Exercises: Kinematics of the β Spectrum

Lecture 04 · Introduction to Neutrino Physics

Author

Dmitry V. Naumov

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Exercises

Exercise 4.1. Spectrum of \(\beta\) decay

Consider an allowed \(\beta\) decay with a single neutrino mass state. Neglect the nuclear recoil and radiative corrections. Take \(F(Z,E)=1\).

Let \(E\) be the kinetic energy of the electron. Use

\[ \frac{d\Gamma}{dE} = C\,p_e(E+m_e)(E_0-E) \sqrt{(E_0-E)^2-m_\nu^2}\, \Theta(E_0-E-m_\nu), \tag{1}\]

where

\[ p_e=\sqrt{E^2+2m_eE}, \qquad m_e=511~\mathrm{keV}, \qquad E_0=18.575~\mathrm{keV}. \tag{2}\]

  1. Write a function that computes the unnormalized spectrum \(d\Gamma/dE\).

  2. For \(m_\nu=0\) and \(m_\nu=1~\mathrm{eV}\) make two panels: the entire allowed range and the last \(5~\mathrm{eV}\). Use the same normalization for both masses.

  3. Determine the maximum kinetic energy of the electron as a function of \(m_\nu\) and mark both endpoints of the spectrum on the zoomed plot.

  4. For \(m_\nu=0\) compute the fraction of all decays falling into the last \(1~\mathrm{eV}\) and \(5~\mathrm{eV}\). Compare the numerical result with the estimate

    \[ \frac{N(E>E_0-\Delta E)}{N_{\mathrm{all}}} \sim\left(\frac{\Delta E}{E_0}\right)^3, \tag{3}\]

    obtained from the \(d\Gamma/dE\propto(E_0-E)^2\) approximation.

  5. Convolve the spectrum with a Gaussian resolution function with \(\sigma_E=0.5~\mathrm{eV}\). Show that the reconstructed energy can lie above the physical endpoint \(E_0\), even though the true spectrum is zero there.

Exercise 4.2. Effective mass and the Kurie plot

An electron antineutrino is produced as a mixture of mass states. Near the endpoint of the spectrum, set \(E_\nu=E_0-E\) and use

\[ \frac{d\Gamma}{dE} =C(E)\sum_i|U_{ei}|^2E_\nu \sqrt{E_\nu^2-m_i^2}\,\Theta(E_\nu-m_i). \tag{4}\]

  1. For \(E_\nu\gg m_i\) expand the square root to first order in \(m_i^2/E_\nu^2\).

  2. Using \(\sum_i|U_{ei}|^2=1\), show that the result has the form of a spectrum with a single effective mass

    \[ m_\beta^2=\sum_i|U_{ei}|^2m_i^2. \tag{5}\]

  3. Explain why the probabilities \(|U_{ei}|^2\) add up here, rather than the amplitudes \(U_{ei}\).

  4. For a single mass state, construct the Kurie function

    \[ K(E)\equiv \sqrt{\frac{1}{C(E)}\frac{d\Gamma}{dE}} =\sqrt{E_\nu\sqrt{E_\nu^2-m_\nu^2}}. \tag{6}\]

    Compare \(m_\nu=0\) and \(m_\nu=1~\mathrm{eV}\) in the last \(5~\mathrm{eV}\).

  5. Why are the individual kinks at \(E_\nu=m_i\) not resolved if the mass differences are much smaller than the energy resolution of the experiment?

Exercise 4.3. Coulomb focusing and the Fermi function

In the point-Coulomb-field approximation, the Fermi function can be written in terms of the Sommerfeld parameter

\[ F(\eta)=\frac{2\pi\eta}{1-e^{-2\pi\eta}}, \qquad \eta=\pm Z_d\alpha\frac{\mathcal E_e}{p_e}, \tag{7}\]

where \(\mathcal E_e=E+m_e\) is the total energy of the charged lepton, the sign “\(+\)” refers to \(\beta^-\) decay, and the sign “\(-\)” to \(\beta^+\) decay. Use \(\alpha=1/137\) and \(m_e=511~\mathrm{keV}\).

