Chapter 28

1. Chapter 28 Conceptual questions: 2,4,8,9,16 Quick quizzes: 1,2,3,4 Problems: 18, 39 Examples: 1,3,4 Atomic Physics

2. Rutherford’s Model of the Atom • Planetary model • Positive charge is concentrated in the center of the atom, called the nucleus • Electrons orbit the nucleus like planets orbit the sun

3. Examples of Spectra a) emission b) absorption

4. Absorption Spectra • An element can also absorb light at specific wavelengths • The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum • The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum

5. Spectral Lines of Hydrogen • The Balmer Series has lines whose wavelengths are given by the preceding equation • Examples of spectral lines • n = 3, λ = 656.3 nm • n = 4, λ = 486.1 nm

6. Balmer’s Equation • The wavelengths of hydrogen’s spectral lines can be found from • RH is the Rydberg constant • RH = 1.0973732 x 107 m-1 • n is an integer, n = 1, 2, 3, … • The spectral lines correspond to different values of n

7. Bohr’s Assumptions for Hydrogen • Circular orbits around the proton under the influence of the Coulomb force of attraction • Only certain electron orbits are stable • Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state

8. Mathematics of Bohr’s Assumptions and Results • Electron’s orbital angular momentum • me v r = n ħ where n = 1, 2, 3, … • The total energy of the atom • Coulomb’s force provides centripetal acceleration • Radiation emitted, Ei – Ef = hf

9. Radii and Energy of Orbits • The radii of the Bohr orbits are quantized • n = 1, the orbit has the smallest radius, called the Bohr radius, ao • ao = 0.0529 nm • A general expression for the radius of any orbit in a hydrogen atom is • rn = n2 ao • The energy of any orbit is • En = - 13.6 eV/ n2

10. Energy Level Diagram • The value of RH from Bohr’s analysis is in excellent agreement with the experimental value • A more generalized equation can be used to find the wavelengths of any spectral lines

11. Generalized Equation • For the Balmer series, nf = 2 • For the Lyman series, nf = 1 • Whenever an transition occurs between a state, ni to another state, nf (where ni > nf), a photon is emitted • The photon has a frequency f = (Ei – Ef)/h and wavelength λ

12. Successes of the Bohr Theory • Explained several features of the hydrogen spectrum • Accounts for Balmer and other series • Predicts a value for RH that agrees with the experimental value • Gives an expression for the radius of the atom • Predicts energy levels of hydrogen • Gives a model of what the atom looks like and how it behaves • Can be extended to “hydrogen-like” atoms • Those with one electron • Ze2 needs to be substituted for e2 in equations • Z is the atomic number of the element

13. de Broglie Waves • In this example, three complete wavelengths are contained in the circumference of the orbit • In general, the circumference must equal some integer number of wavelengths • 2  r = n λ n = 1, 2, …

14. Electron Clouds

15. Modifications of the Bohr Theory – Elliptical Orbits • Sommerfeld extended the results to include elliptical orbits • Retained the principle quantum number, n • Added the orbital quantum number, ℓ • ℓ ranges from 0 to n-1 in integer steps • All states with the same principle quantum number are said to form a shell • The states with given values of n and ℓ are said to form a subshell

16. Modifications of the Bohr Theory – Zeeman Effect • Another modification was needed to account for the Zeeman effect • The Zeeman effect is the splitting of spectral lines in a strong magnetic field • This indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic field • A new quantum number, mℓ, called the orbital magnetic quantum number, had to be introduced • m ℓ can vary from - ℓ to + ℓ in integer steps

17. Quantum numbers for the hydrogen atom

18. QUICK QUIZ 28.1 In an analysis relating Bohr's theory to the de Broglie wavelength of electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the wavelength of the electron becomes 9 times as long, or (d) the electron is moving 9 times as fast.

19. Quantum Number Summary • The values of n can range from 1 to  in integer steps • The values of ℓ can range from 0 to n-1 in integer steps • The values of mℓ can range from -ℓ to ℓ in integer steps

20. QUICK QUIZ 28.2 How many possible orbital states are there for (a) the n = 3 level of hydrogen? (b) the n = 4 level?

21. QUICK QUIZ 28.3 When the principal quantum number is n = 5, how many different values of (a)  and (b) m are possible?

22. Modifications of the Bohr Theory – Fine Structure • High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field • This splitting is called fine structure • Another quantum number, ms, called the spin magnetic quantum number, was introduced to explain the fine structure

23. Spin Magnetic Quantum Number • It is convenient to think of the electron as spinning on its axis • The electron is not physically spinning • There are two directions for the spin • Spin up, ms = ½ • Spin down, ms = -½ • There is a slight energy difference between the two spins and this accounts for the Zeeman effect

24. The Pauli Exclusion Principle • No two electrons in an atom can ever be in the same quantum state • In other words, no two electrons in the same atom can have exactly the same values for n, ℓ, mℓ, and ms • This explains the electronic structure of complex atoms as a succession of filled energy levels with different quantum numbers

26. The Periodic Table • The outermost electrons are primarily responsible for the chemical properties of the atom • Mendeleev arranged the elements according to their atomic masses and chemical similarities • The electronic configuration of the elements explained by quantum numbers and Pauli’s Exclusion Principle explains the configuration

28. QUICK QUIZ 28.4 Krypton (atomic number 36) has how many electrons in its next to outer shell (n = 3)?(a) 2 (b) 4 (c) 8 (d) 18

29. Problems 18. A particle of charge q and mass m, moving with a constant speed v, perpendicular to a constant magnetic field, B, follows a circular path. If the angular momentum about the center of this circle is quantized so that mvr = nħ , show that the allowed radii for the particle are where n = 1, 2, 3, . . . 39. Zirconium (Z = 40) has two electrons in an incomplete d subshell. (a) What are the values of n and l for each electron? (b) What are all possible values of ml and ms? (c) What is the electron configuration in the ground state of zirconium?

