Peaks and diffraction

The background
The background curve
Subtracting the background
Background fraction
Background variation
The Background Points table
Diffraction angles: Theta, 2 Theta and sin(Theta)/lambda
Wavelength
Sample diffraction profiles
A first fit of peaks
Limits on the peaks, the backgrounds and your own formulas
The Residual Spectrum
User-defined curves
Writing the formula of a user-defined curve
Parameter roles and starting values
Selecting and deleting user-defined curves

The background

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

The background is the slowly varying level under the peaks; the program can fit it as a curve of the model, or mark it with background points and subtract it.

There are three ways to deal with it, and the first leaves the profile as it was measured: a background curve in the model (Model > Background > Curve), fitted with the peaks; the quadratic that Model > Background > Enable Variation adds inside a fit; or subtracting it from the profile (Model > Background > Subtract). Fit > Automatically uses the first.

Background points are points of the profile that lie on the background rather than on a peak. Once they are marked, the background they describe can be subtracted from the profile. The Model > Background > Points submenu sets them:

Model > Background > Points > Start Manual Selection, shown as Start Visual Selection, or the Pick button under Background - pick background points on the chart, as many as you like.
Model > Background > Points > Compute Automatically, or Auto - enters the same picking mode and replaces the background points with ones the program proposes. It starts from the lowest point of the profile and walks outwards in both directions, each step taking the lowest point not below the last one taken. You can then add or remove points by clicking.
Model > Background > Points > Remove All, or Clear - discards the background points.

The points are listed in the Background Points table (Position and Amplitude) and drawn as the Background points series. The small toolbar above that table has buttons for computing the points, subtracting automatically, subtracting by the selected points and removing them all. Typing into the table does not change the points.

The Tools tab carries the same commands as buttons under Background: Pick, Auto and Clear run Model > Background > Points > Start Manual Selection, Compute Automatically and Remove All; Subtract runs Model > Background > Subtract > Automatically. The Pick button stays pressed while its picking mode runs, and the menu entry reads Stop Visual Selection.

The Background submenu and the Background buttons are greyed until a profile is loaded, and while a fit runs.

Limitations

The Subtract button always subtracts automatically, ignoring the background points you picked. To subtract through your own points use Model > Background > Subtract > By Selected Points.
The automatic proposal assumes a background that is lowest in one place and rises on either side of it, as in a neutron diffractogram. On a background of another shape it still proposes points, but they need not lie on the background; look at them on the chart before subtracting.

The background curve

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

Model > Background > Curve puts the background under the peaks as a curve of the model, fitted with them, drawn and saved with the project - and leaves the profile as it was measured.

The submenu lists None and every background shape: Constant, Linear and Quadratic backgrounds, an Exponential background for a tail that decays, and a Power-law background. The tick shows the one the model has. Choosing a shape puts one curve of it in every fit interval, after the peaks, starting from the level of the data's own baseline there; Model > Background > Curve > None takes it out again.

It is fitted with the curves by every fit, drawn with them, listed in the Model tab and the Curve Attributes table like them, and saved in the project with what the fit found - so a project reopens with the same background, still fitted. Selecting its row in the Model tab and deleting it (Model > Delete Curve) removes the background from every interval, as None does.

Its coefficients are measured from xr, the start of the fit interval: b0 is the background there, b1 the slope, b2 the curvature; tau is an exponential's decay length and p a power law's exponent. Reducing the number of curves never removes it, and a model of only a background can be drawn and fitted.

A background is not below zero anywhere in its fit interval when the data there are not: one that would dip below is lifted until its lowest point touches zero, and its level b0 carries the lift. Over data that go below zero it may follow them. See Limits on parameter values.

A curve type may say that its curves take no separate background - its curves describing the whole of the data, level included. Model > Background > Curve then offers only None, and says why.

Fit > Automatically adds a Quadratic background when the model has none, and keeps the one you chose when it has. It subtracts nothing from the profile.

The shapes are greyed while Model > Background > Enable Variation is on, and the variation is greyed while the model has a background curve: both put a background under the peaks, and with both it would be counted twice. The hint says so.

