Exercises: The KATRIN Experiment

Lecture 07 · Introduction to Neutrino Physics

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Exercises

Exercise 7.1. Final states and scattering in the source

After the molecular decay, the daughter system is left in the state \(j\) with excitation energy \(V_j\) and probability \(P_j\).

1. Write the observed spectrum as a sum of spectra with endpoints \(E_0-V_j\). Check that \(\sum_jP_j=1\) conserves the total number of decays.

2. Let the mean number of inelastic collisions of an electron in the full gas column be \(\tau=1.4\). Compute

   \[ P_s=e^{-\tau}\frac{\tau^s}{s!} \tag{1}\]

   for \(s=0,1,2\) and the probability of at least one collision.

3. The decays are uniformly distributed along the source. If the optical depth \(u\) remaining after production is uniform on \([0,\tau]\), show that

   \[ \langle P_0\rangle =\frac{1}{\tau}\int_0^\tau e^{-u}\,du =\frac{1-e^{-\tau}}{\tau}. \tag{2}\]

   Compute this number and explain its physical meaning.

4. Let \(g(\Delta E)\) be the energy-loss distribution in a single collision. Show that the loss function has the form

   \[ R(\Delta E)=\sum_{s=0}^{\infty}P_s\,g^{*s}(\Delta E), \qquad g^{*0}(\Delta E)=\delta(\Delta E), \tag{3}\]

   where \(g^{*s}\) denotes the \(s\)-fold convolution.

Exercise 7.2. What KATRIN measures

In the ideal approximation, treat the MAC-E filter as a step function:

\[ \mathcal T(E,qU)=\Theta(E-qU). \tag{4}\]

Then the expected count rate at retarding energy \(qU\) is

\[ R(qU)=A\int_{qU}^{E_0}\frac{d\Gamma}{dE}\,dE+b. \tag{5}\]

1. Introduce \(x=E_0-E\) and \(w=E_0-qU\). For a massless neutrino and \(d\Gamma/dE=Cx^2\), obtain

   \[ R(w)-b=\frac{AC}{3}w^3. \tag{6}\]

2. Show that in the ideal case

   \[ -\frac{dR}{d(qU)} =A\left.\frac{d\Gamma}{dE}\right|_{E=qU}. \tag{7}\]

   Explain why scanning the threshold recovers information about the shape of the differential spectrum.

3. Plot on a single graph \(R(qU)\) for \(m_\beta^2=0\) and \(m_\beta^2=1~\mathrm{eV}^2\) over the last \(10~\mathrm{eV}\). Use the numerical integration of the spectrum from Exercise 04.1.

4. Replace the step function with the known transmission function \(\mathcal T(E,qU)\) and write the general integral for \(R(qU)\). Then include the loss function \(R_{\mathrm{loss}}(\Delta E)\) from Exercise 07.1 as one more convolution.

5. Explain why a single point \(qU\) does not allow one to simultaneously determine \(m_\beta^2\), \(E_0\), the normalization \(A\) and the background \(b\), whereas a set of thresholds allows them to be separated by their different dependence on \(qU\).

Exercise 7.3. Why a fit can give \(m_\beta^2<0\)

This exercise links the physics of the spectrum endpoint with the course on statistical analysis. Use a deliberately simple model: \(E_0\), the normalization and the background are assumed known, and the only fit parameter is \(m_\beta^2\).

Divide the interval \(1<x<10~\mathrm{eV}\), where \(x=E_0-E\), into bins of width \(\Delta x=0.5~\mathrm{eV}\). For the center \(x_i\) set

\[ \mu_i(m_\beta^2) =A\Delta x\, \left(x_i^2-\frac{m_\beta^2}{2}\right)+b, \qquad A=3~\mathrm{eV}^{-3},\quad b=1. \tag{8}\]

This is a linearized model of the spectrum endpoint, not the full KATRIN fit. The chosen ranges of \(x\) and \(m_\beta^2\) preserve \(\mu_i>0\) at all scan points.