  1. Show that for \(|\eta|\ll1\)

    \[ F(\eta)=1+\pi\eta+O(\eta^2). \tag{8}\]

    Explain the sign of the first correction for the electron and the positron.

  2. Obtain the asymptotics

    \[ F(\eta)\simeq2\pi\eta \quad (\eta\gg1), \qquad F(\eta)\simeq2\pi|\eta|e^{-2\pi|\eta|}\quad(\eta\ll-1). \tag{9}\]

    Relate them, respectively, to Coulomb focusing and defocusing of the wave in the nuclear region.

  3. For \(Z_d=29\) compute \(F(+Z_d,\mathcal E_e)\) and \(F(-Z_d,\mathcal E_e)\) at \(E=10~\mathrm{keV}\), \(100~\mathrm{keV}\), and \(1~\mathrm{MeV}\). In which energy range is the plane wave a particularly poor approximation?

  4. Prove the identity

    \[ \frac{F(+|\eta|)}{F(-|\eta|)}=e^{2\pi|\eta|} \tag{10}\]

    and explain why it does not imply a difference between the weak matrix elements for \(\beta^-\) and \(\beta^+\).

  5. In the nonrelativistic limit \(E\to0\) show that for \(\beta^-\) the product \(F\,p_e\mathcal E_e\) tends to a finite limit, although \(F\) itself diverges as \(1/p_e\). What happens to the same product for \(\beta^+\)?

  6. Why does a dependence on \(F(Z_d,\mathcal E_e;A)\) appear in the exact expression, although the formula used contains no \(A\)? Name the physical scale through which the mass number enters.

Exercise 4.4. How the Fermi function changes the \(\beta^-\) and \(\beta^+\) spectra

Consider an illustrative allowed transition with

\[ Z_d=29,\qquad A=64,\qquad E_0=2.0~\mathrm{MeV},\qquad m_\nu=0. \tag{11}\]

Use the point-Coulomb approximation from Exercise 04.3 and the spectra

\[ S_\mp(E)\equiv\frac{d\Gamma_\mp}{dE} \propto F(\pm Z_d,\mathcal E_e)\,p_e\mathcal E_e(E_0-E)^2, \tag{12}\]

as well as the plane-wave result \(S_{\mathrm{pw}}\), obtained for \(F=1\).

  1. Plot \(F(+Z_d,\mathcal E_e)\) and \(F(-Z_d,\mathcal E_e)\) as functions of \(E\) from \(0.1~\mathrm{keV}\) to \(E_0\).

  2. Plot \(S_{\mathrm{pw}}\), \(S_-\), and \(S_+\), normalizing each curve to unit area. Explain why the Coulomb field enhances the low-energy part of the \(\beta^-\) spectrum and suppresses it for \(\beta^+\).

  3. Separately plot the ratios \(S_-/S_{\mathrm{pw}}\) and \(S_+/S_{\mathrm{pw}}\) without renormalization. Why is this exactly the Fermi function?

  4. For each spectrum, construct the correctly corrected Kurie function

    \[ K_\mp(E)= \left[ \frac{S_\mp(E)}{F(\pm Z_d,\mathcal E_e)p_e\mathcal E_e} \right]^{1/2}. \tag{13}\]

    Verify that both curves are proportional to \(E_0-E\). Then repeat the construction, mistakenly setting \(F=1\) in the denominator. What curvature does this error produce for \(\beta^-\) and \(\beta^+\)?

  5. Why does the Fermi function change the probability distribution over energies but not shift the kinematic boundary \(E_{\max}=E_0-m_\nu\)? Formulate the answer in terms of the Coulomb wave function with an already specified asymptotic energy.

