Конец β-спектра и энергетическое разрешение

След массы нейтрино и его размытие детектором

Сначала измените массу нейтрино и найдите сдвиг конечной точки. Затем увеличьте ширину отклика детектора и проследите, как приборное размытие маскирует физический эффект.

viewof fermiMass = Inputs.range([0, 3], {
  value: 1,
  step: 0.1,
  label: html`
    <span class="spectrum-slider-label">
      ${tex`m_\nu,\ \text{эВ}`}
    </span>
  `
})

fermiElectronMass = 511000
fermiEndpoint = 18575
fermiWindow = 3

fermiSpectrum = (energy, mass) => {
  const neutrinoEnergy = fermiEndpoint - energy;
  if (neutrinoEnergy < mass) return 0;
  const electronMomentum = Math.sqrt(
    energy * energy + 2 * fermiElectronMass * energy
  );
  const neutrinoMomentum = Math.sqrt(Math.max(
    0,
    neutrinoEnergy * neutrinoEnergy - mass * mass
  ));
  return electronMomentum
    * (energy + fermiElectronMass)
    * neutrinoEnergy
    * neutrinoMomentum;
}

fermiIntegrate = (lower, upper, mass, steps = 1000) => {
  if (upper <= lower) return 0;
  const step = (upper - lower) / steps;
  let sum = (
    fermiSpectrum(lower, mass)
    + fermiSpectrum(upper, mass)
  ) / 2;
  for (let i = 1; i < steps; i += 1) {
    sum += fermiSpectrum(lower + i * step, mass);
  }
  return sum * step;
}

fermiNorm = fermiSpectrum(fermiEndpoint - fermiWindow, 0)
fermiTotalIntegral = fermiIntegrate(0, fermiEndpoint, 0, 8000)
fermiShadedIntegral = fermiIntegrate(
  fermiEndpoint - fermiMass,
  fermiEndpoint,
  0,
  400
)
fermiShadedFraction = fermiShadedIntegral / fermiTotalIntegral
fermiFractionExponent = fermiShadedFraction === 0
  ? 0
  : Math.floor(Math.log10(fermiShadedFraction))
fermiFractionMantissa = fermiShadedFraction === 0
  ? "0"
  : (
    fermiShadedFraction / 10 ** fermiFractionExponent
  ).toFixed(2)

fermiX = Array.from({length: 501}, (_, i) =>
  -fermiWindow + fermiWindow * i / 500
)
fermiMassless = fermiX.map(x => ({
  x,
  y: fermiSpectrum(fermiEndpoint + x, 0) / fermiNorm
}))
fermiMassive = fermiX
  .filter(x => -x >= fermiMass)
  .map(x => ({
    x,
    y: fermiSpectrum(fermiEndpoint + x, fermiMass) / fermiNorm
  }))
fermiShaded = fermiMassless.filter(d => d.x >= -fermiMass)

fermiPlot = Plot.plot({
  width: 1120,
  height: 300,
  marginTop: 18,
  marginRight: 25,
  marginBottom: 45,
  marginLeft: 72,
  style: {
    background: "#ffffff",
    color: "#111111",
    fontSize: "20px"
  },
  x: {
    axis: "bottom",
    label: null,
    domain: [-fermiWindow, 0],
    ticks: 6,
    grid: true
  },
  y: {
    axis: "left",
    label: null,
    domain: [0, 1.05],
    ticks: 5,
    grid: true,
    tickFormat: value => value.toFixed(1)
  },
  marks: [
    Plot.areaY(fermiShaded, {
      x: "x",
      y: "y",
      fill: "#999999",
      fillOpacity: 0.65
    }),
    Plot.ruleY([0], {
      stroke: "#111111",
      strokeWidth: 1.5
    }),
    Plot.line(fermiMassless, {
      x: "x",
      y: "y",
      stroke: "#4ea1ff",
      strokeWidth: 4
    }),
    Plot.line(fermiMassive, {
      x: "x",
      y: "y",
      stroke: "#ffb347",
      strokeWidth: 4
    }),
    Plot.ruleX([-fermiMass], {
      stroke: "#ffb347",
      strokeDasharray: "7,5",
      strokeWidth: 2.5
    }),
    Plot.frame({stroke: "#111111"})
  ]
})

html`
<div class="spectrum-panel">
  <div class="spectrum-legend">
    <span>
      <i class="spectrum-swatch spectrum-swatch-massless"></i>
      ${tex`m_\nu=0`}
    </span>
    <span>
      <i class="spectrum-swatch spectrum-swatch-massive"></i>
      ${tex`m_\nu=${fermiMass.toFixed(1)}\ \mathrm{eV}`}
    </span>
  </div>
  <div class="spectrum-plot-row">
    <div class="spectrum-y-label">
      ${tex`\dfrac{d\Gamma}{dE}\;(\mathrm{a.u.})`}
    </div>
    <div class="spectrum-plot-host">
      ${fermiPlot}
      <div class="spectrum-integral-overlay">
        <span>
          ${tex`\dfrac{\int_{E_0-m_\nu}^{E_0}(d\Gamma/dE)\,dE}
          {\int_0^{E_0}(d\Gamma/dE)\,dE}`}
        </span>
        <span>
          ≈ ${fermiFractionMantissa}
          ${fermiShadedFraction === 0
            ? ""
            : html`× 10<sup>${fermiFractionExponent}</sup>`}
        </span>
      </div>
    </div>
  </div>
  <div class="spectrum-x-label">
    ${tex`E-E_0\;(\mathrm{eV})`}
  </div>
  <div class="spectrum-summary">
    <span>${tex`E_0=18.575\ \mathrm{keV}`}</span>
    <span>${tex`E_{\max}=E_0-m_\nu`}</span>
  </div>
</div>`

