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# SANE - Services for Advanced Neutron Environment

Our equipments are controlled with IGOR Pro, an extraordinarily powerful and extensible scientific graphing, data analysis, image processing and programming software tool for scientists and engineers.

# Flipping Ratios Calculations

Here is the list of functions used to calculate flipping ratios that are provided by the "Neutron Scattering" XOP:

#### FlippingRatio(Rpp, Rbp, Rpm, Rbm)

The FlippingRatio function returns the flipping ratio from the count rates Rpp, Rbp, Rpm and Rbm. Rpp and Rbp are the peak and background count rates for the [+] polarisation state. Rpm and Rbm are the peak and background count rates for the [-] polarisation state. The flipping ratio is defined by:

$R = \frac{r_{p,+} - r_{b,+}}{r_{p,-} - r_{b,-}}$

#### FlippingRatioSDev(Rpp, Rbp, Rpm, Rbm, RppSDev, RbpSDev, RpmSDev, RbmSDev)

The FlippingRatioSDev function returns the flipping ratio standard deviation from the count rates Rpp, Rbp, Rpm, Rbm and the corresponding standard deviations. Rpp and Rbp are the peak and background count rates for the [+] polarisation state. Rpm and Rbm are the peak and background count rates for the [-] polarisation state. The flipping ratio standard deviation is defined by:

$\sigma_R^2 = \frac{\left( r_{p,+} - r_{b,+} \right)^2 \, \left(\sigma_{r_{p,-}}^2 + \sigma_{r_{b,-}}^2 \right) + \left( r_{p,-} - r_{b,-} \right) ^2 \, \left( \sigma_{r_{p,+}}^2 + \sigma_{r_{b,+}}^2 \right)} {\left( r_{p,-} - r_{b,-} \right) ^4}$

#### FlippingRatioOpt(Npp, Nbp, Npm, Nbm, Tpp, Tbp, Tpm, Tbm, Mp, Mb)

The FlippingRatioOpt function returns the flipping ratio from the counts Npp, Nbp, Npm, Nbm, the counting times Tpp, Tbp, Tpm, Tbm and the monitor values Mp and Mb in the case of cyclic measurements. The first indices [p] or [b] stand for peak or background measurements. The second indices stand for the [+] and [-] polarisation states. The flipping ratio is defined by:

$R_{opt} = \left( \frac{N_{p,+} \cdot t_p} {t_{p,+} \cdot M_p} - \frac{N_{b,+} \cdot t_b} {t_{b,+} \cdot M_b} \right) / \left( \frac{N_{p,-} \cdot t_p} {t_{p,-} \cdot M_p} - \frac{N_{b,-} \cdot t_b} {t_{b,-} \cdot M_b} \right)$

#### FlippingRatioOptSDev(Npp, Nbp, Npm, Nbm, Tpp, Tbp, Tpm, Tbm, Mp, Mb)

The FlippingRatioOptSDev function returns the flipping ratio standard deviation from the counts Npp, Nbp, Npm, Nbm, the counting times Tpp, Tbp, Tpm, Tbm and the monitor values Mp and Mb in the case of cyclic measurements. The first indices [p] or [b] stand for peak or background measurements. The second indices stand for the [+] and [-] polarisation states. The flipping ratio standard deviation is defined by:

$\sigma_{R_{opt}}^2 = \, {\left( \frac{t_b} {M_b \, r_{-}} \right)}^2 \left[ \frac{1}{M_b} { \left( \frac{N_{b,+}} {t_{b,+}} - \frac{N_{b,-}} {t_{b,-}} \, R_{opt} \right) }^2 + {\left( \frac{N_{b,+}} {t_{b,+}} \right)}^2 + {\left( \frac{N_{b,-}} {t_{b,-}}\,R_{opt} \right)}^2 \right] + \ldots$
$\ldots {\left( \frac{t_p} {M_p \, r_{-}} \right)}^2 \left[ \frac{1}{M_p} { \left( \frac{N_{p,+}} {t_{p,+}} - \frac{N_{p,-}} {t_{p,-}} \, R_{opt} \right) }^2 + {\left( \frac{N_{p,+}} {t_{p,+}} \right) }^2 + {\left( \frac{N_{p,-}} {t_{p,-}}\,R_{opt} \right)}^2 \right]$
$\mathrm{with \, {\it r_{-}} = denominator \ of \ {\it R_{opt}^{cor}}}$