1. Generate independent event counts \(n_i\sim\operatorname{Poisson}[\mu_i(0)]\).

2. Scan \(m_\beta^2\) from \(-4\) to \(4~\mathrm{eV}^2\) with a step no larger than \(0.02~\mathrm{eV}^2\). For each point compute the Poisson deviance

   \[ D(m_\beta^2)=2\sum_i \left[\mu_i-n_i+n_i\ln\frac{n_i}{\mu_i}\right], \tag{9}\]

   taking the \(n_i\ln(n_i/\mu_i)\) term to be zero for \(n_i=0\).

3. Find the minimum of the grid scan. Repeat steps 1–2 at least 1000 times and construct the distribution of \(\widehat m_\beta^2\). Find the mean, the standard deviation and the fraction of results with \(\widehat m_\beta^2<0\).

4. Repeat the analysis, artificially forbidding \(m_\beta^2<0\). Compare the mean value of the estimate in the two cases and explain the bias that arises at the physical boundary.

5. Now generate pseudo-data with an unaccounted resolution, replacing the true expectation by

   \[ \mu_i^{\mathrm{true}} =A\Delta x\,(x_i^2+\sigma_E^2)+b, \qquad \sigma_E=0.5~\mathrm{eV}, \tag{10}\]

   but fit with the previous model without \(\sigma_E\). Show numerically that the center of the distribution shifts by approximately

   \[ \Delta m_\beta^2\simeq-2\sigma_E^2. \tag{11}\]

6. Explain the result: why does a negative minimum of the likelihood function not imply a tachyonic neutrino, and why does the experiment allow \(m_\beta^2\) to take negative values at the fitting stage?

Exercise 7.4. From calibration to the spectrum

1. The electron gun passes through the full gas column with optical depth \(\tau=1.4\). Find the fraction of electrons on the plateau without losses. Compare it with the fraction of β electrons produced inside the same column and moving along the axis. Why do the two fractions differ?
2. For \(E_{\mathrm{gun}}=18\,600~\mathrm{eV}\) and \(e|U|=18\,585~\mathrm{eV}\), find the excess energy \(E_s\). Which electrons will pass the ideal filter at loss \(0\), \(12.8\) and \(20~\mathrm{eV}\)? How does changing \(U\) help to recover the loss function?
3. For a divider with \(M=1972\), find the voltage on the voltmeter at \(U=-18.6~\mathrm{kV}\). What absolute stability of the high voltage corresponds to the relative requirement \(3\cdot10^{-6}\)?
4. Suppose that, because of the source potential, the electron energies are additionally distributed with zero mean and rms spread \(\sigma_E=0.2~\mathrm{eV}\). Use the estimate from Exercise 7.3 for the shift \(m_\beta^2\). What measurement inside the gas can constrain this spread?
5. Explain why the energy selection of pulses in the silicon detector does not remove all electrons produced after the analyzing barrier.

Exercise 7.5. What one needs to know about the source

The windowless source is a tube \(10~\mathrm{m}\) long and \(90~\mathrm{mm}\) in diameter with a molecular column density \(\rho d=4\cdot10^{17}~\mathrm{cm}^{-2}\). The fraction of tritium atoms among all hydrogen atoms in the gas is \(\varepsilon_T=0.98\), and the half-life of tritium is \(T_{1/2}=12.32\) years. An electron near the endpoint of the spectrum has kinetic energy \(E=18.6~\mathrm{keV}\), \(m_ec^2=511~\mathrm{keV}\), \(1~\mathrm{u}=931.5~\mathrm{MeV}\).

1. Find the number of tritium atoms in the gas column above the cross-section of the tube and the activity of the source. Compare with the characteristic activity of \(\sim10^{11}\) decays per second. Why does an instability of \(\rho d\) change not only the normalization of the spectrum but also its shape?

2. At the very endpoint of the spectrum the neutrino carries away almost no momentum. Show that the daughter molecule of mass \(M\) receives the recoil energy

   \[ E_{\mathrm{recoil}}\simeq\frac{p_e^2}{2M}, \qquad p_e^2c^2=E\,(E+2m_ec^2). \tag{12}\]

   Compute \(E_{\mathrm{recoil}}\) for \(\mathrm T_2\), \(\mathrm{DT}\) and \(\mathrm{HT}\) (\(M\simeq6.03\), \(5.03\) and \(4.02~\mathrm{u}\)). By how much do the endpoints of the decay spectra of \(\mathrm T_2\) and \(\mathrm{HT}\) differ?

