Quantum Physics

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28.4 Modification of the Bohr Theory 911Although many attempts were made to extend the Bohr theory to more complex,multi-electron atoms, the results were unsuccessful. Even today, only approximatemethods are available for treating multi-electron atoms.Quick Quiz 28.1Consider a hydrogen atom and a singly-ionized helium atom. Which atom has thelower ground state energy? (a) hydrogen (b) helium (c) the ground state energy isthe same for bothQuick Quiz 28.2Consider once again a singly-ionized helium atom. Suppose the remaining electronjumps from a higher to a lower energy level, resulting in the emission of photon,which we’ll call photon-He. An electron in a hydrogen atom then jumpsbetween the same two levels, resulting in an emitted photon-H. Which photon hasthe shorter wavelength? (a) photon-He (b) photon-H (c) The wavelengths are thesame.EXAMPLE 28.2GoalSingly Ionized HeliumApply the modified Bohr theory to a hydrogen-like atom.Problem Singly ionized helium, He , a hydrogen-like system, has one electron in the 1s orbit when the atom is inits ground state. Find (a) the energy of the system in the ground state in electron volts, and (b) the radius of theground-state orbit.Strategy Part (a) requires substitution into the modified Bohr model, Equation 28.18. In part (b), modify Equation28.9 for the radius of the Bohr orbits by replacing e 2 by Ze 2 , where Z is the number of protons in the nucleus.Solution(a) Find the energy of the system in the ground state.Write Equation 28.18 for the energies of a hydrogen-likesystem:E n m ek e 2 Z 2 e 42 2 1 n 2Substitute the constants and convert to electron volts: E n Z 2 (13.6)n 2 eVSubstitute Z 2 (the atomic number of helium) andn 1 to obtain the ground state energy:(b) Find the radius of the ground state.Generalize Equation 28.9 to a hydrogen-like atom bysubstituting Ze 2 for e 2 :For our case, n 1 and Z 2:E 1 4(13.6) eV 54.4 eVr n n2 2m e k e Ze 2 n2Z (a 0) n2(0.052 9 nm)Zr 1 0.026 5 nmRemarks Notice that for higher Z the energy of a hydrogen-like atom is lower, which means that the electron ismore tightly bound than in hydrogen. This results in a smaller atom, as seen in part (b).Exercise 28.2Repeat the problem for the first excited state of doubly-ionized lithium (Z 3, n 2).Answers (a) E 2 30.6 eV (b) r 2 0.070 5 nm
912 Chapter 28 Atomic <strong>Physics</strong>TABLE 28.1Shell and Subshell NotationShellSubshelln Symbol Symbol1 K 0 s2 L 1 p3 M 2 d4 N 3 f5 O 4 g6 P 5 h. . . . . .Figure 28.9 A single line (A) cansplit into three separate lines (B) in amagnetic field.ABWithin a few months following the publication of Bohr’s paper, Arnold Sommerfeld(1868–1951) extended the Bohr model to include elliptical orbits. We examinehis model briefly because much of the nomenclature used in this treatment is still inuse today. Bohr’s concept of quantization of angular momentum led to the principalquantum number n, which determines the energy of the allowed states of hydrogen.Sommerfeld’s theory retained n, but also introduced a new quantum number called the orbital quantum number, where the value of ranges from 0 to n 1 in integersteps. According to this model, an electron in any one of the allowed energystates of a hydrogen atom may move in any one of a number of orbits correspondingto different values. For each value of n, there are n possible orbits corresponding todifferent values. Because n 1 and 0 for the first energy level (ground state),there is only one possible orbit for this state. The second energy level, with n 2, hastwo possible orbits, corresponding to 0 and 1. The third energy level, withn 3, has three possible orbits, corresponding to 0, 1, and 2.For historical reasons, all states with the same principal quantum number n aresaid to form a shell. Shells are identified by the letters K, L, M, . . . , which designatethe states for which n 1, 2, 3, . . . . Likewise, the states with given values of n and are said to form a subshell. The letters s, p, d, f, g, . . . are used to designate the statesfor which 0, 1, 2, 3, 4, . . . . These notations are summarized in Table 28.1.States that violate the restriction 0 n 1, for a given value of n, can’t exist.A 2d state, for instance, would have n 2 and 2, but can’t exist because thehighest allowed value of is n 1, or 1 in this case. For n 2, 2s and 2p are allowedsubshells, but 2d, 2f, . . . are not. For n 3, the allowed states are 3s, 3p, and 3d.Another modification of the Bohr theory arose when it was discovered that thespectral lines of a gas are split into several closely spaced lines when the gas is placedin a strong magnetic field. (This is called the Zeeman effect, after its discoverer.) Figure28.9 shows a single spectral line being split into three closely spaced lines. This indicatesthat the energy of an electron is slightly modified when the atom is immersed ina magnetic field. In order to explain this observation, a new quantum number, m ,called the orbital magnetic quantum number, was introduced. The theory is in accordwith experimental results when m is restricted to values ranging from to ininteger steps. For a given value of , there are 2 1 possible values of m .Finally, very high resolution spectrometers revealed that spectral lines of gasesare in fact