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1 Nuclear Fusion and Radiation Lecture 2 (Meetings 3 & 4) Eugenio Schuster Mechanical Engineering and Mechanics Lehigh University Nuclear Fusion and Radiation p. 1/40

2 Modern Physics Concepts First three decades of the 20th century Modern Physics Theory of special relativity, which extended Newtonian mechanics. Wave-particle duality, which says that both electromagnetic waves and atomic particles have dual wave and particle properties. Quantum mechanics, which revealed that the microscopic atomic world is far different from our everyday macroscopic world. The results and insights provided by these three advances in physics are fundamental to an understanding of nuclear science and technology. Nuclear Fusion and Radiation p. 2/40

3 Theory of Relativity Newton s second law, in the form originally stated by Newton, says that the rate of change of a body s momentum p equals the force F applied to it, i.e., (1) F = dp dt = d(mv) dt For a constant mass m, as assumed by Newton, this equation immediately reduces to the modern form of the second law, (2) F = ma, where a = dv/dt is the acceleration of the body. In 1905 Einstein showed that Eq. (1) is still correct, but that the mass of a body is not constant, but increases with the body s speed v. Eq. (2) is thus incorrect. Nuclear Fusion and Radiation p. 3/40

4 Theory of Relativity Einstein showed that m varies with the body s speed as (3) m = m o 1 v 2 /c 2 where m o is the body s rest mass, i.e., the body s mass when it is at rest, and c m/s is the speed of light. Amazingly, Einstein s correction to the laws of motion was produced theoretically before being discovered experimentally, which is uncommon in physics. This may be explained by the fact that in our everyday world the difference between m and m o is incredibly small. Thus for engineering problems in our macroscopic world, relativistic effects can safely be ignored. However, at the atomic and nuclear level, these effects can be very important. Nuclear Fusion and Radiation p. 4/40

5 Theory of Relativity Newton s words: The motions of bodies included in a given space are the same amongst themselves, whether the space is at rest or moves uniformly forward in a straight line. This means that experiments made in a laboratory in uniform motion (i.e., non-accelerating motion) produce the same results as when the laboratory is at rest. But do all the laws of physics indeed remain the same in all non-accelerating (inertial) coordinate systems? Figure 1: Coordinate Systems Nuclear Fusion and Radiation p. 5/40

6 Theory of Relativity Consider the two coordinate systems shown in Fig. 1. System k is at rest, while system k is moving uniformly to the right with speed v. At t = 0, the origin of k is at the origin of k. The coordinates of some point P are (x,y,z) in k and (x,y,z ) in k. Clearly, the primed and unprimed coordinates are related by (4) x = x vt; y = y; z = z; t = t. Consider a force in the x-direction, F x, acting on some mass m, F x = m d2 x dt 2 = (x vt) md2 dt 2 = m d2 x dt 2. Thus the second law has the same form in both systems. Nuclear Fusion and Radiation p. 6/40

7 Theory of Relativity Since the laws of motion are the same in all inertial coordinate systems, it follows that it is impossible to tell, from results of mechanical experiments, whether or not the system is moving. However, when Eqs. (4) are used to transform Maxwell s equations of electromagnetism (1870) to another inertial system, they assume a different form. Thus from optical experiments in a moving system, one should be able to determine the speed of the system. Indeed, Lorentz observed in 1904 that if the transformation (5) x = x vt 1 v 2 /c 2; y = y; z = z; t = t vx/c2 1 v 2 /c 2; is used, Maxwell s equations become the same in all inertial coordinate systems. Nuclear Fusion and Radiation p. 7/40

8 Theory of Relativity The Lorentz transformation is indeed strange, since it indicates that space and time are not independent quantities. Time in the k system, as measured by an observer in the k system, is different from the time in the observer s system! Einstein showed in 1905 that the Lorentz transformation (5) was indeed the correct transformation relating all inertial coordinate systems. He based his analysis on two postulates: The laws of physics are expressed by equations that have the same form in all coordinate systems moving at constant velocities relative to each other. The speed of light in free space is the same for all observers and is independent of the relative velocity between the source and the observer. Nuclear Fusion and Radiation p. 8/40

