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Physics 202 Professor P. Q. Hung 311B, Physics Building Physics 202 – p. 1/4 Atomic Physics The Hydrogen Atom: Spectrum Pre-Bohr Observation: Emission of light by atoms in a gas. Let the light pass through a narrow-slit aperture ⇒ Discrete set of lines of different colors (or wavelengths). Physics 202 – p. 2/4 Atomic Physics The Hydrogen Atom: Spectrum Pre-Bohr Observation: Emission of light by atoms in a gas. Let the light pass through a narrow-slit aperture ⇒ Discrete set of lines of different colors (or wavelengths). For hydrogen, Balmer empirically fitted the spectrum with n2 λ = (364.5 nm) n2 −4 n = 3, 4, 5, .... Physics 202 – p. 2/4 Atomic Physics The Hydrogen Atom: Spectrum Physics 202 – p. 3/4 Atomic Physics The Hydrogen Atom: Spectrum Physics 202 – p. 4/4 Atomic Physics The Hydrogen Atom: Spectrum One can invert the previous equation to give: 1 1 1 = R( − λ 22 n2 ) where R = 10.97373 µ m−1 is the Rydberg constant. Physics 202 – p. 5/4 Atomic Physics The Hydrogen Atom: Spectrum One can invert the previous equation to give: 1 1 1 = R( − λ 22 n2 ) where R = 10.97373 µ m−1 is the Rydberg constant. More general empirical formula which also applies to heavier atoms: The Rydberg-Ritz formula 1 1 2 1 = R Z ( − ) λ n22 n21 where n1 > n2 and Z = 1 for hydrogen. Physics 202 – p. 5/4 Atomic Physics The Hydrogen Atom: Spectrum; Special cases n2 = 2, n1 = n: Balmer series. This series covers the visible spectrum. The longest wavelength is when n = 3 ⇒ λ = 656.3 nm, on the reddish side. The shortest wavelength is when n = ∞ ⇒ λ = 364.6 nm, on the ultraviolet side. Physics 202 – p. 6/4 Atomic Physics The Hydrogen Atom: Spectrum; Special cases n2 = 1, n1 = 2, 3, ...: Lyman series. n2 λ = (91.13 nm) n2 −1 n = 2, 3, ... This is in the ultraviolet regime. The longest wavelength is when n = 2 ⇒ λ = 121.5 nm. The shortest wavelength is when n = ∞ ⇒ λ = 91.13 nm. Physics 202 – p. 7/4 Atomic Physics The Hydrogen Atom: Spectrum; Special cases n2 = 3, n1 = 4, 5, ...: Paschen series’ n2 λ = (820.14 nm) n2 −9 n = 4, 5, .... This is in the infrared region. The longest wavelength is when n = 4 ⇒ λ = 1874.6 nm. The shortest wavelength is when n = ∞ ⇒ λ = 820.14 nm. Physics 202 – p. 8/4 Atomic Physics The Hydrogen Atom: Spectrum Physics 202 – p. 9/4 Atomic Physics The Bohr Model Two crucial elements: The Rutherford’s picture of the atom and the Balmer formula (or more generally the Rydberg-Ritz formula). Physics 202 – p. 10/4 Atomic Physics The Bohr Model Two crucial elements: The Rutherford’s picture of the atom and the Balmer formula (or more generally the Rydberg-Ritz formula). Defect with the Rutherford’s picture: Classically, the electron in orbit around the nucleus will lose energy through radiation and will collapse into the nucleus. Something else: Bohr’s model in 1913. Physics 202 – p. 10/4 Atomic Physics The Bohr Model: Bohr’s 2 postulates Electrons move in certain stationary, non-radiating, circular orbits consistent with Coulomb’s law and Newton’s law and specified by the quantization of angular momentum L = mvr = n~ n: an integer. Physics 202 – p. 11/4 Atomic Physics The Bohr Model: Bohr’s 2 postulates Electrons move in certain stationary, non-radiating, circular orbits consistent with Coulomb’s law and Newton’s law and specified by the quantization of angular momentum L = mvr = n~ n: an integer. Radiation of frequency Ei −Ef f= h occurs when the electrom jumps from orbit i of energy Ei to orbit f of energy Ef . Physics 202 – p. 11/4 Atomic Physics The Bohr Atom Physics 202 – p. 12/4 Atomic Physics The Bohr Atom Physics 202 – p. 13/4 Atomic Physics The Bohr Model: Consequences of Bohr’s 2 postulates Circular orbit: Take an atom with a positively charged nucleus of charge Ze at the center and one electron going around it. The potential energy is U = −kZe2 /r. The total energy is E = 12 mv 2 + U . For a circular orbit, kZe2 /r2 = mv 2 /r ⇒ 12 mv 2 = kZe2 /2r ⇒ E = −kZe2 /2r. But how do we know what r is for various orbits? Physics 202 – p. 14/4 Atomic Physics The Bohr Model: Consequences of Bohr’s 2 postulates Orbital angular momentum is quantized: From kZe2 /r2 = mv 2 /r, one finds 2 ~2 rn = n