Exercises: MAC-E Spectrometer

Lecture 06 · Introduction to Neutrino Physics

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Exercises

Exercise 6.1. MAC-E spectrometer numbers

Let the source have \(B_{\mathrm{src}}=3.6~\mathrm{T}\), the maximum field be \(B_{\max}=6.0~\mathrm{T}\), and in the analyzing plane \(B_{\mathrm A}=0.35~\mathrm{mT}\).

1. From the conservation of \(p_\perp^2/B\) obtain the condition for passing the magnetic mirror and the maximum initial angle

   \[ \sin^2\vartheta_{\mathrm{acc}}=\frac{B_{\mathrm{src}}}{B_{\max}}. \tag{1}\]

   Calculate \(\vartheta_{\mathrm{acc}}\).

2. What fraction of isotropically emitted electrons in the forward hemisphere satisfies the condition \(\vartheta<\vartheta_{\mathrm{acc}}\)?

3. For an electron with \(E=18.6~\mathrm{keV}\) estimate the maximum residual transverse energy in the analyzing plane and the resolution

   \[ \Delta E\simeq E\frac{B_{\mathrm A}}{B_{\max}}. \tag{2}\]

4. Explain in one sentence the role of magnetic adiabatic collimation and of the electrostatic filter.

Exercise 6.2. Two adiabaticity conditions

An electron moves in a stationary inhomogeneous magnetic field. On the chosen section the following are known

\[ B=0.10~\mathrm{T}, \qquad L_\perp=0.50~\mathrm{m}, \qquad L_\parallel=2.0~\mathrm{m}, \tag{3}\]

and its kinetic energy and angle to the field are

\[ E=10~\mathrm{keV}, \qquad \alpha=45^\circ. \tag{4}\]

1. Find \(p_\perp\), \(v_\parallel\), the Larmor radius \(\rho_L\) and the pitch \(h=v_\parallel T_c\) with the relativistic correction to the cyclotron frequency.
2. Calculate \(\rho_L/L_\perp\) and \(h/L_\parallel\). In which direction is the inhomogeneity more dangerous for the adiabatic approximation?
3. Explain why \(dB/dt\) along the trajectory can be a nonzero quantity, although \(\partial B/\partial t=0\) throughout the whole setup.
4. What changes in the argument about the energy if the current in the magnet starts to be varied over a time comparable to \(T_c\)?

Exercise 6.3. Loss cone in a dipole field

Consider the dipole field line \(L=4\) and a conventional boundary of the atmosphere at altitude \(h_{\mathrm{atm}}=100~\mathrm{km}\). Use \(R_\oplus=6371~\mathrm{km}\) and the formulas

\[ r(\lambda)=LR_\oplus\cos^2\lambda, \qquad \frac{B(L,\lambda)}{B_{\mathrm{eq}}(L)} =\frac{\sqrt{1+3\sin^2\lambda}}{\cos^6\lambda}. \tag{5}\]

1. To what altitude above the equator does \(L=4\) correspond?

2. Find the magnetic latitude \(\lambda_{\mathrm{atm}}\) at which the line crosses the boundary of the atmosphere.

3. Calculate the ratio \(B_{\mathrm{atm}}(L)/B_{\mathrm{eq}}(L)\) and the loss-cone angle

   \[ \alpha_{\mathrm{loss}} =\arcsin\sqrt{\frac{B_{\mathrm{eq}}(L)} {B_{\mathrm{atm}}(L)}}. \tag{6}\]

4. Compare particles with equatorial angles \(5^\circ\), \(20^\circ\) and \(60^\circ\). Which one will be reflected above the atmosphere, and which one will be lost?

5. Draw the sphere of velocity directions and the two loss cones. Why is the loss region not a ring or a flat sector?

Exercise 6.4. Stationary mirror and Fermi acceleration

A particle is elastically reflected from a flat magnetic inhomogeneity moving toward it with speed \(u\ll v\). Consider the motion along the normal to the inhomogeneity and use the nonrelativistic addition of velocities.

1. Go to the reference frame of the inhomogeneity, change the sign of the particle’s velocity upon reflection, and return to the laboratory frame.
2. Show that for a head-on collision the change in velocity is \(\Delta v\simeq2u\).
3. Estimate the relative change in energy for \(u\ll v\).
4. Why is the energy before and after reflection the same in a stationary magnetic mirror, while in this exercise it is not?
5. What energy does the system lose when the particle is accelerated?

