Exercises: Guiding the Electron to the Detector

Lecture 05 · Introduction to Neutrino Physics

Author

Dmitry V. Naumov

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Exercises

Exercise 5.1. Time of flight along a helical trajectory

An electron with kinetic energy \(E\) is born at \(z=0\) in a uniform field \(\mathbf B=B\mathbf e_z\). The angle between the momentum and the direction of \(+\mathbf B\) is \(\alpha\). The detector is located in the plane \(z=L\).

  1. Using the constancy of \(p_\parallel\), obtain \(z(t)\), the time to reach the detector \(t_L\), and the total trajectory length \(\ell=vt_L\).

  2. Express the velocity relativistically:

    \[ \gamma=1+\frac{E}{m_ec^2}, \qquad v=c\sqrt{1-\gamma^{-2}}. \tag{1}\]

  3. For \(E=18.6~\mathrm{keV}\), \(B=2.0~\mathrm{mT}\) and \(L=5.0~\mathrm{m}\) compute, at \(\alpha=15^\circ\), \(45^\circ\) and \(75^\circ\):

    \[ t_L,\qquad \ell,\qquad T_c=\frac{2\pi\gamma m_e}{eB}, \qquad N=\frac{t_L}{T_c}. \tag{2}\]

    Use \(m_ec^2=511~\mathrm{keV}\), \(c=2.998\cdot10^8~\mathrm{m/s}\) and \(e/m_e=1.759\cdot10^{11}~\mathrm{C/kg}\).

  4. What happens to \(t_L\), \(\ell\) and \(N\) as \(\alpha\to90^\circ\)? Which electrons will reach the plane \(z=L>0\) at all?

  5. Which of the quantities found depend on \(B\), and which do not? Explain the result physically.

Exercise 5.2. Electrostatic threshold and angular acceptance

In the non-relativistic approximation, an electron with kinetic energy \(E\) and angle \(\alpha\) to the spectrometer axis has

\[ E_\parallel=E\cos^2\alpha, \qquad E_\perp=E\sin^2\alpha. \tag{3}\]

A retarding potential \(U\) is set up in front of the detector. The electron passes if \(E_\parallel>e|U|\).

  1. For given \(E\) and \(U\), find the maximum transmission angle \(\alpha_{\max}\).

  2. Consider isotropic emission into the forward hemisphere. Show that the fraction of transmitted directions is equal to

    \[ f_\Omega=1-\cos\alpha_{\max} =1-\sqrt{\frac{e|U|}{E}}. \tag{4}\]

  3. Compute \(\alpha_{\max}\) and \(f_\Omega\) for \(E=18.6~\mathrm{keV}\) and \(e|U|=18.0~\mathrm{keV}\).

  4. Why do electrons of the same total energy but with different \(\alpha\) pass different electrostatic thresholds?

  5. Explain why a uniform magnetic field does not fix this problem and why collimation is needed in front of the filter.

Exercise 5.3. Guiding center and the sign of the charge

Let

\[ \mathbf B=B\mathbf e_z, \qquad \Omega=\frac{qB}{m}, \tag{5}\]

and at the initial moment

\[ x_0=y_0=0, \qquad \mathbf v_0=(v_\perp,0,v_\parallel). \tag{6}\]

  1. From the equations

    \[ \dot v_x=\Omega v_y, \qquad \dot v_y=-\Omega v_x \tag{7}\]

    show that

    \[ X=x+\frac{v_y}{\Omega}, \qquad Y=y-\frac{v_x}{\Omega} \tag{8}\]

    are constant.

  2. Find \((X,Y)\) and the orbit radius for an electron \(q=-e\).

  3. Determine the direction of rotation when viewed along \(+\mathbf B\). Repeat the answer for a positive charge.

  4. How does the sign of \(v_\parallel\) determine the motion of the guiding center along the field? Does this answer depend on the sign of the charge?