Exercise 4.5. Film, energy losses, and a false neutrino mass

An electron is produced at depth \(z\) in a film and exits at an angle \(\theta\) to the normal. In a simple model

\[ \ell=\frac{z}{\cos\theta}, \qquad \Delta E=a\ell, \qquad E_{\mathrm{det}}=E-\Delta E, \tag{14}\]

where \(a\) is a constant stopping power. Assume that \(z\) is uniformly distributed on \([0,d]\), and that the directions are isotropic within the accepted solid angle \(0\leq\theta\leq\theta_{\max}\). Hence \(u=\cos\theta\) is uniformly distributed on \([c,1]\), where \(c=\cos\theta_{\max}\).

  1. Find \(\langle\ell\rangle\) and \(\operatorname{Var}(\ell)\). Show that

    \[ \langle\ell\rangle =\frac d2\frac{\ln(1/c)}{1-c}, \tag{15}\]

    \[ \operatorname{Var}(\ell) =d^2\left[ \frac{1}{3c} -\frac14\left(\frac{\ln(1/c)}{1-c}\right)^2 \right]. \tag{16}\]

    From this, find the mean and variance of the energy loss.

  2. Let the true spectrum near the endpoint for \(m_\nu=0\) have the form \(S_0(E)\propto(E_0-E)^2\). Show that the measured spectrum is

    \[ S_{\mathrm{obs}}(E_{\mathrm{det}}) =\int_0^\infty L(\Delta E)S_0(E_{\mathrm{det}}+\Delta E)\,d\Delta E, \tag{17}\]

    where the integrand automatically vanishes for \(E_{\mathrm{det}}+\Delta E>E_0\). Denote \(\varepsilon=E_0-E_{\mathrm{det}}\). In the region where \(\varepsilon\) is larger than the characteristic losses, show that

    \[ S_{\mathrm{obs}}(\varepsilon) \propto (\varepsilon-\overline{\Delta E})^2 +\sigma_{\Delta E}^2. \tag{18}\]

    Which parameter of the endpoint fit absorbs the mean loss \(\overline{\Delta E}\), and why is a correction to \(E_0\) alone insufficient?

  3. For \(\varepsilon\gg m_\nu\) the massive spectrum has the expansion

    \[ \varepsilon\sqrt{\varepsilon^2-m_\nu^2} \simeq \varepsilon^2-\frac{m_\nu^2}{2}. \tag{19}\]

    Let the true variance of the losses be \(\sigma_{\mathrm{true}}^2\), while the analysis used the value \(\sigma_{\mathrm{mod}}^2\). After the endpoint fit, show that

    \[ m_{\nu,\mathrm{fit}}^2 \simeq 2\left( \sigma_{\mathrm{mod}}^2- \sigma_{\mathrm{true}}^2 \right). \tag{20}\]

    In which case does a false positive \(m_\nu^2\) arise, and in which a negative best-fit value of \(m_\nu^2\)?

  4. Let \(ad=40~\mathrm{eV}\). The true acceptance corresponds to \(\theta_{\max}=60^\circ\), but the model uses \(75^\circ\); the mean loss is freely absorbed by the \(E_0\) fit. Estimate the false \(m_{\nu,\mathrm{fit}}^2\) and \(m_{\nu,\mathrm{fit}}\). Compare the scale with the historical ITEP claim of a mass of about \(30~\mathrm{eV}\).

  5. Optional. Carry out a numerical simulation without a moment expansion: generate \(z\) and \(\cos\theta\), convolve the massless spectrum with the resulting loss distribution, and perform a fit with a model that has the wrong \(\theta_{\max}\). Investigate how the result depends on the width of the fitted window near the endpoint of the spectrum.

