Introduction to Modern Physics NE 131 Physics for Nanotechnology Engineering

Transcription

1 Introduction to Modern Physics NE 131 Physics for Nanotechnology Engineering Dr. Jamie Sanchez-Fortún Stoker Department of Physics, University of Waterloo Fall 2005

2 1 Introduction to Modern Physics 1.1 The Breakdown of Classical Physics The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is extremely remote... A A Michelson, 1894 So what could have led Michelson to say this? By 1900: Atomic theory of matter: the idea that matter is considered to be made up of atoms had become widely accepted. The sizes and masses were known with reasonable accuracy. The concept of the periodic table (Mendeleev, 1869) i.e. the apparent periodicity in the physical/chemical properties of the elements had been established, but was poorly understood. Electrons were known to exist (J J Thomson, 1897), but they were thought to be classical particles of known mass and charge. Classical mechanics: firmly established for hundreds of years (e.g. Newton, Galileo, Hooke). Given sufficient information about the dynamics of a macroscopic system (e.g. ball rolling down a slope, planetary orbits about the sun), in theory it is possible to calculate the history and future of the system. Is there any reason to suspect that the motion of atoms and electrons wouldn t obey classical mechanics too? Wave motion: well established for over a century due to the efforts of, for example, Huygens, Young, Doppler and Fresnel. Thermodynamics: well established for over a century due to efforts of, for example, L Boltzmann, J W Gibbs and S Carnot. Applications to combustion systems. 1

3 Electromagnetism: well established for over a century due to efforts of, for example, A Ampére, M Faraday. Electricity and magnetism unified into a single theory, described by Maxwell s equations (J C Maxwell, 1864). Light considered as a wave phenomenon described by Maxwell s equations. Newton s corpuscular theory of light long since dismissed by mainstream physicists. However, within just a few years, many incredible experimental and theoretical discoveries appeared that simply could not be explained using these classical theories. Special Relativity (1905): the nature of space and time (or 4-dimensional spacetime ) is different for different observers moving relative to each other. At speeds v c, the motion of objects is described very well by Newtonian mechanics. At speeds v c, Newtonian mechanics breaks down, while special relativity appears to be consistent with observations. Special relativity leads to astonishing effects such as length contraction and time dilation. General Relativity (1915): gravity is described not as a force (as in Newtonian gravity), but as a warping of spacetime around massive bodies. Newtonian gravity fails to predict accurately the precession of the perihelion of Mercury general relativity does. Newtonian gravity fails to predict accurately the bending of starlight in the gravity of the sun general relativity does (Eddington, 1919). Quantum Mechanics ( ): accurately predicts physics of atoms, sub-atomic particles. With respect to atomic and molecular physics, the following are examples of observations which posed a serious dilemma for classical physics: Blackbody radiation (Planck, 1900): Classically, energy is continuous. Reality: energy exists in small packets (so-called quanta). 2

4 Photoelectric effect (Einstein, 1905): by illuminating a metal surface with electromagnetic radiation, a number of so-called photoelectrons are emitted. Classically, experimental results should show A time delay. Effect should occur at any frequency. Intensity of the incident radiation is proportional to the energy of the photoelectrons. In reality, the experimental results show No significant time delay. Existence of a cut-off frequency, below which the effect does not occur. Intensity of incident radiation is proportional to the number of photoelectrons. Line Spectra from Atoms and Molecules (Ritz, Rydberg, 1908): classical physics cannot explain the existence of spectral lines in atomic and molecular spectra. Bohr Model of the Hydrogen Atom (Bohr, 1915): modeling the hydrogen atom as an electron in a circular orbit around the positive nucleus (proton), Bohr fixes the physics to account for stationary states of the electron. Compton Effect (Compton, 1923): when x-rays are scattered by a graphite target, Classically, the radiation should be re-radiated at the same wavelength and frequency. Reality: in addition to this incident wavelength, radiation is also emitted at a longer wavelength. 1.2 Introduction to Blackbody Radiation and Planck s Law What happens as you apply heat to a body? The body emits radiation. As the temperature is increased, it glows initially a dull-red, and then orange-yellow, and finally it becomes white hot. Classically, such thermal radiation is produced by the accelerations of electrons and the oscillations of molecules. 3