9 Theory of Relativity Einstein demonstrated several amazing properties: 1. The passage of time appears to slow in a system moving with respect to a stationary observer. The time t required for some physical phenomenon (e.g., the interval between two heart beats) in a moving inertial system appears to be longer (dilated) than the time t o for the same phenomenon to occur in the stationary system. The relation between t and t o is (6) t = t o 1 v 2 /c 2 Proof: In class. Nuclear Fusion and Radiation p. 9/40

10 Theory of Relativity Figure 2: Time Dilation: t o is proper time (stationary frame), t is apparent time (moving frame). Timing device emits a pulse of light from source (S) and records time to travel distance d to a detector (D). The speed of light c is the same to all observers. The travel time for the pulse of light appears to dilate as the timing device moves with respect to a stationary observer. Like all other relativity effects, this effect is reciprocal. Nuclear Fusion and Radiation p. 10/40

11 Theory of Relativity 2. The length of a moving object in the direction of its motion appears smaller to an observer at rest, namely, (7) L = L o 1 v 2 /c 2 where L o is the proper length or length of the object when at rest. Proof: In class. Nuclear Fusion and Radiation p. 11/40

12 Theory of Relativity Figure 3: Length Contraction: L o is proper length (t = L o /v), L is apparent length (L = vt o ). Left: Since the bar is moving with respect to the reference frame, we measure apparent length L. Since the transit time is measured at the same position of the reference frame, we measure the proper time t o. Right: Since the bar is stationary with respect to the reference frame, we measure proper length L o. Since the transit time is measured at a different position of the reference frame, we measure the apparent time t. Nuclear Fusion and Radiation p. 12/40

13 Theory of Relativity 3. The laws of motion are correct, as stated by Newton, if the mass of an object is made a function of the object s speed v, i.e., m o (8) m(v) = 1 v 2 /c 2 This result also shows that: No object can travel faster than the speed of light since the relativistic mass m(v) must always be real. An object with a rest mass (m o > 0) cannot travel at the speed of light; otherwise, its relativistic mass (and kinetic energy) would become infinite. Proof: In class. Nuclear Fusion and Radiation p. 13/40

14 Theory of Relativity Figure 4: Mass Increase: m o is the rest mass. By considering elastic collision between two bodies, we can infer that the mass of a body varies with its speed. Both experiments are at rest w.r.t. S and S, which is moving with speed v relative to S in the x-direction. For launch speed u << c, each ball has mass m o. For observer in S, the ball from S takes t o = h/u to reach collision while the ball from S appears to take longer to reach collision, namely the apparent time t. Momentum is conserved (elastic collision). Nuclear Fusion and Radiation p. 14/40

15 Theory of Relativity 4. Energy and mass can be converted to each other, i.e., (9) T = mc 2 m o c 2 E = m o c 2 +T = mc 2 Indeed, all changes in energy of a system result in a corresponding change in the mass of the system. This equivalence of mass and energy plays a critical role in the understanding of nuclear technology. Proof: In class. Nuclear Fusion and Radiation p. 15/40

16 Theory of Relativity Figure 5: Force F accelerates particle along s. By conservation of energy, the work done on this particle as it moves along the path of length s must equal the kinetic energy T at the end of the path. The work done by F ( a vector) on the particle as it moves through a displacement ds (a vector) is F ds. The kinetic energy is then given by T = s 0 F ds Nuclear Fusion and Radiation p. 16/40

17 Theory of Relativity Reduction to Classical Mechanics: Through a Taylor Series expansion we can note that 1 1 v 2 /c = 2 (1 v2 /c 2 ) 1/2 = 1+ v2 2c 2 + 3v4 v c4 2c 2 Then, T = mc 2 m o c 2 = m o c 2 ( ) 1 1 v 2 /c 1 2 ([ ] m o c 2 1+ v2 2c 2 1 ) = 1 2 m 0v 2 Thus the relativistic kinetic energy reduces to the classical expression for kinetic energy if v << c, a reassuring result since the validity of classical mechanics is well established in the macroscopic world. Nuclear Fusion and Radiation p. 17/40

