1. I Early Experiments Fig 1. 4 The kinematics of Conservation of energy dictates that hv+me2=hv+vp2c2+m2c4 (1.11) where me is the mass of the electron. We(necessarily )used the relativistic formula for the energy of the electron. We can isolate p/2 in Equation 1. 11 by squaring (2-2)+m To eliminate p? we note from Fig. 1.4 that p2=(p = Pp+Pp-2PpPp cos e Substituting Equation 1. 13 into Equation 1. 12 and writing the resulting equation terms of the difference in wavelengths between the incident and scattered photons △λ we have △A=入′-入 where Ac is known as the Compton wavelength of the electron 2.43×10 (1.15) Equation 1. 15 can be put in another form by multiplying the numerator and denom- inator by c, the speed of light. The denominator is thus the rest mass of the electron, 0.51 x 10 MeV, while the numerator is 1240(see Equation 1.10) Equation 1. 14 is known as the Compton equation. One of the remarkable features of it is that the change in wavelength of the photon does not depend upon its
1.1 Early Experiments 9 Fig. 1.4 The kinematics of Compton scattering Conservation of energy dictates that hν + mec2 = hν + p2 e c2 + m2 e c4 (1.11) where me is the mass of the electron. We (necessarily) used the relativistic formula for the energy of the electron. We can isolate p2 e in Equation 1.11 by squaring: p2 e = hν c − hν c + mec 2 − m2 e c2 (1.12) To eliminate p2 e we note from Fig. 1.4 that p2 e = pp − p p 2 = p2 p + p2 p − 2 pp p p cos θ = hν c 2 + hν c 2 − 2 hν c hν c cos θ (1.13) Substituting Equation 1.13 into Equation 1.12 and writing the resulting equation in terms of the difference in wavelengths between the incident and scattered photons λ we have λ = λ − λ = λc (1 − cos θ) (1.14) where λc is known as the Compton wavelength of the electron: λc = h mec ≈ 2.43 × 10−3 nm (1.15) Equation 1.15 can be put in another form by multiplying the numerator and denominator by c, the speed of light. The denominator is thus the rest mass of the electron, 0.51 × 106 MeV, while the numerator is 1240 (see Equation 1.10). Equation 1.14 is known as the Compton equation. One of the remarkable features of it is that the change in wavelength of the photon does not depend upon its
I Introduction incident wavelength. The maximum difference in wavelength that can be detected is twice(when 6= I)the Compton wavelength, 5x 10-nm. For this reason it is very difficult to perform Compton scattering experiments using visible light (A a400-700 nm) because the Ai would be only a tiny fraction of the wavelength of the incident photon wavelength. For much shorter wavelengths, as short as w Ac however, A/i can be large enough to measure. Thus, an incident photon of energy comparable with the rest energy of the electron, N 500 ke V, is required. Compton used x-rays having wavelength 0.071nm, roughly 17 keV in his experiments. While 17 kev is more than an order of magnitude lower than the rest mass of the electron, the effect was indeed detectable The Compton wavelength is often seen written as The reason for this is that the actual value of the Compton wavelength is not really important. It is the order of magnitude of it that is significant. This will be discussed later in this chapter. Equation 1. 14 shows that the wavelength of the scattered photon is always longer than the wavelength of the incident photon because cos B is always less than unity ton is elasicll scattered by the electron, imparting momentum and kimnedpho- ergy to the electron. Because conservation of energy dictates that the photon loses energy, it must, in accord with Equations 1. 2, have lower frequency and longer wavelength 1.2 Early Theory 1.2.1 The Bohr Atom and the Correspondence principle Confronted with overwhelming evidence that the amounts of internal energy that could be stored in an atom were not arbitrary, but were, instead, quantized, physicists attempted to explain the origin of these quantum levels. The experiments performed in Great Britain by Lord Rutherford clearly established that the atom consisted of a tiny massive positively charged nucleus surrounded by very light negatively charged electrons that orbited this nucleus. A major problem was that, according to classical electroma gnetic theory, accelerating charges emit electromagnetic energy (light) Thus, an orbiting electron should lose energy as it revolves about the nucleus, thus spiraling into the nucleus. If that spiraling process were to take a very long time, say 10 years, then there would be no problem because that is longer than the age of the universe. on the other hand if the lifetime of these atoms is short then the planetary model of the atom had to be reconciled with classical electromagnetic heory. Because it is important to understand the problem that presented itself to these pioneers of quantum physics it is worthwhile to do a simple calculation to
