Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

Answers without the blur.

Just sign up for free and you're in.

Short Answer

An electron is emitted from a middle-mass nuclide (A=150, say) with a kinetic energy of 1.0 MeV. (a) What is its de-Broglie wavelength? (b) Calculate the radius of the emitting nucleus. (c) Can such an electron be confined as a standing wave in a “box” of such dimensions? (d) Can you use these numbers to disprove the (abandoned) argument that electrons actually exist in nuclei?

The de-Broglie wavelength of the electron is $9 \times 10^{2} fm$ .
The radius of the emitting nucleus is 6.4 fm .
No, the electron cannot be confined as a standing wave in a “box” of such dimensions due to their large wavelengths.
Yes, these numbers can be used to disprove the argument that electrons actually exist.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

Mass number of the nuclide A = 150.
Kinetic energy of the emitted electron.

Step 2: Understanding the concept of size and confinement

The electron can only be confined within a given structure only if its wavelength is smaller than the atomic radius of the nuclide. Now, the wavelength of an electron is associated with the momentum according to the de-Broglie concept. Thus, by comparing the wavelength with the calculated atomic radius of the nuclide, consider the strong argument for the case of confinement.

The kinetic energy of a particle in motion:

$K = m c^{2}$ ….. (i)

The energy and momentum relation according to relativistic concept,

$p c = \sqrt{K^{2} + m^{2} c^{4}}$ …… (ii)

The atomic radius of a nuclide using its nucleon or mass number,

$r = r_{0} A^{\frac{1}{3}}$ …… (iii)

The de-Broglie wavelength of a particle of smaller size:

$λ = \frac{h}{p}$ …… (iv)

Step 3: a) Calculate the de-Broglie wavelength

Consider the known data, ${mc}^{2} = 0.511 MeV$ and hc = 1240 MeV.fm

Substitute the value of momentum from equation (ii) in equation (iv), consider de-Broglie wavelength of the electron as follows:

$λ = \frac{h c}{\sqrt{K^{2} + m^{2} c^{4}}} = \frac{h c}{\sqrt{K^{2} + 2 K m c^{2}}} (from equation (i), K = {mc}^{2})$

Substitute the values as:

$λ = \frac{1240 MeV . fm}{\sqrt{{(1.0 MeV)}^{2} + 2 (1.0 MeV) (0.511 MeV)}} = 9 \times 10^{2} fm$

Hence, the value of the wavelength is $9 \times 10^{2} fm$ .

Step 4: b) Calculate the radius of the emitting nucleus

Using the given data in equation (iii), determine the radius of the emitting nucleus as follows:

$r = (1.2 fm) {(150)}^{1 / 3} = 6.4 fm$

Hence, the value of the radius is 6.4 fm .

Step 5: c) Calculate whether the electron of the given nuclide can be confined as a standing wave

Since, $λ > > r$ from parts (a) and (b) calculations the electron cannot be confined in the nuclide. Recall that at least $\frac{λ}{2}$ is needed in any particular direction, to support a standing wave in an “infinite well.” A finite well is able to support slightly less than $\frac{λ}{2}$ (as one can infer from the ground state wave function in Fig. 39-6), but in the present case $\frac{λ}{r}$ is far too big to be supported.

Step 6: d) Determine whether an argument can be raised for part (c)

A strong cas e can be made on the basis of the remarks in part (c), above.

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of size and confinement

Step 3: a) Calculate the de-Broglie wavelength

Step 4: b) Calculate the radius of the emitting nucleus

Step 5: c) Calculate whether the electron of the given nuclide can be confined as a standing wave

Step 6: d) Determine whether an argument can be raised for part (c)

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of size and confinement

Step 3: a) Calculate the de-Broglie wavelength

Step 4: b) Calculate the radius of the emitting nucleus

Step 5: c) Calculate whether the electron of the given nuclide can be confined as a standing wave

Step 6: d) Determine whether an argument can be raised for part (c)

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics