# Tutorial for `musrfit`

¶

## Single-histogram-fit tutorial¶

The μSR-data-analysis process using musrfit is based on so-called msr files. These files contain all
information needed for the analysis such as names of the data files, a theory function, fit and plot parameters,
and so on. It is the idea of this page to explain the basic use of an msr file and the different programs
of the `musrfit`

suite using the example of a single-histogram fit to time-differential transverse-field
μSR data. For a complete description of all options please refer to the manual.

The example deals with a diamagnetic sample that has been measured in an applied field of approximately 150 G in the GPS spectrometer at PSI using the “transverse geometry”. In this geometry the muon spin is rotated about 50° up and the field is applied parallel to the muon momentum. Hence, the relevant positron counters (to detect the spin precession) are placed above (histogram 3), below (histogram 4) and right of (histogram 5) the sample. The run numbers start from 3110.

To analyze these data (in a simple way) one starts out from the msr file `3110_tutorial.msr`

provided together
with the source-code distribution in the sub-directory `doc/examples/`

. This ASCII file can be edited using any
text editor. For convenience the editor `musredit`

is provided which offer some msr-file-specific functionalities
and additionally serve as front ends to the underlying programs. In the following it is assumed that the file is
open within `musredit`

.

### The msr file¶

The msr file itself is divided into different blocks; a full description of the format can be found here.
In the file `3110_tutorial.msr`

these blocks are successively:

```
sample XYZ
```

A descriptive title of the file.

```
FITPARAMETER
# No Name Value Step Pos_Error Boundaries
1 NormUp 4500 5 none 0 none
2 BgUp 200 1 none 0 none
3 PhaseUp 15 1 none
4 NormDown 4500 5 none 0 none
5 BgDown 200 1 none 0 none
6 PhaseDown 195 1 none
7 NormRight 600 5 none 0 none
8 BgRight 40 1 none 0 none
9 PhaseRight 285 1 none
10 AsymSig1 0.17 0.01 none 0 0.33
11 RateSig1 2.5 0.1 none 0 none
12 FieldSig1 100 1 none
13 AsymSig2 0.02 0.01 none 0 0.33
14 RateSig2 0.5 0.1 none
15 FieldSig2 150 1 none
```

The list of parameters used in the theory function to describe the set of data.
Each parameter has a number, a name, an initial value and an initial step (for
the fitting process). If a parameter should be fixed, the initial step is set
to 0. After a fit (see below), this block contains the determined parameter values
and uncertainties (in the `Step`

column). If asymmetric errors are determined, these
will be listed in the `Step`

column (negative) and the `Pos_Error`

column (positive).
Optionally, lower and upper boundaries for the parameters can be specified as
shown above.

The meaning of the parameters above is explained in the following:

Since a single-histogram fit should be done, some
histogram-specific parameters are needed. These are a normalization constant (parameter 1),
a parameter describing the background of uncorrelated events (parameter 2) and the initial
phase of the spin precession with respect to the detector (parameter 3). As stated above,
there are three histograms containing useful information (“Up”, “Down”, “Right”); hence,
this set of parameters has to be present for each of them (parameters 1–9).
The remaining parameters are used to model the decay asymmetry which is assumed to be equal
for all histograms. In this example one has two signals (*e.g.* from the sample and the
sample holder), each with an amplitude (parameters 10 and 13), a depolarization rate
(parameters 11 and 14) and a mean field (here given in Gauss, parameters 12 and 15).

```
THEORY
asymmetry 10
simplExpo 11 (rate)
TFieldCos map1 fun1 (phase frequency)
+
asymmetry 13
simpleGss 14 (rate)
TFieldCos map1 fun2 (phase frequency)
```

The THEORY block is used to define a fit-parameter-dependent theory function used to model
the decay asymmetry. Different predefined and user-defined
functions can be combined here. Theory lines following each other are *multiplied* and the **+** sign
is used to add different (here: two) signal contributions. The numbers are the parameter numbers
according to the `FITPARAMETER block`

. `map`

and `fun`

are used to refer to
histogram-dependent parameters and to interrelate fit parameters, respectively (see below).

```
FUNCTIONS
fun1 = gamma_mu * par12
fun2 = gamma_mu * par15
```

Here functions interrelating different fit parameters and predefined constants can be defined for
the use in the `THEORY block`