Exercise 6.5. Slow or fast: a numerical test of the invariant

Transverse cyclotron motion in a slowly varying field is an oscillator with a variable frequency. Let us check numerically when the action is conserved. Set \(m=1\) and consider

\[ \ddot Q=-\omega^2(t)\,Q, \qquad J=\frac{1}{\omega}\left(\frac{\dot Q^2}{2}+\frac{\omega^2Q^2}{2}\right). \tag{7}\]

The frequency decreases from \(\omega_0=1\) to \(\omega_1=1/4\), as when the field decreases by a factor of four, over a time \(\tau\) (in units of \(1/\omega_0\)).

1. Sudden change, \(\tau\to0\): \(Q\) and \(P=\dot Q\) have no time to change. For the initial state \(Q=\sqrt{2J_0/\omega_0}\sin\theta_0\), \(P=\sqrt{2\omega_0J_0}\cos\theta_0\) show that

   \[ \frac{J_1}{J_0} =\frac{\omega_0}{\omega_1}\cos^2\theta_0 +\frac{\omega_1}{\omega_0}\sin^2\theta_0. \tag{8}\]

   Find the smallest, the largest and the phase-averaged values.

2. Numerically (Python) integrate the equation for the linear profile \(\omega(t)=\omega_0+(\omega_1-\omega_0)t/\tau\), \(0\le t\le\tau\), for \(\tau=1\), \(10\) and \(100\). For \(64\) uniformly distributed initial phases find the smallest and the largest \(J_1/J_0\).

3. Repeat the calculation for the smooth profile

   \[ \omega(t)=\omega_0+(\omega_1-\omega_0)\, \frac{1+\tanh(t/\tau)}{2}, \qquad -8\tau\le t\le8\tau, \tag{9}\]

   for \(\tau=1\), \(3\) and \(10\). Why does the smooth profile conserve \(J\) much better than the linear one? Where is slowness violated in the linear profile?

4. For the linear profile compute the parameter \(|\dot\omega|/\omega^2\) at the end of the frequency change. Is it consistent with the spread of \(J_1/J_0\) you found?

5. What is the energy \(E_1=\omega_1J_1\) in the adiabatic limit? What does this mean for the transverse energy of an electron if the field along its path decreases by a factor of four? Where does the energy difference go in a stationary magnetic field?

State the main physical conclusion: what exactly must be small for \(J\) to be conserved, and why is the conservation of \(J\) independent of the phase only in the adiabatic limit?

Exercise 6.6. Magnetic bottle: oscillations between mirrors

Near the minimum on the trap axis the field is

\[ B(s)=B_0\left(1+\frac{s^2}{\ell^2}\right). \tag{10}\]

A non-relativistic electron with energy \(E\) passes the point \(s=0\) at an angle \(\alpha_0\) to the field.

1. Using the guiding-center Hamiltonian \(\mathcal H_{\mathrm{gc}}=p_\parallel^2/(2m)+\mu B(s)\), show that the guiding center performs harmonic oscillations along the field with frequency

   \[ \Omega_b=\frac{v_{\perp0}}{\ell}, \tag{11}\]

   and amplitude \(s_m=\ell\cot\alpha_0\). Check that at the point \(s_m\) the field equals \(B_0/\sin^2\alpha_0\).

2. Why is the oscillation frequency determined by \(v_{\perp0}\) rather than by the total speed? Does the position of the mirror point depend on the electron energy?

3. For \(E=1.0~\mathrm{keV}\), \(B_0=0.10~\mathrm{T}\), \(\ell=0.50~\mathrm{m}\) and \(\alpha_0=30^\circ\), \(60^\circ\) compute \(\rho_L\), \(T_c\), \(T_b=2\pi/\Omega_b\) and \(s_m\). Is the condition \(T_c\ll T_b\) satisfied?

4. A real trap is finite: at its ends the field reaches \(B_{\max}=RB_0\). Show that a particle from the center escapes through an end if \(\sin^2\alpha_0<1/R\), and that for an isotropic distribution the fraction of escaping particles is

   \[ f_{\mathrm{loss}}=1-\sqrt{1-\frac1R}. \tag{12}\]

   Compute \(f_{\mathrm{loss}}\) for \(R=2\) and \(R=10\).

5. In a tokamak the field is \(B\propto1/R_{\mathrm{maj}}\), where \(R_{\mathrm{maj}}\) is the distance to the axis of the torus. On a magnetic surface of minor radius \(r\) around a circle of radius \(R_0\) the ratio of the largest and smallest fields equals \((1+\varepsilon)/(1-\varepsilon)\), \(\varepsilon=r/R_0\). Show that for an isotropic distribution on the outer side of the torus the fraction of trapped particles is

   \[ f_{\mathrm{trap}}=\sqrt{\frac{2\varepsilon}{1+\varepsilon}}. \tag{13}\]

   Compute it for \(\varepsilon=0.1\) and \(0.3\). Why are there so many trapped particles although the mirror ratio is close to unity?

