The problem with the conservation of energy and Pauli’s solution of the neutrino

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Extract from the book ‘Neutrino’ by Frank Close (2010) Isaac Newton in the 17th century had realised the importance of energy. Push something and, in the absence of any friction, it will start to move. Keep pushing and it will speed up. Newton defined energy of motion, kinetic energy, as proportional to the amount of force that you pushed with, and the distance over which you kept pushing the object. He was also aware that energy could have different manifestations. A body on top of a cliff has potential energy – the potential to gain kinetic energy if it falls over the edge. Potential energy is in proportion to height, above some ground level: the higher you are the more potential energy you have. As you fall towards ground, the force of gravity accelerates you. You gain kinetic energy at the same rate that you lose potential energy; the sum is preserved. This is a simple example of energy conservation, and of the change from one form of energy to another, in this case from potential to kinetic. There are many other ways that energy can be redistributed. In the 19th century, thermodynamics – the science of heat and motion – matured. Energy in the form of heat can be converted into kinetic energy. The steam engine works on this principle. When water boils, it turns into steam and expands. If the expansion is in a closed cylinder whose end can move, the pressure of the steam can force the piston into motion. Attach the moving end to a rod, which in turn is connected to a wheel, off-centre, and the result will be that the wheel turns. By this means, steam power enabled trains, weighing hundreds of tons, to travel at over 100 kilometres an hour.

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In the steam engine, as in countless other examples, energy is being changed from one form to another, but overall is conserved. That is the first law of thermodynamics on which whole industries have been built. It is one of the most fundamental and far-reaching laws of nature. While all the excitement about radioactivity was happening, and independent of it, in 1905 Albert Einstein announced his Theory of Special Relativity. Its most famous equation, E=mc2, implies a profound link between energy and mass: that mass (m) and energy (E) can be converted one into the other at an exchange rate governed by the speed of light (c). Einstein’s equation expressed a new and profound way of storing and transferring energy, but here again, energy overall is conserved. Radioactivity is an example of E=mc2 at work. When the matter in the nucleus of an atom spontaneously rearranges itself, the energy that had been, a moment earlier, locked within some of the original mass is suddenly released. It may be radiated as light-gamma rays; it may be taken up as kinetic energy as pieces of the precious nucleus shoot off, as in alpha decay; or it may congeal into new forms of matter, as in beta decay. In alpha and gamma decay, the energy accounts were straightforward; in beta decay, however, they seemed not to work. If there is only the one particle emitted each time that a radioactive nucleus decays, energy conservation enforces a single value for its energy. That is what was seen in alpha and gamma decays, but in 1914, James Chadwick discovered that the energy of beta radiation varied from one measurement to the next. Instead of having the same energy, electrons emerged with a continuous range of energies, sometimes almost no energy at all, and on other occasions amounts all the way up to a maximum value. Neils Bohr, who earlier had fathered the model of the atom as electrons orbiting Rutherford’s central nucleus, put his authority behind a radical suggestion: energy is not conserved in beta decay.

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This ran counter the centuries of experience, and was an act of desperation. The Austrian theorist, Wolfgang Pauli, refused to accept it, and put forward another explanation. He proposed that the beta particle was accompanied by an ‘additional very penetrating radiation that consists of new neutral particles’. In such an eventuality, energy is conserved but is being shared between two particles rather than carried off entirely by just one. In Pauli’s theory the visible particle, the beta, sometimes carried all of the available energy leaving nothing for the invisible neutral partner, while on other occasions the invisible one took away some of the energy leaving less for the beta particle. As a result the energy carried by the visible beta particle could be anywhere within a range, rather than being restricted to a single value. This sounds like a conservative idea, and fitted the facts, but at the time it was greeted with little more enthusiasm than Bohr’s proposal. The reason was that it ran counter to the prevailing beliefs about the nature of atoms. The rich tapestry of Nature at that time appeared to be made of just two particles: electrons and protons. This fundamental simplicity promised a beautiful unification at the core of matter, whereas introducing a third particle for no reason other than to fix up one esoteric puzzle, seemed to many to be unwarranted.

