Exercises: Other Ways to Weigh the Neutrino

Lecture 08 · Introduction to Neutrino Physics

Author

Dmitry V. Naumov

← Back to the lecture · All course exercises

Exercises

Exercise 8.1. From phase modulation to the CRES spectrum

Consider the model of the received signal

\[ V(t)=V_0\cos\!\left[2\pi f_ct+h\sin(2\pi f_zt)\right]. \tag{1}\]

  1. Compute the instantaneous frequency

    \[ f_{\mathrm{inst}}(t)=\frac{1}{2\pi}\frac{d\Phi}{dt} \tag{2}\]

    and show that the amplitude of its deviation is \(\Delta f=h f_z\).

  2. Using the Jacobi–Anger identity

    \[ e^{ih\sin\theta}=\sum_{n=-\infty}^{\infty}J_n(h)e^{in\theta}, \tag{3}\]

    obtain the frequencies \(f_n=f_c+n f_z\) and the relative powers \(J_n^2(h)\).

  3. In the limit \(h\ll1\) use

    \[ J_0(h)\simeq1-\frac{h^2}{4}, \qquad J_{\pm1}(h)\simeq\pm\frac h2 \tag{4}\]

    and find the fraction of power in the carrier and in the first sidebands.

  4. Using scipy.special.jv construct the line weights for \(h=0.2\), \(1.2\) and \(3.0\). Check \(\sum_nJ_n^2(h)=1\) numerically.

Exercise 8.2. A real trap and a finite CRES track

Suppose that near the center of the magnetic trap

\[ B(z)=B_0\left(1+\frac{z^2}{L^2}\right), \qquad z(t)=z_{\max}\cos(2\pi f_zt), \tag{5}\]

and that the energy loss over one longitudinal period can be neglected.

  1. Substitute \(z(t)\) into \(B(z)\) and show that the magnetic modulation of the frequency occurs at twice the longitudinal frequency \(2f_z\).

  2. Find the mean cyclotron frequency and the amplitude of its modulation \(\Delta f_B\) in terms of \(f_{c0}\) and \(z_{\max}/L\).

  3. Integrate the instantaneous frequency and show that the index of the magnetic phase modulation is equal to

    \[ h_B=\frac{\Delta f_B}{2f_z}. \tag{6}\]

  4. Explain why the signal propagation phase, proportional to \(z(t)\), also creates sidebands with spacing \(f_z\).

  5. Compute the Fourier transform of a harmonic observed only on the interval \(0<t<T_{\mathrm{obs}}\), and obtain the shape \(T_{\mathrm{obs}}^2\operatorname{sinc}^2[\pi(f-f_n)T_{\mathrm{obs}}]\).

  6. For what \(T_{\mathrm{obs}}\) does the full width at half maximum of the Fourier peak correspond to \(\Delta E=0.10~\mathrm{eV}\), that is, \(\Delta f_c=5.1~\mathrm{kHz}\) near the endpoint of the tritium spectrum? Use a rectangular time window. Explain why the center of the line can be determined more precisely than its width, and how the signal-to-noise ratio enters this precision.

Exercise 8.3. The calorimetric spectrum of \({}^{163}\mathrm{Ho}\)

In a calorimetric experiment the entire atomic excitation energy \(E_c\) is measured, and the neutrino carries away

\[ E_\nu=Q-E_c. \tag{7}\]

Use the model

\[ \frac{d\Gamma}{dE_c} =C\,(Q-E_c)\sqrt{(Q-E_c)^2-m_\nu^2}\, \Theta(Q-E_c-m_\nu) \sum_j w_jL_j(E_c), \tag{8}\]

where the atomic line has the shape

\[ L_j(E_c)=\frac{\Gamma_j/(2\pi)} {(E_c-E_j)^2+\Gamma_j^2/4}. \tag{9}\]

  1. Show that \(\int_{-\infty}^{\infty}L_j(E_c)\,dE_c=1\).

  2. Find the maximum calorimetric energy and explain why the neutrino mass is again measured from the shape of the spectrum near the endpoint.

  3. Construct a toy spectrum for two lines with \((E_j,\Gamma_j,w_j)\) of your choice at \(m_\nu=0\) and \(m_\nu=1~\mathrm{eV}\).