. In the example, the functions are used to calculate the
muon-spin-precession frequencies for the given fields [\(\nu = \gamma_\mu B / (2\pi)\)]. One
function is used for each signal. Altogether, the theory function defined above is
\(A(t) = p_{10} e^{-p_{11} t} \cos(\varphi_i \pi/180 + \gamma_\mu p_{12} t) + p_{13} e^{-(p_{14} t)^2/2} \cos(\varphi_i \pi/180 + \gamma_\mu p_{15} t)\), where the *p* are the parameters in the `FITPARAMETER block`

and \(\varphi_i = p_3, p_6, p_9\)
depending on the histogram as shall be seen later.

```
RUN data/deltat_pta_gps_3110 PIM3 PSI PSI-BIN (name beamline institute data-file-format)
fittype 0 (single histogram fit)
norm 1
backgr.fit 2
map 3 0 0 0 0 0 0 0 0 0
forward 3
data 1 8000
t0 1
fit 0 4.9
packing 20
```

The RUN blocks are used to collect information on the data to be analyzed. Specifically, these are:

RUN data/deltat_pta_gps_3110 PIM3 PSI PSI-BIN (name beamline institute data-file-format)The path to the data file and the file format (NEXUS, ROOT-NPP, ROOT-PPC, PSI-BIN, PSI-MDU, WKM, MUD, MDU-ASCII).

fittype 0 (single histogram fit)the fit type (0 = single-histogram fit)

norm 1the number of the fit parameter representing the normalization constant

backgr.fit 2the number of the fit parameter representing the background

map 3 0 0 0 0 0 0 0 0 0the definition of the

mapsused in the`THEORY block`

— RUN-block-specific parameters are given here; in this example,map1is substituted byparameter 3in the`THEORY block`

for this RUN block andmap2,map3, and so on are undefined.forward 3the histogram number; in this example 3 corresponds to the histogram of the “Up” positron counter

data 1 8000start and end bins of the range containing useful data in the histogram (to be adjusted,

e.g.by using`musrt0`

, see below)t0 1histogram bin corresponding to the time zero (muon implantation time) (to be adjusted,

e.g.by`musrt0`

, see below)fit 0 4.9start and end times (in μs) defining the fit range.

In case the fit range exceeds the range of useful data (specified above using the data tag), eventually this data range is used as fit range.packing 20the packing of the histograms (in histogram bins)

Since the data of three histograms is to be analyzed, the file contains not only one but three RUN blocks — each defining the histogram-specific information following the example given above.

```
COMMANDS
MINIMIZE
MINOS
SAVE
```

In the `COMMANDS block`

, a sequence of operations that should be performed is defined.
Here, the requested operations are the minimization of \(\chi^2\) (MINIMIZE), the
calculation of *asymmetric errors* (MINOS) as well as saving the found parameter values
and uncertainties to the msr file (SAVE). A full description of the possible commands can
be found here.

```
FOURIER
units Gauss # units either 'Gauss', 'MHz', or 'Mc/s'
fourier_power 10
apodization WEAK # NONE, WEAK, MEDIUM, STRONG
plot POWER # REAL, IMAG, REAL_AND_IMAG, POWER, PHASE
phase par3
range 0.0 600.0
```

The `FOURIER block`

is used to define basic settings for the Fourier transform available
in musrview. These are:

units Gauss # units either 'Gauss', 'MHz', or 'Mc/s'the units of the Fourier domain

fourier_power 10the number of data points used for the discrete transform, here \(2^{10}=1024\).

apodization WEAK # NONE, WEAK, MEDIUM, STRONGthe apodization to be used

plot POWER # REAL, IMAG, REAL_AND_IMAG, POWER, PHASEwhat should be plotted (real part, imaginary part, and so on)

phase par3the initial phase of the input data is given here in degrees. Optionally, a phase parameter from the

`FITPARAMETER block`

can be given, herepar3takes the value ofparameter 3.range 0.0 600.0the start and end points of the range of the Fourier transform in the units specified above

```
PLOT 0 (single histo plot)
runs 1 2
range 0 4 -0.2 0.2
```

The `PLOT block`

defines which data (corresponding to the given RUN blocks) is plotted when
musrview is called. In the given example, a canvas would be drawn containing
the (life-time-corrected) data of the first two `RUN blocks`

(“Up” and “Down” positron counters).
The abscissa would range from 0 to 4 μs, the axis of ordinates from -0.2 to 0.2.
It is possible to define more than one `PLOT block`

