<p>We investigate coherent diffraction patterns from an amorphous SiO<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>2</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> across a range of electron counting regimes, from sparse single-event detection to cumulative high-flux exposures. Low electron count rates, with a mean of approximately 0.017 electrons per pixel, reveal discrete electron events dominated by shot noise. In contrast, cumulative patterns acquired over 60&#xa0;s exhibit pronounced speckle features reflecting atomic arrangements in the material. The count contrast <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C_k\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> as a function of mean electron count <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\overline{K}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>K</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> transitions from a Poisson-dominated <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/\sqrt{ \overline{K} }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <msqrt> <mover> <mi>K</mi> <mo>¯</mo> </mover> </msqrt> </mrow> </math></EquationSource> </InlineEquation> trend at low <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\overline{K}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>K</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> to a stabilized <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(C_k = 0.23\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>0.23</mn> </mrow> </math></EquationSource> </InlineEquation> beyond <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\overline{K} = 10\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>, of which behavior is modeled effectively with a spatial degree of freedom <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40543_2025_503_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(M = 32\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>=</mo> <mn>32</mn> </mrow> </math></EquationSource> </InlineEquation>. This <i>M</i> value indicates partially coherent illumination, deviating from single-mode expectations of the electron beam. Our finding bridges a gap between statistical optics and applied TEM and introduces a framework with direct practical relevance for modern detector technologies.</p>

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Coherence and statistical insights for low electron count regimes in transmission electron microscopy

  • Joohyun Lee,
  • Ji-Hwan Kwon,
  • Sooheyong Lee

摘要

We investigate coherent diffraction patterns from an amorphous SiO \(_2\) 2 across a range of electron counting regimes, from sparse single-event detection to cumulative high-flux exposures. Low electron count rates, with a mean of approximately 0.017 electrons per pixel, reveal discrete electron events dominated by shot noise. In contrast, cumulative patterns acquired over 60 s exhibit pronounced speckle features reflecting atomic arrangements in the material. The count contrast \(C_k\) C k as a function of mean electron count \(\overline{K}\) K ¯ transitions from a Poisson-dominated \(1/\sqrt{ \overline{K} }\) 1 / K ¯ trend at low \(\overline{K}\) K ¯ to a stabilized \(C_k = 0.23\) C k = 0.23 beyond \(\overline{K} = 10\) K ¯ = 10 , of which behavior is modeled effectively with a spatial degree of freedom \(M = 32\) M = 32 . This M value indicates partially coherent illumination, deviating from single-mode expectations of the electron beam. Our finding bridges a gap between statistical optics and applied TEM and introduces a framework with direct practical relevance for modern detector technologies.