3. The decaying molecule moves with a thermal velocity. The projection \(u_\parallel\) of its velocity on the direction of the emitted electron is Gaussian distributed with variance \(k_BT/M\). Show that in the laboratory frame the electron energy is shifted by \(\delta E\simeq p_eu_\parallel\), and obtain

   \[ \sigma_E=p_ec\,\sqrt{\frac{k_BT}{Mc^2}}. \tag{13}\]

   Compute \(\sigma_E\) for \(\mathrm T_2\) at \(T=30~\mathrm{K}\) and \(T=80~\mathrm{K}\).

4. Use the estimate \(\Delta m_\beta^2\simeq-2\sigma_E^2\) from Exercise 7.3. What shift would the Doppler broadening at \(80~\mathrm{K}\) produce if it were left out of the model? Why does a known broadening that is included in the model not produce such a shift?

5. The gas composition is measured from the Raman lines of molecular vibrations. In the harmonic approximation the vibrational frequency is proportional to \(\mu^{-1/2}\), where \(\mu\) is the reduced mass of the nuclei. For \(\mathrm T_2\) the line lies at \(2464~\mathrm{cm}^{-1}\). Predict the positions of the \(\mathrm{DT}\) and \(\mathrm{HT}\) lines (\(\mu\simeq1.51\), \(1.21\) and \(0.755~\mathrm{u}\) for \(\mathrm T_2\), \(\mathrm{DT}\), \(\mathrm{HT}\)) and compare with the measured values \(\simeq2743\) and \(\simeq3435~\mathrm{cm}^{-1}\). In which direction do the estimate and the measurement differ, and why is the discrepancy larger for the lighter molecule? Find the wavelength of the \(\mathrm T_2\) Stokes line for a laser with \(\lambda=532~\mathrm{nm}\).

State the main conclusion: which properties of the source change only the number of counts, and which change the shape of the spectrum near the endpoint and therefore enter \(m_\beta^2\) directly?

Exercise 7.6. The krypton line as an energy standard

In the \(^{83\mathrm m}\mathrm{Kr}\) nucleus the transition energy \(\Delta E_{\mathrm{nucl}}=32.152~\mathrm{keV}\) is transferred to an electron of the atomic shell. The binding energies are \(14.326~\mathrm{keV}\) for the \(K\) shell and \(1.678~\mathrm{keV}\) for the \(L_3\) subshell. The atomic mass is \(M\simeq82.91~\mathrm{u}\), and the natural width of the \(K\)-32 line is \(\Gamma=2.7~\mathrm{eV}\).

1. Find the energies of the \(K\)-32 and \(L_3\)-32 conversion electrons neglecting recoil. Then estimate the recoil energy of the atom \(E_{\mathrm{recoil}}=p_e^2/2M\) for both lines. Compare it with the requirement \(|\delta E|\lesssim56~\mathrm{meV}\) on the energy scale.

2. The edges of the two lines are observed at voltmeter readings \(u_K\) and \(u_L\) behind a divider with ratio \(M_{\mathrm{div}}\). Show that

   \[ M_{\mathrm{div}} =\frac{E_{L_3}-E_K}{e\,\bigl(|u_L|-|u_K|\bigr)}, \tag{14}\]

   and that both the nuclear transition energy and the shift due to the potentials, common to the two lines, cancel in the difference. How precisely must the energy difference be known to obtain \(M_{\mathrm{div}}\) with a relative precision of \(5\cdot10^{-6}\)? Can the difference of the recoil energies then be neglected?

3. The line has a Lorentzian shape

   \[ L(E)=\frac{\Gamma/(2\pi)}{(E-E_K)^2+\Gamma^2/4}. \tag{15}\]

   For an ideal step filter, show that the fraction of transmitted electrons is

   \[ \frac{R(qU)}{R_{\max}} =\frac12-\frac1\pi\arctan\frac{2(qU-E_K)}{\Gamma}. \tag{16}\]

   Find the width of the edge between the \(90\%\) and \(10\%\) levels.