mkZe2 = n2 aZ0 (1) ~2 a0 = mke2 = 0.0529 nm is the Bohr’s radius. Physics 202 – p. 15/4 Atomic Physics The Bohr Model: Consequences of Bohr’s 2 postulates How did Bohr derive the Rydberg-Ritz formula? Ei −Ef Using f = h and E = −kZe2 /2r, one finds kZe2 f = ( 2hr )( r1f − r1i ). Using (1) and f = c/λ, one finds the Rydberg-Ritz formula with mk 2 e4 R = 4πc~3 In very good agreement with the empirical determination of the Rydberg constant. Physics 202 – p. 16/4 Atomic Physics The Bohr Model: Consequences of Bohr’s 2 postulates How does one write the total energy of an electron in an orbit rn ? From E = −kZe2 /2r and Eq. (1), one finds En = −Z 2 En20 (2) where k 2 e4 m E0 = 2~2 ≈ 13.6 eV is the ionization or binding energy of the electron in the ground state of the hydrogen atom. It is the minimum energy to remove the electron from the hydrogen atom. Physics 202 – p. 17/4 Atomic Physics The Bohr Model: Consequences of Bohr’s 2 postulates For the hydrogen atom, one has En = − 13.6n2eV E1 = −13.6 eV is the ground state energy of the hydrogen atom. Caveat: Bohr’s model only applies to the case where there is one electron. It fails for atoms having several electrons! Physics 202 – p. 18/4 Atomic Physics The Bohr Atom: Consequences Physics 202 – p. 19/4 Atomic Physics The Bohr Atom: Example 1 The Lithium atom Li has Z = 3. Since it is neutral, it also has 3 electrons. Bohr’s model is not applicable. If two electrons are stripped away, one ends up with the ion Li2+ . What is the ionization energy of Li2+ ? Solution: Since one has only one electron left, the Bohr’s model can be used. The ionization energy is found by putting n = 1 and Z = 3 in Eq. (2). E1 = −32 (13.6 eV ) = −122 eV . So the minimum ionization energy needed to remove the remaining electron is 122 eV . Physics 202 – p. 20/4 Atomic Physics The Bohr Atom: Example 2 Use the Bohr’s model to calculate the quatum number n of the Earth in its orbit about the Sun. Solution: 1) Replace ke2 by GME MS in Eq. (1) with Z and using rn = 6 × 106 m. ~2 2 2) rn = n GM 2 MS ⇒ √ E ME rn GMS ~ = 2 × 1072 . A large number! 3) n = Lesson: For a very large quantum number n, we get back the results of classical physics! Physics 202 – p. 21/4 Atomic Physics Origin of angular momentum quantization Crucial ingredient of Bohr’s model: the quantization of angular momentum, L = mvr = n~. Origin? De Broglie’s wave-particle duality Physics 202 – p. 22/4 Atomic Physics Origin of angular momentum quantization Crucial ingredient of Bohr’s model: the quantization of angular momentum, L = mvr = n~. Origin? De Broglie’s wave-particle duality Electrons behave like waves. On an orbit, standing wave phenomena ⇒ condition for standing wave: 2πr = nλ λ = hp p = mv h ⇒ L = mvr = n 2π . Physics 202 – p. 22/4 Atomic Physics Origin of angular momentum quantization Physics 202 – p. 23/4 Atomic Physics Origin of angular momentum quantization Shortcomings of Bohr’s model: No prediction for relative intensities of spectral lines. Physics 202 – p. 24/4 Atomic Physics Origin of angular momentum quantization Shortcomings of Bohr’s model: No prediction for relative intensities of spectral lines. Cannot describe complex atoms. Physics 202 – p. 24/4 Atomic Physics Origin of angular momentum quantization Shortcomings of Bohr’s model: No prediction for relative intensities of spectral lines. Cannot describe complex atoms. Lack of theoretical foundation. Physics 202 – p. 24/4 Atomic Physics Origin of angular momentum quantization Better picture: wave mechanics or quantum mechanics: Physics 202 – p. 25/4 Atomic Physics Origin of angular momentum quantization Better picture: wave mechanics or quantum mechanics: Propagation of matter waves is governed by the Schroedinger equation. Physics 202 – p. 25/4 Atomic Physics Origin of angular momentum quantization Better picture: wave mechanics or quantum mechanics: Propagation of matter waves is governed by the Schroedinger equation. Matter wave is described by a wave function whose absolute square represents the probability to find the particle within a range of position (roughly speaking) Physics 202 – p. 25/4 Atomic Physics Quantum Mechanics In Bohr’s model, only one quantum number: n In quantum