Chapter 1: Q86E (page 1)

Prove that fur any sine function $s i n (k x + ϕ)$ of wavelength shorter than 2a, where ais the atomic spacing. there is a sine function with a wavelength longer than 2a that has the same values at the points x = a , 2a , 3a . and so on. (Note: It is probably easier to work with wave number than with wavelength. We sick to show that for every wave number greater than there is an equivalent less than $π / a$ .)

Short Answer

Expert verified

It is proved that there is a sine function with a wavelength less than 2a.

Step by step solution

Sum of angle and expression for θ:

The atomic spacing is a.

The expression of sum of angles is given by,

$θ_{1} + θ_{2} = n π$

The expression for is given by,

$θ = k x + ϕ$

Determine sum of angles:

The expression of sum of angles is calculated as,

$θ_{1} + θ_{2} = n π n (k_{1} x + ϕ) + (k_{2} x + ϕ) = n π n k_{1} x + k_{2} x + 2 ϕ = n π n . . . . . . . . . . (1)$

For

$\begin{array}{rcl} (\frac{π}{a}) (n a) + ϕ & = & \frac{n π}{2} ϕ \\ = & \frac{n π}{2} - n π \\ = & - \frac{n π}{2} \end{array}$

solve further:

Substitute $\begin{array}{rcl} - \frac{n π}{2} \end{array}$ for $ϕ$ in equation (1).

$\begin{array}{rcl} (k_{1} + k_{2}) x + 2 (- \frac{n π}{2}) & = & n π \\ (k_{1} + k_{2}) (na) & = & 2 nπ \\ k_{1} + k_{2} & = & \frac{2 π}{a} \\ k_{1} & = & \frac{2 π}{a} - k_{2} \end{array}$

Now since,

$k_{1} > \frac{π}{a}, \frac{2 π}{a} - k_{2} > \frac{π}{a}, k_{2} < \frac{π}{a}$

Therefore, it is proved that there is sine function with wavelength less than 2a.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Sum of angle and expression for θ:

Determine sum of angles:

solve further:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Electromagnetism

Energy Physics

Astrophysics

Linear Momentum

Thermodynamics

Study anywhere. Anytime. Across all devices.