State the main physical conclusion: what determines whether a magnetic mirror confines a particle, and what does not?

Exercise 6.7. Transmission function of an ideal MAC-E filter

Monoenergetic electrons with kinetic energy \(E=18.6~\mathrm{keV}\) are emitted isotropically in the source. Use the non-relativistic relation \(E_\perp=E\sin^2\alpha\), the conservation of \(p_\perp^2/B\), and

\[ B_{\mathrm{src}}=3.6~\mathrm{T}, \qquad B_{\max}=6.0~\mathrm{T}, \qquad B_{\mathrm A}=0.30~\mathrm{mT}. \tag{14}\]

1. Show that in the analyzing plane \(E_{\perp,\mathrm A}=E\sin^2\alpha_{\mathrm{src}}\,B_{\mathrm A}/B_{\mathrm{src}}\) and that the electron passes the barrier if \(\sin^2\alpha_{\mathrm{src}}<\sin^2\alpha_T\), where

   \[ \sin^2\alpha_T =\frac{E-e|U|}{E}\,\frac{B_{\mathrm{src}}}{B_{\mathrm A}}. \tag{15}\]

2. Because of the magnetic mirror only angles \(\alpha_{\mathrm{src}}<\alpha_{\mathrm{acc}}\) are accepted. Obtain the transmitted fraction of the accepted electrons

   \[ T(E,U)=\frac{1-\cos\alpha_T}{1-\cos\alpha_{\mathrm{acc}}}, \qquad e|U|\le E\le e|U|+\Delta E, \tag{16}\]

   with \(T=0\) below and \(T=1\) above this interval. Show that \(\Delta E=E\,B_{\mathrm A}/B_{\max}\) and that \(T\) is continuous at both ends of the interval.

3. Introduce \(u=(E-e|U|)/\Delta E\) and write \(T(u)\) in terms of the ratio \(B_{\mathrm{src}}/B_{\max}\). Compute \(T\) at \(u=0.25\), \(0.5\), \(0.75\) and the slopes \(dT/du\) for \(u\to0\) and \(u\to1\). Plot it. Is the transmission edge linear?

4. If the integral spectrum is almost linear over the width \(\Delta E\), the smooth edge acts as a step shifted by

   \[ \delta=\Delta E\int_0^1\bigl[1-T(u)\bigr]\,du. \tag{17}\]

   Prove this and compute \(\delta/\Delta E\).

5. Include the relativistic correction: \(\Delta E\) is multiplied by \((\gamma+1)/2\). Assuming the shape of \(T(u)\) unchanged, find \(\Delta E\) and \(\delta\) in electronvolts.

State the main physical conclusion: why does an edge about \(1~\mathrm{eV}\) wide not prevent a search for a neutrino mass below \(1~\mathrm{eV}\), provided the shape of the edge is known?

Exercise 6.8. Why the main spectrometer is the size of a house

The electrons accepted by KATRIN fill a magnetic tube with flux \(\Phi_{\mathrm{tube}}=191~\mathrm{T\,cm^2}\). Use \(E=18.6~\mathrm{keV}\), \(\gamma=1.036\), \(B_{\mathrm{src}}=3.6~\mathrm{T}\), \(B_{\max}=6.0~\mathrm{T}\), \(B_{\mathrm A}=0.30~\mathrm{mT}\). The main vessel has a length of \(23.3~\mathrm{m}\) and an inner diameter of \(9.8~\mathrm{m}\).

1. From flux conservation find the diameter of the tube in the source, at the field maximum and in the analyzing plane.

2. Show that the largest Larmor radius in the analyzing plane is

   \[ \rho_{L,\mathrm A}^{\max}=\frac{p}{e\sqrt{B_{\mathrm A}B_{\max}}}, \tag{18}\]

   and compute it. What sets the diameter of the vessel: the size of an individual orbit or the cross section of the tube?

3. Compute the pitch \(h=v_\parallel T_c\) for an electron moving almost along the field without retardation (\(v_\parallel\simeq v\)) at \(B=6.0~\mathrm{T}\), \(0.10~\mathrm{T}\), \(1.0~\mathrm{mT}\) and \(0.30~\mathrm{mT}\). Where is adiabaticity at risk?

4. Require \(h/L_\parallel\le\varepsilon\) along the whole path. Show that this condition has the form

   \[ \left|\frac{d}{ds}\frac{1}{B}\right| \le\frac{\varepsilon e}{2\pi\gamma m_ev}, \tag{19}\]

   and that the distance from the field maximum to the analyzing plane is at least \(s_{\min}\simeq h(B_{\mathrm A})/\varepsilon\). Estimate \(s_{\min}\) in order of magnitude for \(\varepsilon\sim1\) and compare it with half the length of the vessel.

