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Ch. 2 Problems

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Basic

Medium

Advanced

Basic

B-1. In Compton scattering, suppose the wavelength of the incident X-ray is nm. For a scattering angle (back

In Compton scattering, suppose the wavelength of the incident X-ray is \(\lambda = 0.0711\) nm. Find the wavelength \(\lambda'\) of the scattered X-ray when the scattering angle is \(\theta = 180°\) (backscattering). You may use the Compton wavelength \(h/(m_e c) = 2.43 \times 10^{-12}\) m.

Hint

Substitute \(\theta = 180°\) into Eq. (2.1) \(\lambda' - \lambda = \dfrac{h}{m_e c}(1 - \cos\theta)\). Note that \(\cos 180° = -1\).

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B-2. In the Compton scattering formula (2.1), find the wavelength shift for scattering angle

In the Compton scattering formula (2.1), find the wavelength shift \(\Delta\lambda = \lambda' - \lambda\) for a scattering angle of \(\theta = 60°\).

Hint

Substitute \(\cos 60° = 0.5\) and calculate \(\Delta\lambda = \dfrac{h}{m_e c}(1 - \cos 60°)\).

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B-3. Find the de Broglie wavelength of a proton with mass kg moving at velocity m/s

Find the de Broglie wavelength \(\lambda = h/(mv)\) of a proton with mass \(m = 1.67 \times 10^{-27}\) kg moving at velocity \(v = 3.0 \times 10^{4}\) m/s.

Hint

Substitute \(h = 6.626 \times 10^{-34}\) J·s in the numerator and \(mv\) in the denominator to calculate. Verify that the units come out to m.

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B-4. de Broglie wavelength of an electron accelerated through voltage V, using the simplified formula

Find the de Broglie wavelength of an electron accelerated through an accelerating voltage \(V_{\mathrm{acc}} = 200\) V, using the simplified formula

\[\lambda \approx \frac{1.226}{\sqrt{V_{\mathrm{acc}}}}\ \mathrm{nm}\]

Also, convert the result to Å (angstrom) units.

Hint

Calculate \(\sqrt{200}\) first, then divide \(1.226\) by it. Use the relation 1 nm = 10 Å.

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B-5. When an electron (mass kg) is accelerated through an accelerating voltage V, find the electron's velocity using Eq. (2.9)

When an electron (mass \(m_e = 9.109 \times 10^{-31}\) kg) is accelerated through an accelerating voltage \(V_{\mathrm{acc}} = 54\) V, find the electron's velocity \(v\) using Eq. (2.9)

\[\frac{1}{2}m_e v^2 = eV_{\mathrm{acc}}\]

where \(e = 1.602 \times 10^{-19}\) C.

Hint

Calculate \(v = \sqrt{2eV_{\mathrm{acc}}/m_e}\). First compute the numerator \(2eV_{\mathrm{acc}}\) and the denominator \(m_e\) numerically, then divide and take the square root.

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B-6. Finding the wavelength of a diffracted wave given crystal plane spacing in Å, angle of incidence, and order in the Bragg condition

In the Bragg condition \(2d\sin\theta = n\lambda\), find the wavelength \(\lambda\) of the diffracted wave in Å when the crystal plane spacing is \(d = 0.91\) Å, the angle of incidence is \(\theta = 65°\), and the order is \(n = 1\).

Hint

Substitute the numerical values into \(\lambda = 2d\sin\theta / n\). Use \(\sin 65° \approx 0.906\).

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B-7. Find the de Broglie wavelength of a neutron (mass in kg) with kinetic energy in eV, given the conversion in J

Find the de Broglie wavelength of a neutron (mass \(m_n = 1.675 \times 10^{-27}\) kg) with kinetic energy \(K = 1.0\) eV. Assume \(1\ \mathrm{eV} = 1.602 \times 10^{-19}\) J.

Hint

The momentum can be obtained from \(p = \sqrt{2m_n K}\). Convert \(K\) to units of J, then calculate \(\lambda = h/p\).

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B-8. Using de Broglie's relations expressed in terms of angular frequency and wavenumber, E = ℏω and p = ℏk

Using de Broglie's relations expressed in terms of angular frequency \(\omega\) and wavenumber \(k\), namely \(E = \hbar\omega\) and \(p = \hbar k\), find the wavenumber \(k\) in SI units (m\(^{-1}\)) corresponding to an electron with wavelength \(\lambda = 2.0\) Å.

Hint

Use the relation \(k = 2\pi/\lambda\). Convert \(\lambda\) to meters before substituting (\(1\ \mathrm{Å} = 10^{-10}\) m).

