Chapter 4: Structure of the atomfrom plum pudding to Bohr's quantized orbits

4.1The atomic models of Thomson and Rutherford

In J. J. Thomson's model, electrons are embedded in a homogeneous, positively charged mass, like raisins in a plum pudding, which makes the atom electrically neutral. Geiger and Marsden (1909) directed α particles (ionized helium atoms) at a gold foil \(6.0\times10^{-7}\) m thick and found, surprisingly, that some were scattered backward, through angles greater than 90°. That is impossible if the α particles hit only electrons.

Example 4.1. In a head-on elastic collision of an α particle (mass \(M_\alpha\approx7000m_e\)) with an electron at rest, the α particle hardly slows down and the electron leaves at about \(2v_\alpha\), so the largest possible momentum transfer is \(\Delta p_{\max} = 2m_ev_\alpha\). Even if all of it were perpendicular to the motion, the deflection would be at most

Multiple scattering from \(N\) electrons gives an average angle \(\langle\theta\rangle_{\rm total}\approx\sqrt N\,\theta\). Gold has \(5.9\times10^{28}\) atoms/m³, so neighboring atoms are \(d = (5.9\times10^{28})^{-1/3} = 2.6\times10^{-10}\) m apart and an α particle crosses about \(N = 6\times10^{-7}/2.6\times10^{-10} = 2300\) atoms, giving \(\sqrt{2300}\times0.016^\circ = 0.8^\circ\); even scattering from all 79 electrons of each atom gives only 6.8°. Rutherford: "It seems reasonable to suppose that the deflection through a large angle is due to a single atomic encounter." He proposed that an atom consists mostly of empty space with a central charge distributed through a very small volume: the nucleus (원자핵), which contains most of the mass and is positively charged, surrounded by the negative electrons (the "planetary" model).

4.2Rutherford scattering

A projectile of mass \(m\) and charge \(Z_1e\) approaches a scatterer of charge \(Z_2e\) with speed \(v_0\) and impact parameter \(b\), and leaves at the scattering angle \(\theta\). Rutherford assumed: (1) the scatterer is so massive that it does not recoil, so the initial and final kinetic energies are practically equal; (2) the target is so thin that only a single scattering occurs; (3) the particles can be treated as point masses and charges; (4) only the Coulomb force acts. Particles with small impact parameters approach the nucleus most closely and scatter to the largest angles.

The Coulomb force \(\vec F = \frac{1}{4\pi\epsilon_0}\frac{Z_1Z_2e^2}{r^2}\hat e_r\) is central, so the angular momentum \(mv_0b\) is conserved and the trajectory lies in a plane. The change in momentum equals the impulse, \(\Delta\vec p = \int\vec F\,dt\), and by symmetry it points along the bisector, with magnitude \(\Delta p = 2mv_0\sin(\theta/2)\) and component of force \(F_{\Delta p} = F\cos\phi\). With \(mv_0b = mr^2\,d\phi/dt\), so \(r^2 = v_0b/(d\phi/dt)\),

using \(\phi_i = -\tfrac12(\pi-\theta)\) and \(\phi_f = \tfrac12(\pi-\theta)\). With \(K = \tfrac12mv_0^2\), this gives the fundamental relation between impact parameter and scattering angle,

We can't select \(b\) in an experiment, but every particle with impact parameter less than \(b_0\) scatters through an angle greater than \(\theta_0\). The cross section \(\sigma = \pi b^2\) for scattering through \(\theta\) or more is related to the probability of being scattered by a nucleus. A thin target of thickness \(t\) and area \(A\) with \(n\) atoms per unit volume contains \(ntA\) nuclei (\(n = \rho N_AN_M/M_g\)), which expose a total area \(ntA\sigma\). The fraction scattered by \(\theta\) or more is therefore

Detectors cover a small range \(\theta\) to \(\theta+d\theta\). Differentiating, \(df = -\pi nt\big(\frac{Z_1Z_2e^2}{8\pi\epsilon_0K}\big)^2\cot\frac\theta2\csc^2\frac\theta2\,d\theta\), and of \(N_i\) incident particles, \(N_i|df|\) land in the ring of area \(dA = (r\,d\theta)(2\pi r\sin\theta)\) at distance \(r\). The number per unit area is \(N(\theta) = N_i|df|/dA\), which with \(\sin\theta = 2\sin\frac\theta2\cos\frac\theta2\) becomes

the famous Rutherford scattering equation. Scattering is proportional to \(Z_1^2\) and \(Z_2^2\), inversely proportional to \(K^2\) and to \(\sin^4(\theta/2)\), and proportional to the target thickness. Geiger and Marsden confirmed these predictions in 1913, and the angular dependence can be verified in an undergraduate laboratory.

Closest approach (Conceptual Example 4.4). In a head-on collision the projectile turns around at \(r_{\min}\), where all its kinetic energy has become Coulomb potential energy: \(K = Z_1Z_2e^2/4\pi\epsilon_0r_{\min}\), so \(r_{\min} = Z_1Z_2e^2/4\pi\epsilon_0K\). Example 4.5: Rutherford found deviations from his formula when 7.7 MeV α particles (\(Z_1 = 2\)) scattered backward from aluminum (\(Z_2 = 13\)). Then \(r_{\min} = 4.9\times10^{-15}\) m, roughly the sum of the helium and aluminum nuclear radii: the α particle gets close enough to feel the nuclear force (Chapter 12).

