Exercises: Neutrino Oscillations in Matter. Refractive Index

Lecture 11 · Introduction to Neutrino Physics

Author

Dmitry V. Naumov

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Exercises

Exercise 11.1. The first Fresnel zone

The rings of the layer with boundaries \(r_m=z+m\lambda/2\) are called Fresnel zones. Find the radius \(b_1\) of the first zone for \(z\gg\lambda\) and the number of scatterers in it for a layer of thickness \(dz\) and concentration \(\mathcal N\). Explain why the growth of the ring area compensates the factor \(1/r\) of the spherical wave, and check that for \(z\gg ka^2\) the scattering amplitude hardly changes over the first zone.

Exercise 11.2. Phase and attenuation

Let \(f(0)=a+ib\), where \(a,b>0\), and the medium is homogeneous over the length \(D\).

  1. Find the additional phase of the transmitted wave.
  2. Find the intensity ratio \(I(D)/I(0)\).
  3. Using the optical theorem, express the result in terms of the total cross section.

Exercise 11.3. Two refractive indices

The matrix refractive index has the form

\[ \mathbf n=n_0\mathbb I+ \begin{pmatrix}\delta&\kappa\\\kappa&-\delta\end{pmatrix}. \tag{1}\]

  1. Find the two eigen refractive indices.
  2. Find the rotation angle from the original basis to the eigenstates.
  3. Show that for \(\delta=0\) the original first state passes completely into the second after a suitable path length, if \(\kappa\ne0\).

Exercise 11.4. When coherence is lost

For each process, indicate whether it can be included in the real refractive index of the transmitted wave, or whether it requires a description of the loss of coherence:

  • elastic scattering strictly forward without a change of the atom’s state;
  • excitation of the atom;
  • scattering through a finite angle;
  • elastic scattering on two indistinguishable centers;
  • absorption of a neutrino.

For each answer, formulate the physical criterion of distinguishability of the alternatives.

Exercise 11.5. A plate and the accumulated phase

Consider a transparent layer with a real refractive index. Neglect reflection at the boundaries.

  1. At fixed frequency, obtain the ratios \(\lambda_{\mathrm{med}}/\lambda_0\) and \(v_{\mathrm{ph}}/c\).
  2. Show that after exiting a layer of thickness \(D\) the additional phase is equal to \(\Delta\varphi=2\pi(n-1)D/\lambda_0\), even though the wavelength is again equal to \(\lambda_0\).
  3. Calculate the phase for \(n=1.08\) and \(D=1.5\lambda_0\). What must the thickness be, for the same \(n\), to obtain a shift \(\pi\)?
  4. Divide the layer into \(M\) parts and find the limit \([1+i\Delta\varphi/M]^M\) as \(M\to\infty\). Why does the small contribution of each layer allow a large total phase?
  5. Check the calculation with the applet (in Russian). Find two pairs \(n,D\) with the same phase at the exit. Are the waves inside the plate the same?

Exercise 11.6. Two eigenstates

Let, at constant density,

\[ \begin{aligned} |\nu_e\rangle&=c_m|\nu_{\mathrm{mat},1}\rangle+s_m|\nu_{\mathrm{mat},2}\rangle,\\ |\nu_x\rangle&=-s_m|\nu_{\mathrm{mat},1}\rangle+c_m|\nu_{\mathrm{mat},2}\rangle, \end{aligned} \qquad c_m=\cos\theta_m,\quad s_m=\sin\theta_m. \tag{2}\]

Denote \(\Phi=E(n_{\mathrm{mat},1}-n_{\mathrm{mat},2})z\).

  1. Write down the state at distance \(z\) for the initial \(\nu_e\). By projecting onto \(\nu_x\), obtain the amplitude and the probability \(P_{e\to x}=\sin^22\theta_m\sin^2(\Phi/2)\).
  2. Show that the common shift \(n_{\mathrm{mat},i}\to n_{\mathrm{mat},i}+C\) changes only the overall phase. Decompose the increment \(\operatorname{diag}(\delta n_e,\delta n_x)\) into a common part and a part that distinguishes the flavors.
  3. Calculate \(P_{e\to x}\) for \(\theta_m=30^\circ\) and \(\Phi=\pi\). How does the answer change for \(\theta_m=0\), \(\theta_m=45^\circ\), or equal refractive indices?
  4. Find the first maximum and the spatial period of the probability. What does the angle change, and what does the difference of the refractive indices change?
  5. Check your answers with the applet (in Russian). Which parameters are specified independently? How can they be found for a layer of given density from the weak interaction?

Exercise 11.7. A half-wave plate and neutrinos

The input is a pure state \(\nu_e\), the mass axes \(\nu_1,\nu_2\) are rotated by \(-\theta\) relative to the flavor axes. The phase difference of the waves \(\nu_1\) and \(\nu_2\) is \(\pi\).

  1. Find the output state in the basis \((\nu_e,\nu_x)\) and verify that it is rotated by \(2\theta\) relative to the input.
  2. Calculate \(P_{e\to x}\) for \(\theta=33.7^\circ\) and compare with the fractions of light intensity \(I_V\) and \(I_H\) for the same plate.
  3. Does the output coincide with one of the crystal axes? For which \(\theta\) will it coincide with \(\nu_x\)?

Exercise 11.8. The refractive index operator

  1. Starting from \(\hat n_{\mathrm{vac}}=\sum_i n_i|\nu_i\rangle\langle\nu_i|\) and the matrix \(U\), obtain the matrix \([\hat n_{\mathrm{vac}}]_f\) in the flavor basis.
  2. Show that its trace and determinant do not depend on \(\theta\).
  3. Add \(\delta\hat n=\mathrm{diag}(\delta n_e,\delta n_x)\) and find the angle \(\theta_m\) as a function of \(\theta\), \(n_2-n_1\), and \(\delta n_e-\delta n_x\). For what relation among these quantities is \(\theta_m=45^\circ\)?

