

The solution is not unique: multiple minima may be equally valid solutions to the inversion problem.

The measurement is sensitive to nuisance parameters such as sample alignment. That means the analysis program must include data reduction steps, making data handling complicated.

The models are complex. Since the ideal profile is not unique and is difficult to locate, we often constrain our search to feasible physical models to limit the search space, and to account for information from other sources.

The reflectivity is dependent on the type of radiation used to probe the sample and even its energy.

Model scripts associate a sample description with data and fitting options to define the system you wish to refine.

The adjustable values in each component of the system are defined by

`Parameter`

objects. When you set the range on a parameter, the system will be able to automatically adjust the value in order to find the best match between theory and data.

Data is loaded from instrument specific file formats into a generic

`Probe`

. The probe object manages the data view and by extension, the view of the theory. The probe object also knows the measurement resolution, and controls the set of theory points that must be evaluated in order to computed the expected value at each point.

The strength of the interaction can be represented either in terms of their scattering length density using

`SLD`

, or by their chemical formula using`Material`

, with scattering length density computed from the information in the probe.`Mixture`

can be used to make a composite material whose parts vary be mass or by volume.

Materials are composed into samples, usually as a

`Stack`

of`Slabs`

layers, but more specific profiles such as`PolymerBrush`

are available. Freeform sections of the profile can be described using`FreeLayer`

, allowing arbitrary scattering length density profiles within the layer, or`FreeInterface`

allowing arbitrary transitions from one SLD to another. New layer types can be defined by subclassing`Layer`

.

Sample descriptions and data sets are combined into an

`Experiment`

object, allowing the program to compute the expected reflectivity from the sample and the probability that reflectivity measured could have come from that sample. For complex cases, where the sample varies on a length scale larger than the coherence length of the probe, you may need to model your measurement with a`CompositeExperiment`

.

One or more experiments can be combined into a

`FitProblem`

. This is then given to one of the many fitters, such as`PTFit`

, which adjust the varying parameters, trying to find the best fit. PTFit can also be used for Bayesian analysis in order to estimate the confidence in which the parameter values are known.