6.6 Wave-particle duality  (Page 8/12)

Find the maximum change in X-ray wavelength that can occur due to Compton scattering. Does this change depend on the wavelength of the incident beam?

A photon of wavelength 700 nm is incident on a hydrogen atom. When this photon is absorbed, the atom becomes ionized. What is the lowest possible orbit that the electron could have occupied before being ionized?

What is the maximum kinetic energy of an electron such that a collision between the electron and a stationary hydrogen atom in its ground state is definitely elastic?

Singly ionized atomic helium ${\text{He}}^{+1}$ is a hydrogen-like ion. (a) What is its ground-state radius? (b) Calculate the energies of its four lowest energy states. (c) Repeat the calculations for the ${\text{Li}}^{2+}$ ion.

A triply ionized atom of beryllium ${\text{Be}}^{3+}$ is a hydrogen-like ion. When ${\text{Be}}^{3+}$ is in one of its excited states, its radius in this n th state is exactly the same as the radius of the first Bohr orbit of hydrogen. Find n and compute the ionization energy for this state of ${\text{Be}}^{3+}.$

In extreme-temperature environments, such as those existing in a solar corona, atoms may be ionized by undergoing collisions with other atoms. One example of such ionization in the solar corona is the presence of ${\text{C}}^{5+}$ ions, detected in the Fraunhofer spectrum. (a) By what factor do the energies of the ${\text{C}}^{5+}$ ion scale compare to the energy spectrum of a hydrogen atom? (b) What is the wavelength of the first line in the Paschen series of ${\text{C}}^{5+}$ ? (c) In what part of the spectrum are these lines located?

(a) Calculate the ionization energy for ${\text{He}}^{+}.$ (b) What is the minimum frequency of a photon capable of ionizing ${\text{He}}^{+}$ ?

Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. Find the wavelength of such an ultracold neutron and its kinetic energy.

Find the velocity and kinetic energy of a 6.0-fm neutron. (Rest mass energy of neutron is $E_0=940\text{MeV}\text{.)}$

The spacing between crystalline planes in the NaCl crystal is 0.281 nm, as determined by X-ray diffraction with X-rays of wavelength 0.170 nm. What is the energy of neutrons in the neutron beam that produces diffraction peaks at the same locations as the peaks obtained with the X-rays?

What is the wavelength of an electron accelerated from rest in a 30.0-kV potential difference?

Calculate the velocity of a $1.0\text{-μ}\text{m}$ electron and a potential difference used to accelerate it from rest to this velocity.

In a supercollider at CERN, protons are accelerated to velocities of 0.25 c . What are their wavelengths at this speed? What are their kinetic energies? If a beam of protons were to gain its kinetic energy in only one pass through a potential difference, how high would this potential difference have to be? (Rest mass energy of a proton is $E_0=938\text{MeV).}$

Find the de Broglie wavelength of an electron accelerated from rest in an X-ray tube in the potential difference of 100 keV. (Rest mass energy of an electron is $E_0=511\text{keV}\text{.)}$

The cutoff wavelength for the emission of photoelectrons from a particular surface is 500 nm. Find the maximum kinetic energy of the ejected photoelectrons when the surface is illuminated with light of wavelength 450 nm.

Compare the wavelength shift of a photon scattered by a free electron to that of a photon scattered at the same angle by a free proton.

The spectrometer used to measure the wavelengths of the scattered X-rays in the Compton experiment is accurate to $5.0\times {10}^{\mathrm{-4}}\text{nm}.$ What is the minimum scattering angle for which the X-rays interacting with the free electrons can be distinguished from those interacting with the atoms?

Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge $q=+Ze.$ Derive the formulas for the energy $E_n$ of the electron in n th orbit and the orbital radius $r_n.$

Assume that a hydrogen atom exists in the $n=2$ excited state for ${10}^{\mathrm{-8}}\text{s}$ before decaying to the ground state. How many times does the electron orbit the proton nucleus during this time? How long does it take Earth to orbit the sun this many times?

An atom can be formed when a negative muon is captured by a proton. The muon has the same charge as the electron and a mass 207 times that of the electron. Calculate the frequency of the photon emitted when this atom makes the transition from $n=2$ to the $n=1$ state. Assume that the muon is orbiting a stationary proton.