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10/7/2019 - 7:14 AM

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<li><a href="#黒体放射">黒体放射</a></li>
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<h2 id="黒体放射">黒体放射</h2>
<h3 id="プランクの法則">プランクの法則</h3>
<p>黒体から輻射される電磁波の分光放射輝度：</p>
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml">$<semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>ν</mi><mo separator="true">,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle mathsize="1.2em"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>2</mn><mi>h</mi><msup><mi>v</mi><mn>3</mn></msup></mrow><msup><mi>C</mi><mn>2</mn></msup></mfrac></mstyle><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><msup><mi>e</mi><mrow><mi>h</mi><mi>ν</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>T</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mstyle></mstyle></mrow><annotation encoding="application/x-tex">I(ν,T)={\large\cfrac{2hv^3}{C^2}\times\cfrac{1}{e^{hν/kT}-1}}</annotation></semantics>$</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathdefault" style="margin-right: 0.07847em;">I</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right: 0.06366em;">ν</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.166667em;"></span><span class="mord mathdefault" style="margin-right: 0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 2.8312em; vertical-align: -0.923196em;"></span><span class="mord"><span class="mord sizing reset-size6 size7"><span class="mopen nulldelimiter sizing reset-size7 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.59em;"><span class="" style="top: -2.514em;"><span class="pstrut" style="height: 3.2em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.718627em;"><span class="" style="top: -3.089em; margin-right: 0.0416667em;"><span class="pstrut" style="height: 2.8em;"></span><span class="sizing reset-size7 size4 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.43em;"><span class="pstrut" style="height: 3.2em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.94em;"><span class="pstrut" style="height: 3.2em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault">h</span><span class="mord"><span class="mord mathdefault" style="margin-right: 0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.792627em;"><span class="" style="top: -3.163em; margin-right: 0.0416667em;"><span class="pstrut" style="height: 2.8em;"></span><span class="sizing reset-size7 size4 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class=""></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin sizing reset-size6 size7">×</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord sizing reset-size6 size7"><span class="mopen nulldelimiter sizing reset-size7 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.59em;"><span class="" style="top: -2.514em;"><span class="pstrut" style="height: 3.2em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.789em;"><span class="" style="top: -3.089em; margin-right: 0.0416667em;"><span class="pstrut" style="height: 2.8em;"></span><span class="sizing reset-size7 size4 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="mord mathdefault mtight" style="margin-right: 0.06366em;">ν</span><span class="mord mtight">/</span><span class="mord mathdefault mtight" style="margin-right: 0.03148em;">k</span><span class="mord mathdefault mtight" style="margin-right: 0.13889em;">T</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mord">1</span></span></span><span class="" style="top: -3.43em;"><span class="pstrut" style="height: 3.2em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.94em;"><span class="pstrut" style="height: 3.2em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.76933em;"><span class=""></span></span></span></span></span><span class=""></span></span></span></span></span></span></span></p>
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml">$<semantics><mrow><mstyle mathsize="0.8em"><mi>h</mi><mo>=</mo><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>k</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><mi mathvariant="normal">‎</mi><mo>=</mo><mn>6.62</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mi mathvariant="normal">−</mi><mn>34</mn></mrow></msup><mi>J</mi><mi>s</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\footnotesize h=Planck Constant(‎=6.62×10^{−34} J s)}</annotation></semantics>$</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.931464em; vertical-align: -0.2em;"></span><span class="mord"><span class="mord mathdefault sizing reset-size6 size4">h</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mrel sizing reset-size6 size4">=</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.13889em;">P</span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.01968em;">l</span><span class="mord mathdefault sizing reset-size6 size4">a</span><span class="mord mathdefault sizing reset-size6 size4">n</span><span class="mord mathdefault sizing reset-size6 size4">c</span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.03148em;">k</span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault sizing reset-size6 size4">o</span><span class="mord mathdefault sizing reset-size6 size4">n</span><span class="mord mathdefault sizing reset-size6 size4">s</span><span class="mord mathdefault sizing reset-size6 size4">t</span><span class="mord mathdefault sizing reset-size6 size4">a</span><span class="mord mathdefault sizing reset-size6 size4">n</span><span class="mord mathdefault sizing reset-size6 size4">t</span><span class="mopen sizing reset-size6 size4">(</span><span class="mord sizing reset-size6 size4">‎</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mrel sizing reset-size6 size4">=</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mord sizing reset-size6 size4">6</span><span class="mord sizing reset-size6 size4">.</span><span class="mord sizing reset-size6 size4">6</span><span class="mord sizing reset-size6 size4">2</span><span class="mspace" style="margin-right: 0.260222em;"></span><span class="mbin sizing reset-size6 size4">×</span><span class="mspace" style="margin-right: 0.260222em;"></span><span class="mord sizing reset-size6 size4">1</span><span class="mord sizing reset-size6 size4"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.91433em;"><span class="" style="top: -3.031em; margin-right: 0.0625em;"><span class="pstrut" style="height: 2.6em;"></span><span class="sizing reset-size4 size2 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.09618em;">J</span><span class="mord mathdefault sizing reset-size6 size4">s</span><span class="mclose sizing reset-size6 size4">)</span></span></span></span></span></span><br>