  5. Is there a transverse drift of the guiding center in a uniform magnetic field without an electric field?

Exercise 5.4. From the phase ellipse to the variables \((\theta,J)\)

For transverse cyclotron motion the following are introduced:

\[ Q=x-X=-\frac{v_y}{\Omega}, \qquad P=m\dot Q=mv_x. \tag{9}\]

  1. Show that

    \[ H_\perp(Q,P) =\frac{P^2}{2m} +\frac{m\omega_c^2Q^2}{2} =E_\perp, \qquad \omega_c=|\Omega|. \tag{10}\]

  2. Find the semi-axes of the phase ellipse and compute

    \[ J=\frac{1}{2\pi}\oint P\,dQ =\frac{E_\perp}{\omega_c}. \tag{11}\]

  3. Introduce

    \[ \theta= \operatorname{atan2}(m\omega_cQ,P) \tag{12}\]

    and obtain the inverse transformation

    \[ Q=\sqrt{\frac{2J}{m\omega_c}}\sin\theta, \qquad P=\sqrt{2m\omega_cJ}\cos\theta. \tag{13}\]

  4. By direct computation, verify the Poisson bracket \(\{\theta,J\}_{Q,P}=1\).

  5. Write \(H_\perp\) in terms of \(J\) and obtain from Hamilton’s equations \(\dot J\) and \(\dot\theta\). Why is the angle \(\theta\) undefined at \(J=0\)?

Exercise 5.5. Four physical meanings of the action

For a non-relativistic cyclotron orbit use

\[ \rho_L=\frac{p_\perp}{eB}, \qquad J=\frac{p_\perp^2}{2eB}, \qquad L_{\mathrm{orb}}=\rho_Lp_\perp, \tag{14}\]

\[ \mu=\frac{p_\perp^2}{2m_eB}, \qquad \Phi=\pi\rho_L^2B. \tag{15}\]

  1. Prove the chain of equalities

    \[ \boxed{ J=\frac{L_{\mathrm{orb}}}{2} =\frac{m_e}{e}\mu =\frac{e}{2\pi}\Phi }. \tag{16}\]

  2. Check the dimensions of all four expressions for \(J\).

  3. For

    \[ E=1.00~\mathrm{keV}, \qquad \alpha=60^\circ, \qquad B=0.100~\mathrm{T} \tag{17}\]

    in the non-relativistic approximation compute \(\rho_L\), \(\omega_c\), \(J\), \(\mu\) and \(\Phi\). Use

    \[ e=1.602\cdot10^{-19}~\mathrm{C}, \qquad m_e=9.109\cdot10^{-31}~\mathrm{kg}. \tag{18}\]

  4. Numerically verify the equalities from part 1.

  5. Find \(J/\hbar\) for \(\hbar=1.055\cdot10^{-34}~\mathrm{J\,s}\). What does a large value of this ratio tell us about the applicability of the classical description of the orbit?

Exercise 5.6. Relativistic transverse energy

The formula \(E_\parallel=E\cos^2\alpha\) from Exercise 5.2 is non-relativistic. Let us check how accurate it is for an electron near the endpoint of the tritium spectrum, where the neutrino mass is sought at the electronvolt level.

An electron with kinetic energy \(E\) is emitted at an angle \(\alpha\) to the \(z\) axis and moves in an electrostatic potential \(\Phi(z)\) that depends only on \(z\). At the starting point \(\Phi=0\); at the top of the barrier the potential energy of the electron equals \(e|U|\).

  1. Show that \(p_\perp\) is conserved and that the total energy satisfies

    \[ \sqrt{p^2c^2+m_e^2c^4}+q\Phi(z)=\mathrm{const}. \tag{19}\]

  2. The electron passes the barrier if \(p_\parallel^2>0\) at its top. Show that the exact transmission condition has the form

    \[ E-e|U|>E_\perp, \qquad E_\perp\equiv\sqrt{p_\perp^2c^2+m_e^2c^4}-m_ec^2, \qquad p_\perp c=\sin\alpha\sqrt{E^2+2Em_ec^2}. \tag{20}\]

  3. Expand \(E_\perp\) to second order in \(E/(m_ec^2)\) and obtain

    \[ E_\perp\simeq E\sin^2\alpha \left[1+\frac{E}{2m_ec^2}\cos^2\alpha\right]. \tag{21}\]

    Why does the correction vanish at \(\alpha=0\) and at \(\alpha=90^\circ\)?

  4. For \(E=18.6~\mathrm{keV}\) and \(m_ec^2=511~\mathrm{keV}\) compute the exact \(E_\perp\) and the non-relativistic \(E\sin^2\alpha\) at \(\alpha=10^\circ\), \(30^\circ\) and \(60^\circ\). Express the difference in electronvolts.

