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

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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

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Examples of Spectra a) emission b) absorption

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Absorption Spectra An element can also absorb light at specific wavelengths 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 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 The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum

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

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Balmer’s Equation The wavelengths of hydrogen’s spectral lines can be found from The wavelengths of hydrogen’s spectral lines can be found from R H is the Rydberg constant R H is the Rydberg constant R H = 1.0973732 x 10 7 m -1 R H = 1.0973732 x 10 7 m -1 n is an integer, n = 1, 2, 3, … n is an integer, n = 1, 2, 3, … The spectral lines correspond to different values of n The spectral lines correspond to different values of n

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

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Mathematics of Bohr’s Assumptions and Results Electron’s orbital angular momentum Electron’s orbital angular momentum m e v r = n ħ where n = 1, 2, 3, … m e v r = n ħ where n = 1, 2, 3, … The total energy of the atom The total energy of the atom Coulomb’s force provides centripetal acceleration Coulomb’s force provides centripetal acceleration Radiation emitted, E i – E f = hf Radiation emitted, E i – E f = hf

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Radii and Energy of Orbits The radii of the Bohr orbits are quantized The radii of the Bohr orbits are quantized n = 1, the orbit has the smallest radius, called the Bohr radius, a o n = 1, the orbit has the smallest radius, called the Bohr radius, a o a o = 0.0529 nm a o = 0.0529 nm A general expression for the radius of any orbit in a hydrogen atom is A general expression for the radius of any orbit in a hydrogen atom is r n = n 2 a o r n = n 2 a o The energy of any orbit is The energy of any orbit is E n = - 13.6 eV/ n 2 E n = - 13.6 eV/ n 2

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

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Generalized Equation For the Balmer series, n f = 2 For the Balmer series, n f = 2 For the Lyman series, n f = 1 For the Lyman series, n f = 1 Whenever an transition occurs between a state, n i to another state, n f (where n i > n f ), a photon is emitted Whenever an transition occurs between a state, n i to another state, n f (where n i > n f ), a photon is emitted The photon has a frequency f = (E i – E f )/h and wavelength λ The photon has a frequency f = (E i – E f )/h and wavelength λ

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Successes of the Bohr Theory Explained several features of the hydrogen spectrum Explained several features of the hydrogen spectrum Accounts for Balmer and other series Accounts for Balmer and other series Predicts a value for R H that agrees with the experimental value Predicts a value for R H that agrees with the experimental value Gives an expression for the radius of the atom Gives an expression for the radius of the atom Predicts energy levels of hydrogen Predicts energy levels of hydrogen Gives a model of what the atom looks like and how it behaves Gives a model of what the atom looks like and how it behaves Can be extended to “hydrogen-like” atoms Can be extended to “hydrogen-like” atoms Those with one electron Those with one electron Ze 2 needs to be substituted for e 2 in equations Ze 2 needs to be substituted for e 2 in equations Z is the atomic number of the element Z is the atomic number of the element

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

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Electron Clouds

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

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

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Quantum numbers for the hydrogen atom

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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. QUICK QUIZ 28.1

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Quantum Number Summary The values of n can range from 1 to in integer steps 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 ℓ can range from 0 to n-1 in integer steps The values of m ℓ can range from -ℓ to ℓ in integer steps The values of m ℓ can range from -ℓ to ℓ in integer steps

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

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

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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 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 This splitting is called fine structure Another quantum number, m s, called the spin magnetic quantum number, was introduced to explain the fine structure Another quantum number, m s, called the spin magnetic quantum number, was introduced to explain the fine structure

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

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

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The Periodic Table The outermost electrons are primarily responsible for the chemical properties of the atom 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 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 The electronic configuration of the elements explained by quantum numbers and Pauli’s Exclusion Principle explains the configuration

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

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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 m l and m s ? (c) What is the electron configuration in the ground state of zirconium?

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Characteristic X-Rays

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Explanation of Characteristic X-Rays The details of atomic structure can be used to explain 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 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 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 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 The transition is accompanied by the emission of a photon whose energy is equal to the difference between the two levels

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Atomic Transitions – Energy Levels An atom may have many possible 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 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 Only photons with energies corresponding to differences between energy levels can be absorbed

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Atomic Transitions – Stimulated Absorption The blue dots represent electrons The blue dots represent electrons When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level These higher levels are called excited states These higher levels are called excited states ΔE = hƒ = E 2 – E 1 ΔE = hƒ = E 2 – E 1 In general, ΔE can be the difference between any two energy levels In general, ΔE can be the difference between any two energy levels

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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 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 This process is called spontaneous emission

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Atomic Transitions – Stimulated Emission An atom is in an excited stated and a photon is incident on it 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 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 There are two emitted photons, the incident one and the emitted one The emitted photon is in exactly in phase with the incident photon The emitted photon is in exactly in phase with the incident photon

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Lasers To achieve laser action, three conditions must be met To achieve laser action, three conditions must be met The system must be in a state of population inversion The system must be in a state of population inversion The excited state of the system must be a metastable state 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 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 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 This is achieved by using reflecting mirrors

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Production of a Laser Beam

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Laser Beam – He Ne Example The energy level diagram for Ne The energy level diagram for Ne The mixture of helium and neon is confined to a glass tube sealed at the ends by mirrors 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 A high voltage applied causes electrons to sweep through the tube, producing excited states When the electron falls to E 2 in Ne, a 632.8 nm photon is emitted When the electron falls to E 2 in Ne, a 632.8 nm photon is emitted

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Conceptual questions 2.Does a light emitted by a neon sign constitute a continuous spectrum or only a few colors. 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? 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? 8. If matter has a wave nature, why is this not observable in our daily experiences? 9. Discuss consequences of the exclusion principle. 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? 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?

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

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

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Conductors When a voltage is applied to a conductor, the electrons accelerate and gain energy 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 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 For example, it takes very little energy for electrons to jump from the partially filled to one of the nearby empty states

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

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

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Semiconductors, cont Holes are empty states in the valence band created by electrons that have jumped to the conduction band 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 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 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 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

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

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Doping in Semiconductors Doping is the adding of impurities to a semiconductor Doping is the adding of impurities to a semiconductor Generally about 1 impurity atom per 10 7 semiconductor atoms Generally about 1 impurity atom per 10 7 semiconductor atoms Doping results in both the band structure and the resistivity being changed Doping results in both the band structure and the resistivity being changed

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

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

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

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