91Ó°ÊÓ

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression \(\lambda=h / p\) still holds, but we must use the rclativistic cxpression for momcntum, \(p=m v / \sqrt{1-v^{2} / c^{2}}\) (a) Show that the speed of an electron that has de Broglie wavelength \(\lambda\) is $$ v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}} $$ (b) The quantity \(h / m c\) equals \(2426 \times 10^{-12} \mathrm{m}\) (As we saw in Section 38.7 , this same quantity appears in Eq. (38.23), the expression for Compton scattering of photons by electrons) If \(\lambda\) is small compared to \(h / m c,\) the denominator in the expression found in part (a) is close to unity and the speed \(v\) is very close to \(c\) . In this case it is convenient to write \(v=(1-\Delta) c\) and express the speed of the electron in terms of \(\Delta\) rather than \(v\) . Find an expression for \(\Delta\) valid when \(\lambda \ll h / m c\) . [Hine. Use the binomial expansion \((1+z)^{n}=1+n z+n(n-1) z^{2} / 2+\cdots,\) valid for the case \(|z| < 1 .\) (c) How fast must an electron move for its de Broglie wavelength to be \(1.00 \times 10^{-15} \mathrm{m}\) , comparable to the size of a proton? Express your answer in the form \(v=(1-\Delta) c,\) and state the value of \(\Delta .\)

Step by step solution

01

Expression for speed using de Broglie wavelength

To show that the speed of an electron with a given de Broglie wavelength \(\lambda\) is \(v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}}\), we start by recognizing the de Broglie wavelength formula \(\lambda = \frac{h}{p}\). Substituting the relativistic momentum equation \(p = \frac{mv}{\sqrt{1 - v^2 / c^2}}\) into the de Broglie formula, we have \(\lambda = \frac{h \sqrt{1 - v^2 / c^2}}{mv}\). Rearranging for \(v\), multiply both sides by \(mv\) and divide by \(h\), we get \(mv\lambda = h \sqrt{1 - v^2 / c^2}\). Square both sides: \((mv\lambda)^2 = h^2 (1 - v^2 / c^2)\). Solving for \(v\), after some algebraic manipulation, we arrive at \(v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}}\).

02

Expression for \(\Delta\) when \(\lambda \ll h/mc\)

Given the expression for speed from part (a), the denominator \(\sqrt{1+(m c \lambda / h)^{2}}\) is close to unity when \(\lambda \ll h/mc\). We set \(v=(1-\Delta)c\) and use the relation \(v=\frac{c}{\sqrt{1+(m c \lambda/h)^2}}\), leading to \((1-\Delta) = \frac{1}{\sqrt{1+(m c \lambda/h)^2}}\). Rearranging, \(\Delta = 1 - \frac{1}{\sqrt{1+(m c \lambda/h)^2}}\). Using binomial expansion, \(\sqrt{1+z} \approx 1 + \frac{z}{2}\) for small z. Here \(z = (m c \lambda/h)^2\), hence \(\sqrt{1+z} \approx 1 + \frac{(m c \lambda/h)^2}{2}\). Therefore, \(\Delta \approx \frac{(m c \lambda/h)^2}{2}\).

03

Calculate speed for \(\lambda = 1.00 \times 10^{-15} \mathrm{m}\)

For \(\lambda = 1.00 \times 10^{-15} \mathrm{m}\), since \(\lambda\) is small compared to \(h/mc\), use the expression for \(\Delta\) derived in Step 2. With \(h/mc = 2426 \times 10^{-12} \, \mathrm{m}\), calculate \((m c \lambda/h)^2 = \left(\frac{1.00 \times 10^{-15}}{2426 \times 10^{-12}}\right)^2\). Simplify to find \(\Delta\) using \(\Delta \approx \frac{(m c \lambda/h)^2}{2}\). Calculate \(\Delta\) and express \(v\) in the form \(v=(1-\Delta)c\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

de Broglie Wavelength

The concept of the de Broglie wavelength connects the wave-like and particle-like properties of matter, specifically how particles like electrons can exhibit wave-like behavior. The de Broglie wavelength (\( \lambda \)) is given by the formula \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the particle. This expression shows that as the momentum of a particle increases, its wavelength decreases, meaning it behaves more like a particle than a wave.
- The de Broglie wavelength is crucial in understanding phenomena at microscopic scales, such as electron diffraction and the quantum mechanical nature of particles.
The challenge arises when dealing with particles moving at relativistic speeds, where classical momentum doesn't hold, and the relativistic momentum equation \( p = \frac{mv}{\sqrt{1 - v^2/c^2}} \) is necessary. At these speeds, incorporating the relativistic expression ensures accurate calculations for situations involving high-speed electrons, like probing atomic nuclei.

Atomic Nucleus

The atomic nucleus is the central part of an atom, consisting of protons and neutrons, and holds most of the atom's mass. Electrons orbit the nucleus at various energy levels. Understanding the structure and behavior of the atomic nucleus is essential in fields like nuclear physics and chemistry.
- Probing the atomic nucleus requires high-energy particles, like electrons, to overcome the electric forces and provide insight into the internal structure.
- Electron scattering experiments use the de Broglie wavelength of high-speed electrons to measure the dimensions and shape of the nucleus.
Given the extremely small size of a typical nucleus (on the order of 10^-15 meters), understanding how electrons can be used to investigate these scales helps in designing experiments to study nuclear properties and interactions.

Electron Speed

Electron speed becomes a significant factor when dealing with particles moving close to the speed of light, particularly in the context of probing atomic nuclei. The relativistic speed of an electron can be derived from its de Broglie wavelength, as shown in the exercise solution:\[ v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}} \]
- In this equation, \( v \) is the speed of the electron, \( c \) is the speed of light, and \( \lambda \) is the de Broglie wavelength.
- This formula accounts for relativistic effects, ensuring the measured speed doesn't exceed the speed of light.
For cases where the wavelength \( \lambda \) is much smaller than \( h/mc \), \( v \) approaches \( c \), reflecting the increased momentum and energy required to achieve such wavelengths. Calculating the speed accurately is crucial for determining the kinetic energy and behavior of electrons in various scientific applications, including accelerators and particle physics.

Binomial Expansion

The binomial expansion is a mathematical technique used to approximate expressions raised to a power, especially useful when dealing with small quantities. It is given by:\[(1+z)^n = 1 + nz + \frac{n(n-1)z^2}{2} + \cdots\]
This series is particularly helpful when \(|z| < 1\).
- In the context of electron speed, binomial expansion simplifies the expression \((1+z)^{-1/2}\) when \( z = (mc\lambda/h)^2 \) is very small.
- It allows us to approximate \( \frac{1}{\sqrt{1+z}} \) as \( 1 - \frac{z}{2} \), leading to the calculation of \( \Delta \) in the exercise.
Using the binomial expansion in this way enables physicists and engineers to simplify complex relativistic equations in high-speed scenarios, making analysis and computation more manageable, while maintaining accuracy for small values of \( \lambda \).