Resonance Processing

Abstract

In this chapter, we will study the Doppler broadening of resonances. Doppler effect improves reactor stability. Broadened resonance or heating of a fuel results in a higher probability of absorption, thus causing negative reactivity insertion or reduction of reactor power. One of the most important virtues of the optical model is that it takes into account the existence of giant or broad resonances in the total cross section as part of neutronic analysis for nuclear reactor systems. For resonances of energy levels, which are spaced widely apart, we can describe the energy dependency of the absorption cross section via Breit-Wigner single-level resonance formula.

Problems

Problem 7.1

Explain how the temperature coefficient of reactivity is largely determined by the resonances in U²³⁸.

Problem 7.2

Why does a PWR need to be refueled well before all the U²³⁵ in the fuel rod is used up?

Problem 7.3

The 129 keV gamma ray transitioning in Ir¹⁹¹ was used in a Mӧsbauer experiment in which a line shift equivalent to the full width at half maximum (Γ) was observed for a source speed of 1 cm/s. Estimate the value of Γ and the mean lifetime of the excited state in Ir¹⁹¹.

Problem 7.4

An excited atom of total mass M at rest with respect to a certain inertial system emits a photon, thus going over into a lower state with an energy smaller by Δw. Calculate the frequency of the photon emitted.

Problem 7.5

Calculate the spread in energy of the 661 keV internal conversion line of Cs¹³⁷ due to the thermal motion of the source. Assume that all atoms move with the root mean square velocity for a temperature of 15 °C.

Problem 7.6

Pound and Rebka at Harvard performed an experiment to verify the red shift predicted by general theory of relativity. The experiment consisted of the use of 14 keV γ-ray of Fe⁵⁷ source placed on the top of a tower 22.6 meter high and the absorber at the bottom. The red shift was detected by the Mӧsbauer technique. What velocity of the absorber foil was required to compensate the red shift, and in which direction?

Problem 7.7

Obtain an expression for the Doppler line width for a spectral line of wavelength λ emitted by an atom of mass m at a temperature T.

Problem 7.8

For the 2P_3/2 → 2S_1/2 transition of an alkali atom, sketch the splitting of the energy levels and the resulting Zeeman spectrum for atoms in a weak external magnetic field. (Express your results in terms of the frequency v₀ of the transition, in the absence of an applied magnetic field.)

$$ \left\{\mathrm{The}\ \mathrm{Lande}\ \mathrm{g}\hbox{-} \mathrm{factor}\ \mathrm{is}\ \mathrm{given}\ \mathrm{by}\kern0.24em g=1+\frac{j\left(j+1\right)+s\left(s+1\right)-l\left(l+1\right)}{2j\left(j+1\right)}\right\} $$

Problem 7.9

The spacings of adjacent energy levels of increasing energy in a calcium triplet are 30 × 10–4 and 60 × 10⁻⁴ eV. What are the quantum numbers of the three levels? Write down the levels using the appropriate spectroscopic notation.

Problem 7.10

An atomic transition line with wavelength 350 nm is observed to be split into three components, in a spectrum of light from a sunspot. Adjacent components are separated by 1.7 pm. Determine the strength of the magnetic field in the sunspot. μ_B = 9.17 × 10⁻²⁴ J T⁻¹.

Problem 7.11

Calculate the energy spacing between the components of the ground-state energy level of hydrogen when split by a magnetic field of 1.0 T. What frequency of electromagnetic radiation could cause a transition between these levels? What is the specific name given to this effect?

Problem 7.12

Consider the transition 2P_1/2 → 2S_1/2, for sodium in the magnetic field of 1.0 T, given that the energy splitting ΔE = gμ_BBm_j, where μ_B is the Bohr magneton. Draw the sketch.

Problem 7.13

Find the Doppler shift in wavelength of H line at 6563 ⁰A emitted by a star receding with a relative velocity of 3 × 10⁶ ms⁻¹.

