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1 Scales of the microworld 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 10 2 y ≈ 10 9 s, since 1 year approximately equals .10 7 s – check it for yourself) we are capable of carrying and lifting our weight with some load, i.e. about 10 2 kg to the peaks with a speed of about 500 m per hour, what means the power of mg h/ t = 10 2.10.500/3600 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”... Jiří Dolejší, Olga Kotrbová, Charles University Prague

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2 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... What about atomic scales? Let us try to get there! 100 g = 10 -1 kg 100/15 g 1 1/2 1/2 2 1/2 3 1/2 4 1/2 5 1/2 6 1/2 7 1/2 8 1/2 9 1/2 10 1/2 11 1/2 12 1/2 13 1/2 14 100/15. 1/2 14 g = = 4.10 -4 g = = 0.4 mg How close to atoms we are???

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3 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 10 3 1 10 -3 10 -6 10 -9 10 -12 10 -15 10 -18 10 -21 10 -24 10 -27 10 -30 electron the smallest piece of chocolate I can see Length scale kg 1 10 -3 10 -6 10 -9 10 -12 10 -15 10 -18 m our body atom the smallest piece of chocolate I can see electron the atomic nucleus the light wavelength atom our body

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4 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 ( 12 C) 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 = mc 2 : We can express mass in terms of energy divided by c 2. The most frequently used units for measuring energy in the microworld are electronvolts: 1 eV = 1.602 176 46 × 10 -19 J, 1 eV/c 2 = 1.782 661 73 × 10 -36 kg 1 u = 931.494 01 MeV/c 2 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: m proton ≈ u ≈ 1 GeV/c 2, m electron ≈ 0,5 MeV/c 2 1 eV ≈ 1,6 × 10 -19 J, c ≈ 3 × 10 8 m/s

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5 The mass and length scale again 10 3 1 10 -3 10 -6 10 -9 10 -12 10 -15 10 -18 10 -21 10 -24 10 -27 10 -30 electron our body atom the smallest piece of chocolate I can see kg TeV GeV MeV /c 2 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). T G M k m n p f tera giga mega kilo mili micro nano pico femto 10 12 10 9 10 6 10 3 1 10 -3 10 -6 10 -9 10 -12 10 -15

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6 As the atoms are so small, there is plenty of them in any piece of matter – the Avogadro number (6.022 142 0 × 1O 23 ) 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 (2×1+16=18 g for H 2 O) Two H atoms per H 2 O molecule What is the average volume occupied by one water molecule? The are 0,67 × 10 25 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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7 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 Expert pages! You don´t need to understand them, but it is a challenge! This is the acceleration of gravity g The minus sign in the gravitational potential says that a body with mass m has the negative potential energy E = 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: 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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8 The energy scale As we already said the most frequently used unit in the microworld is the electronvolt. Rest energy of an atom Rest energy of an electron Thermal energy of an atom Energy of an electron in the TV 1joule Highest proton energy from current accelerator (Tevatron in FNAL) Proton energy from “free fall example” 10 30 10 27 10 24 10 21 10 18 10 15 10 12 10 9 10 6 10 3 1 10 -3 eV (TeV) (GeV) (MeV) (keV) (eV) (meV) Energy contained in a glass of beer (0.5 liter) Human daily power consumption Highest energy of a single particle observed in cosmic radiation Rest energy of a mosquito Binding energies of nucleons in nuclei Binding energies of electrons in atoms Photons energies in visible light My kinetic energy when walking Kinetic energy of a flying mosquito

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9 To be continued

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Consider a force like gravitation which varies as but 1) billion-billion-billion-billion times stronger; 2) there are two kinds of “matter”: positive.

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