10 1 Introduction incident wavelength. The maximum difference in wavelength that can be detected is twice (when θ = π) the Compton wavelength, ∼ 5 × 10−3nm. For this reason it is very difficult to perform Compton scattering experiments using visible light (λ ≈ 400−700 nm) because the λ would be only a tiny fraction of the wavelength of the incident photon wavelength. For much shorter wavelengths, as short as ∼ λc, however, λ/λ can be large enough to measure. Thus, an incident photon of energy comparable with the rest energy of the electron, ∼ 500 keV, is required. Compton used x-rays having wavelength 0.071nm, roughly 17 keV in his experiments. While 17 keV is more than an order of magnitude lower than the rest mass of the electron, the effect was indeed detectable. The Compton wavelength is often seen written as λc = mec ≈ 0.39 × 10−3 nm (1.16) The reason for this is that the actual value of the Compton wavelength is not really important. It is the order of magnitude of it that is significant. This will be discussed later in this chapter. Equation 1.14 shows that the wavelength of the scattered photon is always longer than the wavelength of the incident photon because cos θ is always less than unity. Thus, λ > 0. The process can thus be envisioned as one in which the photon is elastically scattered by the electron, imparting momentum and kinetic energy to the electron. Because conservation of energy dictates that the photon loses energy, it must, in accord with Equations 1.2, have lower frequency and longer wavelength. 1.2 Early Theory 1.2.1 The Bohr Atom and the Correspondence Principle Confronted with overwhelming evidence that the amounts of internal energy that could be stored in an atom were not arbitrary, but were, instead, quantized, physicists attempted to explain the origin of these quantum levels. The experiments performed in Great Britain by Lord Rutherford clearly established that the atom consisted of a tiny massive positively charged nucleus surrounded by very light negatively charged electrons that orbited this nucleus. A major problem was that, according to classical electromagnetic theory, accelerating charges emit electromagnetic energy (light). Thus, an orbiting electron should lose energy as it revolves about the nucleus, thus spiraling into the nucleus. If that spiraling process were to take a very long time, say 1050 years, then there would be no problem because that is longer than the age of the universe. On the other hand, if the “lifetime” of these atoms is short, then the planetary model of the atom had to be reconciled with classical electromagnetic theory. Because it is important to understand the problem that presented itself to these pioneers of quantum physics it is worthwhile to do a simple calculation to
estimate t, the classical lifetime for a hydrogen atom, that is, the decay time due to radiation of an electron in orbit around a proton From electromagnetic theory the famous Larmor formula gives the instantaneous power P radiated by an electron undergoing acceleration a. In SI units, which we will use throughout this book unless otherwise state where the minus sign indicates that power is being radiated away. If we assume that each successive loop of the spiral toward the nucleus is a circle of radius r, then we may compute the acceleration a using Coulomb's law 1 The total mechanical energy (TME)E of the electron in the orbit is the sum of kinetic energy and the Coulomb potential energy: The motion is assumed to be circular so we can eliminate the velocity by equatin the centripetal force to the Coulomb force between the electron and proton. This results in E 2(4 (1.20) Now, P is the rate of loss of energy d E/dt so we may differentiate Equation 1.20 with respect to time and equate it to Equation 1. 17. We obtain d=-3m2c3r2 (1.21) which, when integrated from the initial radius R to the nucleus, yields t 4e4 (1.22) From Rutherford's experiments it was known that R N 0. Inm. The other parameters in this equation for t were reasonably well known. When inserted in Equation 1. 22 the result is T 10-Is, hardly comparable with the age of the universe. There was clearly a problem
1.2 Early Theory 11 estimate τ , the classical lifetime for a hydrogen atom, that is, the decay time due to radiation of an electron in orbit around a proton. From electromagnetic theory the famous Larmor formula gives the instantaneous power P radiated by an electron undergoing acceleration a. In SI units, which we will use throughout this book unless otherwise stated, P = − e2a2 6π 0c3 (1.17) where the minus sign indicates that power is being radiated away. If we assume that each successive loop of the spiral toward the nucleus is a circle of radius r, then we may compute the acceleration a using Coulomb’s law: a = 1 me 1 4π 0 e2 r 2 (1.18) The total mechanical energy (TME) E of the electron in the orbit is the sum of kinetic energy and the Coulomb potential energy: E = 1 2 mev2 − 1 4π 0 e2 r (1.19) The motion is assumed to be circular so we can eliminate the velocity by equating the centripetal force to the Coulomb force between the electron and proton. This results in E = −1 2 1 4π 0 e2 r (1.20) Now, P is the rate of loss of energy d E/dt so we may differentiate Equation 1.20 with respect to time and equate it to Equation 1.17. We obtain dr dt = −4 3 e4 m2 e c3r 2 (1.21) which, when integrated from the initial radius R to the nucleus, yields τ : τ = m2 e c3R3 4e4 (1.22) From Rutherford’s experiments it was known that R ∼ 0.1nm. The other parameters in this equation for τ were reasonably well known. When inserted in Equation 1.22 the result is τ ∼ 10−11s, hardly comparable with the age of the universe. There was clearly a problem