#### FlippingRatioOptDDT(Npp, Nbp, Npm, Nbm, Tpp, Tbp, Tpm, Tbm, Mp, Mb, a, b)

The FlippingRatioOptDDT function returns the flipping ratio from the counts Npp, Nbp, Npm, Nbm, the counting times Tpp, Tbp, Tpm, Tbm and the monitor values Mp and Mb in the case of cyclic measurements. This function corrects the counts for detector dead-time using a and b , the first and second order correction coefficients expressed in seconds. The first indices [p] or [b] stand for peak or background measurements. The second indices stand for the [+] and [-] polarisation states. The flipping ratio is defined by:

$R_{opt}^{cor} = \left( \frac{N_{p,+}^{cor} \cdot t_p} {t_{p,+} \cdot M_p} - \frac{N_{b,+}^{cor} \cdot t_b} {t_{b,+} \cdot M_b} \right) / \left( \frac{N_{p,-}^{cor} \cdot t_p} {t_{p,-} \cdot M_p} - \frac{N_{b,-}^{cor} \cdot t_b} {t_{b,-} \cdot M_b} \right)\ \mathrm{where}\ N_{\alpha,\beta}^{cor} = \frac{N_{\alpha,\beta}} {1 - a\left( \frac{N_{\alpha,\beta}}{T_{\alpha,\beta}} \right) - b \left(\frac{N_{\alpha,\beta}}{T_{\alpha,\beta}}\right)^2}$

#### FlippingRatioOptDDTSDev(Npp, Nbp, Npm, Nbm, Tpp, Tbp, Tpm, Tbm, Mp, Mb, a, b , aSDev, bSDev)

The FlippingRatioOptDDTSDev function returns the flipping ratio standard deviation from the counts Npp, Nbp, Npm, Nbm, the counting times Tpp, Tbp, Tpm, Tbm and the monitor values Mp and Mb in the case of cyclic measurements. This function corrects the counts for detector dead-time using a and b , the first and second order correction coefficients expressed in seconds, and aSDev, bSDev the corresponding standard deviations. The first indices [p] or [b] stand for peak or background measurements. The second indices stand for the [+] and [-] polarisation states. The flipping ratio standard deviation is defined by:

$\sigma_{R_{opt}^{cor}}^2 = \,{\left( \frac{t_b} {M_b \, r_{-}} \right)}^2 \left[ \frac{1}{M_b} { \left( \frac{N_{b,+}^{cor}} {t_{b,+}} - \frac{N_{b,-}^{cor}} {t_{b,-}} \, R_{opt}^{cor} \right) }^2 + {\left( \frac{N_{b,+}^{cor}} {t_{b,+}} \right)}^2 + {\left( \frac{N_{b,-}^{cor}} {t_{b,-}}\,R_{opt}^{cor} \right)}^2 \right] + \ldots$
$\ldots {\left( \frac{t_p} {M_p \, r_{-}} \right)}^2 \left[ \frac{1}{M_p} { \left( \frac{N_{p,+}^{cor}} {t_{p,+}} - \frac{N_{p,-}^{cor}} {t_{p,-}} \, R_{opt}^{cor} \right) }^2 + {\left( \frac{N_{p,+}^{cor}} {t_{p,+}} \right) }^2 + {\left( \frac{N_{p,-}^{cor}} {t_{p,-}}\,R_{opt}^{cor} \right)}^2 \right]$
$\mathrm{with \, {\it r_{-}} = denominator \ of \ {\it R_{opt}^{cor}}}$

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