Q065 - Physics For Engineers And Scientists - Solutions

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Question 65 - Problems - Chapter 1

Problem:

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

Pollished crystal-grid

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.

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

In [1]:
# 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

سُوۡرَةُ حٰمٓ السجدة / فُصّلَت
ثُمَّ اسْتَوَىٰ إِلَى السَّمَاءِ وَهِيَ دُخَانٌ فَقَالَ لَهَا وَلِلْأَرْضِ ائْتِيَا طَوْعًا أَوْ كَرْهًا قَالَتَا أَتَيْنَا طَائِعِينَ ﴿١١
فَقَضَاهُنَّ سَبْعَ سَمَاوَاتٍ فِي يَوْمَيْنِ وَأَوْحَىٰ فِي كُلِّ سَمَاءٍ أَمْرَهَا ۚ وَزَيَّنَّا السَّمَاءَ الدُّنْيَا بِمَصَابِيحَ وَحِفْظًا ۚ ذَٰلِكَ تَقْدِيرُ الْعَزِيزِ الْعَلِيمِ ﴿١٢


Surah Fussilat: Thereafter turned He to the heaven and it was as smoke, and said Unto it and Unto the earth: come ye twain, willingly or loth. They said: we come willingly. (11) Then He decreed them as seven heavens in two days, and revealed Unto each heaven the command thereof; and We bedecked the nether heaven with lamps and placed therein a guard. That is the ordinance of the Mighty, the Knower. (12)

De hemelen en aarde (waren) een samenhangende massa. Wij hebben ze toen van elkaar gescheiden… Wij hebben de hemel tot een beschermend dak gemaakt … en de dag en de nacht, de zon en de maan geschapen. (Koran 41 11:12)


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