Exercise 11.9. Vacuum and the Sun: two crystals of the same scale

A neutrino with energy \(E=10\) MeV, two flavors, \(\Delta m^2=m_2^2-m_1^2= 7.5\times10^{-5}\) eV\(^2\), \(\theta=33.7^\circ\). Constants: \(\hbar c=1.973\times10^{-7}\) eV\(\cdot\)m, \(G_F=1.166\times10^{-5}\) GeV\(^{-2}\), solar radius \(R_\odot=6.96\times10^{8}\) m. At the center of the Sun the electron density is \(\mathcal N_e=6.08\times10^{25}\) cm\(^{-3}\), and the charged current gives the difference of the additions (with \(\hbar=c=1\); take the formula as given)

\[ \delta n_e-\delta n_x=-\frac{\sqrt2\,G_F\mathcal N_e}{E}. \tag{3}\]

  1. Find the neutrino wavelength \(\lambda=2\pi\hbar c/E\). How many wavelengths fit between the center of the Sun and its surface?
  2. The vacuum crystal. From \(n_i\simeq1-m_i^2/(2E^2)\) find \(n_1-n_2\). Which of the two indices is larger? Over what length \(L_{\mathrm{vac}}\) does the phase difference \(E(n_1-n_2)z\) reach \(2\pi\), and how many wavelengths does it contain?
  3. Compare with optics: for calcite \(n_o-n_e=0.172\) at \(\lambda_0=589\) nm. Find the thickness of a calcite half-wave plate and the length of a neutrino “half-wave plate” in vacuum. Express both in wavelengths.
  4. Matter. Compute \(V=\sqrt2\,G_F\mathcal N_e\) in eV (convert the density to natural units with the factor \((\hbar c)^3\)) and \(\delta n_e-\delta n_x\). Find the ratio \[ A=\frac{\delta n_x-\delta n_e}{n_1-n_2}. \tag{4}\]
  5. If the central density persisted up to the surface, what phase difference \(E|\delta n_e-\delta n_x|R_\odot\) would the neutrino accumulate? In the real Sun the density falls toward the edge, and about \(4\times10^{3}\) rad accumulates. What path length at the central density does this correspond to?
  6. At constant central density the eigenstates of the sum are determined by \[ \tan2\theta_m=\frac{\sin2\theta}{\cos2\theta-A}, \qquad \frac{n_{\mathrm{mat},1}-n_{\mathrm{mat},2}}{n_1-n_2} =\sqrt{(\cos2\theta-A)^2+\sin^22\theta}. \tag{5}\] Find \(\theta_m\) (in which quadrant does \(2\theta_m\) lie?), the amplitude \(\sin^22\theta_m\) and the period \(L_{\mathrm{mat}}\). Compare with vacuum. What fraction of \(\nu_e\) belongs to \(\nu_{\mathrm{mat},2}\)?

State the main physical conclusion: why does a difference of indices of order \(10^{-19}\) not mean a small effect, and why can neither vacuum nor matter be neglected at the center of the Sun?

Exercise 11.10. A refractive index below one

An electron in an atom is an oscillator with natural frequency \(\omega_0\) and damping \(\gamma\). In Gaussian units, \(r_e=e^2/(mc^2)=2.818\times10^{-13}\) cm,

\[ f(0)=r_e\,\frac{\omega^2}{\omega_0^2-\omega^2-i\gamma\omega}, \qquad n-1=\frac{2\pi\mathcal N f(0)}{k^2}, \tag{6}\]

where \(\mathcal N\) is the electron density. Consider silicon: \(\rho=2.33\) g/cm\(^3\), molar mass \(M=28.09\) g/mol, \(Z=14\), the X-ray line \(\lambda=0.154\) nm (\(\hbar\omega\approx8.05\) keV, all \(\omega_0\) of the atom are much smaller than \(\omega\)).

  1. Neglecting \(\gamma\), find the sign of \(n-1\) for \(\omega<\omega_0\) and for \(\omega>\omega_0\). Show that for \(\omega<\omega_0\) the index grows with frequency. Which light does a prism deflect more strongly, red or blue?
  2. Show that for \(\omega\gg\omega_0\) \[ n=1-\delta,\qquad \delta=\frac{\mathcal N r_e\lambda^2}{2\pi}, \tag{7}\] and \(\omega_0\) drops out of the answer. Find \(\mathcal N\) and \(\delta\) for silicon.
  3. A “paradox”: for \(n<1\) the phase velocity \(c/n\) exceeds \(c\). Write \(\delta=\omega_p^2/(2\omega^2)\), where \(\omega_p^2=4\pi\mathcal Ne^2/m\), and from \(k=n\omega/c\) find the group velocity \(v_g=d\omega/dk\). Show that \(v_g<c\) and \(v_{\mathrm{ph}}v_g\simeq c^2\). Compute \(\hbar\omega_p\).
  4. An X-ray beam falls from vacuum onto a smooth silicon surface at a grazing angle \(\alpha\) to it. Using Snell’s law, show that for \(\alpha<\alpha_c\), \(\cos\alpha_c=n\), there is no refracted beam and total external reflection occurs. Find \(\alpha_c\approx\sqrt{2\delta}\) in mrad and in degrees. Why are the mirrors of X-ray telescopes set almost parallel to the beam?

State the main conclusion: what sets the sign of \(n-1\), and why does a neutrino refractive index below one not contradict causality?