18 Theory of Relativity Relationship Between Kinetic Energy and Momentum: Both classically and relativistically, the momentum p of a particle is given by p = mv. Classical physics: (10) T = mv2 2 = p2 2m p = 2mT Relativistic physics: Squaring (8) we obtain (11) p 2 (mv) 2 = (mc) 2 (m o c) 2 = 1 c 2[(mc2 ) 2 (m o c 2 ) 2 ] Finally, using (9) for mc 2 we obtain (12) p = 1 c T 2 +2Tm o c 2 Nuclear Fusion and Radiation p. 18/40

19 Wave-particle Duality Matter (e.g., electrons) and radiation (e.g., X rays) both have wave like and particle like properties. This dichotomy, known as the wave particle duality principle, is a cornerstone of modern physics. For some phenomena, a wave description works best; for others, a particle model is appropriate. Three pioneering experiments helped to establish the wave-particle nature of matter: Photoelectric Effect Compton Scattering Electron Scattering Nuclear Fusion and Radiation p. 19/40

20 Wave-particle Duality Photoelectric Effect: In 1887, Heinrich Hertz discovered that, when metal surfaces were irradiated with light, electricity was emitted. In 1898, Joseph John Thomson showed that these emissions were electrons (thus the term photoelectrons). Figure 6: Photoelectric Effect Nuclear Fusion and Radiation p. 20/40

21 Wave-particle Duality Photoelectric Effect: Classical (wave theory) description of light: light energy absorbed by the metal surface, a bound electron is freed, a photoelectron boils off the surface. If light were a wave: Photoelectrons should be produced by light of all frequencies. At low intensities, a time lag would be expected between the start of irradiation and the emission of a photoelectron (time to absorb sufficient energy). As the light intensity (i.e., wave amplitude) increases, more energy is absorbed per unit time and, hence, the photoelectron emission rate should increase. The kinetic energy of the photoelectron should increase with the light intensity since more energy is absorbed. Nuclear Fusion and Radiation p. 21/40

22 Wave-particle Duality Photoelectric Effect: However, experimental results differed dramatically with these results. It was observed: For each metal there is a minimum light frequency below which no photoelectrons are emitted no matter how high the intensity. There is no time lag between the start of irradiation and the emission of photoelectrons, no matter how low the intensity. The intensity of the light affects only the emission rate of photoelectrons. The kinetic energy of the photoelectron depends only on light frequency and not on its intensity. The higher the frequency, the more energetic is the photoelectron. Nuclear Fusion and Radiation p. 22/40

23 Wave-particle Duality Photoelectric Effect: In 1905, Albert Einstein assumed that light energy consists of photons or quanta of energy, each with an energy (13) E = hν where h is Planck s constant ( Js) and ν is the light frequency. He further assumed that the energy associated with each photon interacts as a whole, i.e., either all the energy is absorbed by an atom or none is. With this particle model for the light, the maximum kinetic energy of a photoelectron would be (14) E = hν A, where A is the amount of energy (the so-called work function) required to free an electron from the metal. Nuclear Fusion and Radiation p. 23/40

24 Wave-particle Duality Photoelectric Effect: Thus if hν < A, no photoelectrons are produced. Increasing the light intensity only increases the number of photons hitting the current collector metal surface per unit time and, thus, the rate of photoelectron emission. Electromagnetic waves traveling at the speed of light c with wavelength λ and frequency ν satisfy c = λν ν = c/λ. Figure 7: Photoelectric Effect Nuclear Fusion and Radiation p. 24/40

25 Wave-particle Duality Photoelectric Effect: Figure 8: Photoelectric Effect At a critical voltage V o (no current), the kinetic energy of the photoelectron equals the potential energy V o e (e is the electron charge) the photoelectron must overcome, i.e., (15) V o e = hν A, V o = hν e A e. Nuclear Fusion and Radiation p. 25/40

