#### Transcript Chapter 42

Chapter 42 Atomic Physics Application of Quantum Mechanics to Atomic Physics A large portion of the chapter focuses on the hydrogen atom. Reasons for the importance of the hydrogen atom include. The hydrogen atom is the only atomic system that can be solved exactly. Much of what was learned in the twentieth century about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+. The hydrogen atom is an ideal system for performing precision tests of theory against experiment. Also for improving our understanding of atomic structure. The quantum numbers that are used to characterize the allowed states of hydrogen can also be used to investigate more complex atoms. This allows us to understand the periodic table. The basic ideas about atomic structure must be well understood before we attempt to deal with the complexities of molecular structures and the electronic structure of solids. Introduction Other Ideas in Atomic Physics The full mathematical solution of the Schrödinger equation applied to the hydrogen atom gives a complete and beautiful description of the atom’s properties. Quantum numbers are used to characterize various allowed states in the atom. The quantum numbers have physical significance. Certain quantum states are affected by a magnetic field. The exclusion principle is important for understanding the properties of multielectron atoms and the arrangement of the elements in the periodic table. Introduction Atomic Spectra All objects emit thermal radiation characterized by a continuous distribution of wavelengths. A discrete line spectrum is observed when a low-pressure gas is subjected to an electric discharge. Observation and analysis of these spectral lines is called emission spectroscopy. The simplest line spectrum is that for atomic hydrogen. Section 42.1 Emission Spectra Examples Section 42.1 Uniqueness of Atomic Spectra Other atoms exhibit completely different line spectra. Because no two elements have the same line spectrum, the phenomena represents a practical and sensitive technique for identifying the elements present in unknown samples. Section 42.1 Absorption Spectroscopy An absorption spectrum is obtained by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed. The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source. Section 42.1 Balmer Series In 1885, Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen. Hα is red, λ = 656.3 nm Hβ is green, λ = 486.1 nm Hγ is blue, λ = 434.1 nm Hδ is violet, λ = 410.2 nm Section 42.1 Emission Spectrum of Hydrogen – Equation The wavelengths of hydrogen’s spectral lines can be found from 1 1 1 RH 2 2 λ 2 n RH is the Rydberg constant RH = 1.097 373 2 x 107 m-1 n is an integer, n = 3, 4, 5,… The spectral lines correspond to different values of n. Values of n from 3 to 6 give the four visible lines. Values of n > 6 give the ultraviolet lines in the Balmer series. The series limit is the shortest wavelength in the series and corresponds to n →∞. Section 42.1 Other Hydrogen Series Other series were also discovered and their wavelengths can be calculated: Lyman series: 1 1 RH 1 2 n 2, 3, 4, λ n Paschen series: 1 1 1 RH 2 2 n 4 , 5 , 6 , λ 3 n Brackett series: 1 1 1 RH 2 2 n 5 , 6 , 7 , λ 4 n Section 42.1 Joseph John Thomson 1856 – 1940 English physicist Received Nobel Prize in 1906 Usually considered the discoverer of the electron Worked with the deflection of cathode rays in an electric field Opened up the field of subatomic particles Section 42.2 Early Models of the Atom, Thomson’s J. J. Thomson established the charge to mass ratio for electrons. His model of the atom A volume of positive charge Electrons embedded throughout the volume The atom as a whole would be electrically neutral. Section 42.2 Rutherford’s Thin Foil Experiment Experiments done in 1911 A beam of positively charged alpha particles hit and are scattered from a thin foil target. Large deflections could not be explained by Thomson’s model. Section 42.2 Early Models of the Atom, Rutherford’s Rutherford Planetary model Based on results of thin foil experiments Positive charge is concentrated in the center of the atom, called the nucleus Electrons orbit the nucleus like planets orbit the sun Section 42.2 Difficulties with the Rutherford Model Atoms emit certain discrete characteristic frequencies of electromagnetic radiation. The Rutherford model is unable to explain this phenomena. Rutherford’s electrons are undergoing a centripetal acceleration. It should radiate electromagnetic waves of the same frequency. The radius should steadily decrease as this radiation is given off. The electron should eventually spiral into the nucleus. It