two very closely spaced lines even in the absence of an external magneticfield. This splitting was referred to as fine structure. In 1925 Samuel Goudsmit andGeorge Uhlenbeck introduced the idea of an electron spinning about its own axisto explain the origin of fine structure. The results of their work introduced yet anotherquantum number, m s , called the spin magnetic quantum number.For each electron there are two spin states. A subshell corresponding to a givenfactor of can contain no more than 2(2 1) electrons. This number comesfrom the fact that electrons in a subshell must have unique pairs of the quantumnumbers (m , m s ). There are 2 1 different magnetic quantum numbers m ,and two different spin quantum numbers m s , making 2(2 1) unique pairs(m , m s ). For example, the p subshell ( 1) is filled when it contains 2(2 1 1) 6electrons. This fact can be extended to include all four quantum numbers, as willbe important to us later when we discuss the Pauli exclusion principle.All these quantum numbers (addressed in more detail in upcoming sections)were postulated to account for the observed spectra of elements. Only later werecomprehensive mathematical theories developed that naturally yielded the sameanswers as these empirical models.28.5 DE BROGLIE WAVES ANDTHE HYDROGEN ATOMOne of the postulates made by Bohr in his theory of the hydrogen atom was thatthe angular momentum of the electron is quantized in units of , orm e vr n
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29.7 Medical Applications of Radiat
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29.8 Radiation Detectors 963Figure
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Summary 965Photo Researchers, Inc./
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Problems 967CONCEPTUAL QUESTIONS1.
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Problems 96924. A building has beco
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Problems 97157. A by-product of som
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This photo shows scientist MelissaD
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30.1 Nuclear Fission 975Applying Ph
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30.2 Nuclear Reactors 977Courtesy o
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30.2 Nuclear Reactors 979events in
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30.3 Nuclear Fusion 981followed by
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30.3 Nuclear Fusion 983VacuumCurren
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30.6 Positrons and Other Antipartic
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30.7 Mesons and the Beginning of Pa
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30.9 Conservation Laws 989LeptonsLe
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30.10 Strange Particles and Strange
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30.12 Quarks 993n pΣ _ Σ 0 Σ + S
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30.12 Quarks 995charm C 1, its anti
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30.14 Electroweak Theory and the St
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30.15 The Cosmic Connection 999prot
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30.16 Problems and Perspectives 100
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Problems 100330.12 Quarks &30.13 Co
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Problems 1005particles fuse to prod
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Problems 100740. Assume binding ene
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A.1 MATHEMATICAL NOTATIONMany mathe
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A.3 Algebra A.3by 8, we have8x8 32
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A.3 Algebra A.5EXERCISESSolve the f
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A.5 Trigonometry A.7When natural lo
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APPENDIX BAn Abbreviated Table of I
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An Abbreviated Table of Isotopes A.
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An Abbreviated Table of Isotopes A.
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Some Useful Tables A.15TABLE C.3The
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Answers to Quick Quizzes,Odd-Number
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Answers to Quick Quizzes, Odd-Numbe
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IndexPage numbers followed by “f
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Current, 568-573, 586direction of,
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Index I.5Fissionnuclear, 973-976, 9
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Index I.7Magnetic field(s) (Continu
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Polarizer, 805-806, 805f, 806-807Po
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South poleEarth’s geographic, 626
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CreditsPhotographsThis page constit
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PEDAGOGICAL USE OF COLORDisplacemen
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PHYSICAL CONSTANTSQuantity Symbol V
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