Limitations

With curve scaling on, the coefficients shown carry the one factor that scales the whole model onto the data - as a peak's amplitude does - so the curve drawn is the background, and the ratios of its coefficients are what the fit found.
Intensity the model has no curve for - small peaks nobody placed - is taken up by the background, which then rises under them. Place the peaks, or narrow the fit interval.
One shape serves every fit interval. The Power-law background is refused for data that reaches x = 0 or below.

Subtracting the background

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

Model > Background > Subtract joins the background points by straight lines and subtracts those lines from the profile.

Model > Background > Subtract > By Selected Points uses the background points you picked or had proposed. It needs at least two; with fewer it says so and does nothing.
Model > Background > Subtract > Automatically ignores any points you picked, proposes its own as Compute Automatically does, and subtracts through them. The Subtract button under Background in the Tools tab runs this one.

Between each two neighbouring background points the straight line through them is subtracted from the amplitudes, so every background point itself ends at zero. The profile on the chart and in the Data table is replaced by the result and the background points are cleared. With a data interval selected, only that interval is changed.

Subtract is greyed until a profile is loaded, and while a fit runs.

Limitations

Points outside the outermost background points are left as they were, so place background points near both ends of the stretch you fit.
Subtraction cannot be undone, and it can be repeated: each run subtracts again from what the last one left. File > Reload Profile brings back the data as measured.
It changes the data, not the model. To keep the profile as measured and fit the background instead, use Model > Background > Curve - which is what Fit > Automatically does.

Background fraction

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

The background fraction decides how small a peak the automatic searches still count: a point below 1/N of the highest point, for the factor N you set, is treated as background.

Model > Background > Set Background Fraction opens the Background Factor dialog, whose field is labelled Background factor value 1 /. It must be a number greater than 1; anything else is refused with a message. The default is 30, so anything lower than one thirtieth of the highest point is not searched for peaks.

The factor is used by Model > Curve Positions > Compute Automatically, by Model > Fit Intervals > Compute Automatically, and by a fit that proposes positions or intervals itself, Fit > Automatically included. A larger factor finds smaller peaks; a smaller one keeps only the tall ones.

The factor is saved with the project.

Limitations

Despite its name, the factor has no effect on the automatic background points, which are proposed without it.
The threshold is a fraction of the highest point, not a level above the background: where the background is high, the background itself may stand above it. Fit > Automatically searches a copy of the profile with the background taken off for exactly this reason; the Compute Automatically commands search the profile itself, so subtract the background first when using them on a high one.

Background variation

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

With Model > Background > Enable Variation ticked, a fit also adjusts a smooth background of its own alongside the curves.

The background the fit varies is a quadratic curve about a centre, described by four coefficients: curvature, slope, offset and centre. The curvature and the offset are kept from going negative, so the background never bends downwards and never sits below zero.

It is off until you turn it on. Choosing the entry turns it on or off, and a tick shows that it is on. The setting is saved with the project.

Use it when a background is left under the peaks that subtraction did not remove, or when you would rather fit the background than subtract it.

It is greyed while the model has a background curve (Model > Background > Curve), which puts a background under the peaks already - with both it would be counted twice. With it on, Fit > Automatically adds no background curve and subtracts nothing, and says that the variation is acting as the background.

Limitations

The varied background is always a quadratic. A background of another shape - one with a step or a separate hump - cannot be described by it.

The Background Points table

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

Lists the background points by Position and Amplitude, beside the buttons that find, subtract and remove them.

Each row is one point of the background, picked on the chart or found from the data.

The toolbar, from the top:

Compute background points automatically - Model > Background > Points > Compute Automatically.
Subtract a background found from the data - Model > Background > Subtract > Automatically.
Subtract the background through the points in the table - Model > Background > Subtract > By Selected Points. With fewer than two points, it finds them from the data first.
Remove the background points - Model > Background > Points > Remove All.

A point can also be moved by typing: change its Position or its Amplitude and leave the cell, and the point moves, on the chart and in the engine.