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Medium

M-1. Derivation of the Relationship Between Accelerating Voltage and de Broglie Wavelength

Derivation of the Relationship Between Accelerating Voltage and de Broglie Wavelength

When an electron (mass \(m_e\), charge \(e\)) is accelerated from rest through a voltage \(V_{\mathrm{acc}}\), follow the steps below to derive that the de Broglie wavelength is expressed as

\[\lambda = \frac{h}{\sqrt{2m_e eV_{\mathrm{acc}}}}\]

(a) From energy conservation, express the electron's momentum \(p\) in terms of \(m_e\), \(e\), and \(V_{\mathrm{acc}}\).

(b) Substitute into the de Broglie relation \(\lambda = h/p\) to obtain the above expression.

(c) Substitute numerical values of the constants to verify that \(\lambda \approx 1.226/\sqrt{V_{\mathrm{acc}}}\) nm is obtained.

Hint

(a) Find \(v\) from \(\frac{1}{2}m_e v^2 = eV_{\mathrm{acc}}\) and set \(p = m_e v\). Alternatively, you may directly obtain \(p\) from \(p^2/(2m_e) = eV_{\mathrm{acc}}\). (c) Substitute the numerical values of \(h\), \(m_e\), and \(e\) all in SI units.

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M-2. Reproduction Calculation of the Davisson-Germer Experiment

Reproduction Calculation of the Davisson-Germer Experiment

In the Davisson-Germer experiment, electrons accelerated through a voltage of \(V_{\mathrm{acc}} = 54\) V were directed at a nickel single crystal (surface atomic spacing \(d = 2.15\) Å), and a strong diffraction peak was observed at a scattering angle of \(\phi = 50°\).

(a) Using Eq. (2.12), find the de Broglie wavelength \(\lambda_{\mathrm{dB}}\) of electrons accelerated through 54 V.

(b) Substituting \(n = 1\) and \(\phi = 50°\) into the diffraction condition \(d\sin\phi = n\lambda\), find the wavelength \(\lambda_{\mathrm{exp}}\) obtained from the experiment.

(c) Compare \(\lambda_{\mathrm{dB}}\) and \(\lambda_{\mathrm{exp}}\), and discuss the validity of the de Broglie hypothesis.

Hint

(a) Calculate \(\lambda \approx 1.226/\sqrt{54}\) nm. (b) Use \(\sin 50° \approx 0.766\). (c) It is helpful to evaluate the difference between the two values as a percentage.

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M-3. Structural Analysis of the Compton Scattering Formula

Structural Analysis of the Compton Scattering Formula

Regarding the Compton scattering formula

\[\lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)\]

answer the following questions.

(a) Determine the range of possible values of \(\Delta\lambda = \lambda' - \lambda\) as a function of \(\theta\) (for \(0 \le \theta \le 180°\)).

(b) Calculate the Compton wavelength \(h/(m_p c)\) for the case where the scattering target is a proton (mass \(m_p = 1.673 \times 10^{-27}\) kg) instead of an electron, and compare it with the electron case.

(c) Using the result from (b), explain why "the wavelength shift in Compton scattering is larger for lighter particles."

Hint

(a) Consider the minimum and maximum values of \(1 - \cos\theta\). (b) Take the ratio with the electron Compton wavelength \(2.43 \times 10^{-12}\) m. (c) Note that the Compton wavelength is inversely proportional to mass.

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M-4. Analysis of Electron Diffraction Using the Bragg Condition

Analysis of Electron Diffraction Using the Bragg Condition

Consider a situation modeled after G. P. Thomson's experiment. Electrons accelerated through a voltage of \(V_{\mathrm{acc}} = 10{,}000\) V are incident on an aluminum crystal with interplanar spacing \(d = 2.34\) Å.

(a) Find the de Broglie wavelength of the electrons.

(b) Using the Bragg condition \(2d\sin\theta = n\lambda\), find the incident angle \(\theta\) at which \(n = 1\) diffraction occurs.

(c) When the accelerating voltage is doubled (\(20{,}000\) V), how does the diffraction angle \(\theta\) for \(n = 1\) change? Provide a qualitative explanation and then calculate the specific value.

Hint

(a) Use Eq. (2.12). \(V_{\mathrm{acc}} = 10{,}000\) V. (b) Find \(\theta\) from \(\sin\theta = \lambda/(2d)\). Consider whether the small-angle approximation \(\sin\theta \approx \theta\) (in radians) is applicable. (c) Use the fact that \(\lambda \propto 1/\sqrt{V_{\mathrm{acc}}}\).

