8.S: Atomic Structure (Summary)
- Page ID
- 10321
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Terms
| angular momentum orbital quantum number (l) | quantum number associated with the orbital angular momentum of an electron in a hydrogen atom |
| angular momentum projection quantum number (m) | quantum number associated with the z-component of the orbital angular momentum of an electron in a hydrogen atom |
| atomic orbital | region in space that encloses a certain percentage (usually 90%) of the electron probability |
| Bohr magneton | magnetic moment of an electron, equal to \(\displaystyle 9.3×10^{−24}J/T\) or \(\displaystyle 5.8×10^{−5}eV/T\) |
| braking radiation | radiation produced by targeting metal with a high-energy electron beam (or radiation produced by the acceleration of any charged particle in a material) |
| chemical group | group of elements in the same column of the periodic table that possess similar chemical properties |
| coherent light | light that consists of photons of the same frequency and phase |
| covalent bond | chemical bond formed by the sharing of electrons between two atoms |
| electron configuration | representation of the state of electrons in an atom, such as \(\displaystyle 1s^22s^1\) for lithium |
| fine structure | detailed structure of atomic spectra produced by spin-orbit coupling |
| fluorescence | radiation produced by the excitation and subsequent, gradual de-excitation of an electron in an atom |
| hyperfine structure | detailed structure of atomic spectra produced by spin-orbit coupling |
| ionic bond | chemical bond formed by the electric attraction between two oppositely charged ions |
| laser | coherent light produced by a cascade of electron de-excitations |
| magnetic orbital quantum number | another term for the angular momentum projection quantum number |
| magnetogram | pictoral representation, or map, of the magnetic activity at the Sun’s surface |
| metastable state | state in which an electron “lingers” in an excited state |
| monochromatic | light that consists of photons with the same frequency |
| Moseley plot | plot of the atomic number versus the square root of X-ray frequency |
| Moseley’s law | relationship between the atomic number and X-ray photon frequency for X-ray production |
|
orbital magnetic dipole moment |
measure of the strength of the magnetic field produced by the orbital angular momentum of the electron |
| Pauli’s exclusion principle | no two electrons in an atom can have the same values for all four quantum numbers \(\displaystyle (n,l,m,ms)\) |
| population inversion | condition in which a majority of atoms contain electrons in a metastable state |
| principal quantum number (n) | quantum number associated with the total energy of an electron in a hydrogen atom |
| radial probability density function | function use to determine the probability of a electron to be found in a spatial interval in r |
| selection rules | rules that determine whether atomic transitions are allowed or forbidden (rare) |
| spin projection quantum number (\(\displaystyle m_s\)) | quantum number associated with the z-component of the spin angular momentum of an electron |
| spin quantum number (s) | quantum number associated with the spin angular momentum of an electron |
| spin-flip transitions | atomic transitions between states of an electron-proton system in which the magnetic moments are aligned and not aligned |
| spin-orbit coupling | interaction between the electron magnetic moment and the magnetic field produced by the orbital angular momentum of the electron |
| stimulated emission | when a photon of energy triggers an electron in a metastable state to drop in energy emitting an additional photon |
| transition metal | element that is located in the gap between the first two columns and the last six columns of the table of elements that contains electrons that fill the d subshell |
| valence electron | electron in the outer shell of an atom that participates in chemical bonding |
| Zeeman effect | splitting of energy levels by an external magnetic field |
Key Equation
| Orbital angular momentum | \(\displaystyle L=\sqrt{l(l+1)}ℏ\) |
| z-component of orbital angular momentum | \(\displaystyle L_z=mℏ\) |
| Radial probability density function | \(\displaystyle P(r)dr=∣ψ_{n00}∣^24πr^2dr\) |
| Spin angular momentum | \(\displaystyle S=\sqrt{s(s+1)}ℏ\) |
| z-component of spin angular momentum | \(\displaystyle S_z=m_sℏ\) |
| Electron spin magnetic moment | \(\displaystyle \vec{μ_s}=(\frac{e}{m_e})\vec{S}\) |
| Electron orbital magnetic dipole moment | \(\displaystyle \vec{μ}=−(\frac{e}{2m_e})\vec{L}\) |
| Potential energy associated with the magnetic interaction between the orbital magnetic dipole moment and an external magnetic field \(\displaystyle vec{B}\) | \(\displaystyle U(θ)=−μ_zB=mμ_BB\) |
| Maximum number of electrons in a subshell of a hydrogen atom | \(\displaystyle N=4l+2\) |
| Selection rule for atomic transitions in a hydrogen-like atom | \(\displaystyle Δl=±1\) |
| Moseley’s law for X-ray production | \(\displaystyle (Z−1)=constant\sqrt{f}\) |
Summary
8.1 The Hydrogen Atom
- A hydrogen atom can be described in terms of its wave function, probability density, total energy, and orbital angular momentum.
- The state of an electron in a hydrogen atom is specified by its quantum numbers (n, l, m).
- In contrast to the Bohr model of the atom, the Schrödinger model makes predictions based on probability statements.
- The quantum numbers of a hydrogen atom can be used to calculate important information about the atom.
8.2 Orbital Magnetic Dipole Moment of the Electron
- A hydrogen atom has magnetic properties because the motion of the electron acts as a current loop.
- The energy levels of a hydrogen atom associated with orbital angular momentum are split by an external magnetic field because the orbital angular magnetic moment interacts with the field.
- The quantum numbers of an electron in a hydrogen atom can be used to calculate the magnitude and direction of the orbital magnetic dipole moment of the atom.
8.3 Electron Spin
- The state of an electron in a hydrogen atom can be expressed in terms of five quantum numbers.
- The spin angular momentum quantum of an electron is = \(\displaystyle +½\). The spin angular momentum projection quantum number is \(\displaystyle ms =+½\) or \(\displaystyle −½\) (spin up or spin down).
- The fine and hyperfine structures of the hydrogen spectrum are explained by magnetic interactions within the atom.
8.4 The Exclusion Principle and the Periodic Table
- Pauli’s exclusion principle states that no two electrons in an atom can have all the same quantum numbers.
- The structure of the periodic table of elements can be explained in terms of the total energy, orbital angular momentum, and spin of electrons in an atom.
- The state of an atom can be expressed by its electron configuration, which describes the shells and subshells that are filled in the atom.
8.5 Atomic Spectra and X-rays
- Radiation is absorbed and emitted by atomic energy-level transitions.
- Quantum numbers can be used to estimate the energy, frequency, and wavelength of photons produced by atomic transitions.
- Atomic fluorescence occurs when an electron in an atom is excited several steps above the ground state by the absorption of a high-energy ultraviolet (UV) photon.
- X-ray photons are produced when a vacancy in an inner shell of an atom is filled by an electron from the outer shell of the atom.
- The frequency of X-ray radiation is related to the atomic number Z of an atom.
8.6 Lasers
- Laser light is coherent (monochromatic and “phase linked”) light.
- Laser light is produced by population inversion and subsequent de-excitation of electrons in a material (solid, liquid, or gas).
- CD and Blu-Ray players uses lasers to read digital information stored on discs.