4.3The classical atomic model

Consider the "planetary" model of hydrogen, the simplest atom: an electron of mass \(m\) in uniform circular motion around the nucleus, with the Coulomb force supplying the centripetal force,

(Example 4.6: for \(r = 0.5\times10^{-10}\) m, \(v\approx2.2\times10^6\) m/s \(\lt 0.01c\), so a nonrelativistic treatment is justified.) The proton is 1836 times heavier and is considered at rest. With \(K = \tfrac12mv^2\) and \(V = -e^2/4\pi\epsilon_0r\), the total mechanical energy is

negative for a bound system (with \(E = 0\) at \(r\to\infty\)). The problem: an accelerated electric charge continuously radiates energy as electromagnetic radiation, and so loses energy. The electron spirals inward and crashes into the nucleus: the planetary model is doomed. This failure was a decisive turning point, like Planck's quantum hypothesis of 1900.

4.4The Bohr model of the hydrogen atom

Niels Bohr (1885–1962) stated his "general assumptions" as follows:

1. Certain stationary states exist in atoms, which differ from the classical stable states in that the orbiting electrons do not continuously radiate electromagnetic energy. The stationary states are states of definite total energy.
2. The emission or absorption of electromagnetic radiation can occur only in conjunction with a transition between two stationary states. The frequency of the radiation is proportional to the difference in energy of the two states: \(E = E_1 - E_2 = hf\), with \(h\) Planck's constant.
3. The dynamical equilibrium of the system in the stationary states is governed by classical laws of physics, but these laws do not apply to transitions between stationary states.
4. The mean kinetic energy of the electron–nucleus system is \(K = nhf_{\rm orb}/2\), where \(f_{\rm orb} = v/2\pi r\) is the frequency of rotation. For a circular orbit this is equivalent to the angular momentum of the system being an integral multiple of \(h/2\pi\equiv\hbar\) ("h bar").

So \(L = |\vec r\times\vec p| = mvr = n\hbar\), with \(n\) the principal quantum number. Combining \(v = n\hbar/mr\) with the classical \(v = e/\sqrt{4\pi\epsilon_0mr}\) gives \(e^2/4\pi\epsilon_0mr = n^2\hbar^2/m^2r^2\), the quantized radius

the Bohr radius, matching the suspected size of the hydrogen atom (\(2a_0\approx10^{-10}\) m). The state \(n=1\) is the lowest energy ("ground") state, and \(n>1\) are excited states. The quantized energy is

and \(E_0\) is exactly the experimentally measured ionization energy of hydrogen. In a transition from an upper level \(E_u\) to a lower level \(E_\ell\) the atom emits a photon of energy \(hf = hc/\lambda = E_u - E_\ell\) (the emission spectrum); absorbing a photon of the same energy drives the reverse transition (the absorption spectrum). Thus

the Rydberg equation found experimentally in Chapter 3, now derived, with the Rydberg constant for an infinitely heavy nucleus. Transitions down to \(n=1\) form the Lyman series (ultraviolet), to \(n=2\) the Balmer series (visible), and to \(n=3\) the Paschen series (infrared). The electron's speed in orbit \(n\) is

the fine structure constant. Example 4.7: the Paschen series (\(n_\ell = 3\)) runs from \(\lambda_{\max} = 1875\) nm (\(n_u = 4\)) down to \(\lambda_{\min} = 820\) nm (\(n_u = \infty\)), all in the infrared.

The correspondence principle

Bohr's correspondence principle: in the limits where classical and quantum theories should agree, the quantum theory must reduce to the classical result. Classically the radiated frequency is the orbital frequency, \(f_{\rm classical} = \omega/2\pi = v/2\pi r\); with \(v = e/\sqrt{4\pi\epsilon_0mr}\) and \(r_n = n^2a_0\),

Classically the "continuous" radiation of an inward spiral would be a cascade of transitions \(n+1\to n\to n-1\to\cdots\). For one step, with \(E_0 = me^4/8h^2\epsilon_0^2\),

The frequencies of the radiation agree for large \(n\): the correspondence principle holds for large orbits, where classical and quantum physics should agree.

As Bohr's model gained acceptance, the assumption of quantized angular momentum \(L_n = n\hbar\) led to the quantization of the other quantities: \(r_n = n^2a_0\), \(v_n = n\hbar/mr_n\), and \(E_n = -e^2/8\pi\epsilon_0a_0n^2\).