Diffraction angles: Theta, 2 Theta and sin(Theta)/lambda

Convention — Common practice or a guideline in the field, not a rule its sources require.

For a diffraction pattern the stored argument is read as the scattering angle 2 Theta in degrees, and the axis can show it as 2 Theta, as Theta, or as sin(Theta)/lambda.

Diffraction patterns are usually recorded against the scattering angle 2 Theta, and the program takes the stored argument to be 2 Theta in degrees whenever one of these three rules is in force:

Data > Argument Transformation > Use Rule > 2 * Theta - the stored value itself, labelled 2*Theta [deg].
Data > Argument Transformation > Use Rule > Theta - half of it, labelled Theta [deg].
Data > Argument Transformation > Use Rule > Sin Theta / Lambda - sin(Theta) divided by the wavelength lambda in angstrom, labelled Sin(Theta)/Lambda [1/A].

The value of a diffraction profile is the measured Intensity, and Data > Value Transformation > Use Rule > Intensity names it so; a model of the peak shapes the program ships is shown against it by Automatic.

By Bragg's law, lambda = 2 d sin(Theta), the value sin(Theta)/lambda equals 1/(2d), where d is the spacing of the diffracting planes. Peaks recorded at different wavelengths therefore line up on this axis, which is why it is common in crystallography.

Only Sin Theta / Lambda needs a wavelength. The first time you choose it with no wavelength known, the program asks for one before switching; Theta and 2 * Theta need none.

Limitations

The program cannot tell whether your argument really is 2 Theta in degrees. On data recorded against anything else these three rules show numbers that mean nothing; use General Position or Automatic instead.

Wavelength

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

The wavelength of the radiation, in angstrom, is the number the Sin Theta / Lambda axis divides by.

Set it with Data > Argument Transformation > Set Rule Parameters > Wavelength. A dialog titled Wavelength asks for Wavelength (angstrom). It accepts only a positive number - 1.5406, for example - and keeps asking until it gets one or you cancel. On OK the wavelength is stored and the axis switches to Sin Theta / Lambda.

Set Rule Parameters is greyed unless the axis in force is a diffraction angle - Theta, 2 * Theta or Sin Theta / Lambda, or Automatic while it resolves to one. To give a wavelength for the first time you can also choose Data > Argument Transformation > Use Rule > Sin Theta / Lambda: with none known, it asks for one first.

The wavelength is saved with the project.

Limitations

Only what is shown depends on the wavelength. The fit works on the stored argument, so a wrong wavelength mislabels the axis without changing any fitted value.

Sample diffraction profiles

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

Nine diffraction profiles come with the peaks, in the Data folder, to try a fit on.

1.dat - a whole diffraction pattern, 1692 points with arguments from 3.0 to 172.1 in steps of 0.1.
2.dat to 9.dat - short stretches of 20 to 70 points, each around one or a few peaks. 2.dat, with two overlapping peaks between 116 and 121, is a good first fit.
test_data.zip - the nine .dat files packed in one archive.

A .dat file is read as a diffraction profile - its argument an angle - and File > Import Profile names the format so.

Limitations

They are examples to learn on, not reference measurements: nothing about the sample or the instrument they came from is recorded with them.

A first fit of peaks

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

Two overlapping peaks in 2.dat, fitted automatically in a few clicks.

File > Import Profile, and choose Data/2.dat: two overlapping peaks in about fifty points.
Choose Pseudo-Voigt under Model > Curve Type, a common choice for diffraction peaks.
Run Fit > Automatically. Fit puts a background curve under the peaks, places curves, marks a fit interval around each peak and removes curves for as long as the fit stays good enough.
Read the result: a systematic bump in the Difference series is a peak the model has missed; the Curve Attributes table gives each peak's position, width and amplitude.

Limitations

Two peaks this close are resolved by the shape the model assumes: another curve type can split the same data differently, and the program cannot say which is physically right.

Limits on the peaks, the backgrounds and your own formulas

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

How wide each peak shape may be, how the backgrounds keep above zero, and what the parameters of a formula you define may take.