Exercise 4.6. Kink from a heavy mass state

Suppose that the electron antineutrino contains, in addition to the light states (whose masses can be taken as zero on the scale of this problem), a heavy state with mass \(m_4=5~\mathrm{keV}\) and admixture \(|U_{e4}|^2=0.01\). For tritium \(E_0=18.575~\mathrm{keV}\); denote \(\varepsilon=E_0-E\). The spectrum has the form

\[ \frac{d\Gamma}{dE} =C(E)\,\varepsilon \left[ \bigl(1-|U_{e4}|^2\bigr)\varepsilon +|U_{e4}|^2\sqrt{\varepsilon^2-m_4^2}\,\Theta(\varepsilon-m_4) \right]. \tag{21}\]

  1. Why do the contributions of the mass states enter with weights \(|U_{ei}|^2\) rather than through a sum of amplitudes? Show that the ratio to the spectrum without the heavy state,

    \[ \mathcal R(\varepsilon) =1-|U_{e4}|^2 +|U_{e4}|^2\sqrt{1-\frac{m_4^2}{\varepsilon^2}}\;\Theta(\varepsilon-m_4), \tag{22}\]

    does not depend on \(C(E)\) and hence not on the Fermi function.

  2. Find \(\mathcal R\) for \(\varepsilon<m_4\) and at \(E=0\), and its asymptotics for \(\varepsilon\gg m_4\). At what electron energy \(E\) is the singularity located?

  3. Set \(\varepsilon=m_4+\delta\), \(\delta\ll m_4\), and show that

    \[ \mathcal R-\bigl(1-|U_{e4}|^2\bigr) \simeq|U_{e4}|^2\sqrt{\frac{2\delta}{m_4}}. \tag{23}\]

    Why is this a kink with an infinite derivative rather than a jump?

  4. Plot \(\mathcal R(E)\) over the whole spectrum and mark the point \(E=E_0-m_4\). For the massless spectrum with \(F=1\) and \(m_e=511~\mathrm{keV}\), compute numerically the fraction of decays with \(\varepsilon>m_4\) and the ratio of the total decay rates with and without the heavy state.

  5. In a real fit the normalization of the spectrum is free. Why is the constant factor \(1-|U_{e4}|^2\) by itself unobservable? Which feature of the shape carries the information on \(m_4\) and \(|U_{e4}|^2\)? Why is a heavy state searched for deep inside the spectrum, where the statistics is huge, while the masses of the light states are searched for only at the very endpoint?

State the main physical conclusion: each mass state gives its own threshold \(E_0-m_i\), and the visibility of the threshold is set by the weight \(|U_{ei}|^2\) and by the statistics in the neighborhood of the kink.

Exercise 4.7. What can \(m_\beta\) be?

Oscillations measure only differences of squared masses. Use the rounded values

\[ \sin^2\theta_{12}=0.31, \quad \sin^2\theta_{13}=0.022, \quad \Delta m^2_{21}=7.5\times10^{-5}~\mathrm{eV}^2, \quad |\Delta m^2_{31}|\simeq2.5\times10^{-3}~\mathrm{eV}^2 \tag{24}\]

and

\[ |U_{e1}|^2=c_{12}^2c_{13}^2, \qquad |U_{e2}|^2=s_{12}^2c_{13}^2, \qquad |U_{e3}|^2=s_{13}^2. \tag{25}\]

  1. Compute \(|U_{ei}|^2\) and check their sum.

  2. In the normal ordering the lightest state is \(m_1\), and \(m_2^2=m_1^2+\Delta m^2_{21}\), \(m_3^2=m_1^2+\Delta m^2_{31}\). Show that

    \[ m_\beta^2=m_1^2+|U_{e2}|^2\Delta m^2_{21}+|U_{e3}|^2\Delta m^2_{31}. \tag{26}\]

    Derive the analogous formula for the inverted ordering, where the lightest state is \(m_3\), \(m_1^2=m_3^2+|\Delta m^2_{31}|\), \(m_2^2=m_1^2+\Delta m^2_{21}\).

  3. Find the smallest possible \(m_\beta\) for each ordering.

  4. Plot \(m_\beta\) as a function of the mass of the lightest state from \(10^{-4}\) to \(1~\mathrm{eV}\) on a logarithmic scale for both orderings. Above what lightest mass do the two curves differ by less than \(10\%\)? What lightest mass corresponds to \(m_\beta=0.45~\mathrm{eV}\)?

