Scales of the microworld

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Scales of the microworld
Jirí Dolejší, Olga Kotrbová, Charles University
Prague

• We look at the world from our human point of view
  and the basic scale is related to human
  dimensions
• We are born about 0.5 meter big and we gradually
  grow to about 1.5 2 meters, interesting
  dimensions are e.g. 0.9-0.6-0.9 m etc.
• We start with a mass of few kilograms and we
  gradually reach tens or maximum few hundreds kg
• The typical time intervals which we can perceive
  range from fractions of seconds (sometimes
  deciding between life and death on the streets)
• to tens of years of our life (i.e. from 10-1 s
  to about 102 y 109 s, since 1 year
  approximately equals p.107 s check it for
  yourself)
• we are capable of carrying and lifting our
  weight with some load, i.e. about 102 kg to the
  peaks with a speed of about 500 m per hour, what
  means the power of mgDh/Dt 102.10.500/3600
  watt 140 W. This is just one fifth of the
  horse power (745 W) and twice the power which is
  usually called the manpower (1/10 of HP). 5 hour
  climb with this rate means the work of 2 500 000
  joule 2.5 MJ. We need about 10 MJ per day in
  food even if we do almost nothing...

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What about atomic scales? Let us try to get there!
We can try to cut some macroscopic thing into
microscopic pieces I decided to cut something
eatable - a piece of chocolate. I proceeded by
halving ...
100 g 10-1 kg
1 1/2 1/22 1/23 1/24
100/15 g
1/25 1/26 1/27 1/28 1/29
100/15 . 1/214 g 4.10-4 g 0.4 mg How
close to atoms we are???
1/210 1/211 1/212 1/213 1/214
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It took centuries to learn atomic dimensions and
properties. Today we know, that a typical length
scale for atoms is 10-10 m and their mass is of
the order 10-27 10-25 kg. The lightest particle
is the electron with mass 10-30 kg. Mass scale
the smallest piece of chocolate I can see
electron
atom
our body
103 1 10-3 10-6 10-9
10-12 10-15 10-18 10-21
10-24 10-27 10-30
kg
Length scale
the smallest piece of chocolate I can see
the atomic nucleus
the light wavelength
our body
electron
atom
m
1 10-3 10-6 10-9 10-12
10-15 10-18
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The lesson we may learn from the chocolate
cutting is that atoms are far much smaller and
lighter than we can imagine. We can hardly get
oriented in this world by our common sense, we
should rather rely on different estimates. One
important help are smart units. For mass we can
use a quite natural unit close to the mass of the
lightest atom (H) ... atomic mass unit u, which
is defined as 1/12 of the mass of carbon (12C)
atom. 1 u 1.660 538 7 10-27 kg Another
useful mass unit is introduced with the help of
the Einstein energy-mass relation E mc2 We can
express mass in terms of energy divided by
c2. The most frequently used units for measuring
energy in the microworld are electronvolts 1 eV
1.602 176 46 10-19 J, 1 eV/c2 1.782 661
73 10-36 kg 1 u 931.494 01 MeV/c2 We do not
expect that anybody will memorize these awkward
numbers. But it is helpful to remember the proton
and electron mass, c and eV to J
conversion mproton u 1 GeV/c2, melectron
0,5 MeV/c2 1 eV 1,6 10-19 J, c 3 108
m/s
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The mass and length scale again
the smallest piece of chocolate I can see
electron
our body
atom
kg
103 1 10-3 10-6 10-9
10-12 10-15 10-18 10-21
10-24 10-27 10-30
TeV GeV MeV /c2
It is rather easy to accommodate the length scale
to the microworld it is sufficient to use the
appropriate prefixes fractions of nanometer for
atoms and femtometers for nuclei. Look to chapter
2 for the detailed description of the experiment
revealing the structure of an atom. You may also
meet angström (1 Å 10-10 m) and fermi (1 F 1
fm 10-15 m).
1012 109 106 103 1
10-3 10-6 10-9 10-12
10-15
T G M k
m m n p
f tera giga mega kilo
mili micro nano pico femto
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As the atoms are so small, there is plenty of
them in any piece of matter the Avogadro number
(6.022 142 0 1O23) in each mol. Let us
calculate how many atoms are in a glass of water
(say 0.2 liter).
Volume density mass of the water
mass divided by molar mass (211618 g for H2O)
Two H atoms per H2O molecule
What is the average volume occupied by one water
molecule? The are 0,67 1025 water
molecules in the mentioned glass, so
If the volume has a form of a cube, its edge will
have a length 0.3 nm.
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Expert pages! You dont need to understand them,
but it is a challenge!
Could you calculate the energy of a proton
falling from the infinity to the Earth surface
(neglecting air)? Comment We have in mind that
the gravity is an effective accelerator, at least
for stones, planes, suicides etc. and so we
expect quite significant energy... Maybe you
remember that potential of the field is the
helpful quantity to solve our question, you met
the potential of the central gravitational field
and/or of the central Coulombic field. This
potential is equal zero at infinity and at given
distance r from the source it has a value
The minus sign in the gravitational potential
says that a body with mass m has the negative
potential energy E f(r) m. The body is bounded
by the gravitation, we should supply the energy
E to free it. We can call E -E the binding
energy of this body in the field. In our case we
consider proton at rest at infinity (zero
kinetic, potential and total energy), which will
be accelerated by the attractive force (it gains
positive kinetic energy which compensates
negative potential energy keeping the total
energy zero). The kinetic energy of the proton we
can use for experiments, this is the quantity we
are interested in
This is the acceleration of gravity g
So the electrical field created from the AA cell
from your walkman accelerates proton more than
the Earth's gravitational field!!!
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The energy scale
As we already said the most frequently used unit
in the microworld is the electronvolt.
Photons energies in visible light
Rest energy of an atom
Rest energy of a mosquito
My kinetic energy when walking
Kinetic energy of a flying mosquito
Rest energy of an electron
Thermal energy of an atom
1joule
1030 1027 1024 1021 1018
1015 1012 109 106 103
1 10-3 eV

(TeV) (GeV) (MeV) (keV) (eV) (meV)
Highest energy of a single particle observed in
cosmic radiation
Binding energies of nucleons in nuclei
Binding energies of electrons in atoms
Human daily power consumption
Highest proton energy from current accelerator
(Tevatron in FNAL)
Energy of an electron in the TV
Proton energy from free fall example
Energy contained in a glass of beer (0.5 liter)
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To be continued