doesn’t Section 42.2 Niels Bohr 1885 – 1962 Danish physicist An active participant in the early development of quantum mechanics Headed the Institute for Advanced Studies in Copenhagen Awarded the 1922 Nobel Prize in physics For structure of atoms and the radiation emanating from them Section 42.3 The Bohr Theory of Hydrogen In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory. His model includes both classical and non-classical ideas. He applied Planck’s ideas of quantized energy levels to Rutherford’s orbiting electrons. This model is now considered obsolete. It has been replaced by a probabilistic quantum-mechanical theory. The model can still be used to develop ideas of energy quantization and angular momentum quantization as applied to atomic-sized systems. Section 42.3 Bohr’s Postulates for Hydrogen, 1 The electron moves in circular orbits around the proton under the electric force of attraction. The Coulomb force produces the centripetal acceleration. Section 42.3 Bohr’s Postulates, 2 Only certain electron orbits are stable. Bohr called these stationary states. These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation, even though it is accelerating. Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion. This representation claims the centripetally accelerated electron does not continuously emit energy and therefore does not eventually spiral into the nucleus. Section 42.3 Bohr’s Postulates, 3 Radiation is emitted by the atom when the electron makes a transition from a more energetic initial stationary state to a lower-energy stationary state. The transition cannot be treated classically. The frequency emitted in the transition is related to the change in the atom’s energy. The frequency is independent of frequency of the electron’s orbital motion. The frequency of the emitted radiation is given by Ei – Ef = hƒ If a photon is absorbed, the electron moves to a higher energy level. Section 42.3 Bohr’s Postulates, 4 The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum. The allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of h Section 42.3 Bohr’s Postulates, Notes These postulates were a mixture of established principles and completely new ideas. Postulate 1 – from classical mechanics Treats the electron in orbit around the nucleus in the same way we treat a planet in orbit around a star. Postulate 2 – new idea It was completely at odds with the understanding of electromagnetism at the time. Postulate 3 – Principle of Conservation of Energy Postulate 4 – new idea Had no basis in classical physics Section 42.3 Mathematics of Bohr’s Assumptions and Results Electron’s orbital angular momentum mevr = nħ where n = 1, 2, 3,… The total energy of the atom is 1 e2 2 E K U mev ke 2 r The electron is modeled as a particle in uniform circular motion, so the electrical force on the electron equals the product of its mass and its centripetal acceleration. 2 k e e The total energy can also be expressed as E . 2r The total energy is negative, indicating a bound electron-proton system. Section 42.3 Bohr Radius The radii of the Bohr orbits are quantized n2 2 rn mekee 2 n 1, 2, 3, This shows that the radii of the allowed orbits have discrete values—they are quantized. This result is based on the assumption that the electron can exist only in certain allowed orbits determined by n (Bohr’s postulate 4). When n = 1, the orbit has the smallest radius, called the Bohr radius, ao ao = 0.052 9 nm Section 42.3 Radii and Energy of Orbits A general expression for the radius of any orbit in a hydrogen atom is rn = n2ao The energy of any orbit is ke e 2 1 En n 1, 2,3, 2ao n2 This becomes En = - 13.606 eV / n2 Section 42.3 Specific Energy Levels Only energies satisfying the energy equation are allowed. The lowest energy state is called the ground state. This corresponds to n = 1 with E = –13.606 eV The ionization energy is the energy needed to completely remove the electron from the ground state in the atom. The ionization energy for hydrogen is 13.6 eV. Section 42.3 Energy Level Diagram Quantum numbers are given on the left and energies on the right. The uppermost level, E = 0, represents the state for which the electron is removed from the atom. Adding more energy than this amount ionizes the atom. Section 42.3 Frequency and Wavelength of Emitted Photons The frequency of the photon emitted when the electron makes a transition from an outer orbit to an inner orbit is Ei Ef kee 2 1 1 ƒ 2 2 h 2aoh nf ni It is convenient to look at the wavelength instead. The wavelengths are found by 1 1 1 ƒ k ee 2 1 1 R 2 H 2 2 2 λ c 2aohc nf ni nf ni The value of RH from Bohr’s analysis is in excellent