4. Numerical experiment. Draw \(N\) energies from the Lorentzian with \(\Gamma=2.7~\mathrm{eV}\) for \(N=10^2\), \(10^3\), \(10^4\), \(10^5\) and estimate the center of the line in two ways: by the arithmetic mean and by the median. Repeat the sampling at least \(300\) times and compare the spread of the estimates. Why does the mean not improve as \(N\) grows? Using the asymptotic formula \(\sigma_{\mathrm{med}}\simeq\pi\Gamma/(4\sqrt N)\), find how many electrons are needed to determine the center to within \(50~\mathrm{meV}\).

5. The monitor spectrometer sees that the line of a stable source has shifted by \(30~\mathrm{meV}\). What relative drift of the \(18.6~\mathrm{kV}\) high voltage does this correspond to? Does it meet the requirement \(3\cdot10^{-6}\)?

State the main conclusion: why does a line \(2.7~\mathrm{eV}\) wide calibrate the scale at the level of tens of millielectronvolts?

Exercise 7.7. Where measuring pays off most

Take a simplified model of the integral spectrum near the endpoint: a sharp filter, a single final state, all parameters except \(m_\beta^2\) known, and \(w=E_0-qU\):

\[ R(w)=\kappa\left(w^3-\frac32m_\beta^2w\right)+b, \qquad \kappa=10^{-3}~\mathrm{s}^{-1}\mathrm{eV}^{-3}, \quad b=0.2~\mathrm{s}^{-1}. \tag{17}\]

The values of \(\kappa\) and \(b\) are illustrative; they only set the scale.

1. Show that the relative change of the signal due to the mass is \(-3m_\beta^2/(2w^2)\). Compute it for \(m_\beta^2=0.2~\mathrm{eV}^2\) at \(w=2\), \(7\) and \(20~\mathrm{eV}\). Why is the point right at the endpoint, where the effect is largest, not necessarily the most favorable one?

2. At the point \(w\) one measures for a time \(t\) and obtains \(n\sim\operatorname{Poisson}(tR)\). Show that the error of the estimate of \(m_\beta^2\) from a single point at \(m_\beta^2\simeq0\) is

   \[ \sigma^2(m_\beta^2)=\frac{1}{t\,I(w)}, \qquad I(w)=\frac{1}{R}\left(\frac{\partial R}{\partial m_\beta^2}\right)^2 =\frac{9\kappa^2w^2}{4(\kappa w^3+b)}. \tag{18}\]

3. Find the value \(w_{\mathrm{opt}}\) at which \(I(w)\) is maximal, and show that at this point the signal is twice the background: \(\kappa w_{\mathrm{opt}}^3=2b\). Compute \(w_{\mathrm{opt}}\).

4. Let the whole time \(T=259\) days be spent at \(w_{\mathrm{opt}}\). Find \(\sigma(m_\beta^2)\) and compare with the statistical error \(0.108~\mathrm{eV}^2\) of the first five KATRIN campaigns. Name two reasons why the toy estimate is too optimistic.

5. Show that at the optimal point

   \[ \sigma(m_\beta^2)\propto\kappa^{-2/3}\,b^{1/6}\,T^{-1/2}. \tag{19}\]

   By what factor does the error decrease if the background is halved? If the signal is doubled?

6. The shifted analyzing region reduced the background roughly by half but widened the transmission edge from \(0.95\) to \(2.7~\mathrm{eV}\). Why does a known edge width, unlike the unaccounted broadening of Exercise 7.3, not shift \(m_\beta^2\), yet still reduce the sensitivity? Why is the choice between the two configurations made on the basis of the total error rather than of either of these numbers alone?

State the main conclusion: why does a real scan distribute the time over many points below and above the endpoint instead of spending it at a single optimal point?

Exercise 7.8. Calibration in the likelihood function

Suppose that near the minimum the spectral data are sensitive only to a combination of the squared mass and the optical depth: a shift of \(\tau\) by \(\delta\tau\) can be compensated by a shift of \(m^2\equiv m_\beta^2\) by \(-k\,\delta\tau\). Then

\[ D_\beta(m^2,\tau)= \frac{\bigl[m^2+k(\tau-\tau_0)-\mu\bigr]^2}{\sigma_s^2}, \tag{20}\]

while the electron gun gave \(\tau_{\mathrm{cal}}\pm\sigma_\tau\).

