\( {\rm Instrumental \, function \, for\,the \, Soller} \)-\( {\rm slits \, angles },\,\Phi_{\rm SS}^{\rm (i)}\, {\rm on \, the\, incident\,beam\,side } \),
\( {\rm and}\, \Phi_{\rm SS}^{\rm (d) }\,{\rm on\,the\,diffracted\,beam\,side,\,is\,given\,by} \)
\[ \omega^{\rm (A)} \left( \Delta 2\Theta ; 2\Theta, \Phi_{\rm SS}^{\rm (i)}, \Phi_{\rm SS}^{\rm (d)} \right) = \int\limits_{ -\Phi_{\rm SS}^{\rm (d)} }^{ \Phi_{\rm SS}^{\rm (d)} } \int\limits_{ -\Phi_{\rm SS}^{\rm (i)} }^{ \Phi_{\rm SS}^{\rm (i)} } \delta\left( \Delta 2\Theta – f(2\Theta,\alpha,\beta ) \right)
g\left( \alpha, \Phi_{\rm SS}^{\rm (i)} \right)
g \left( \beta, \Phi_{\rm SS}^{\rm (d)} \right)
\, {\rm d}\alpha \,{\rm d}\beta
\]
\[ f(2\Theta,\alpha,\beta) \equiv 2\Theta – \arccos\left( \cos 2\Theta \cos\alpha\cos\beta + \sin\alpha \sin\beta \right) \]
\[ g( \phi , \Phi) \equiv \frac{ 1 }{ \Phi } \left( 1 – \frac{ |\phi| }{ \Phi } \right) \]
\( {\rm where \,} \delta(x)\,{\rm is\,the\,Dirac\,delta\,function} \).
\( { \rm It \, is \, complicated \, to \, derive \, a \, formula \, of \, the \, instrumental \, function } \).
\( {\rm On \, the \, other \, hand , \,it \, is \, not \, too \, difficult \, to \, derive \, the \, formula }\)
\( {\rm for \, evaluating \, the \, lower \, order \, cumulants \, of \, the \, instrumental \, function \, by \, numerical \, calculation } \).
\( {\rm The \,1st \, to \, 3rd \, order \, cumulants,} \, \kappa_1 \, { \rm to } \,\kappa_3, \, {\rm are \, calculated \, by} \)
\[ \kappa_1 = \frac{s_1}{s_0} \]
\[ \kappa_2 = \frac{s_2}{s_0} – \frac{s_1^2}{s_0^2} \]
\[ \kappa_3 = \frac{s_3}{s_0} – \frac{ 3 s_2 s_1 }{ s_0^2 } + \frac{ 2 s_1^3 }{ s_0^3 } \]
\[ s_\nu = \int\limits_{ -\Phi_{\rm SS}^{\rm (d)} }^{ \Phi_{\rm SS}^{\rm (d)} } \int\limits_{ -\Phi_{\rm SS}^{\rm (i)} }^{ \Phi_{\rm SS}^{\rm (i)} } \left[ f(2\Theta,\alpha,\beta ) \right]^\nu
g\left( \alpha, \Phi_{\rm SS}^{\rm (i)} \right)
g \left( \beta, \Phi_{\rm SS}^{\rm (d)} \right)
\, {\rm d}\alpha \,{\rm d}\beta\]
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