  5. For \(m_\beta=0.45~\mathrm{eV}\) estimate \(m_3-m_1\) and \(m_2-m_1\) and compare them with the energy resolution \(\sigma_E=0.5~\mathrm{eV}\). Why is the spectrum described by a single effective mass in this regime?

State the main physical conclusion: what information on the mass spectrum does the kinematic limit on \(m_\beta\) provide, and what would a reliable measurement \(m_\beta<0.04~\mathrm{eV}\) mean?

Exercise 4.8. Nuclear recoil and the endpoint of the tritium spectrum

In the decay \({}^3\mathrm H\to{}^3\mathrm{He}^++e^-+\bar\nu_e\) take \(E_0=18.575~\mathrm{keV}\), \(m_e=511~\mathrm{keV}\),

\[ M({}^3\mathrm{He})c^2\simeq2.81~\mathrm{GeV}, \qquad M({}^3\mathrm{HeT}^+)c^2\simeq5.62~\mathrm{GeV}, \qquad k_B=8.62\times10^{-5}~\mathrm{eV/K}. \tag{27}\]

Set \(c=1\).

  1. The recoil momentum equals \(|\mathbf p_e+\mathbf p_\nu|\le p_e+p_\nu\). Show that the largest recoil energy is reached at the endpoint of the spectrum, where \(p_\nu\to0\):

    \[ E_{\mathrm{rec}}^{\max}\simeq\frac{E_0(E_0+2m_e)}{2M}. \tag{28}\]

    Compute it for a free \({}^3\mathrm{He}\) atom and compare it with the scale \(m_\nu\sim0.45~\mathrm{eV}\).

  2. In the last \(10~\mathrm{eV}\) of the spectrum \(p_\nu\lesssim10~\mathrm{eV}\). Estimate the spread of the recoil energy due to the angle between \(\mathbf p_e\) and \(\mathbf p_\nu\), and its change when \(E\) is shifted by \(10~\mathrm{eV}\). Why does the recoil near the endpoint only shift the endpoint and get absorbed by the free parameter \(E_0\)?

  3. In a molecule the recoil momentum is received instantaneously by the helium nucleus. Show that the center-of-mass motion of \({}^3\mathrm{HeT}^+\) carries away the energy \(p^2/(2M_{\mathrm{mol}})\), while the remainder \(p^2/(2M_{\mathrm{He}})-p^2/(2M_{\mathrm{mol}})\) goes into the internal motion of the molecule. Estimate this remainder at the endpoint of the spectrum. Why does the decay of a molecule inevitably leave the daughter ion in a rotationally and vibrationally excited state?

  4. The parent \(\mathrm T_2\) molecules (mass about \(6.03~u\), \(uc^2=931.5~\mathrm{MeV}\)) are in thermal motion at \(T=30~\mathrm K\). The electron energy in the laboratory is shifted by approximately \(p_ev_x\). Show that the Doppler broadening equals \(\sigma\simeq p_e\sqrt{k_BT/M}\), and compute it. Using the result of Exercise 4.5 for an unaccounted Gaussian variance, estimate the shift of \(m^2_{\nu,\mathrm{fit}}\) if this broadening is not included in the model.

State the main physical conclusion: which recoil effects only shift the endpoint, and which change the shape of the spectrum and must therefore enter the model?

Exercise 4.9. How many decays are needed to see the mass?

For tritium \(T_{1/2}=12.32\) years, the molar mass is \(3.016~\text{g/mol}\), \(E_0=18\,575~\mathrm{eV}\). Assume ideal resolution, zero background and a known \(E_0\).

  1. Compute the specific activity of tritium in becquerels per gram.

  2. Use the estimate of the fraction of decays in the last interval \(f(\Delta E)\simeq(\Delta E/E_0)^3\) (an exact calculation with the slowly varying factor \(C(E)\) gives about twice as much; this does not matter for the estimate). Let the spectrometer collect electrons from an effective activity \(A_{\mathrm{eff}}=10^{11}~\text{Bq}\). What mass of tritium does this correspond to? Find the count rate in the last \(1\) and \(2~\mathrm{eV}\).