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Advanced

A-1. Derivation of the Compton Scattering Formula

Derivation of the Compton Scattering Formula

Consider the case where an X-ray photon of wavelength \(\lambda\) collides with an electron at rest (mass \(m_e\)). Let \(\lambda'\) be the wavelength of the scattered photon, \(\theta\) be the scattering angle, \(p_e\) be the magnitude of the recoil electron's momentum, and \(\varphi\) be the recoil angle (the angle between the incident direction and the electron's direction of motion).

Following the steps below, derive the Compton scattering formula

\[\lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)\]

The photon energy is given by \(E = pc\) (the relativistic relation for a massless particle), and the electron energy is given relativistically by \(E_e = \sqrt{(p_e c)^2 + (m_e c^2)^2}\).

(a) Write down the energy conservation equation. Express it using \(h c/\lambda\) for the incident photon energy, \(h c/\lambda'\) for the scattered photon energy, and \(m_e c^2\) for the rest energy of the recoil electron.

(b) Write down the momentum conservation equations, separating them into the component along the incident direction (\(x\) direction) and the perpendicular component (\(y\) direction).

(c) Eliminate the recoil angle \(\varphi\) from the two equations in (b), and express \(p_e^2\) in terms of \(\lambda\), \(\lambda'\), and \(\theta\).

(d) From the energy conservation in (a), express \(p_e^2 c^2\) in terms of \(\lambda\) and \(\lambda'\), and by equating it with the result from (c), derive the Compton scattering formula.

Hint

(b) \(x\) component: \(h/\lambda = (h/\lambda')\cos\theta + p_e \cos\varphi\); \(y\) component: \(0 = (h/\lambda')\sin\theta - p_e \sin\varphi\). (c) Rearrange each component, square both sides, and use \(\cos^2\varphi + \sin^2\varphi = 1\). (d) Rewrite the energy conservation equation in the form \((E_e)^2 = (p_e c)^2 + (m_e c^2)^2\) and solve for \(p_e^2 c^2\). Terms \(1/\lambda - 1/\lambda'\) and \(1/\lambda^2 + 1/\lambda'^2 - 2\cos\theta/(\lambda\lambda')\) will appear. Ultimately, the terms involving the product \(\lambda'\lambda\) remain, and the expression simplifies to the form \(\lambda' - \lambda\).

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A-2. Relativistic Electron de Broglie Wavelength and Application to Electron Microscopy

Relativistic Electron de Broglie Wavelength and Application to Electron Microscopy

Equation (2.11) in this chapter is based on the non-relativistic approximation (\(v \ll c\)). When the accelerating voltage is very high (e.g., \(V_{\mathrm{acc}} = 200\) kV as used in electron microscopes), the kinetic energy of the electron is no longer negligible compared to the rest energy \(m_e c^2 \approx 511\) keV, and relativistic corrections become necessary.

(a) In the relativistic energy-momentum relation

\[E^2 = (pc)^2 + (m_e c^2)^2\]

use the fact that the total energy is \(E = m_e c^2 + eV_{\mathrm{acc}}\) (rest energy + kinetic energy) to express the momentum \(p\) in terms of \(m_e\), \(c\), \(e\), and \(V_{\mathrm{acc}}\).

(b) Show that the relativistic de Broglie wavelength can be written as

\[\lambda = \frac{h}{\sqrt{2m_e eV_{\mathrm{acc}}\left(1 + \dfrac{eV_{\mathrm{acc}}}{2m_e c^2}\right)}}\]

(c) For \(V_{\mathrm{acc}} = 200\) kV, calculate the wavelength numerically using both the non-relativistic formula (2.11) and the relativistic formula from (b), and evaluate the difference between them as a percentage.

(d) Assume that the resolution of an electron microscope is approximately equal to the wavelength used. Discuss whether an electron microscope with \(V_{\mathrm{acc}} = 200\) kV can resolve atomic-level structures (\(\sim 1\) Å).

Hint

(a) Substitute \(E = m_e c^2 + eV_{\mathrm{acc}}\) into \(E^2 = (pc)^2 + (m_e c^2)^2\) and solve for \(p\). Expanding \((m_e c^2 + eV_{\mathrm{acc}})^2 - (m_e c^2)^2\) yields \(2m_e c^2 \cdot eV_{\mathrm{acc}} + (eV_{\mathrm{acc}})^2\). (b) Calculate \(\lambda = h/p\) from the result of (a) and factor out common terms. (c) Use \(eV_{\mathrm{acc}} = 200\) keV and \(m_e c^2 = 511\) keV to evaluate the magnitude of the correction factor \(eV_{\mathrm{acc}}/(2m_e c^2)\).

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