  4. Let the activity of a single pixel be \(a\), and let two pulses be unresolved if the interval between them is smaller than \(\tau_r\). For \(a\tau_r\ll1\) show that the fraction of unresolved pile-up events is of order

    \[ f_{\mathrm{pu}}\simeq a\tau_r, \tag{10}\]

    and that their energy spectrum is proportional to the convolution \((d\Gamma/dE_c)*(d\Gamma/dE_c)\).

  5. Estimate \(f_{\mathrm{pu}}\) for \(a=10~\mathrm{s}^{-1}\) and \(\tau_r=1~\mathrm{\mu s}\). Why does increasing the activity improve the statistics but at the same time strengthen this background?

  6. In one paragraph, compare the systematics of the KATRIN spectrometer and of the calorimeter: where the source is located, which energy is measured, and why the calorimeter eliminates the losses on the electron’s exit from the source but acquires the problem of unresolved pile-up.

Exercise 8.4. The radio chain: cutoff, cavity, cascade

Femtowatt-level power must be brought to the ADC without losing anything in frequency. The formulas of the chain are given in the lecture without derivation; derive them yourself.

  1. The field in a waveguide vanishes at the metal walls. From the condition that at least a half-wavelength must fit across a tube of diameter \(d\), obtain an estimate of the cutoff frequency

    \[ f_{\mathrm{cr}}\simeq\frac{c}{2d} \tag{11}\]

    and compute it for \(d=10\) mm. The exact solution for a circular tube of radius \(a\) gives \(f_{\mathrm{cr}}=1.841\,c/2\pi a\). Compare with the estimate and explain the sign of the discrepancy.

  2. The quality factor \(Q=f/\Delta f\) is equal to \(2\pi\) times the ratio of the energy stored in the cavity to the losses per period. Show that the field decays over a time \(\tau_Q=Q/2\pi f\), and compute \(\tau_Q\) and the bandwidth \(\Delta f\) for \(Q=10^4\) at \(26\) GHz. Compare \(\tau_Q\) with the track duration \(100\) μs.

  3. An amplifier with gain \(G\) and noise temperature \(T\) adds noise power \(kT\Delta f\). Refer the noise of two successive stages to the input of the first and obtain

    \[ T_{\mathrm{sys}}=T_1+\frac{T_2}{G_1}, \tag{12}\]

    then generalize to \(n\) stages.

  4. Let the first stage be cooled (\(T_1=20\) K, \(G_1=20\) dB), and the second operate at room temperature (\(T_2=300\) K, \(G_2=20\) dB). Find \(T_{\mathrm{sys}}\). Swap the stages and find it again. By what factor does the noise power in the band change?

  5. A mixer multiplies \(\cos(2\pi f_ct)\) and \(\cos(2\pi f_{\mathrm{ref}}t)\). Write down the result and state which of the two frequencies the filter keeps. For what \(f_{\mathrm{ref}}\) do the signals \(26.0\) and \(26.1\) GHz end up at the same intermediate frequency?

  6. The ADC records a band \(B\) around the intermediate frequency. Formulate the Nyquist condition and estimate the sampling frequency for \(B=100\) MHz. Where will the line \(1.52\) GHz end up if it is digitized at a frequency of \(1\) GHz?

Exercise 8.5. A femtowatt: power, noise, and observation time

  1. From the Larmor formula for an accelerated charge obtain the power of the cyclotron radiation

    \[ P_{\mathrm{rad}} =\frac{e^4B^2}{6\pi\varepsilon_0m_e^2c}\,\gamma^2\beta^2\sin^2\alpha \tag{13}\]

    and compute it for \(E=18.6\) keV, \(B=1\) T, \(\alpha=90^\circ\).

  2. How much energy does the electron lose in \(T_{\mathrm{obs}}=100\) μs? Express the answer in eV and compare with the target precision \(0.1\) eV.

  3. The receiver noise gives a power \(kT_{\mathrm{n}}\) per hertz of bandwidth, and the bandwidth of a single Fourier line is \(\Delta f=0.886/T_{\mathrm{obs}}\). Find the noise power in the line for \(T_{\mathrm{n}}=132\) K and \(T_{\mathrm{obs}}=100\) μs and the signal-to-noise ratio in the ideal case where all the radiated power reaches the amplifier.

  4. Show that for a fixed signal power the signal-to-noise ratio in a single line grows in proportion to \(T_{\mathrm{obs}}\). What prevents lengthening the recording indefinitely? Take into account item 2 and the inelastic collisions with gas molecules from Lecture 07.

