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| .. algorithm:: | ||
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| .. summary:: | ||
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| .. alias:: | ||
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| .. properties:: | ||
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| Description | ||
| ----------- | ||
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| The measured spectrum :math:`I(Q, \omega)` is proportional to the four | ||
| dimensional convolution of the scattering law :math:`S(Q, \omega)` with the | ||
| resolution function :math:`R(Q, \omega)` of the spectrometer via :math:`I(Q, | ||
| \omega) = S(Q, \omega) ⊗ R(Q, \omega)`, so :math:`S(Q, \omega)` can be obtained, | ||
| in principle, by a deconvolution in :math:`Q` and :math:`\omega`. The method | ||
| employed here is based on the Fourier Transform (FT) technique [6,7]. On Fourier | ||
| transforming the equation becomes :math:`I(Q, t) = S(Q, t) x R(Q, t)` where the | ||
| convolution in :math:`\omega`-space is replaced by a simple multiplication in | ||
| :math:`t`-space. The intermediate scattering law :math:`I(Q, t)` is then | ||
| obtained by simple division and the scattering law :math:`S(Q, \omega)` itself | ||
| can be obtained by back transformation. The latter however is full of pitfalls | ||
| for the unwary. The advantage of this technique over that of a fitting procedure | ||
| such as SWIFT is that a functional form for :math:`I(Q, t)` does not have to be | ||
| assumed. On IRIS the resolution function is close to a Lorentzian and the | ||
| scattering law is often in the form of one or more Lorentzians. The FT of a | ||
| Lorentzian is a decaying exponential, :math:`exp(-\alpha t)` , so that plots of | ||
| :math:`ln(I(Q, t))` against t would be straight lines thus making interpretation | ||
| easier. | ||
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| In general, the origin in energy for the sample run and the resolution run need | ||
| not necessarily be the same or indeed be exactly zero in the conversion of the | ||
| RAW data from time-of-flight to energy transfer. This will depend, for example, | ||
| on the sample and vanadium shapes and positions and whether the analyser | ||
| temperature has changed between the runs. The procedure takes this into account | ||
| automatically, without using an arbitrary fitting procedure, in the following | ||
| way. From the general properties of the FT, the transform of an offset | ||
| Lorentzian has the form :math:`(cos(\omega_{0}t) + isin(\omega_{0}t))exp(-\Gamma | ||
| t)` , thus taking the modulus produces the exponential :math:`exp(-\Gamma t)` | ||
| which is the required function. If this is carried out for both sample and | ||
| resolution, the difference in the energy origin is automatically removed. The | ||
| results of this procedure should however be treated with some caution when | ||
| applied to more complicated spectra in which it is possible for :math:`I(Q, t)` | ||
| to become negative, for example, when inelastic side peaks are comparable in | ||
| height to the elastic peak. | ||
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| The interpretation of the data must also take into account the propagation of | ||
| statistical errors (counting statistics) in the measured data as discussed by | ||
| Wild et al [1]. If the count in channel :math:`k` is :math:`X_{k}` , then | ||
| :math:`X_{k}=<X_{k}>+\Delta X_{k}` where :math:`<X_{k}>` is the mean value and | ||
| :math:`\Delta X_{k}` the error. The standard deviation for channel :math:`k` is | ||
| :math:`\sigma k` :math:`2=<\Delta X_{k}>2` which is assumed to be given by | ||
| :math:`\sigma k=<X_{k}>`. The FT of :math:`X_{k}` is defined by | ||
| :math:`X_{j}=<X_{j}>+\Delta X_{j}` and the real and imaginary parts denoted by | ||
| :math:`X_{j} I` and :math:`X_{j} I` respectively. The standard deviations on | ||
| :math:`X_{j}` are then given by :math:`\sigma 2(X_{j} R)=1/2 X0 R + 1/2 X2j R` | ||
| and :math:`\sigma 2(X_{j} I)=1/2 X0 I - 1/2 X2j I`. | ||
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| Note that :math:`\sigma 2(X_{0} R) = X_{0} R` and from the properties of FT | ||
| :math:`X_{0} R = X_{k}`. Thus the standard deviation of the first coefficient | ||
| of the FT is the square root of the integrated intensity of the spectrum. In | ||
| practice, apart from the first few coefficients, the error is nearly constant | ||
| and close to :math:`X_{0} R`. A further point to note is that the errors make | ||
| the imaginary part of :math:`I(Q, t)` non-zero and that, although these will be | ||
| distributed about zero, on taking the modulus of :math:`I(Q, t)`, they become | ||
| positive at all times and are distributed about a non-zero positive value. When | ||
| :math:`I(Q, t)` is plotted on a log-scale the size of the error bars increases | ||
| with time (coefficient) and for the resolution will reach a point where the | ||
| error on a coefficient is comparable to its value. This region must therefore be | ||
| treated with caution. For a true deconvolution by back transforming, the data | ||
| would be truncated to remove this poor region before back transforming. If the | ||
| truncation is severe the back transform may contain added ripples, so an | ||
| automatic back transform is not provided. | ||
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| References: | ||
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| 1. U P Wild, R Holzwarth & H P Good, Rev Sci Instr 48 1621 (1977) | ||
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| .. categories:: |
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