A peak's width limit is on the peak, not on the number, because a width means different things in different curve types. A pseudo-Voigt's or a Pearson VII's sigma is its full width at half maximum, so it may be as wide as the interval; a Gaussian's sigma is a standard deviation, and the peak is 2.35 of them wide at half its maximum, so it may be the interval divided by 2.35. A Voigt's gamma and a Doniach-Sunjic sigma are half widths; an exponentially modified Gaussian's tail tau falls to half in ln 2 of its length; how wide a Moffat or a Doniach-Sunjic line is per unit of sigma depends on its shape, and the limit follows the shape the fit has reached. A Voigt's two widths are held together, so the peak they make fits. Both widths of a two-branch pseudo-Voigt are held, and the difference of an asymmetric pseudo-Voigt's two side widths too.

Where it is held, the level of an Exponential or a Power-law background is kept at zero or above. A Constant, Linear or Quadratic background, whose sign no single coefficient decides, is lifted after every calculation until its lowest point in the fit interval touches zero, and its level b0 carries the lift - so the coefficients in the table describe the curve that is drawn. A lift rather than a cut at zero, because a cut background has parts no coefficient can move, and rather than a penalty, because a penalty only discourages going below zero.

A curve type you define yourself keeps the range of the role you give each parameter in its dialog: an Amplitude is not below zero, and a Width is above zero and held where the peak your formula draws is as wide as the fit interval - measured from the formula, since the program cannot know whether it reads the Width as a standard deviation, a half width or a full width. A parameter with any other role takes any value.

Limitations

A Width's limit is measured from the formula as drawn, so a formula whose width does not follow its Width parameter is held where it does not matter.

The Residual Spectrum

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

The Residual Spectrum tab of the chart area maps where, and at which length of rhythm, the difference between the data and the model holds more than noise.

It draws the wavelet transform the program judges the difference with after every fit (see "Is the difference only noise?") as a map: the model's positions across, the length of a rhythm upwards - 4, 8, 16 points per cycle and so on - and at each place and rhythm what the difference holds there. A cell in the colour of the shaded stretches is a wave the model misses: exactly the cells that judgement named, so the map and the stretches always agree, and a patch says both where the model falls short and what length of wave it left out. A dark grey cell is as strong as noise is only one time in twenty, but among hundreds of cells some are that strong by chance, so it is not named; a light grey one is noise above its average, and the background is noise. The key under the map names every colour, and the line under its title says how to read it.

THE MODEL'S CHART AND THE RESIDUAL SPECTRUM SHOW ONE VIEW of the model's x. Zooming or scrolling either does the same to the other, and making either show everything again shows everything on both, values included. Up and down is each chart's own: the model's values on its chart, the wave lengths on the map.

Below the map a scroll bar moves the view along x, always agreeing with the one below the model's chart; the one to its right moves it along the wave lengths.

Limitations

Near the ends of a fit interval the edge shapes what is seen, so those cells, like the gaps between intervals, are greyed as not judged.

User-defined curves

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

You can fit a shape of your own by typing its formula; the program turns it into a curve type that is saved and offered like the built-in ones.

Choose Model > Curve Type > User > New User Curve, or New User Curve in the Tools list. Two dialogs follow: Create New Curve Type, where you name the curve and write its formula, and Set Curve Type Properties, where you say which parameter plays which role and give starting values.

When you press Done the curve type is saved and selected. It appears under Model > Curve Type > User, below New User Curve, and is ticked while it is in use. It is kept between sessions: each one is stored as a file in the program's settings folder.

A user-defined curve is fitted like any other: place positions, set fit intervals, fit. Its formula is saved with the project. It says nothing about its axes, so under the Automatic axis rules it is shown against what the data says, or a plain Position.

New User Curve does not name a shape: it opens the dialogs that define one. Cancelling them leaves the previous type in use. The User group comes last in Model > Curve Type and in the Tools list.

Cancelling either dialog creates nothing: a curve type saved when its formula was accepted is removed again if you then cancel the second dialog.