  5. A smartphone antenna receives about \(10^{-10}\) W from a base station, and about \(1.4\cdot10^{-16}\) W from a GPS satellite. Place on one logarithmic scale these quantities, the power of the phone’s transmitter (\(0.2\) W), the CRES signal, and the noise from item 3. Which of the comparisons determines the detectability of the signal?

Exercise 8.6. Missing momentum: the pion and the tau

  1. For the decay of a pion at rest \(\pi^+\to\mu^++\nu_\mu\) obtain

    \[ m_\nu^2=m_\pi^2+m_\mu^2-2m_\pi\sqrt{p_\mu^2+m_\mu^2}. \tag{14}\]

  2. Set \(m_\nu=0\) and find \(p_\mu=(m_\pi^2-m_\mu^2)/2m_\pi\) numerically for \(m_\pi=139.57\) MeV and \(m_\mu=105.66\) MeV.

  3. Differentiate Equation 14 and obtain

    \[ \delta m_\nu^2=-2m_\pi\frac{p_\mu}{E_\mu}\,\delta p_\mu. \tag{15}\]

    What relative precision of the measurement of \(p_\mu\) is needed to be sensitive to \(m_\nu=1\) eV? Compare with the limit achieved, \(170\) keV.

  4. For the decay \(\tau^-\to X^-+\nu_\tau\) in the τ rest frame obtain

    \[ E_X^*=\frac{m_\tau^2+m_X^2-m_\nu^2}{2m_\tau}, \qquad m_X\le m_\tau-m_\nu. \tag{16}\]

  5. In the lab the τ moves with \(\gamma_\tau\), \(\beta_\tau\). Show that for a given \(m_X\) the energy of the visible system fills the band \(E_X=\gamma_\tau\left(E_X^*\pm\beta_\tau p_X^*\right)\), where \(p_X^*=\sqrt{(E_X^*)^2-m_X^2}\). Why is the mass sought from the two-dimensional boundary in the \((m_X,E_X)\) plane?

  6. By what factor are the limits \(m_{\nu_\mu}<170\) keV and \(m_{\nu_\tau}<18.2\) MeV weaker than the tritium limit \(m_\beta<0.45\) eV? Explain the loss in one sentence about the scale of the energy release.

Exercise 8.7. Field, frequency and track jumps

A CRES cell sits in a field \(B=0.959~\mathrm{T}\); \(m_ec^2=511.0~\mathrm{keV}\), the electron energy near the endpoint of the tritium spectrum is \(E=18.6~\mathrm{keV}\), and the radiated power is \(P_{\mathrm{rad}}=1.2~\mathrm{fW}\). The cross section for inelastic scattering of an electron on a \(\mathrm T_2\) molecule at this energy is \(\sigma_{\mathrm{inel}}\simeq3.4\cdot10^{-18}~\mathrm{cm}^2\), and the typical energy loss in a single collision is \(12.8~\mathrm{eV}\).

  1. Find \(f_0=eB/(2\pi m_e)\) and the cyclotron frequency

    \[ f_c=\frac{f_0}{1+E/m_ec^2} \tag{17}\]

    near the endpoint of the spectrum and for the krypton line \(E=17.83~\mathrm{keV}\). How many kilohertz correspond to \(1~\mathrm{eV}\)?

  2. Show that for simultaneous errors in the field and in the frequency

    \[ \delta E=(m_ec^2+E)\left(\frac{\delta B}{B}-\frac{\delta f_c}{f_c}\right). \tag{18}\]

    What \(\delta B/B\) gives \(\delta E=0.10~\mathrm{eV}\)? Express \(\delta B\) in microtesla and compare with the Earth’s magnetic field (\(\sim50~\mathrm{\mu T}\)). Why is the field calibrated with the electrons of known energy themselves, and not only with a magnetometer?

  3. Find the frequency drift rate \(\dot f_c=f_cP_{\mathrm{rad}}/(m_ec^2+E)\) and the frequency shift over \(100~\mathrm{\mu s}\). In which direction does the frequency drift, and why?

  4. Find the frequency jump after a collision with a loss of \(12.8~\mathrm{eV}\). Compare it with the drift from item 3 and with the linewidth \(0.886/T_{\mathrm{obs}}\) for \(T_{\mathrm{obs}}=100~\mathrm{\mu s}\). Why do collisions show up on the track as separate steps?