<span class="katex--inline"><span class="katex"><span class="katex-mathml">$<semantics><mrow><mstyle mathsize="0.8em"><mi>k</mi><mo>=</mo><mi>B</mi><mi>o</mi><mi>l</mi><mi>t</mi><mi>z</mi><mi>m</mi><mi>a</mi><mi>n</mi><mi>n</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><mo>=</mo><mn>1.38</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mi mathvariant="normal">−</mi><mn>23</mn></mrow></msup><mi>J</mi><msup><mi>K</mi><mrow><mi mathvariant="normal">−</mi><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\footnotesize k=Boltzmann Constant(=1.38×10^{−23}J K^{−1})}</annotation></semantics>$</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.931464em; vertical-align: -0.2em;"></span><span class="mord"><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.03148em;">k</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mrel sizing reset-size6 size4">=</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.05017em;">B</span><span class="mord mathdefault sizing reset-size6 size4">o</span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.01968em;">l</span><span class="mord mathdefault sizing reset-size6 size4">t</span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.04398em;">z</span><span class="mord mathdefault sizing reset-size6 size4">m</span><span class="mord mathdefault sizing reset-size6 size4">a</span><span class="mord mathdefault sizing reset-size6 size4">n</span><span class="mord mathdefault sizing reset-size6 size4">n</span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.07153em;">C</span><span class="mord mathdefault sizing reset-size6 size4">o</span><span class="mord mathdefault sizing reset-size6 size4">n</span><span class="mord mathdefault sizing reset-size6 size4">s</span><span class="mord mathdefault sizing reset-size6 size4">t</span><span class="mord mathdefault sizing reset-size6 size4">a</span><span class="mord mathdefault sizing reset-size6 size4">n</span><span class="mord mathdefault sizing reset-size6 size4">t</span><span class="mopen sizing reset-size6 size4">(</span><span class="mrel sizing reset-size6 size4">=</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mord sizing reset-size6 size4">1</span><span class="mord sizing reset-size6 size4">.</span><span class="mord sizing reset-size6 size4">3</span><span class="mord sizing reset-size6 size4">8</span><span class="mspace" style="margin-right: 0.260222em;"></span><span class="mbin sizing reset-size6 size4">×</span><span class="mspace" style="margin-right: 0.260222em;"></span><span class="mord sizing reset-size6 size4">1</span><span class="mord sizing reset-size6 size4"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.91433em;"><span class="" style="top: -3.031em; margin-right: 0.0625em;"><span class="pstrut" style="height: 2.6em;"></span><span class="sizing reset-size4 size2 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mord mathdefault sizing reset-size6 size4" style="margin-right: 0.09618em;">J</span><span class="mord sizing reset-size6 size4"><span class="mord mathdefault" style="margin-right: 0.07153em;">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.91433em;"><span class="" style="top: -3.031em; margin-right: 0.0625em;"><span class="pstrut" style="height: 2.6em;"></span><span class="sizing reset-size4 size2 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose sizing reset-size6 size4">)</span></span></span></span></span></span></p>
<h3 id="ヴィーンの変位則-wiens-displacement-law">ヴィーンの変位則 (Wien’s displacement law)</h3>
<p>黒体からの輻射のピーク波長が温度に反比例するという法則</p>
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml">$<semantics><mrow><mstyle mathsize="1.2em"><msub><mi>λ</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>b</mi><mi mathvariant="normal">/</mi><mi>T</mi></mstyle></mrow><annotation encoding="application/x-tex">{\large \lambda_{max} = b/T}</annotation></semantics>$</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.2em; vertical-align: -0.3em;"></span><span class="mord"><span class="mord sizing reset-size6 size7"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.13704em;"><span class="" style="top: -2.65em; margin-left: 0em; margin-right: 0.0416667em;"><span class="pstrut" style="height: 2.8em;"></span><span class="sizing reset-size7 size4 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel sizing reset-size6 size7">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathdefault sizing reset-size6 size7">b</span><span class="mord sizing reset-size6 size7">/</span><span class="mord mathdefault sizing reset-size6 size7" style="margin-right: 0.13889em;">T</span></span></span></span></span></span></p>
<p><span class="katex--inline"><span class="katex"><span class="katex-mathml">$<semantics><mrow><mstyle mathsize="0.8em"><mi>b</mi><mo>=</mo><mn>2.897</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">{\footnotesize b=2.897 \times 10^{-3}}</annotation></semantics>$</span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.798128em; vertical-align: -0.066664em;"></span><span class="mord"><span class="mord mathdefault sizing reset-size6 size4">b</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mrel sizing reset-size6 size4">=</span><span class="mspace" style="margin-right: 0.325278em;"></span><span class="mord sizing reset-size6 size4">2</span><span class="mord sizing reset-size6 size4">.</span><span class="mord sizing reset-size6 size4">8</span><span class="mord sizing reset-size6 size4">9</span><span class="mord sizing reset-size6 size4">7</span><span class="mspace" style="margin-right: 0.260222em;"></span><span class="mbin sizing reset-size6 size4">×</span><span class="mspace" style="margin-right: 0.260222em;"></span><span class="mord sizing reset-size6 size4">1</span><span class="mord sizing reset-size6 size4"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.91433em;"><span class="" style="top: -3.031em; margin-right: 0.0625em;"><span class="pstrut" style="height: 2.6em;"></span><span class="sizing reset-size4 size2 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>※短波長領域での近似式<br>
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