12 I Introduction Niels Bohr attempted to explain the quantized levels using a combination of clas sical ideas, quantal hypotheses, and postulates of his own [1]. This pioneering work was published in 1913 and Bohr was awarded the Nobel Prize in Physics in 1922 for his services in the investigation of the structure of atoms and of the radiatio emanating from them To deal with the problem of radiation by an accelerating charge Bohr simply avoided it by postulating his way out of it. Paraphrasing the first of his postulates I An atom exists in a series of energy states such that the accelerating electron does not ate energy when in these states. These states are designated as stationary states. he designation as"stationary states"has survived time and is used today. Why the accelerating electron ignored the classical laws of electromagnetic theory by not radiating was simply finessed, that is, ignored. Bohr's second postulate accounted for the emitted and absorbed radiation in terms of the stationary states II. Radiation is absorbed or emitted during a transition between two stationary states. The frequency of the absorbed or emitted radiation is given by Planck's theory. Bohr's reference here to "Plancks theory" is the relationship between the energy and the frequency, Equation 1. 1, that was used by Planck to explain blackbody ra- diation. The energy was taken to be the difference in the energies of the two states involved in the transition. Thus, the frequency, v, of this radiation is given by where h is Planck's constant and e and e are the energies of the two states in- Bohr had a third postulate, although he did not state it as such. It is the famous and ingenious correspondence principle. Loosely stated, the correspondence princi- ple states that when quantum systems become large they behave in a manner that th classical physics. Bohi although many derivations of the consequences of the Bohr model of the atom of- ten ignore the correspondence principle. Instead, these treatments postulate that the angular momentum must be quantized in units of h. Bohr made no such postulate, although it does lead to the correct answers without appealing to the correspondence principle. These derivations usually then present the correspondence principle as a con equence of this erroneous postulate It is a simple matter to obtain the relationship between the TMe of the electron E and the circular orbital radius r using elementary classical mechanics and elec tromagnetic theory. Equating the centripetal force to the Coulomb force we have mev (1.24) where v is the speed of the electron in the orbit of radius r. For simplicity and convenience we are assuming that the reduced mass of the electron-proton system
12 1 Introduction Niels Bohr attempted to explain the quantized levels using a combination of classical ideas, quantal hypotheses, and postulates of his own [1]. This pioneering work was published in 1913 and Bohr was awarded the Nobel Prize in Physics in 1922 “for his services in the investigation of the structure of atoms and of the radiation emanating from them.” To deal with the problem of radiation by an accelerating charge Bohr simply avoided it by postulating his way out of it. Paraphrasing the first of his postulates: I. An atom exists in a series of energy states such that the accelerating electron does not radiate energy when in these states. These states are designated as stationary states. The designation as “stationary states” has survived time and is used today. Why the accelerating electron ignored the classical laws of electromagnetic theory by not radiating was simply finessed, that is, ignored. Bohr’s second postulate accounted for the emitted and absorbed radiation in terms of the stationary states. II. Radiation is absorbed or emitted during a transition between two stationary states. The frequency of the absorbed or emitted radiation is given by Planck’s theory. Bohr’s reference here to “Planck’s theory” is the relationship between the energy and the frequency, Equation 1.1, that was used by Planck to explain blackbody radiation. The energy was taken to be the difference in the energies of the two states involved in the transition. Thus, the frequency, ν, of this radiation is given by hν = E − E (1.23) where h is Planck’s constant and E and E are the energies of the two states involved in the emission or absorption. Bohr had a third postulate, although he did not state it as such. It