viewof resolutionSigma = Inputs.range([0, 2], {
  value: 0.5,
  step: 0.1,
  label: html`
    <span class="spectrum-slider-label">
      ${tex`\sigma_E,\ \text{эВ}`}
    </span>
  `
})

resolutionMass = 1
resolutionXMin = -3
resolutionXMax = 1.5
resolutionGaussianNorm = Math.sqrt(2 * Math.PI)

resolutionConvolvedSpectrum = (xReco, mass, sigma) => {
  if (sigma === 0) {
    return fermiSpectrum(fermiEndpoint + xReco, mass) / fermiNorm;
  }

  const lower = Math.max(-fermiEndpoint, xReco - 6 * sigma);
  const upper = Math.min(-mass, xReco + 6 * sigma);
  if (upper <= lower) return 0;

  const steps = 240;
  const step = (upper - lower) / steps;
  const integrand = xTrue => {
    const pull = (xReco - xTrue) / sigma;
    const response = Math.exp(-0.5 * pull * pull)
      / (resolutionGaussianNorm * sigma);
    return response
      * fermiSpectrum(fermiEndpoint + xTrue, mass)
      / fermiNorm;
  };

  let sum = (integrand(lower) + integrand(upper)) / 2;
  for (let i = 1; i < steps; i += 1) {
    sum += integrand(lower + i * step);
  }
  return sum * step;
}

resolutionX = Array.from({length: 501}, (_, i) =>
  resolutionXMin
    + (resolutionXMax - resolutionXMin) * i / 500
)

resolutionData = resolutionX.map(x => ({
  x,
  massless: resolutionConvolvedSpectrum(x, 0, resolutionSigma),
  massive: resolutionConvolvedSpectrum(
    x,
    resolutionMass,
    resolutionSigma
  )
}))

resolutionPlot = Plot.plot({
  width: 1120,
  height: 300,
  marginTop: 18,
  marginRight: 25,
  marginBottom: 45,
  marginLeft: 72,
  style: {
    background: "#ffffff",
    color: "#111111",
    fontSize: "20px"
  },
  x: {
    axis: "bottom",
    label: null,
    domain: [resolutionXMin, resolutionXMax],
    ticks: 7,
    grid: true
  },
  y: {
    axis: "left",
    label: null,
    domain: [0, 1.5],
    ticks: 6,
    grid: true,
    tickFormat: value => value.toFixed(1)
  },
  marks: [
    Plot.areaY(resolutionData, {
      x: "x",
      y1: "massive",
      y2: "massless",
      fill: "#999999",
      fillOpacity: 0.55
    }),
    Plot.ruleY([0], {
      stroke: "#111111",
      strokeWidth: 1.5
    }),
    Plot.line(resolutionData, {
      x: "x",
      y: "massless",
      stroke: "#4ea1ff",
      strokeWidth: 4
    }),
    Plot.line(resolutionData, {
      x: "x",
      y: "massive",
      stroke: "#ffb347",
      strokeWidth: 4
    }),
    Plot.ruleX([0], {
      stroke: "#4ea1ff",
      strokeDasharray: "7,5",
      strokeWidth: 2.5
    }),
    Plot.ruleX([-resolutionMass], {
      stroke: "#ffb347",
      strokeDasharray: "7,5",
      strokeWidth: 2.5
    }),
    Plot.frame({stroke: "#111111"})
  ]
})

html`
<div class="spectrum-panel">
  <div class="spectrum-legend">
    <span>
      <i class="spectrum-swatch spectrum-swatch-massless"></i>
      ${tex`m_\nu=0`}
    </span>
    <span>
      <i class="spectrum-swatch spectrum-swatch-massive"></i>
      ${tex`m_\nu=1\ \text{эВ}`}
    </span>
  </div>
  <div class="spectrum-plot-row">
    <div class="spectrum-y-label">
      ${tex`\dfrac{d\Gamma_{\mathrm{rec}}}{dE_{\mathrm{rec}}}
      \;(\text{отн. ед.})`}
    </div>
    <div class="spectrum-plot-host">
      ${resolutionPlot}
    </div>
  </div>
  <div class="spectrum-x-label">
    ${tex`E_{\mathrm{rec}}-E_0\;(\text{эВ})`}
  </div>
</div>`

Первый график показывает истинный спектр, второй — реконструированную энергию после свёртки с гауссовым откликом. Масса и разрешение меняют форму по-разному, хотя оба эффекта сосредоточены у конечной точки.