Scales of the microworld Ji Dolej Olga Kotrbov

Scales of the microworld Jiří 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 mg. Dh/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”. . . 1

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 100/15 g 1 1/22 1/23 1/24 1/25 1/26 1/27 1/28 1/29 1/210 1/211 1/212 1/213 1/214 100/15. 1/214 g = = 4. 10 -4 g = = 0. 4 mg How close to atoms we are? ? ? 2

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 our body 103 1 10 -3 10 -6 Length scale 10 -9 10 -12 atom 10 -15 10 -18 the smallest piece of chocolate I can see the light wavelength our body 1 10 -3 10 -6 10 -9 10 -21 10 -24 electron 10 -27 10 -30 kg the atomic nucleus electron atom 10 -12 10 -15 10 -18 m 3

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 energymass 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 e. V = 1. 602 176 46 × 10 -19 J, 1 e. V/c 2 = 1. 782 661 73 × 10 -36 kg 1 u = 931. 494 01 Me. V/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 e. V to J conversion: mproton ≈ u ≈ 1 Ge. V/c 2, melectron ≈ 0, 5 Me. V/c 2 1 e. V ≈ 1, 6 × 10 -19 J, c ≈ 3 × 108 m/s 4

The mass and length scale again the smallest piece of chocolate I can see our body 103 1 10 -3 10 -6 10 -9 10 -12 atom 10 -15 10 -18 10 -21 10 -24 Te. V electron 10 -27 Ge. V 10 -30 kg Me. V /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). 1012 T tera 109 G giga 106 103 M k mega kilo 1 10 -3 m mili 10 -6 10 -9 m n micro nano 10 -12 p pico 10 -15 f femto 5

As the atoms are so small, there is plenty of them in any piece of matter – the Avogadro number (6. 022 142 0 × 1 O 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 × 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. 6

Expert pages! You don´t 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 7 than the Earth's gravitational field!!!

The energy scale As we already said the most frequently used unit in the microworld is the electronvolt. Rest energy of a mosquito My kinetic energy when walking Kinetic energy of a flying mosquito Rest energy of an atom Rest energy of an electron Photons energies in visible light Thermal energy of an atom 1 joule 1030 1027 1024 Human daily power consumption 1021 1018 1015 Highest energy of a single particle observed in cosmic radiation Energy contained in a glass of beer (0. 5 liter) 1012 109 106 103 (Te. V) (Ge. V) (Me. V) (ke. V) Binding energies of nucleons in nuclei Highest proton energy from current accelerator (Tevatron in FNAL) 1 (e. V) 10 -3 e. V (me. V) Binding energies of electrons in atoms Energy of an electron in the TV Proton energy from “free fall example” 8

To be continued 9