is the famous and ingenious correspondence principle. Loosely stated, the correspondence principle states that when quantum systems become large they behave in a manner that is consistent with classical physics. Bohr essentially used this as his third postulate, although many derivations of the consequences of the Bohr model of the atom often ignore the correspondence principle. Instead, these treatments postulate that the angular momentum must be quantized in units of . Bohr made no such postulate, although it does lead to the correct answers without appealing to the correspondence principle. These derivations usually then present the correspondence principle as a consequence of this erroneous postulate. It is a simple matter to obtain the relationship between the TME of the electron E and the circular orbital radius r using elementary classical mechanics and electromagnetic theory. Equating the centripetal force to the Coulomb force we have mev2 r = e2 4π 0 1 r 2 (1.24) where v is the speed of the electron in the orbit of radius r. For simplicity and convenience we are assuming that the reduced mass of the electron–proton system
is the same as me. From Equation 1. 24 we can solve for the kinetic energy of the electron so the tme is E 2(4x60)r-(4xe)F If we now apply Postulate II assuming a transition from state n to state m, we note that the only variable in the expression for the energy, Equation 1. 25, is the orbital radius r. We must therefore attach a subscript to r to designate to which state it belongs For definiteness we assume that n> m and, applying Postulate Il, we h where Vnm is the frequency of the photon emitted in the transition from the higher state n to the lower state m At this point there were two ingenious steps taken by Bohr. The first was to note the similarity between Equation 1.26 and the generalized Balmer formula, Equation 1.9(recall the reciprocal relationship between v and A). The orbital radius is the only variable in Equation 1. 26 so it is clear that it is In that is quantized. That is, each of the stationary states must have a unique orbital radius. Moreover, to be consistent with Equation 1.9 these orbital radii must be such that where ao has units of length. It is called the Bohr radius. To find it Bohr imposed he correspondence principle We had noted that accelerating charges radiate electromagnetic energy. But that is not the whole story. If these accelerating charges are being accelerated periodically, for example, a harmonically oscillating charge or a circularly moving charge, then the frequency of the emitted radiation is the same as the frequency of the motion Bohr therefore stated that as n and m become very large, the frequency Vnm in Equa tion 1 26 must approach the frequency orbit of the circular motion of the electron at the nth Bohr radius. The orbital frequency is where Un is the orbital speed in the nth Bohr orbit, Equation 1. 24. Working with the uare of orbit for convenience and using Equation 1. 28 we have
1.2 Early Theory 13 is the same as me. From Equation 1.24 we can solve for the kinetic energy of the electron so the TME is E = 1 2 e2 4π 0 1 r − e2 4π 0 1 r = −1 2 e2 4π 0 1 r (1.25) If we now apply Postulate II assuming a transition from state n to state m, we note that the only variable in the expression for the energy, Equation 1.25, is the orbital radiusr. We must therefore attach a subscript to r to designate to which state it belongs For definiteness we assume that n > m and, applying Postulate II, we write hνnm = 1 2 e2 4π 0 1 rm − 1 rn (1.26) where νnm is the frequency of the photon emitted in the transition from the higher state n to the lower state m. At this point there were two ingenious steps taken by Bohr. The first was to note the similarity between Equation 1.26 and the generalized Balmer formula, Equation 1.9 (recall the reciprocal relationship between ν and λ). The orbital radius is the only variable in Equation 1.26 so it is clear that it is rn that is quantized. That is, each of the stationary states must have a unique orbital radius. Moreover, to be consistent with Equation 1.9 these orbital radii must be such that rn = n2 a0 (1.27) where a0 has units of length. It is called the Bohr radius. To find it Bohr imposed the correspondence principle. We had noted that accelerating charges radiate electromagnetic energy. But that is not the whole story. If these accelerating charges are being accelerated periodically, for example, a harmonically oscillating charge or a circularly moving charge, then the frequency of the emitted radiation is the same as the frequency of the motion. Bohr therefore stated that as n and m become very large, the frequency νnm in Equation 1.26 must approach the frequency νorbit of the circular motion of the electron at the nth Bohr radius. The orbital frequency is νorbit = vn 2πrn (1.28) where vn is the orbital speed in the nth Bohr orbit, Equation 1.24. Working with the square of νorbit for convenience and using Equation 1.28 we have