26 Wave-particle Duality Photoelectric Effect: In 1912, Arthur Llewelyn Hughes showed that, for a given metallic surface, V o was a linear function of the light frequency ν. In 1916, Robert Andrews Millikan, verified that plots of V o versus ν had a slope of h/e, from which h could be evaluated. Figure 9: Photoelectric Effect Nuclear Fusion and Radiation p. 26/40

27 Wave-particle Duality Compton Scattering: Other experimental observations showed that light, besides having quantized energy characteristics, must have another particle like property, namely, momentum. According to the wave model of electromagnetic radiation, radiation should be scattered from an electron with no change in wavelength. However, in 1922, Arthur Holly Compton observed that X rays scattered from electrons had a decrease in the wavelength λ = λ f λ i proportional to (1 cos(θ)) where θ was the scattering angle. Nuclear Fusion and Radiation p. 27/40

28 Wave-particle Duality Compton Scattering: To explain this observation, it was necessary to treat X rays as particles with a linear momentum p = h/λ and energy E = hν = pc (electromagnetic waves traveling at the speed of light c with wavelength λ and frequency ν satisfy c = λν). Figure 10: Compton Scattering Nuclear Fusion and Radiation p. 28/40

29 Wave-particle Duality Compton Scattering: In an X-ray scattering interaction, the energy and momentum before scattering must equal the energy and momentum after scattering, i.e., (16) p i c+m o c 2 = p f c+mc 2 (17) p i = p f +p e p 2 e = p 2 i +p 2 f 2p ip f cos(θ) where m o and m are the at-rest and relativistic masses of the electron respectively. Taking into account from (11) that p 2 e = (mc)2 (m o c) 2 mc 2 = c p 2 e +(m oc) 2, we replace this last expression for mc 2 in (16) and solve for p 2 e. Finally, we replace the obtained expression for p 2 e in (17) to obtain (18) 1 p f 1 p i = 1 m o c (1 cos(θ)) Nuclear Fusion and Radiation p. 29/40

30 Wave-particle Duality Compton Scattering: Then since λ = h/p, this result gives the decrease in the scattered wavelength as (19) λ f λ i = h m o c (1 cos(θ)) With the photon relations λ = c/ν and E = hν, we can write (20) 1 E f 1 E i = 1 m o c 2(1 cos(θ)) Thus, Compton was able to predict the wavelength change of scattered X rays by using a particle model for the X rays, a prediction which could not be obtained with a wave model. Nuclear Fusion and Radiation p. 30/40

31 Wave-particle Duality Electron Scattering: In 1924, Louis de Broglie postulated that, since light had particle properties, then particles should have wave properties. Because photons had a discrete energy E = hν and momentum p = h/λ, de Broglie suggested that a particle, because of its momentum, should have an associated wavelength λ = h/p. In 1927, Clinton Davisson and Lester Germer confirmed that electrons did indeed behave like waves with de Broglie s predicted wavelength. In their experiment, they illuminated the surface of a Ni crystal by a perpendicular beam of 54eV electrons and measured the number of electrons N(φ) reflected at different angles φ from the incident beam. Nuclear Fusion and Radiation p. 31/40

32 Wave-particle Duality Electron Scattering: Figure 11: Electron Scattering Nuclear Fusion and Radiation p. 32/40

33 Wave-particle Duality Electron Scattering: According to the particle model, electrons should be scattered by individual atoms isotropically and N(φ) should exhibit no structure. However, N(φ) was observed to have a peak near 50. This observation could only be explained by recognizing the peak as a constructive interference peak a wave phenomenon. Specifically, two reflected electron waves are in phase (constructively interfere) if the difference in their path lengths is an integral number of wavelengths. This experiment and many similar ones clearly demonstrated that electrons (and other particles such as atoms) have wave like properties. Nuclear Fusion and Radiation p. 33/40