mechanics: Principal quantum number n. Physics 202 – p. 26/4 Atomic Physics Quantum Mechanics In Bohr’s model, only one quantum number: n In quantum mechanics: Principal quantum number n. Orbital quantum number l. l = 0, 1, 2, .., (n − 1). Magnitude of angular momentum: p L = l(l + 1)~. Physics 202 – p. 26/4 Atomic Physics Quantum Mechanics Magnetic quantum number ml . ml = −l, .., −1, 0, 1, .., l. Lz = ml ~. Physics 202 – p. 27/4 Atomic Physics Quantum Mechanics Magnetic quantum number ml . ml = −l, .., −1, 0, 1, .., l. Lz = ml ~. Spin quantum number ms . ms = ± 12 . Physics 202 – p. 27/4 Atomic Physics Quantum mechanics: Hydrogen atom Physics 202 – p. 28/4 Atomic Physics Pauli Exclusion Principle Pauli Exclusion principle: No two electrons can have the same set of values for n, l, ml , ms . Physics 202 – p. 29/4 Atomic Physics Pauli Exclusion Principle Pauli Exclusion principle: No two electrons can have the same set of values for n, l, ml , ms . An atom is characterized by energy levels, e.g. n = 1, l = 0 is the lowest energy level or ground state. Physics 202 – p. 29/4 Atomic Physics Pauli Exclusion Principle Pauli Exclusion principle: No two electrons can have the same set of values for n, l, ml , ms . An atom is characterized by energy levels, e.g. n = 1, l = 0 is the lowest energy level or ground state. Ground state of an atom: the electrons are in the lowest available energy levels consistent with the Pauli exclusion principle. Physics 202 – p. 29/4 Atomic Physics Pauli Exclusion Principle Shell: levels with n. K,L,M,N,O,... Physics 202 – p. 30/4 Atomic Physics Pauli Exclusion Principle Shell: levels with n. K,L,M,N,O,... Subshell: For a given shell, levels with different l, ml , ms . For l, one usually call: s, p, d, f, g, h,... Physics 202 – p. 30/4 Atomic Physics Pauli Exclusion Principle Shell: levels with n. K,L,M,N,O,... Subshell: For a given shell, levels with different l, ml , ms . For l, one usually call: s, p, d, f, g, h,... The Pauli Exclusion principle allows only a maximum number of electrons that can fit into an energy level or subshell. Physics 202 – p. 30/4 Atomic Physics Pauli Exclusion Principle For each l, there can be 2(2l + 1) electrons that can fit. (This is because of the number of combinations of ml and ms .) Physics 202 – p. 31/4 Atomic Physics Pauli Exclusion Principle For each l, there can be 2(2l + 1) electrons that can fit. (This is because of the number of combinations of ml and ms .) Notation for the ground state of an atom: n lnumber of electrons . Hydrogen(H): 1 electron,1 s1 . Helium (He): 2 electrons, 1 s2 . Lithium (Li): 3 electrons, 1 s2 2 s1 . Beryllium (Be): 4 electrons, 1 s2 2 s2 etc.. Physics 202 – p. 31/4 Atomic Physics Pauli Exclusion Principle Physics 202 – p. 32/4 Atomic Physics Pauli Exclusion Principle Physics 202 – p. 33/4 Atomic Physics Pauli Exclusion Principle Physics 202 – p. 34/4 Atomic Physics Pauli Exclusion Principle Physics 202 – p. 35/4 Atomic Physics Atomic Radiation Two types of X-rays: bremsstrahlung and characteristic X-rays bremsstrahlung: radiation from energetic electrons which decelerate upon hitting a target. Continuous spectrum. Physics 202 – p. 36/4 Atomic Physics Atomic Radiation Two types of X-rays: bremsstrahlung and characteristic X-rays bremsstrahlung: radiation from energetic electrons which decelerate upon hitting a target. Continuous spectrum. characteristic X-rays: An incoming electron knocks off say a K-shell electron. That void is filled immediately by electrons from outer shells ⇒ X-rays. To do that the incoming electron must have at least (Z−1)2 EK = −13.6 eV 12 . Physics 202 – p. 36/4 Atomic Physics Atomic Radiation Notice that the energy of the electron in the (Z−1)2 other shells is En = −13.6 eV n2 . Physics 202 – p. 37/4 Atomic Physics Atomic Radiation Notice that the energy of the electron in the (Z−1)2 other shells is En = −13.6 eV n2 . Characteristic peaks, e.g. Kα , Kβ . Read the book concerning lasers, etc... Physics 202 – p. 37/4 Atomic Physics Atomic Radiation Physics 202 – p. 38/4 Atomic Physics Atomic Radiation Physics 202 – p. 39/4 Atomic Physics Atomic Radiation Physics 202 – p. 40/4 Atomic Physics Atomic Radiation Physics 202 – p. 41/4