  5. Recompute exactly \(\alpha_{\max}\) and \(f_\Omega\) from Exercise 5.2 for \(e|U|=18.0~\mathrm{keV}\). Is the difference in the angular acceptance noticeable?

State the main physical conclusion: what accuracy can be expected from the non-relativistic split \(E=E_\parallel+E_\perp\), and why does the analysis of the endpoint of the spectrum require relativistic kinematics?

Exercise 5.7. Period from area: a box and a bouncing ball

For one periodic degree of freedom

\[ J(E)=\frac{1}{2\pi}\oint p\,dq, \qquad T=2\pi\frac{dJ}{dE}. \tag{22}\]

Let us test this relation on two systems whose phase orbits are not ellipses.

  1. A particle of mass \(m\) moves between elastic walls at \(q=0\) and \(q=a\). Draw the phase orbit at energy \(E\), find \(J(E)\) and obtain the period from it. Compare with \(2a/v\).

  2. A ball of mass \(m\) bounces elastically above a floor:

    \[ H=\frac{p^2}{2m}+mgq, \qquad q\ge0. \tag{23}\]

    Draw the phase orbit and show that

    \[ J(E)=\frac{2\sqrt2}{3\pi}\,\frac{E^{3/2}}{g\sqrt m}. \tag{24}\]

    Obtain from \(J(E)\) the period \(T=2v_0/g\), where \(v_0\) is the speed at the floor. Check the answer with elementary kinematics.

  3. For the box, the oscillator and the ball \(J\propto E^k\). Find \(k\) in each case and show that \(T\propto E^{k-1}\). Why does the period of the oscillator not depend on the energy, and why is this an exception?

  4. A tennis ball of mass \(m=58~\mathrm{g}\) bounces to a height \(h=1.0~\mathrm{m}\). Find \(J\) and \(J/\hbar\) for \(g=9.81~\mathrm{m/s^2}\) and \(\hbar=1.055\cdot10^{-34}~\mathrm{J\,s}\).

  5. For a neutron (\(m_n=1.675\cdot10^{-27}~\mathrm{kg}\)) find the bounce height at which \(J=\hbar\). What does this result mean for the classical description of the motion of ultracold neutrons above a mirror?

State the main physical conclusion: what does the area of a phase orbit know about the period of the motion, even when the orbit is not an ellipse?

Exercise 5.8. Which changes of variables are canonical?

For one degree of freedom a change \((q,p)\to(Q,P)\) is canonical if

\[ \{Q,P\}_{q,p} =\frac{\partial Q}{\partial q}\frac{\partial P}{\partial p} -\frac{\partial Q}{\partial p}\frac{\partial P}{\partial q} =1. \tag{25}\]

The same expression equals the Jacobian of the change: \(dQ\,dP=\{Q,P\}_{q,p}\,dq\,dp\).

  1. Check whether the following changes are canonical:

    1. \(Q=aq\), \(P=p/a\), \(a>0\);
    2. \(Q=p\), \(P=-q\);
    3. \(Q=q\), \(P=p+kq^2\);
    4. \(Q=q^2\), \(P=p\);
    5. \(Q=q^2\), \(P=p/(2q)\), \(q>0\).

  2. Draw the image of the square \(1\le q\le2\), \(0\le p\le1\) for each change and find its area.

  3. For a free particle \(H=p^2/(2m)\) write \(H(Q,P)\) in cases (d) and (e). Show that in case (d) \(\dot Q\ne\partial H/\partial P\), while in case

    1. both Hamilton’s equations hold.

  4. For the oscillator \(H=p^2/(2m)+m\omega^2q^2/2\) choose \(a\) in change (a) so that the phase orbit becomes a circle. Find its radius and show that the area equals \(2\pi E/\omega\) for any \(a\).

  5. Change (b) rotates the phase plane by \(90^\circ\). Apply it to the oscillator in the variables of part 4. How has the Hamiltonian changed? What does this say about the distinction between “coordinate” and “momentum”?

State the main physical conclusion: why is the action \(J\) a property of the orbit itself rather than of the chosen variables?

Exercise 5.9. The pendulum: from oscillation to rotation

The Hamiltonian of the simple pendulum is

\[ H(\varphi,p_\varphi) =\frac{p_\varphi^2}{2I}-I\omega_0^2\cos\varphi, \qquad I=ml^2, \qquad \omega_0^2=\frac{g}{l}. \tag{26}\]

Use units \(I=\omega_0=1\); then \(E\ge-1\).