34 Quantum Mechanics The demonstration that particles (point objects) also had wave properties led to another major advance of modern physics. Because a material object such as an electron has wave properties, it should obey some sort of wave equation. Indeed, Erwin Rudolf Josef Alexander Shrodinger in 1925 showed that atomic electrons could be well described as standing waves around the nucleus. Further, the electron associated with each wave could have only a discrete energy. The branch of physics devoted to this wave description of particles is called quantum mechanics. Nuclear Fusion and Radiation p. 34/40

35 Quantum Mechanics Schrödinger Wave Equation: To illustrate Schrödinger s wave equation, we begin with an analogy to the standing waves produced by a plucked string anchored at both ends. The wave equation that describes the displacement Ψ(x,t) as a function of position x from one end of the string, which has length L, and at time t is (21) 2 Ψ(x,t) x 2 = 1 u 2 2 Ψ(x,t) t 2. Here u is the wave speed. There are infinitely many discrete solutions to this homogeneous partial differential equation, subject to the boundary condition Ψ(0, t) = Ψ(L, t) = 0. Nuclear Fusion and Radiation p. 35/40

36 Quantum Mechanics Figure 12: Wave solution of a vibrating string (u = 1) Nuclear Fusion and Radiation p. 36/40

37 Quantum Mechanics The first mode is called the fundamental solution. The frequencies ν of the solutions are also discrete. The solution of the wave equation (21) is separable, i.e., it has the form Ψ(x, t) = ψ(x)t(t) = ψ(x) sin(2πνt). Substitution of this separable form into (21) yields (22) 2 ψ(x) x 2 + 4π2 ν 2 u 2 ψ(x) = 0 or, since u = λν, (23) 2 ψ(x) x 2 + 4π2 λ 2 ψ(x) = 0 Nuclear Fusion and Radiation p. 37/40

38 Quantum Mechanics Generalizing the wave equation to describe the dynamics of an electron bound to an atomic nucleus in a three-dimensional space, we obtain the steady-state Schrödinger wave equation, which is the fundamental equation of quantum mechanics. h2 (24) 8π 2 m 2 ψ(x,y,z)+v(x,y,z)ψ(x,y,z) = Eψ(x,y,z) The nucleus produces an electric field or electric force on the electron V(x,y,z). The electron with (rest) mass m has a total energy E, kinetic energy T, and a potential energy V such that T = E V. The wavelength of the electron is λ = h/p = h/ 2mT = h/ 2m(E V) (assuming the electron is nonrelativistic). The electron around a nucleus can have only discrete values of E = E n (verified by experiment). Nuclear Fusion and Radiation p. 38/40

39 Quantum Mechanics The non-trivial solution ψ n (x,y,z) of (24) when E = E n (an eigenvalue of the equation) is called a wave function. In general, this is a complex quantity which extends over all space, and may be thought of as the relative amplitude of a wave associated with the particle described by (24). The wave function is normalized as ψ(x,y,z)ψ (x,y,z)dv = 1. The square of the amplitude of the wave function ψ 2 gives the probability of finding the particle at any position in space. Thus, the probability that the particle is in some small volume dv around the point (x,y,z) is given by ψ(x,y,z) 2 dv = ψ(x,y,z)ψ (x,y,z)dv. The normalization condition requires that the particle be somewhere in space. Nuclear Fusion and Radiation p. 39/40

40 Quantum Mechanics The Uncertainty Principle: Within quantum mechanics it is no longer possible to say that a particle is at a particular location. We can say only that the particle has a probability ψψ dv of being in dv. In 1927, Werner Heisenberg postulated that if one attempts to measure both a particle s position along the x-axis and its momentum, there will be an uncertainty x in the measured position and an uncertainty p in the momentum given by (25) x p h 2π This limitation is a direct consequence of the wave properties of a particle and can be derived rigorously from Schrödinger s wave equation. However, a more phenomenological approach is to consider an attempt to measure the location of an electron with very high accuracy (By observing a system, the system is necessarily altered). Nuclear Fusion and Radiation p. 40/40

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