Problem 7.14

Show that for slow speeds, the Doppler shift can be approximated as (Δλ/λ) = (υ/c) where Δλ is the change in wavelength.

Problem 7.15

A physicist was arrested for going over the railway level crossing on a motorcycle when the lights were red. When he was produced before the magistrate, the physicist declared that he was not guilty as red lights (λ = 670 nm) appeared green (λ = 525 nm) due to Doppler effect. At what speed was he traveling for the explanation to be valid? Do you think such a speed is feasible?

Problem 7.16

Find the wavelength shift in the Doppler effect for the sodium line 589 nm emitted by a source moving in a circle with a constant speed 0.05 c observed by a person fixed at the center of the circle.

Problem 7.17

A π-meson with a kinetic energy of 140 MeV decays in flight into μ-meson and a neutrino. Calculate the maximum energy, which:

• (a) The μ-meson.

• (b) The neutrino may have in the laboratory system.

(Mass of π-meson = 140 MeV/c2, mass of μ-meson = 106 MeV/c, mass of neutrino = 0). Assume ${T}_{\mu}^{\ast }=4.0$ MeV.

Problem 7.18

A linear accelerator produces a beam of excited carbon atoms of kinetic energy of 120 MeV. Light emitted on de-excitation is viewed at right angles to the beam and has a wavelength λ^′. If λ is the wavelength emitted by a stationary atom, what is the value of (λ^′ − λ)/λ? (Take the rest energies of both protons and neutrons to be 10⁹ eV.)

Problem 7.19

A certain spectral line of a star has natural frequency of 5 × 10¹⁷ c/s. If the star is approaching the earth at 300 km/s, what would be the fractional change of frequency?

Problem 7.20

Show that deuteron energy E has twice the range of proton of energy E/2.

Problem 7.21

If the mean range of 10 MeV protons in lead is 0.316 mm, calculate the mean range of 20 MeV deuterons and 40 MeV α-particles.

Problem 7.22

Show that the range of α-particles and protons of energy 1–10 MeV in aluminum is 1/1600 of the range in air at 15 °C, 760 mm of Hg. Hint: Apply the Bragg-Kleeman formula from your modern physics book.

Problem 7.23

Show that except for small ranges, the straggling of a beam of He³ particles is greater than that of a beam of He⁴ particles of equal range.

Problem 7.24

The range of a 15 MeV proton is 1100 ìm in nuclear emulsions. A second particle whose initial ionization is the same as the initial ionization of proton has a range of 165 μm. What is the mass of the particle? (The rate at which a singly ionized particle loses energy E by ionization along its range is given by dE/dR = K/(βc)² MeV μm where βc is the velocity of the particle and K is a constant depending only on emulsion; the mass of proton is 1837 mass of electron.)

Problem 7.25

• (a) Show that the specific ionization of 480 MeV α-particle is approximately equal to that of 30 MeV proton.

• (b) Show that the rate of change of ionization with distance is different for the two particles, and indicate how this might be used to identify one particle, assuming the identity of the other is known.

Problem 7.26

Calculate the 6.67 eV resonance integral for U²³⁸ in a typical PWR. Compare the actual resonance integral with its infinite dilute value.

Fuel	Moderator	Resonance
UO2 = 3% enriched	Water	ER = 6.67 eV
NUO2 = 0.0223 Mol/barn/cm	NH2O = 0.0335 Mol/barn/cm	σ0 = 216,000 barns
${\sigma}_{\mathrm{s}}^{\mathrm{O}}$ = 4.2 barns/atom	${\sigma}_{\mathrm{s}}^{\mathrm{H}}$ = 20.2 barns/atom	Γt = 0.0275 eV
${\sigma}_{\mathrm{p}}^{\mathrm{U}}$ = 8.3 barns/atom	Volume fraction = 0.55	Γγ = 0.026 eV
<R> = 0.94 cm	Square lattice	Γn = 0.0015 eV
Volume fraction = 0.45	Tfuel = 600 K