agreement with the experimental value. Section 42.3 Extension to Other Atoms Bohr extended his model for hydrogen to other elements in which all but one electron had been removed. rn n 2 ao Z k ee 2 Z 2 En 2 n 1, 2, 3, 2ao n Z is the atomic number of the element and is the number of protons in the nucleus. Section 42.3 Difficulties with the Bohr Model Improved spectroscopic techniques found that many of the spectral lines of hydrogen were not single lines. Each “line” was actually a group of lines spaced very close together. Certain single spectral lines split into three closely spaced lines when the atoms were placed in a magnetic field. The Bohr model cannot account for the spectra of more complex atoms. Scattering experiments show that the electron in a hydrogen atom does not move in a flat circle, but rather that the atom is spherical. The ground-state angular momentum of the atom is zero. These deviations from the model led to modifications in the theory and ultimately to a replacement theory. Section 42.3 Bohr’s Correspondence Principle Bohr’s correspondence principle states that quantum physics agrees with classical physics when the differences between quantized levels become vanishingly small. Similar to having Newtonian mechanics be a special case of relativistic mechanics when v << c. Section 42.3 The Quantum Model of the Hydrogen Atom The difficulties with the Bohr model are removed when a full quantum model involving the Schrödinger equation is used to describe the hydrogen atom. The potential energy function for the hydrogen atom is e2 U (r ) k e r ke is the Coulomb constant r is the radial distance from the proton to the electron The proton is situated at r = 0 Section 42.4 Quantum Model, cont. The formal procedure to solve the hydrogen atom is to substitute U(r) into the Schrödinger equation, find the appropriate solutions to the equations, and apply boundary conditions. Because it is a three-dimensional problem, it is easier to solve if the rectangular coordinates are converted to spherical polar coordinates. Section 42.4 Quantum Model, final ψ(x, y, z) is converted to ψ(r, θ, φ) Then, the space variables can be separated: ψ(r, θ, φ) = R(r), ƒ(θ), g(φ) When the full set of boundary conditions are applied, we are led to three different quantum numbers for each allowed state. The three different quantum numbers are restricted to integer values. They correspond to three degrees of freedom. Three space dimensions Section 42.4 Principal Quantum Number The first quantum number is associated with the radial function R(r). It is called the principal quantum number. It is symbolized by n. The potential energy function depends only on the radial coordinate r. The energies of the allowed states in the hydrogen atom are the same En values found from the Bohr theory. Section 42.4 Orbital and Orbital Magnetic Quantum Numbers The orbital quantum number is symbolized by ℓ. It is associated with the orbital angular momentum of the electron. It is an integer. The orbital magnetic quantum number is symbolized by mℓ. It is also associated with the angular orbital momentum of the electron and is an integer. Section 42.4 Quantum Numbers, Summary of Allowed Values The values of n are integers that can range from 1 to The values of ℓ are integers that can range from 0 to n - 1. The values of mℓ are integers that can range from –ℓ to ℓ. Example: If n = 1, then only ℓ = 0 and mℓ = 0 are permitted If n = 2, then ℓ = 0 or 1 If ℓ = 0 then mℓ = 0 If ℓ = 1 then mℓ may be –1, 0, or 1 Section 42.4 Quantum Numbers, Summary Table Section 42.4 Shells Historically, all states having the same principle quantum number are said to form a shell. Shells are identified by letters K, L, M,… for which n = 1, 2, 3, … All states having the same values of n and ℓ are said to form a subshell. The letters s, p, d, f, g, h, .. are used to designate the subshells for which ℓ = 0, 1, 2, 3,… Section 42.4 Shell Notation, Summary Table Section 42.4 Subshell Notation, Summary Table Section 42.4 Wave Functions for Hydrogen The simplest wave function for hydrogen is the one that describes the 1s state and is designated ψ1s(r). ψ1s (r ) 1 π ao3 e r ao As ψ1s(r) approaches zero, r approaches and is normalized as presented. ψ1s(r) is also spherically symmetric This symmetry exists for all s states. Section 42.5 Probability Density The probability density for the 1s state is ψ1s 2 1 2r ao 3 e πao The radial probability density function P(r) is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr. Section 42.5 Radial Probability Density A spherical shell of radius r and thickness dr has a volume of 4πr2 dr The radial probability