4.5Successes and failures of the Bohr model

The Bohr model was a first step in understanding the structure of the atom; the correct, quantum-mechanical description comes in Chapter 7. One refinement is the reduced mass correction: the electron and the nucleus actually revolve about their mutual center of mass. This two-body problem reduces to an equivalent one-body problem for a particle of mass

(Example 4.8: deuterium's H\(_\alpha\) line, slightly shifted from hydrogen's, revealed deuterium in 1932.) The model also applies to any single-electron atom (hydrogen-like ions such as He\(^+\) and Li\(^{2+}\)) by replacing \(e^2\) with \(Ze^2\):

In 1913 Bohr predicted the spectral lines of He\(^+\) before they had been identified in the laboratory, and showed that certain lines attributed to hydrogen were actually from ionized helium: its lines for \(n_\ell = 4\) nearly coincide with hydrogen's for \(n_\ell = 2\), with additional lines in between.

Other limitations. Precise spectroscopy resolved single lines into two or more, the fine structure. Sommerfeld accounted for part of it with relativistic elliptical orbits, but the electron's spin also contributes. External magnetic fields (the Zeeman effect) and electric fields (the Stark effect) split and broaden lines. The Bohr model (1) applies successfully only to single-electron atoms, (2) cannot account for the intensities or fine structure of spectral lines, and (3) cannot explain the binding of atoms into molecules. It was an ad hoc theory that retained too many classical concepts, yet it was the first step from a purely classical to the correct quantum description. Einstein later said Bohr's achievement "appeared to me like a miracle and appears as a miracle even today."

4.6Characteristic x-ray spectra and atomic number

The sharp peaks in x-ray spectra (Chapter 3) can be understood with Bohr's electron shell hypothesis: a shell for each principal quantum number \(n\), with radius \(r_n = n^2a_0\). Adding electrons to a fully ionized many-electron atom, the inner shells (lower \(n\)) fill first (the exclusion principle, Chapter 8). The shells are named K (\(n = 1\), the lowest energy state), L (\(n = 2\)), M (\(n = 3\)), and so on. An x ray is emitted when an electron drops into a vacancy, \(E_{\rm x\,ray} = E_u - E_\ell\). Transitions into the K shell give the K lines (K\(_\alpha\) from L, K\(_\beta\) from M), the two strong peaks in an x-ray tube spectrum, and transitions into the L shell give the L lines.

H. G. J. Moseley (1887–1915) found empirically that the K\(_\alpha\) frequencies of the elements follow an amazingly linear law:

Bohr's model explains it. If a vacancy occurs in the K shell, one electron remains there, and an L-shell electron feels an effective charge \((Z-1)e\): the nuclear \(+Ze\) screened by the remaining K electron. Treating the atom as hydrogen-like,

consistent with Moseley. His plot ordered the elements by atomic number rather than atomic weight, and its gaps marked elements (\(Z = 43, 61, 75\)) not yet discovered. Example 4.10: Moseley found \(f_{L_\alpha} = \tfrac{5}{36}cR(Z-7.4)^2\) for the L\(_\alpha\) line (M → L). With an effective charge \(Z_{\rm eff} = Z - 7.4\), screened by the L electrons, the two K electrons, and others, \(1/\lambda_L = R(Z-7.4)^2(1/4 - 1/n^2)\).

4.7Atomic excitation by electrons

James Franck and Gustav Hertz transferred part of an electron's kinetic energy to an atom. Electrons from a hot filament are accelerated through mercury vapor toward a grid; a small decelerating voltage between grid and collector stops electrons with too little energy from reaching the collector. Below 4.88 eV, no energy is transferred to the Hg atoms (elastic collisions, no loss of kinetic energy). Above 4.88 eV, an electron can transfer exactly that energy to an Hg atom, exciting it from the ground state to its first excited state (an inelastic collision), and the current drops. At 7 or 8 V the remaining energy is again enough to reach the collector; at 9.8 V electrons can excite two Hg atoms in succession, losing \(2\times4.88\) eV, and the current drops again. The peaks are typically 4.9 V apart; the first is not at 4.9 V because of the difference in work functions of the cathode and anode metals. The experiment convincingly proved the quantization of atomic energy levels.

Conceptual Example 4.11. The excited Hg atom decays back to the ground state in about \(10^{-8}\) s. Franck and Hertz looked for the emitted radiation and found that mercury emitted an ultraviolet line at 254 nm only once the electron energy exceeded about 5 eV. Setting \(E = 4.88\ \text{eV} = hf = hc/\lambda\) with \(\lambda = 254\) nm gave a value of \(h\) in good agreement with other methods.

This completes Chapter 4. Chapter 5 gives matter its wave properties.

Further reading

• S. T. Thornton, A. Rex, and C. E. Hood, Modern Physics for Scientists and Engineers, 5th ed. (Cengage Learning), Chapter 4.
• E. Rutherford, "The scattering of α and β particles by matter and the structure of the atom," Phil. Mag. 21, 669 (1911).
• H. Geiger and E. Marsden, "The laws of deflexion of α particles through large angles," Phil. Mag. 25, 604 (1913).
• N. Bohr, "On the constitution of atoms and molecules," Phil. Mag. 26, 1 (1913).
• H. G. J. Moseley, "The high-frequency spectra of the elements," Phil. Mag. 26, 1024 (1913) and 27, 703 (1914).
• J. Franck and G. Hertz, Verh. Dtsch. Phys. Ges. 16, 457 (1914).