  5. At what number density of molecules \(n\) is the mean time between collisions \(1/(n\sigma_{\mathrm{inel}}v)\) equal to \(100~\mathrm{\mu s}\)? Compare with the mean number density in the KATRIN source, where the column density \(4\cdot10^{17}~\mathrm{cm}^{-2}\) is accumulated over a length of \(10~\mathrm{m}\).

  6. Estimate the activity of \(1~\mathrm{cm}^3\) of such a gas (\(T_{1/2}=12.32\) years, two tritium atoms per molecule) and the number of decays per year with the electron in the last \(10~\mathrm{eV}\), if the fraction of decays in the last interval \(\Delta E\) is approximately \(2\cdot10^{-13}\,(\Delta E/1~\mathrm{eV})^3\).

State the main conclusion: why is CRES limited not by its resolution but by the volume of the source, and why do the next stages need a volume of the order of a cubic meter?

Exercise 8.8. The microcalorimeter: heating, noise and the choice of isotope

An absorber with heat capacity \(C=10^{-12}~\mathrm{J/K}\) operates at \(T=20~\mathrm{mK}\); the energy release of the \(^{163}\mathrm{Ho}\) decay is \(Q\simeq2.86~\mathrm{keV}\). Take the heat capacity to be independent of temperature over a small interval. The half-lives are \(4570\) years for \(^{163}\mathrm{Ho}\) and \(4.3\cdot10^{10}\) years for \(^{187}\mathrm{Re}\).

  1. Find the heating \(\Delta T\) from a decay with \(E_c=Q\) and the relative change of the magnetization of a paramagnetic sensor obeying the Curie law \(M\propto1/T\).

  2. In the canonical ensemble the energy of a body fluctuates:

    \[ \langle\delta E^2\rangle=k_BT^2C. \tag{19}\]

    Derive this relation from \(\langle E\rangle=-\partial\ln Z/\partial\beta\), \(\beta=1/k_BT\). Compute \(\sqrt{\langle\delta E^2\rangle}\) in electronvolts and the corresponding full width at half maximum \(2.355\sqrt{\langle\delta E^2\rangle}\). Compare with the resolution of about \(6~\mathrm{eV}\) achieved by HOLMES.

  3. A superconducting thermometer with resistance \(R(T)\) is held at constant voltage, so the Joule power is \(P_J=V^2/R\). Introduce \(\alpha=d\ln R/d\ln T\) and linearize the heat-balance equation \(C\,d\Delta T/dt=-G\Delta T+\Delta P_J\). Show that

    \[ \tau_{\mathrm{eff}}=\frac{C/G}{1+\mathcal L}, \qquad \mathcal L=\frac{\alpha P_J}{GT}. \tag{20}\]

    Why is the pulse shorter than \(C/G\) for \(\alpha>0\), and how does this help at high activity?

  4. Find the number of atoms and the mass of the isotope needed for an activity \(a=10~\mathrm{s}^{-1}\) in a single pixel, for \(^{163}\mathrm{Ho}\) and for \(^{187}\mathrm{Re}\). By what factor do the numbers of atoms differ?

  5. Explain why holmium can be implanted into a small absorber, whereas rhenium would have to be the absorber itself, and how this affects \(C\) and the heating from item 1.

State the main conclusion: why is a small \(Q\) useful only together with a short half-life and a small heat capacity?

Exercise 8.9. What did supernova 1987A tell us about the mass?

A neutrino of mass \(m\) and energy \(E_\nu\) is delayed by

\[ \Delta t\simeq5.15~\mathrm{ms} \left(\frac{L}{10~\mathrm{kpc}}\right) \left(\frac{m}{1~\mathrm{eV}}\right)^2 \left(\frac{10~\mathrm{MeV}}{E_\nu}\right)^2. \tag{21}\]

Supernova 1987A exploded in the Large Magellanic Cloud, \(L\simeq50~\mathrm{kpc}\). The Kamiokande-II detector recorded about a dozen events within \(\simeq12.4~\mathrm{s}\) with energies from roughly \(7.5\) to \(35~\mathrm{MeV}\); take the neutrino energies equal to these values.