5 It is well known that the light emitted by a hot object is independent of the material at a given temperature, the distribution of thermal radiation among the various wavelengths is the same for all bodies. An object placed in a furnace will absorb energy until it reaches the temperature of the furnace. Since radiation continues to be incident upon it, the object must also emit radiation in order to remain in thermal equilibrium. A blackbody is a perfect radiator: That is, a body capable of absorbing and emitting radiation at all wavelengths. There are no known perfect blackbodies, but many good approximations e.g. graphite absorbs 95% of radiation falling upon it. It is possible to make a small area of a body black in practice, a cavity with a tiny opening acts as a blackbody: Any radiation entering into the cavity is unlikely to emerge, and is ultimately absorbed. It has been known for approx. 150 years that a completely black body at a uniform temperature has the same distribution of frequencies of absorbed/emitted radiation no matter what the material making up the blackbody is, and no matter what its shape. For example, Kirchhoff (1859) verified experimentally that the measured frequency distribution (i.e. spectrum) was found to depend only upon the absolute temperature T of the blackbody. The resultant radiation is therefore referred to as thermal radiation. In astrophysics, the spectra of stars approximate blackbody spectra, enabling the temperatures of stellar atmospheres to be estimated by the corresponding blackbody temperature. Figure 1 shows the thermal radiation spectrum, i.e. the distribution of thermal radiation from a blackbody, for three different temperatures. Introducing the radiation density, ρ(v, T), At 300 K: the maximum in ρ(ν, T = 300) occurs in the infra-red region. 4

6 ν / Hz T = K 2 ρ(ν) / J s m T = 300 K ( x 10 4 ) T = 5000 K ( x 10 ) infrared visible ultraviolet Figure 1: The temperature dependence of the distribution of thermal radiation from a blackbody cavity. At 5000 K: the maximum in ρ(ν, T = 5000) begins to enter the visible region. At K: the maximum in ρ(ν, T = 15000) has entered the ultra-violet region. Note: the units of ρ are J s m 3, so ρ(ν, T ) represents A density in space (i.e. per unit volume, m 3 ). A density in frequency (i.e. per unit frequency interval, s = (s 1 ) 1 ). Among the well known experimental observations on blackbody radiation were the following two results: Wien Displacement Law (1893): this gives the position of the radiation maximum for blackbody emission corresponding to different temperatures (figure 2). It is given by λ max T = constant λ max = T. (1.1) Stefan-Boltzmann Law (1884): this relates the total power P radiated per unit area at temperature T by a blackbody, and is given by P = σt 4, (1.2) where σ is known as Stefan s constant (σ = W m 2 K 4 ). 5

7 Figure 2: The temperature dependence of the distribution of thermal radiation from a blackbody cavity. An empirical law, obtained by Wien for the short-wavelength (i.e. high frequency) monochromatic (i.e. single frequency) emissive power of a blackbody, corresponding to an energy density in which a and b are curve-fitting parameters. ρ (λ, T ) = a { λ 5 exp b }, (1.3) λt So figure 1 shows the experimental curves for various temperatures. What happens when we try to derive these results theoretically? Is the resulting theory in agreement with the Wien displacement law and the Stefan-Boltzmann law? Classically: prior to 1900, the theoretical model, the so-called Rayleigh-Jeans law for the frequency distribution of radiation from a blackbody, gave an expression 6

8 ρ classical (ν, T ) = 8πk BT c 3 ν 2 ν 2, (1.4) for the radiation density, where k B is the Boltzmann constant k B = J K 1, T is the temperature and c is the speed of light. The Rayleigh-Jeans law predicts that the classical energy density at a given frequency ν is proportional to ν 2. So this classical theory predicts an infinite radiation density at an infinite frequency this is referred to as the ultra-violet catastrophe. Clearly, this result is indeed a catastrophe for classical physics. New Theory: in 1900, Planck proposed the following description of blackbody radiation: The material of the blackbody are made up of oscillators which could only emit/absorb light in small packets/increments of energy, de, given by where h is some constant of proportionality. de (ν) = hν, (1.5) Planck used his model to obtain the so-called Planck law of radiation ρ (ν, T ) = 8πhν3 c 3 1 exp (hν/k B T ) 1. (1.6) So does Planck s law give Wien s displacement and empirical laws, as well as the Stefan- Boltzmann result? Let us derive the results (1.3) and (1.1) using (1.6). We require, first of all, the Planck expression in terms of wavelength λ instead of frequency ν. Recalling that c = νλ, The energy-density in the frequency range ν to ν + dν must be the same as that in the corresponding wavelength range λ to λ + dλ. Therefore, we obtain ρ (ν, T ) dν = ρ (λ, T ) dλ, (1.7) where the ve sign arises because of the reciprocal relationship between ν and λ an increase in ν leads to a decrease in λ. 7