  3. Show that the number of events in the window \(0<\varepsilon<\Delta\) for mass \(m_\nu\) relates to the massless one as

    \[ \frac{N(\Delta;m_\nu)}{N(\Delta;0)} =\left(1-\frac{m_\nu^2}{\Delta^2}\right)^{3/2} \simeq1-\frac32\frac{m_\nu^2}{\Delta^2}. \tag{29}\]

  4. Require that the deficit of events exceed three standard deviations of the Poisson count. Show that \(N\gtrsim(2\Delta^2/m_\nu^2)^2\) events in the window are needed. Find \(N\) and the measurement time for \(m_\nu=0.45~\mathrm{eV}\) and \(\Delta=2~\mathrm{eV}\). By what factor must the statistics be increased to improve the sensitivity to \(m_\nu\) by a factor of \(10\)?

  5. Show that in this model the measurement time grows as \(\Delta/m_\nu^4\), that is, a narrow window is more favorable. Why can a real experiment not narrow the window to fractions of an electronvolt, and why is the time obtained a strongly optimistic estimate? Name at least three reasons.

State the main physical conclusion: why does the sensitivity to the mass improve only as the fourth root of the statistics, and why is the main resource of the experiment not only the activity of the source but also control of the shape of the spectrum near the endpoint?

Exercises for discussion at the seminar

Seminar 04.1. Why is the \(\beta\) spectrum continuous?

Let the initial nucleus be at rest, and the decay energy \(Q\) be fixed.

  1. What would the electron energy be in a hypothetical two-body decay \(A\to B+e^-\)?
  2. Why can the electron have a continuous set of energies in the three-body decay \(A\to B+e^-+\bar\nu_e\)?
  3. Where is the endpoint of the spectrum if the nuclear recoil is neglected but the neutrino mass is retained?
  4. What will qualitatively change when the nuclear recoil is taken into account?

Seminar 04.2. The price of approaching the endpoint

Denote \(\varepsilon=E_0-E\). For a massless neutrino near the endpoint of the allowed spectrum,

\[ \frac{d\Gamma}{dE}\propto\varepsilon^2. \tag{30}\]

  1. Show that the number of events in the last interval of width \(\Delta\) grows as \(\Delta^3\).
  2. By what factor does the statistics increase when the window is widened from \(1\) to \(5~\mathrm{eV}\)?
  3. Using \(\varepsilon\sqrt{\varepsilon^2-m_\nu^2} \simeq\varepsilon^2-m_\nu^2/2\), estimate how the relative effect of the mass depends on \(\varepsilon\).
  4. Formulate the trade-off between statistics and sensitivity to the mass.

Seminar 04.3. What is visible on the Kurie plot?

For an allowed transition, the function

\[ K(E)= \left[ \frac{d\Gamma/dE}{F(Z,E)p_eE_e} \right]^{1/2}. \tag{31}\]

is defined.

  1. Show that for \(m_\nu=0\) the plot of \(K(E)\) is a straight line \(K(E)=E_0-E\).
  2. How does a nonzero mass change the position of the endpoint and the shape of the last points?
  3. How does this effect differ from an overall shift of the energy scale?
  4. How can electron energy losses in the source mimic a change in the shape of the spectrum near the endpoint?

Seminar 04.4. The Fermi function and causality

An electron in the Coulomb field of the daughter nucleus is described not by a plane wave but by a scattering state with a given asymptotic energy \(\mathcal E_e\). The Fermi function \(F(Z,\mathcal E_e)\) compares the density of this wave function in the nuclear region with the density of a plane wave.

  1. Why, for \(\beta^-\) decay, does the Coulomb attraction enhance the spectrum at low energies, while for \(\beta^+\) decay the repulsion suppresses it?
  2. Why is it incorrect to describe the effect as the emission of an electron with one energy and its subsequent transformation into another?
  3. Explain why the neutrino energy and the asymptotic electron energy are fixed by a single energy conservation law, so that no contradiction with causality arises.