Limitations

The curve types you have defined are listed only in the menu, under Model > Curve Type > User. The Tools list offers New User Curve but not the types already made.

Writing the formula of a user-defined curve

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

In the Create New Curve Type dialog you give the curve a name and a formula in x; every other name in the formula becomes a parameter.

Name is what the curve type is called in the menu (MyCurve until you change it). Expression is the formula, filled in with a bell curve to start from: A*exp(-((x-x0)/SIGMA)^2).

x is the argument, the variable running along the axis. Some names mean something, whatever their case: x0 is the position, where the curve is placed, and the fit does not move it; A is taken as the amplitude and SIGMA as the width, and SIGMA starts at 0.25 rather than 0 so that a formula dividing by it can be evaluated. Every other name is an ordinary fitted value starting at 0. The next dialog can change all of this.

The keypad inserts at the cursor. A function is inserted with its brackets, and the cursor left between them. It offers Sin, Cos, Tg, Ctg, the hyperbolic Sh, Ch, Th, Cth, Sch and Csch, the inverses Arcsin, Arccos, Arctg, Arcctg, Arsh, Arch, Arth and Arcth, Ln (natural logarithm), Log (base 10), Exp, Abs and Sqrt, the digits, the operators + - * / ^, brackets and a decimal point. Names are not case-sensitive, a comma typed on the keyboard becomes a decimal point, and sqr and pi can be typed though they have no button.

Continue >> checks the formula before anything is saved. If it cannot be used a message says why and the dialog opens again for you to correct it:

the Expression is empty;
the formula could not be understood - a typo, an unmatched bracket or an unknown function;
the formula does not use x as its argument;
the formula cannot be evaluated at its starting values - most often a parameter used as a denominator left at 0.

Cancel abandons the definition.

Limitations

Pressing a function button while text is selected replaces the selection with an empty call rather than wrapping it: the selected text is lost.

Parameter roles and starting values

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

The Set Curve Type Properties dialog says which parameter of a user-defined curve is its argument, position, amplitude and width, which are held, and where each starts.

Expression shows the formula. Four lists give the roles, and each role is held by at most one parameter - giving it to one takes it from any other:

Argument - the variable along the axis, x unless you choose another.
Position parameter - the one that places the curve at a pick. The parameter chosen here starts out held.
Amplitude parameter - the height, started from the data peak, and never below zero.
Width parameter - the width, started from the fit interval, and never wider than it (see Limits on parameter values).

Amplitude and Width offer (none), which leaves the role unheld.

Shared parameters lists the position and the ordinary parameters - not the argument, the amplitude or the width. Ticking the position parameter makes the fit leave it where it was picked; unticking it lets the fit move it. Ticking any other parameter makes it shared: one value, varied by the fit and held equal across the curves of an interval.

Held parameters lists the shared and the ordinary parameters. Ticking one holds it at its starting value: the fit does not vary it. The tick is kept with the curve and in the project.

To give a starting value, select a parameter in Shared parameters, type the value in Initial value and press Apply. A value that is not a number is refused with a message.

Done saves the roles and selects the curve type. << Backtrack discards this draft and returns to the formula dialog. Cancel closes the dialog and removes the draft.

Once the curves are placed, a value can also be typed into the Curve Attributes table.

Selecting and deleting user-defined curves

This software's choice — A decision made by this software rather than by the field's sources; the explanation says why it was made.

The curve types you have defined are chosen and deleted under Model > Curve Type > User.

Below New User Curve and a separator, the User group lists every curve type you have defined. Choosing one selects it, and it is ticked while in use. New User Curve itself is never ticked: it names no curve, it makes one.

Model > Curve Type > User > Delete User Curve opens a list of the same curve types. Choosing one deletes it, and its file, at once and without asking. The entry is there only while at least one curve type of your own exists.

If you delete the curve type in use, the program says so: the model is left with no curve type, and a fit is refused until you choose another under Model > Curve Type.

A curve type saved without a formula - by an older version, or by a session interrupted while defining it - cannot be selected; choosing it says so and suggests deleting it and making it again.

Limitations

Deleting cannot be undone.

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