  1. Find the delay for \(E_\nu=10\) and \(30~\mathrm{MeV}\) at \(m=1\) and \(10~\mathrm{eV}\).
  2. Require that the difference between the delays at \(7.5\) and \(35~\mathrm{MeV}\) not exceed the duration of the signal. What upper bound on \(m\) does this give? What assumption about the emission time of neutrinos of different energies is made here? Compare the bound with \(15~\mathrm{eV}\) from ECHo and \(0.45~\mathrm{eV}\) from KATRIN.
  3. Can neutrinos outrun light? The neutrinos arrived about three hours before the light. What relative velocity difference \((v-c)/c\) would this correspond to? Explain why in reality the light simply left the star later. In 2011 a measurement of \((v-c)/c\simeq2.5\cdot10^{-5}\) was reported for accelerator neutrinos; the cause later turned out to be an equipment fault. How many years before the light would the neutrinos from SN 1987A have arrived at such a velocity?
  4. Suppose that for a future supernova at a distance of \(10~\mathrm{kpc}\) the signal is cut off abruptly, and the moment of the cutoff for \(E_\nu=10\) and \(30~\mathrm{MeV}\) can be compared with a precision of \(1~\mathrm{ms}\). What mass could one be sensitive to?
  5. Each mass state is delayed in its own way. Estimate the delay for \(m=0.45~\mathrm{eV}\), \(E_\nu=10~\mathrm{MeV}\), \(L=10~\mathrm{kpc}\) and the difference between the delays of two states if \(|m_3^2-m_1^2|=2.5\cdot10^{-3}~\mathrm{eV}^2\). Can the mass states be separated by their arrival times?

State the main conclusion: why does the time-of-flight method not need a model of the β spectrum, but does need a model of the source?

Exercise 8.10. Three ways to weigh: \(m_\beta\), \(m_{\beta\beta}\) and \(\Sigma\)

Use rounded parameters: \(|V_{e2}|^2=0.30\), \(|V_{e3}|^2=0.022\), \(|V_{e1}|^2=1-|V_{e2}|^2-|V_{e3}|^2\), \(\Delta m_{21}^2=7.5\cdot10^{-5}~\mathrm{eV}^2\), \(|\Delta m_{31}^2|=2.5\cdot10^{-3}~\mathrm{eV}^2\). The three observables are

\[ m_\beta^2=\sum_i|V_{ei}|^2m_i^2, \qquad m_{\beta\beta}=\Bigl|\sum_i|V_{ei}|^2e^{i\phi_i}m_i\Bigr|, \qquad \Sigma=m_1+m_2+m_3, \tag{22}\]

where the phases \(\phi_i\) are unknown.

  1. Find the three masses for the normal ordering with \(m_1=0\) and for the inverted ordering with \(m_3=0\) (for the inverted ordering \(m_1^2=|\Delta m_{31}^2|\), \(m_2^2=m_1^2+\Delta m_{21}^2\)).
  2. Compute \(m_\beta\) and \(\Sigma\) for both orderings.
  3. Find the smallest and the largest \(m_{\beta\beta}\) for arbitrary phases for both orderings. Hint: the modulus of a sum of three vectors of lengths \(a_i\) lies between \(\max\bigl(0,\,2\max a_i-\sum a_i\bigr)\) and \(\sum a_i\).
  4. Plot the three quantities as functions of the lightest mass from \(10^{-4}\) to \(1~\mathrm{eV}\) for both orderings. Find the interval of \(m_1\) in which \(m_{\beta\beta}\) can vanish for the normal ordering. Why does \(m_\beta\) never vanish?
  5. The KATRIN limit \(m_\beta<0.45~\mathrm{eV}\) lies in the region of nearly degenerate masses. What constraint on \(\Sigma\) follows from it? Which mass ordering can an experiment with a sensitivity of \(m_\beta\simeq40~\mathrm{meV}\) test, and why is such a goal insufficient for the normal ordering?

State the main conclusion: why can the three mass measurements not be compared as three measurements of the same number?

Seminar

S8.1. How long to listen to the electron

A long recording narrows the line, but during that time the electron manages to lose energy to radiation. Let us find the compromise.

  1. Convert the linewidth into an energy error, \(\delta E_{\mathrm{lin}}=(m_ec^2+E)\,\Delta f/f_c\) at \(\Delta f=0.886/T_{\mathrm{obs}}\), and fill in the table for \(T_{\mathrm{obs}}=30\), \(100\), \(300\) and \(1000\) μs.
  2. For the same times find the total energy loss to radiation \(\Delta E_{\mathrm{rad}}=P_{\mathrm{rad}}T_{\mathrm{obs}}\).
  3. Suppose the track model reconstructs this loss with a precision of \(10\%\), so that the residual systematic is \(0.1\,\Delta E_{\mathrm{rad}}\). Add it to \(\delta E_{\mathrm{lin}}\) in quadrature and find the optimal \(T_{\mathrm{obs}}\).
  4. Discuss what else terminates the track: inelastic scattering on the gas and the escape of the electron from the trap. How do these processes change the optimum you found?