Kinetic energy: Energy associated with motion: K.E(J)=1/2⋅m(kg)⋅v(m.s-1)2 Potential energy: Energy associated with the gravitational interaction with the Earth: P.E=m⋅g⋅h m(kg) : mass; v(m.s-1): speed ; g(=10 m.s-2): gravitational field or acceleration due to gravity ; h(m): height; energy (J) Power: It is the ratio of the energy transferred to the time taken: P (W)=ΔE (J)/Δt(s) Mass-energy equivalence: Every mass has an energy equivalent and vice versa: E(J)=m(kg).c2 Questions: 1. What is the kinetic energy of a 100-ton train racing at a speed of 100 km/h? 2. A mass of 50 kg is quickly raised by a crane through 1.5 m in 2 s. Find the gain in potential energy of the mass. How much power does the engine develop? 3. Use the energy conservation as mentioned by Close (line 7 to 10) and find out the velocity in km.h-1 of a 150 g object dropped from a height of 20 m as it hits the floor. Ignore air resistance. 4. What kind of energy conversion is achieved in a coal-fired steam engine? How does it work? What is the role of the crankshaft? 5. What kind of energy conversion is achieved in a hydroelectric dam? How does it work? 6. What kind of energy conversion is achieved in a wind turbine? How does it work? 7. What kind of energy conversion is achieved in a nuclear power plant? How does it work? 8. Write down the equation corresponding to the beta minus disintegration of bismuth (Z=83; A=210). This is an extract from the periodic table : Pb ; Bi ; Po. 9. According to the principle of conservation of energy, what is the amount of energy in MeV that you would expect for the emitted electron: mBi = 209,938584 u ; mPo = 209,936790 u ; me = 5,49×10-4 u ; 1u.c2 = 931.5 MeV (1u=mass unit=one twelfth of the mass of a C12 atom=103/NA; 1 MeV=106 eV; 1 eV=1.6×10-19 J). 10. What was the problem posed by the energy spectrum of beta decay electrons from bismuth? What was the solution suggested by Pauli? Why was it greeted with little enthusiasm at the time? Has Pauli been proved to be right since then? How?

Correction: The problem with the conservation of energy and Pauli’s solution of the neutrino

1. Kinetic energy is energy associated with motion=K.E(J)=1/2⋅m(kg)⋅v(m.s-1)2 K.E=0.5×1.00×105×(100/3.6)2=3.86×107 J =38.6 MJ. (Scientific notation with 3 significant figures, as many as given for the value of the speed or the value of the mass) 2. Potential P.E is energy associated with the gravitational attraction=m⋅g⋅h P.E=50×10×1.5=750 J ; Power=P.E/Δt=750/2=375 W 3. The object at the height h of 20 m has gained potential energy: E=m⋅g⋅h=0.15×10×20=30 J This potential energy is converted into kinetic energy therefore: 1 ∙ 𝑚 ∙ 𝑣 ! = 𝑚 ∙ 𝑔 ∙ ℎ ⟹ 𝑣 = 2𝑔 ∙ ℎ ⇒ 𝑣 = 20! = 𝟐𝟎 𝒎. 𝒔!𝟏 = 𝟕𝟐 𝒌𝒎. 𝒉!𝟏 2 Another way to proceed is to consider the conservation of the total mechanical energy, which is the sum of the potential energy and the kinetic energy. It keeps the same value provided that there is no loss of energy due to friction. 1 1 𝑀. 𝐸! = 𝑀. 𝐸! ⇒ ∙ 𝑚 ∙ 0! + 𝑚 ∙ 𝑔 ∙ ℎ = ∙ 𝑚 ∙ 𝑣 ! + 𝑚 ∙ 𝑔 ∙ 0 2 2 4. In a coal-fired engine, coal is burned, the heat turns water into steam, the steam moves a piston, and its translational motion is turned into the rotational motion of the wheels thanks to the crankshaft attached to the piston. We can say that the chemical energy stored in the carbon atoms of coal is mostly turned into the kinetic energy of the steam engine train. 5. In a hydroelectric dam, the potential energy of water is turned into the kinetic energy of a turbine, which is linked to an alternator, which produces electrical energy. 6. A wind turbine is driven by the kinetic energy of the wind captured by its blades, which is turned into electrical energy. 7. In a nuclear power plant, neutrons bombard uranium nuclei inside the reactor; they are broken into smaller nuclei in a chain but controlled reaction and the process releases a lot of energy. This energy turns water into steam, which drives a turbine linked to an alternator. ! !"# 8. !"# !"𝐵𝑖 → !!𝑒 + !"𝑃𝑜 !"# !! 9. Δ𝑚 = 𝑚 !!!𝑒 + 𝑚 !"# 𝑢 !"𝑃𝑜 − 𝑚( !"𝐵𝑖) = −1.245×10 ! !! ! !! 𝐸 = Δ𝑚 ∙ 𝑐 = −1.245×10 𝑢. 𝑐 = −1.245×10 ×931.5 = −𝟏. 𝟏𝟓 𝑴𝒆𝑽 This energy is in the form of the kinetic energy of the electron, as both nuclei are immobile. 10. According to the previous equation, the amount of the kinetic energy of the electron should take only one value whereas the graph shows that the energy of the electron can take a large range of values. To interpret this fact, Pauli considered the emission of another particle he called the neutrino. In this way the energy is shared between these two particles so that the energy of the electron can be maximum (-1.15 MeV as you can read on the graph) if the neutrino has no energy, or it can be minimum (0 MeV) if the neutrino has all the energy. Today the existence of the neutrino has been proved by means of detectors sensitive to them: a large volume of water surrounded by phototubes; an incoming neutrino creates an electron or a muon (an unstable particle with a mean life time of 2.2 𝜇𝑠, same charge but heavier than the electron). Pauli’s claims were greeted with little enthusiasm at the time because physicists thought only two particles existed: protons and electrons.

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