. Each `PLOT block`

corresponds to a
separate canvas. Therefore, the second `PLOT block`

in the file

```
PLOT 0 (single histo plot)
runs 3
use_fit_ranges
```

produces an additional window containing the (not life-time-corrected) data of the third `RUN block`

(“Right” positron counter). The abscissa ranges from 0 to 4.9 μs (according to the fit range given
in the `RUN block`

).

```
STATISTIC --- 2011-07-09 10:58:44
chisq = 1348.1764, NDF = 1146, chisq/NDF = 1.176419
```

This block is the last block of a msr file. It contains some information on the fit: the date and time as well as the absolute and normalized values of \(\chi^2\) and the number of degrees of freedom in the fit. These information only have a meaning if the fitting procedure has been executed at least once and the fit has converged!

### Determine t0 and the data range using musrt0¶

Before the given model can be fitted to the data, the data ranges and time zeros of the different
`RUN blocks`

have to be set correctly. This can be achieved using musrt0.
Starting this program by selecting the `musrt0`

button in `musredit`

or calling from a terminal

```
$ musrt0 3110_tutorial.msr
```

opens an interactive window plotting the data of the first RUN block:

The green line (at bin 1) represents time zero, the blue lines the start (at bin 1) and end (at bin 8000)
of the data range and the optional red lines give the limits of the background range (if the background
shall not be determined by the fit). All lines can be either dragged to reasonable locations or set by
pressing the corresponding keyboard shortcuts. In the example the time zero can be set by pressing **T**
(which moves the green line to the bin containing the maximum number of counts), the start of the data
range should be set to about five bins later by zooming into the graph around bin 250 and pressing **d**
when the cursor is found at a suitable location:

After all lines have been adjusted for the first histogram, one can go on to the second by pressing **q**.
This procedure is repeated until all ranges have been set correctly. When finished with the last histogram
the new t0 and data-range values will be written to the msr file. A full description of `musrt0`

can be
found here.

### Fit the model to the data using `musrfit`

¶

Now that the basics of the msr file have been introduced and the necessary adjustments have been done one
can proceed with fitting the specified model to the data. This is done by selecting the `musrfit`

icon
in `musredit`

or calling from a terminal

```
$ musrfit 3110_tutorial.msr
```

After the fitting procedure has finished the msr file is updated and contains the newly determined values of the fit parameters.

### View the data and the fit using `musrview`

¶

The data and the corresponding fit can be visualized by choosing the `musrview`

icon in `musredit`

or
calling from a terminal

```
$ musrview 3110_tutorial.msr
```

This creates the windows according to the given PLOT blocks in the msr file:

The basic features of `musrview`

can be found in the `Musrfit menu`

. They include:

- export the shown data and the model to an ASCII file with comma-separated values
- presenting the difference between the shown data and the model (keyboard shortcut:
**d**) - calculate and show the Fourier transform of the shown data (keyboard shortcut:
**f**),*e.g.*for the asymmetry of the “Up” and “Down” detectors:

### Process multiple msr files using `msr2data`

¶

This short introduction is concluded by calling attention to the program msr2data.
As the name indicates the main purpose of this program is to extract the `FITPARAMETER`

block
information from multiple msr files with the same structure and to summarize them in a single
ASCII file (either in `TRIUMF DB`

format or simple columns). However, as described in detail
in the manual, it is not only possible to collect data from msr files but also to create new
msr files from a template and to process these files using `musrfit`

on the fly.

Assuming the successive runs **3111** through **3114** can be analyzed using the same model introduced
above for run **3110** (*e.g.* in the case one has done measurements at different temperatures) one can
use the file `3110_tutorial.msr`

as template, generate the files `3111_tutorial.msr`

through
`3114_tutorial.msr`

with the corresponding updates in the RUN blocks, call for each one `musrfit`

and finally summarize the FITPARAMETER block information of all processed files in an ASCII output
file `tutorial-T-dependence.dat`

(simple columns). If further information on the temperature or the
applied field is available in the data files these will be included as well in the resulting ASCII file.
From the terminal this would be done as follows:

```
$ msr2data 3110 3114 _tutorial fit-3110 -o tutorial-T-dependence.dat data
```

To achieve the same in `musredit`

(in this snapshot in the *dark theme*) one selects the `msr2data`

icon and fills the form accordingly:

## Asymmtery-fit tutorial¶

to be written yet.