Question 65 - Problems - Chapter 1

Problem:

Some crystal can be polished at an angle to produce sharp "Atomic Steps" (see fig. below).

Question:

a) Wat angle $\theta$ should be chosen to produce a step every 5 atoms?

b) What angle to produce a step every 10 atoms?

c) How many atoms will there be along a step if a crystal is polished as flat as is normal practical. (usualy arround 0.10°)

We assume that the atoms are spherical.

Solution:

Since the atoms are spherical we can use the ratio without having to know the dimensions.

$$ \theta =\textrm{atan}\left(\frac{\textrm{upright}\;\textrm{side}}{\textrm{lying}\;\textrm{side}}\right) $$

# Calculations
import numpy as np
from astropy import units as u
from astropy.coordinates import Angle

step = 5
angle = np.arctan(1/step)* u.rad
Theta = Angle(angle, u.degree)
print(f'The angle for a step of {step} atoms is {Theta.dms.d:.0f}° {Theta.dms.m:.0f}\' {Theta.dms.s:.2f}\"' )

step = 10
angle = np.arctan(1/step)* u.rad
Theta = Angle(angle, u.degree)
print(f'The angle for a step of {step} atoms is {Theta.dms.d:.0f}° {Theta.dms.m:.0f}\' {Theta.dms.s:.2f}\"' )

angle = Angle(0.10, u.degree)
step = 1/np.tan(angle);
print(f'The step for the lowest possible angle is {step:.0f} atoms')

The angle for a step of 5 atoms is 11° 18' 35.76"
The angle for a step of 10 atoms is 5° 42' 38.14"
The step for the lowest possible angle is 573 atoms