function is P(r) = 4πr2 |ψ|2 Section 42.5 P(r) for 1s State of Hydrogen The radial probability density function for the hydrogen atom in its ground state is 4r 2 2r ao P1s (r ) 3 e ao The peak indicates the most probable location. The peak occurs at the Bohr radius. The average value of r for the ground state of hydrogen is 3/2 ao. Section 42.5 Electron Clouds According to quantum mechanics, the atom has no sharply defined boundary as suggested by the Bohr theory. The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud. This shows the probability density as a function of position in the xy plane. The darkest area, r = ao, corresponds to the most probable region. Section 42.5 Wave Function of the 2s state The next-simplest wave function for the hydrogen atom is for the 2s state. n = 2; ℓ = 0 The normalized wave function is 1 1 ψ2s (r ) 4 2π ao 3 2 r r 2 e ao 2 ao ψ2s depends only on r and is spherically symmetric. Section 42.5 Comparison of 1s and 2s States The plot of the radial probability density for the 2s state has two peaks. The highest value of P corresponds to the most probable value. In this case, r 5ao Section 42.5 Physical Interpretation of ℓ The magnitude of the angular momentum of an electron moving in a circle of radius r is L = mevr The direction of the angular momentum vector is perpendicular to the plane of the circle. The direction is given by the right hand rule. In the Bohr model, the angular momentum of the electron is restricted to multiples of . This incorrectly predicts that the ground state of hydrogen has one unit of angular momentum. If L is taken to be zero in the Bohr model, the electron must be pictured as a particle oscillating along a straight line through the nucleus. Section 42.6 Physical Interpretation of ℓ, cont. Quantum mechanics resolves these difficulties found in the Bohr model. According to quantum mechanics, an atom in a state whose principle quantum number is n can take on the following discrete values of the magnitude of the orbital angular momentum: L 1 0,1, 2, , n 1 L can equal zero, which causes great difficulty when attempting to apply classical mechanics to this system. In the quantum mechanical representation, the electron cloud for the L = 0 state is spherically symmetric and has no fundamental rotation axis. Section 42.6 Physical Interpretation of mℓ The atom possesses an orbital angular momentum. There is a sense of rotation of the electron around the nucleus, so that a magnetic moment is present due to this angular momentum. There are distinct directions allowed for the magnetic moment vector with respect to the magnetic field vector . Because the magnetic moment μ of the atom can be related to the angular momentum vector, L, the discrete direction of the magnetic moment vector translates into the fact that the direction of the angular momentum is quantized. Therefore, Lz, the projection of L along the z axis, can have only discrete values. Section 42.6 Physical Interpretation of mℓ, cont. The orbital magnetic quantum number mℓ specifies the allowed values of the z component of orbital angular momentum. Lz = mℓ The quantization of the possible orientations of L with respect to an external magnetic field is often referred to as space quantization. L does not point in a specific direction Even though its z-component is fixed Knowing all the components is inconsistent with the uncertainty principle Imagine that L must lie anywhere on the surface of a cone that makes an angle θ with the z axis. Section 42.6 Physical Interpretation of mℓ, final θ is also quantized Its values are specified through cos θ Lz L m 1 Section 42.6 Zeeman Effect The Zeeman effect is the splitting of spectral lines in a strong magnetic field. In this case the upper level, with ℓ = 1, splits into three different levels corresponding to the three different directions of µ. Section 42.6 Spin Quantum Number ms Electron spin does not come from the Schrödinger equation. Additional quantum states can be explained by requiring a fourth quantum number for each state. This fourth quantum number is the spin magnetic quantum number ms. Section 42.6 Electron Spins Only two directions exist for electron spins. The electron can have spin up (a) or spin down (b). In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases. Section 42.6 Electron Spins, cont. The concept of a spinning electron is conceptually useful. The electron is a point particle, without any spatial extent. Therefore the electron cannot be considered to be actually spinning. The experimental evidence supports the electron having some intrinsic angular momentum that can be described by ms. Dirac showed