4 Absorption SpectraAn element can also absorb light at specific wavelengthsThe absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrumThe 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 equationExamples of spectral linesn = 3, λ = nmn = 4, λ = nm
6 Balmer’s EquationThe wavelengths of hydrogen’s spectral lines can be found fromRH is the Rydberg constantRH = x 107 m-1n 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 attractionOnly certain electron orbits are stableRadiation 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 momentumme v r = n ħ where n = 1, 2, 3, …The total energy of the atomCoulomb’s force provides centripetal accelerationRadiation emitted, Ei – Ef = hf
9 Radii and Energy of Orbits The radii of the Bohr orbits are quantizedn = 1, the orbit has the smallest radius, called the Bohr radius, aoao = nmA general expression for the radius of any orbit in a hydrogen atom isrn = n2 aoThe energy of any orbit isEn = eV/ n2
10 Energy Level DiagramThe value of RH from Bohr’s analysis is in excellent agreement with the experimental valueA more generalized equation can be used to find the wavelengths of any spectral lines
11 Generalized EquationFor the Balmer series, nf = 2For the Lyman series, nf = 1Whenever an transition occurs between a state, ni to another state, nf (where ni > nf), a photon is emittedThe photon has a frequency f = (Ei – Ef)/h and wavelength λ
12 Successes of the Bohr Theory Explained several features of the hydrogen spectrumAccounts for Balmer and other seriesPredicts a value for RH that agrees with the experimental valueGives an expression for the radius of the atomPredicts energy levels of hydrogenGives a model of what the atom looks like and how it behavesCan be extended to “hydrogen-like” atomsThose with one electronZe2 needs to be substituted for e2 in equationsZ is the atomic number of the element
13 de Broglie WavesIn this example, three complete wavelengths are contained in the circumference of the orbitIn general, the circumference must equal some integer number of wavelengths2 r = n λ n = 1, 2, …
15 Modifications of the Bohr Theory – Elliptical Orbits Sommerfeld extended the results to include elliptical orbitsRetained the principle quantum number, nAdded the orbital quantum number, ℓℓ ranges from 0 to n-1 in integer stepsAll states with the same principle quantum number are said to form a shellThe 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 effectThe Zeeman effect is the splitting of spectral lines in a strong magnetic fieldThis indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic fieldA new quantum number, m ℓ, called the orbital magnetic quantum number, had to be introducedm ℓ can vary from - ℓ to + ℓ in integer steps
18 QUICK QUIZ 28.1In 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 stepsThe values of ℓ can range from 0 to n-1 in integer stepsThe values of m ℓ can range from -ℓ to ℓ in integer steps
20 QUICK QUIZ 28.2How many possible orbital states are there for (a) the n = 3 level of hydrogen? (b) the n = 4 level?
21 QUICK QUIZ 28.3When 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 fieldThis splitting is called fine structureAnother 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 axisThe electron is not physically spinningThere are two directions for the spinSpin up, ms = ½Spin down, ms = -½There 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 stateIn other words, no two electrons in the same atom can have exactly the same values for n, ℓ, m ℓ, and msThis explains the electronic structure of complex atoms as a succession of filled energy levels with different quantum numbers
26 The Periodic TableThe outermost electrons are primarily responsible for the chemical properties of the atomMendeleev arranged the elements according to their atomic masses and chemical similaritiesThe electronic configuration of the elements explained by quantum numbers and Pauli’s Exclusion Principle explains the configuration
28 QUICK QUIZ 28.4Krypton (atomic number 36) has how many electrons in its next to outer shell (n = 3)? (a) 2 (b) 4 (c) 8 (d) 18
29 Problems18. 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 arewhere 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?
31 Explanation of Characteristic X-Rays The details of atomic structure can be used to explain characteristic x-raysA bombarding electron collides with an electron in the target metal that is in an inner shellIf there is sufficient energy, the electron is removed from the target atomThe vacancy created by the lost electron is filled by an electron falling to the vacancy from a higher energy levelThe 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 levelsAt ordinary temperatures, most of the atoms in a sample are in the ground stateOnly photons with energies corresponding to differences between energy levels can be absorbed
33 Atomic Transitions – Stimulated Absorption The blue dots represent electronsWhen a photon with energy ΔE is absorbed, one electron jumps to a higher energy levelThese higher levels are called excited statesΔE = hƒ = E2 – E1In 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 photonThis process is called spontaneous emission
35 Atomic Transitions – Stimulated Emission An atom is in an excited stated and a photon is incident on itThe incoming photon increases the probability that the excited atom will return to the ground stateThere are two emitted photons, the incident one and the emitted oneThe 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 inversionThe excited state of the system must be a metastable stateIts lifetime must be long compared to the normal lifetime of an excited stateThe emitted photons must be confined in the system long enough to allow them to stimulate further emission from other excited atomsThis is achieved by using reflecting mirrors
38 Laser Beam – He Ne Example The energy level diagram for NeThe mixture of helium and neon is confined to a glass tube sealed at the ends by mirrorsA high voltage applied causes electrons to sweep through the tube, producing excited statesWhen the electron falls to E2 in Ne, a nm photon is emitted
39 Conceptual questions2.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 statesGold represents energy bands that are emptyWhite represents energy gapsElectrons can have any energy within the allowed bandsElectrons cannot have energies in the gaps
41 Energy Level Definitions The valence band is the highest filled bandThe conduction band is the next higher empty bandThe 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 ConductorsWhen a voltage is applied to a conductor, the electrons accelerate and gain energyIn quantum terms, electron energies increase if there are a high number of unoccupied energy levels for the electron to jump toFor 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 bandsA 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 gapThe resistivity of semiconductors decreases with increases in temperatureThe white area in the valence band represents holes
45 Semiconductors, contHoles are empty states in the valence band created by electrons that have jumped to the conduction bandSome electrons in the valence band move to fill the holes and therefore also carry currentThe valence electrons that fill the holes leave behind other holesIt 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 suppliedElectrons move toward the positive electrodeHoles move toward the negative electrodeThere is a symmetrical current process in a semiconductor
47 Doping in Semiconductors Doping is the adding of impurities to a semiconductorGenerally about 1 impurity atom per 107 semiconductor atomsDoping results in both the band structure and the resistivity being changed
48 A p-n JunctionA p-n junction is formed when a p-type semiconductor is joined to an n-typeThree distinct regions existA p regionAn n regionA depletion region
49 Diode ActionThe p-n junction has the ability to pass current in only one directionWhen the p-side is connected to a positive terminal, the device is forward biased and current flowsWhen 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 RectifiersChange AC voltage to DC voltageA half-wave rectifier allows current to flow during half the AC cycleA full-wave rectifier rectifies both halves of the AC cycleTransistorsMay be used to amplify small signalsIntegrated circuitA collection of interconnected transistors, diodes, resistors and capacitors fabricated on a single piece of silicon