S8.2. The trap: statistics versus resolution

Let the field at the edges of the trap be \(B_{\max}=B_0(1+\eta)\), \(\eta\ll1\).

  1. From the reflection condition obtain \(|\cos\alpha_0|<\sqrt{\eta/(1+\eta)}\simeq\sqrt\eta\).
  2. For an isotropic source \(\cos\alpha_0\) is uniformly distributed on \([-1,1]\). Show that the fraction of trapped electrons is \(\sqrt\eta\), and compute it for \(\eta=10^{-1}\), \(10^{-2}\), \(10^{-3}\).
  3. A trapped electron passes through fields from \(B_0\) to \(B_{\max}\). Show that without a trajectory correction this smears the reconstructed energy by \(\delta E\simeq(m_ec^2+E)\,\eta\), and compute \(\delta E\) for the same \(\eta\).
  4. Choose \(\eta\) such that the fraction of trapped electrons is at least \(1\%\), and the smearing is at most \(0.1\) eV. Does it work out? What conclusion follows about the need to reconstruct the trajectory from the sidebands?

S8.3. Neutrino mass from arrival times

A continuation of the applet “Mass from arrival times” and a bridge to the course on statistical analysis.

  1. From \(\Delta t=L\,m^2c^4/2cE_\nu^2\) obtain the working form

    \[ \Delta t\simeq5.15~\text{ms} \left(\frac{L}{10~\text{kpc}}\right) \left(\frac{m}{1~\text{eV}}\right)^2 \left(\frac{10~\text{MeV}}{E_\nu}\right)^2. \tag{23}\]

  2. Let \(N\) neutrinos be produced at the moment \(t_0\) with a Gaussian spread \(\sigma_t\), and let pairs \((E_i,t_i)\) be measured. Write down \(-2\ln L(m^2,t_0)\), eliminate \(t_0\) analytically, and show that what remains is the sum of squared deviations from a straight line in the variable \(x_i=(10~\text{MeV}/E_i)^2\).

  3. Obtain the estimate of the slope and its variance:

    \[ \hat m^2=\frac{S_{xt}}{5.15\,S_{xx}}, \qquad \sigma(\hat m^2)=\frac{\sigma_t}{5.15\sqrt{S_{xx}}}, \tag{24}\]

    where \(S_{xx}=\sum_i(x_i-\bar x)^2\), \(S_{xt}=\sum_i(x_i-\bar x)(t_i-\bar t)\). Explain why \(\hat m^2\) can turn out to be negative.

  4. Take \(N=15\) energies uniformly covering \(6\)–\(35\) MeV and \(\sigma_t=2\) ms. Compute \(\sigma(\hat m^2)\). How many events are needed for \(\sigma(\hat m^2)=0.1\) eV²? Compare with the two dozen events from SN 1987A.

  5. Check your formulas with the applet: for zero true mass estimate how often the estimate comes out negative.

S8.4. Holmium: pixels versus pile-up

  1. For a Poisson stream with activity \(a\), show that the fraction of events with an unresolved neighbor within the resolving time \(\tau_r\) is equal to \(1-e^{-a\tau_r}\simeq a\tau_r\).

  2. Let the total activity \(A\) be distributed over \(N\) identical pixels. Obtain the rate of unresolved pairs

    \[ R_{\mathrm{par}}\simeq\frac{A^2\tau_r}{N} \tag{25}\]

    and explain why, at fixed \(A\), splitting into pixels helps.

  3. Calculate \(R_{\mathrm{par}}\) for \(A=10^6~\text{s}^{-1}\), \(\tau_r=1\) μs at \(N=10^3\) and \(N=10^5\).

  4. Estimate the fraction of events in the last \(10\) eV of the holmium spectrum, assuming the phase space \(\propto(Q-E_c)^2\) at \(Q=2.86\) keV. Compare the useful count rate with \(R_{\mathrm{par}}\) from item 3 and say whether splitting into pixels alone is sufficient.

  5. From \(\Delta T=E_c/C\) find the heating at \(E_c=2.86\) keV and \(C=10^{-12}\) J/K. How does the choice of \(C\) and the thermal coupling \(G\) conflict with the requirement of small \(\tau_r\)?