An alternative statistical approach to the collection and processing of Mössbauer spectroscopy data

Abstract

The idea of recording the total history of a Mössbauer spectrum collection in carefully selected time intervals is introduced in this work, providing a potential to wield and interpret the mustered data using the novel-suggested experimental quantity of ‘count rate’, in comparison to the conventionally used experimental quantity of simple accumulated counts. The gain is a rich and detailed statistical set of data described by Poisson functions, that can be analyzed to reveal several characteristics, such as mean values and partial probability sums, providing different proposed experimental Mössbauer spectrum estimations. It is shown that the application of the proposed method affords improvements referring to line-width reduction and increase of signal to noise ratio to the resulting experimental ⁵⁷Fe Mössbauer spectrum estimations, in comparison to the corresponding conventional spectrum recorded by simply accumulating counts. Further improvement of the experimental spectral resolution is accomplished through the implementation of a proposed background filtering algorithm, whose validity is moreover tested on theoretical Mössbauer spectra containing simulated Poisson noise, where it was also found to be effective. The method is proposed to have applications to other similar experimental techniques that are based on estimating statistical mean values of experimental physical quantities.

Appendix

In the present section we will explain why the CDFDAM approximations are valid mean spectra estimations and what kind of interpretation can be given to them. The proof will be made for a continuous symmetric around-the-mean pdf function as for example is the Gaussian-normal function and then try to interpret the results for a discrete PMF case (Poisson function). We will begin the reasoning by noting that a spectrum is just an ordered array of numbers representing mean values, e.g., $\bar{x}_1,\bar{x}_2,\bar{x}_3 \ldots$ and certainly the exact same spectrum is represented by the ordered array of the halved mean values $\frac{\bar{x}_1}{2},\frac{\bar{x}_2}{2},\frac{\bar{x}_3}{2}\ldots$. The definition for a mean value of a known PDF f(x) is simply $\bar{x}=\int _{-\infty }^{\infty }xf(x) \mathrm{d}x$ or for positive only defined functions $\bar{x}=\int _{0}^{\infty }{xf(x) \mathrm{d}x}$ and due to the symmetry around the mean value of the assumed PDF, this means $\frac{\bar{x}}{2}=\int _{0}^{\overline{x}} xf(x) \mathrm{d}x$. So starting from (3) and interpreting it to represent mean values we have $\rightarrow \frac{\overline{N(v)}}{2}=\frac{\overline{\mathrm{BG}}}{2}-\frac{\overline{A(v)}}{2}$

$$\begin{aligned} \rightarrow \int _{0}^{\overline{A(v)}} xf_{A(v)}(x) \mathrm{d}x=\int _{0}^{\overline{\mathrm{BG}}} xf_{\mathrm{BG}}(x) \mathrm{d}x - \int _{0}^{\overline{N(v)}} xf_{N(v)}(x) \mathrm{d}x \end{aligned}$$

(A.1)

This equation is exact but for cases where $\mathrm{PDF}_{N(v)} \approx \mathrm{PDF}_{\mathrm{BG}}$ as that displayed in Fig. 3 we are in the fortunate position to approximate the above equation as

$$\begin{aligned} \rightarrow \int _{0}^{\overline{A(v)}} x \mathrm{PDF}_{A(v)} \mathrm{d}x \approx \int _{0}^{\overline{\mathrm{BG}}} x \mathrm{PDF}_{N(v)} \mathrm{d}x - \int _{0}^{\overline{N(v)}} x \mathrm{PDF}_{N(v)} \mathrm{d}x=\int _{\overline{N(v)}}^{\overline{\mathrm{BG}}} x \mathrm{PDF}_{N(v)} \mathrm{d}x \end{aligned}$$

(A.2)

This is what in reality the CDFDAM method calculates and as we can see, Eq. (A.2) can be interpreted as the mean value of the true signal part A(v), without the BG part contribution. We need to mention that the approximation is valid for low CR values, e.g CR$\lessapprox$10 counts/TI and small effect values $\frac{\overline{A(v)}}{\overline{\mathrm{BL}}}\lessapprox 6\%$, in order to fulfill the approximation condition $\mathrm{PDF}_{N(v)} \approx \mathrm{PDF}_{\mathrm{BG}}$. The smaller the effect, the better the approximation, even for high CR cases.

Form the above we can conclude that an alternative algorithm that can still result in substantial background noise elimination but still preserve the spectrum’s line-widths or even reduce them, could be to implement the exact form of Eq. (A.1) for every channel exceeding a threshold x value and the approximate Eq. (A.2) from every point below that threshold value.