30. Characteristic X-Rays

31. Explanation of Characteristic X-Rays • The details of atomic structure can be used to explain characteristic x-rays • A bombarding electron collides with an electron in the target metal that is in an inner shell • If there is sufficient energy, the electron is removed from the target atom • The vacancy created by the lost electron is filled by an electron falling to the vacancy from a higher energy level • The transition is accompanied by the emission of a photon whose energy is equal to the difference between the two levels

32. Atomic Transitions – Energy Levels • An atom may have many possible energy levels • At ordinary temperatures, most of the atoms in a sample are in the ground state • Only photons with energies corresponding to differences between energy levels can be absorbed

33. Atomic Transitions – Stimulated Absorption • The blue dots represent electrons • When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level • These higher levels are called excited states • ΔE = hƒ = E2 – E1 • In general, ΔE can be the difference between any two energy levels

34. Atomic Transitions – Spontaneous Emission • Once an atom is in an excited state, there is a constant probability that it will jump back to a lower state by emitting a photon • This process is called spontaneous emission

35. Atomic Transitions – Stimulated Emission • An atom is in an excited stated and a photon is incident on it • The incoming photon increases the probability that the excited atom will return to the ground state • There are two emitted photons, the incident one and the emitted one • The emitted photon is in exactly in phase with the incident photon

36. Lasers • To achieve laser action, three conditions must be met • The system must be in a state of population inversion • The excited state of the system must be a metastable state • Its lifetime must be long compared to the normal lifetime of an excited state • The emitted photons must be confined in the system long enough to allow them to stimulate further emission from other excited atoms • This is achieved by using reflecting mirrors

37. Production of a Laser Beam

38. Laser Beam – He Ne Example • The energy level diagram for Ne • The mixture of helium and neon is confined to a glass tube sealed at the ends by mirrors • A high voltage applied causes electrons to sweep through the tube, producing excited states • When the electron falls to E2 in Ne, a 632.8 nm photon is emitted

39. Conceptual questions • 2.Does a light emitted by a neon sign constitute a continuous spectrum or only a few colors. • 4. Must an atom first be ionized before it can emit light? • 8. If matter has a wave nature, why is this not observable in our daily experiences? • 9. Discuss consequences of the exclusion principle. • 16. A 1 mW laser might damage your eye if you look directly at it, but there is no harm at looking directly at a 100 W lightbulb. Why?

40. Energy Bands in Solids • Sodium example • Blue represents energy bands occupied by the sodium electrons when the atoms are in their ground states • Gold represents energy bands that are empty • White represents energy gaps • Electrons can have any energy within the allowed bands • Electrons cannot have energies in the gaps

41. Energy Level Definitions • The valence band is the highest filled band • The conduction band is the next higher empty band • The energy gap has an energy, Eg, equal to the difference in energy between the top of the valence band and the bottom of the conduction band

42. Conductors • When a voltage is applied to a conductor, the electrons accelerate and gain energy • In quantum terms, electron energies increase if there are a high number of unoccupied energy levels for the electron to jump to • For example, it takes very little energy for electrons to jump from the partially filled to one of the nearby empty states

43. Insulators • The valence band is completely full of electrons • A large band gap separates the valence and conduction bands • A large amount of energy is needed for an electron to be able to jump from the valence to the conduction band

44. Semiconductors • A semiconductor has a small energy gap • Thermally excited electrons have enough energy to cross the band gap • The resistivity of semiconductors decreases with increases in temperature • The white area in the valence band represents holes

45. Semiconductors, cont • Holes are empty states in the valence band created by electrons that have jumped to the conduction band • Some electrons in the valence band move to fill the holes and therefore also carry current • The valence electrons that fill the holes leave behind other holes • It is common to view the conduction process in the valence band as a flow of positive holes toward the negative electrode applied to the semiconductor

46. Current Process in Semiconductors • An external voltage is supplied • Electrons move toward the positive electrode • Holes move toward the negative electrode • There is a symmetrical current process in a semiconductor

47. Doping in Semiconductors • Doping is the adding of impurities to a semiconductor • Generally about 1 impurity atom per 107 semiconductor atoms • Doping results in both the band structure and the resistivity being changed

48. A p-n Junction • A p-n junction is formed when a p-type semiconductor is joined to an n-type • Three distinct regions exist • A p region • An n region • A depletion region

49. Diode Action • The p-n junction has the ability to pass current in only one direction • When the p-side is connected to a positive terminal, the device is forward biased and current flows • When the n-side is connected to the positive terminal, the device is reverse biased and a very small reverse current results

50. Applications of Semiconductor Diodes • Rectifiers • Change AC voltage to DC voltage • A half-wave rectifier allows current to flow during half the AC cycle • A full-wave rectifier rectifies both halves of the AC cycle • Transistors • May be used to amplify small signals • Integrated circuit • A collection of interconnected transistors, diodes, resistors and capacitors fabricated on a single piece of silicon