1. Show that \(D_\beta\) has no unique minimum in the \((m^2,\tau)\) plane. What does the region of small \(D_\beta\) look like, and what does it mean physically?

2. Add the calibration term \((\tau-\tau_{\mathrm{cal}})^2/\sigma_\tau^2\) and minimize over \(\tau\) at fixed \(m^2\). Show that

   \[ -2\ln L_{\mathrm{prof}}(m^2) =\frac{\bigl[m^2+k(\tau_{\mathrm{cal}}-\tau_0)-\mu\bigr]^2} {\sigma_s^2+k^2\sigma_\tau^2}. \tag{21}\]

   What is the error of \(m^2\)? Consider the limits \(\sigma_\tau\to0\) and \(\sigma_\tau\to\infty\).

3. What happens if one simply fixes \(\tau=\tau_{\mathrm{cal}}\)? Why does such an analysis underestimate the mass error?

4. The contributions to \(\sigma(m_\beta^2)\) in the first five KATRIN campaigns, in \(\mathrm{eV}^2\), are: statistics \(0.108\); optical depth \(0.052\); energy-loss distribution \(0.034\); background dependent on the step duration \(0.027\); changes of the source potential \(0.022\); non-Poissonian background fluctuations \(0.015\). Add the systematic contributions in quadrature, then add the statistics. Compare with the result \(m_\beta^2=-0.14^{+0.13}_{-0.15}~\mathrm{eV}^2\). What fraction of the total variance comes from statistics? Why is such a sum only an estimate?

5. Assume that for a zero true mass the \(90\%\) sensitivity corresponds to \(m_\beta^2\simeq1.64\,\sigma(m_\beta^2)\). Estimate the sensitivity for the total error from item 4. What total error is needed for a sensitivity of \(0.30~\mathrm{eV}\)? Compare it with the statistical error scaled from \(259\) to \(1000\) days by the \(1/\sqrt t\) law. What, besides the data-taking time, has to improve?

State the main conclusion: why is the calibration included in the joint fit as an additional term rather than simply substituting the measured value?

Exercise 7.9. A kink from a heavy neutrino

Suppose that, in addition to the light states, there is a mass state \(m_4\) of order keV with admixture \(|V_{e4}|^2\ll1\). With the notation \(x=E_0-E\) and the spectrum of Exercise 04.1,

\[ S(E)=\bigl(1-|V_{e4}|^2\bigr)S_0(E)+|V_{e4}|^2S_{m_4}(E), \qquad S_m(E)=C\,p_e(E+m_e)\,x\sqrt{x^2-m^2}\,\Theta(x-m). \tag{22}\]

1. Show that the relative distortion of the spectrum is

   \[ r(E)\equiv\frac{S}{S_0}-1= \begin{cases} -|V_{e4}|^2, & x<m_4,\\[1mm] |V_{e4}|^2\left(\sqrt{1-m_4^2/x^2}-1\right), & x>m_4. \end{cases} \tag{23}\]

   Why does the region \(E>E_0-m_4\) say nothing about the heavy state if the normalization of the spectrum is a free parameter?

2. Show that near the threshold \(S_{m_4}\propto\sqrt{x-m_4}\), and explain why a kink appears in the spectrum. Compute \(r/|V_{e4}|^2\) for \(x/m_4=1.01\), \(1.1\), \(1.5\) and \(2\).

3. Find numerically the fraction of all decays with \(E<E_0-m_4\) for \(m_4=2\), \(5\) and \(10~\mathrm{keV}\) (take the Fermi function equal to one). Where in the spectrum should one look for the kink at \(m_4=10~\mathrm{keV}\), and are there many electrons there?

4. The distortion is of order \(|V_{e4}|^2\). Estimate how many electrons \(N\) are needed to see it at the level of three standard deviations for \(|V_{e4}|^2=10^{-6}\). How long does it take to collect them at a count rate of \(10^8~\mathrm{s}^{-1}\)? Why is statistics not the main difficulty?

5. Why is the integral mode, in which the points are collected by sequential scanning of the voltage, inconvenient for this search, and what does a detector that measures the energy of each electron give?

Compare the two searches: for \(m_\beta\) the effect is noticeable but there are few events near the endpoint; for \(m_4\) there are many events but the effect is small. What limits each of them?