  1. Draw the phase portrait and mark the three regimes: oscillation \(-1<E<1\), the separatrix \(E=1\) and rotation \(E>1\). Find the turning point \(\varphi_0(E)\) for oscillation.

  2. Show that for oscillation

    \[ J(E)=\frac{2}{\pi}\int_0^{\varphi_0}\sqrt{2(E+\cos\varphi)}\,d\varphi, \qquad \cos\varphi_0=-E, \tag{27}\]

    and that at small amplitude \(J\simeq(E+1)/\omega_0\), as for an oscillator.

  3. Numerically (Python) compute \(J\) and \(T=2\pi\,dJ/dE\) for the amplitudes \(\varphi_0=10^\circ\), \(90^\circ\), \(150^\circ\) and \(179^\circ\). Compare the result with the exact formula

    \[ T=\frac{4}{\omega_0}K\!\left(\sin\frac{\varphi_0}{2}\right), \tag{28}\]

    where \(K(k)\) is the complete elliptic integral of the first kind. In scipy.special.ellipk(m) the argument is \(m=k^2\).

  4. Show that on the separatrix \(J=8/\pi\), while the period diverges. How is this seen on the plot of \(J(E)\)?

  5. For rotation \(J(E)=(2\pi)^{-1}\int_0^{2\pi}p_\varphi\,d\varphi\). Show that for \(E\gg1\) one gets \(J\to\sqrt{2E}\), that is, the angular momentum of the rotor. Find \(J\) for \(E\to1^+\) and compare it with \(J\) of oscillation on the separatrix. Where does the factor \(2\) come from?

State the main physical conclusion: what do the actions of the oscillator, the pendulum and the rotor have in common, and what changes on crossing the separatrix?

Exercises for discussion in the seminar

Seminar 05.1. The first bend of the trajectory

Let \(\mathbf B=B\mathbf e_z\), and let the initial velocity of the particle be

\[ \mathbf v=(v_\perp,0,v_\parallel). \tag{29}\]

  1. Find the direction of the Lorentz force \(q\mathbf v\times\mathbf B\) for an electron and a positron.
  2. Which velocity component starts to change at the first moment?
  3. Show directly that the magnetic field does no work.
  4. Sketch the initial segment of the trajectory for both signs of the charge.

Seminar 05.2. Two electrons with the same energy

Two electrons with the same energies enter the same uniform field at angles \(\alpha_1=30^\circ\) and \(\alpha_2=60^\circ\) to the field.

  1. Compare their cyclotron periods.
  2. Find the ratios of the Larmor radii \(\rho_2/\rho_1\) and of the pitches of the helix \(h_2/h_1\).
  3. Find the ratio of the times in which the electrons cover the same distance along the field.
  4. Which of these differences are visible in the animation, and which require time measurement?

Seminar 05.3. Why does a uniform field not collimate?

In a uniform constant magnetic field, \(p_\parallel\) and \(p_\perp\) are conserved.

  1. Show that the pitch angle \(\tan\alpha=p_\perp/p_\parallel\) remains constant.
  2. What does the field do to isotropically emitted electrons: does it keep them near a field line, turn them along the field, or both?
  3. Why is a uniform field alone insufficient if a small transverse momentum is required in front of the electrostatic filter?
  4. Which property of the field will have to be changed in the next exercise of the course?

Seminar 05.4. The action for two periodic motions

For a harmonic oscillator

\[ H=\frac{P^2}{2m}+\frac{m\omega^2Q^2}{2}=E \tag{30}\]

and for a planar rotor

\[ H=\frac{L^2}{2I} \tag{31}\]

compute the action \(J=(2\pi)^{-1}\oint p\,dq\).

  1. Find the area of the oscillator ellipse in the \((Q,P)\) plane and show that \(J=E/\omega\).
  2. Show that for the rotor \(J=L\).
  3. A cyclotron orbit is equivalent to an oscillator with \(E=E_\perp\) and \(\omega=\omega_c\). Obtain \(J=E_\perp/\omega_c=p_\perp^2/(2|q|B)\).
  4. Why for a cyclotron orbit \(J=|L_{\mathrm{mech}}|/2\), whereas for a rotor \(J=L\)? What plays the role of the periodic coordinate in these two exercises?