5. Near the analyzing plane the electron is retarded by the potential. Find the pitch at \(B_{\mathrm A}\) for a longitudinal energy of \(0.1\) and \(1.0~\mathrm{eV}\) (here \(\gamma\simeq1\)). How does the retardation relax the estimate of part 4?

State the main physical conclusion: what two prices does the spectrometer pay for a small ratio \(B_{\mathrm A}/B_{\max}\)?

Exercise 6.9. Where is the electron’s fate decided?

Near the analyzing plane on the spectrometer axis use the model

\[ e|U(s)|=e|U_0|\left[1-\frac{(s-\delta)^2}{a^2}\right], \qquad B(s)=B_{\mathrm A}\left(1+\frac{s^2}{b^2}\right), \tag{20}\]

where \(e|U_0|=18.6~\mathrm{keV}\), \(a=5.0~\mathrm{m}\), \(b=3.0~\mathrm{m}\), and \(\delta\) is the offset of the potential maximum relative to the field minimum. In the source the potential is zero and the kinetic energy of the electron is \(E\). For the accepted electrons \(\mu B_{\mathrm A}\le\Delta E=0.95~\mathrm{eV}\).

1. From the conservation of \(\mathcal E=p_\parallel^2/(2m)+\mu B+q\Phi\) and of \(\mu\) obtain, in non-relativistic form,

   \[ E_\parallel(s)=E-e|U(s)|-\mu B(s). \tag{21}\]

   Explain why the electron passes the filter only if \(\min_sE_\parallel(s)>0\).

2. Let \(\delta=0\). Show that the minimum of \(E_\parallel\) is at \(s=0\) if \(e|U_0|/a^2>\mu B_{\mathrm A}/b^2\), and estimate the ratio of the two coefficients. What is the transmission threshold?

3. Let \(\delta\ne0\). Find the position of the minimum \(s^*\) and show that to leading order

   \[ \min_sE_\parallel \simeq E-e|U_0|-\mu B_{\mathrm A}\left(1+\frac{\delta^2}{b^2}\right). \tag{22}\]

4. Find the width of the threshold for \(\delta=0.5\), \(1.0\) and \(2.0~\mathrm{m}\). For which electrons does the offset not matter?

5. A “paradox”: for \(\delta=1.0~\mathrm{m}\) an electron with energy slightly above \(e|U_0|+\mu B_{\mathrm A}\) has a longitudinal energy of about \(0.7~\mathrm{keV}\) at the point \(s=0\), but can still be reflected. Find this energy and explain where the analyzing plane physically is.

State the main physical conclusion: why must the maximum of the retarding potential coincide with the minimum of the magnetic field?

Exercise 6.10. Whom does the Earth’s magnetic field confine?

In the plane of the magnetic equator the dipole field is

\[ B_{\mathrm{eq}}(r)=B_{\mathrm{eq},\oplus}\left(\frac{R_\oplus}{r}\right)^3, \qquad B_{\mathrm{eq},\oplus}=3.1\cdot10^{-5}~\mathrm{T}, \qquad R_\oplus=6371~\mathrm{km}. \tag{23}\]

Use \(m_ec^2=0.511~\mathrm{MeV}\), \(m_pc^2=938.3~\mathrm{MeV}\) and the relativistic Larmor radius \(\rho_L=p_\perp/(|q|B)\); for the estimate set \(p_\perp\simeq p\).

1. Show that at the equator the scale of the inhomogeneity across the field is \(L_\perp=r/3\).
2. Compute \(\rho_L\) on the lines \(L=1.5\) and \(L=4\) for a \(1~\mathrm{MeV}\) electron, a \(1~\mathrm{MeV}\) proton and a \(100~\mathrm{MeV}\) proton.
3. Find \(\rho_L/L_\perp\) and show that at fixed momentum this ratio grows as \(L^4\). For which particles and lines is the adiabatic approximation reliable?
4. For a \(100~\mathrm{MeV}\) proton on the line \(L=4\) compare the cyclotron period with the transit time \(LR_\oplus/v\). Can one speak of the conservation of \(\mu\)?
5. Protons with energies up to hundreds of megaelectronvolts are observed in the inner belt, while the outer belt consists mainly of electrons. What does your estimate say about where energetic protons can be stably confined at all?

State the main physical conclusion: why is the same magnetic field a good trap for electrons and a poor one for energetic protons at large \(L\)?