this results from the relativistic properties of the electron. Section 42.6 Spin Angular Momentum The total angular momentum of a particular electron state contains both an orbital contribution L and a spin contribution.S Electron spin can be described by a single quantum number s, whose value can only be s = ½ . The spin angular momentum of the electron never changes. The magnitude of the spin angular momentum is S s(s 1) 3 2 The spin angular momentum can have two orientations relative to a z axis, specified by the spin quantum number ms = ± ½ . ms = + ½ corresponds to the spin up case. ms = - ½ corresponds to the spin down case. Section 42.6 Spin Angular Momentum, cont. The z component of spin angular momentum is Sz = msh = ± ½ h Spin angular moment is quantized. Section 42.6 Spin Magnetic Moment The spin magnetic moment µspin is related to the spin angular momentum by μspin e S me The z component of the spin magnetic moment can have values μspin , z e 2me Section 42.6 Quantum States There are eight quantum states corresponding to n = 2. These states depend on the addition of the possible values of ms. Table 42.4 summarizes these states. Section 42.6 Quantum Numbers for n = 2 State of Hydrogen Section 42.6 Wolfgang Pauli 1900 – 1958 Austrian physicist Important review article on relativity At age 21 Discovery of the exclusion principle Explanation of the connection between particle spin and statistics Relativistic quantum electrodynamics Neutrino hypothesis Hypotheses of nuclear spin Section 42.7 The Exclusion Principle The four quantum numbers discussed so far can be used to describe all the electronic states of an atom regardless of the number of electrons in its structure. The exclusion principle states that no two electrons can ever be in the same quantum state. Therefore, no two electrons in the same atom can have the same set of quantum numbers. If the exclusion principle was not valid, an atom could radiate energy until every electron was in the lowest possible energy state and the chemical nature of the elements would be modified. Section 42.7 Filling Subshells The electronic structure of complex atoms can be viewed as a succession of filled levels increasing in energy. Once a subshell is filled, the next electron goes into the lowest-energy vacant state. If the atom were not in the lowest-energy state available to it, it would radiate energy until it reached this state. Section 42.7 Orbitals An orbital is defined as the atomic state characterized by the quantum numbers n, ℓ and mℓ. From the exclusion principle, it can be seen that only two electrons can be present in any orbital. One electron will have spin up and one spin down. Each orbital is limited to two electrons, the number of electrons that can occupy the various shells is also limited. Section 42.7 Allowed Quantum States, Example with n = 3 In general, each shell can accommodate up to 2n2 electrons. Section 42.7 Hund’s Rule Hund’s Rule states that when an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins. Some exceptions to the rule occur in elements having subshells that are close to being filled or half-filled. Section 42.7 Configuration of Some Electron States The filling of electron states must obey both the exclusion principle and Hund’s rule. Section 42.7 Periodic Table Dmitri Mendeleev made an early attempt at finding some order among the chemical elements. He arranged the elements according to their atomic masses and chemical similarities. The first table contained many blank spaces and he stated that the gaps were there only because the elements had not yet been discovered. By noting the columns in which some missing elements should be located, he was able to make rough predictions about their chemical properties. Within 20 years of the predictions, most of the elements were discovered. The elements in the periodic table are arranged so that all those in a column have similar chemical properties. Section 42.7 Periodic Table, Explained The chemical behavior of an element depends on the outermost shell that contains electrons. For example, the inert (or noble) gases (last column) have filled subshells and a wide energy gap occurs between the filled shell and the next available shell. Section 42.7 Hydrogen Energy Level Diagram Revisited The allowed values of ℓ are separated horizontally. Transitions in which ℓ does not change are very unlikely to occur and are called forbidden transitions. Such transitions actually can occur, but their probability is very low compared to allowed transitions. Section 42.8 Selection Rules The selection rules for allowed transitions are Δℓ = ±1 Δmℓ = 0, ±1 The angular momentum of the atom-photon system must be conserved. Therefore, the photon involved in the process must carry angular momentum. The photon has angular momentum equivalent to that of a particle with spin 1. A photon has energy, linear momentum and angular momentum. Section 42.8 Multielectron Atoms For multielectron atoms, the positive nuclear charge Ze is largely shielded by the negative charge of the inner shell electrons. The outer electrons interact with a net charge that is smaller than the nuclear charge. Allowed energies are En 2 13.6 eV Zeff n2 Zeff depends on n and ℓ Section 42.8 X-Ray Spectra These x-rays are a result of the slowing down of high energy electrons as they strike a metal target. The kinetic energy lost can be anywhere from 0 to all of the kinetic energy of the electron. The continuous spectrum is called bremsstrahlung, the German word for “braking radiation”. Section 42.8 X-Ray Spectra, cont. The discrete lines are called characteristic x-rays. These are created when A bombarding electron collides with a target atom. The electron removes an inner-shell electron from orbit. An electron from a higher orbit drops down to fill the vacancy. The photon emitted during this transition has an energy equal to the energy difference between the levels. Typically, the energy is greater than 1000 eV. The emitted photons have wavelengths in the range of 0.01 nm to 1 nm. Section 42.8 Moseley Plot Henry G. J. Moseley plotted the values of atoms as shown. λ is the wavelength of the Kα line of each element. The Kα line refers to the photon emitted when an electron falls from the L to the K shell. From this plot, Moseley developed a periodic table in agreement with the one based on chemical properties. Section 42.8 Stimulated Absorption When a photon has energy hƒ equal to the difference in energy levels, it can be absorbed by the atom. This is called stimulated absorption because the photon stimulates the atom to make the upward transition. At ordinary temperatures, most of the atoms in a sample are in the ground state. The absorption of the photon causes some of the atoms to be raised to excited states. Section 42.9 Spontaneous Emission Once an atom is in an excited state, the excited atom can make a transition to a lower energy level. This process is known as spontaneous emission. Because it happens naturally Section 42.9 Stimulated Emission In addition to spontaneous emission, stimulated emission occurs. Stimulated emission may occur when the excited state is a metastable state. A metastable state is a state whose lifetime is much longer than the typical 10-8 s. An incident photon can cause the atom to return to the ground state without being absorbed. Therefore, you have two photons with identical energy, the emitted photon and the incident photon. They both are in phase and travel in the same direction. Section 42.9 Lasers – Properties of Laser Light Laser light is coherent. The individual rays in a laser beam maintain a fixed phase relationship with each other. Laser light is monochromatic. The light has a very narrow range of wavelengths. Laser light has a small angle of divergence. The beam spreads out very little, even over long distances. Section 42.10 Lasers – Operation It is equally probable that an incident photon would cause atomic transitions upward or downward. Stimulated absorption or stimulated emission If a situation can be caused where there are more electrons in excited states than in the ground state, a net emission of photons can result. This condition is called population inversion. The photons can stimulate other atoms to emit photons in a chain of similar processes. The many photons produced in this manner are the source of the intense, coherent light in a laser. Section 42.10 Conditions for Build-Up of Photons The system must be in a state of population inversion. The excited state of the system must be a metastable state. In this case, the population inversion can be established and stimulated emission is likely to occur before spontaneous emission. The emitted photons must be confined in the system long enough to enable them to stimulate further emission. This is achieved by using reflecting mirrors. Section 42.10 Laser Design – Schematic Energy-Level Diagram for Neon in a Helium-Neon Laser The atoms emit 632.8-nm photons through stimulated emission. The transition is E3* to E2 * indicates a metastable state Section 42.10 Laser Applications Applications include: Medical and surgical procedures Precision surveying and length measurements Precision cutting of metals and other materials Telephone communications Biological and medical research Section 42.10