Because electrons are wave-like, they do not have trajectories but have cloud-like or blurry orbits {orbit, electron} {electron orbit}|. Electron repulsions also spread orbits.
energy
Electron energy has quanta, so electrons have minimum energy. Uncertainty principle requires that energy cannot be zero. Shell, orbital, spin-orbit interaction, and spin angular momentum contribute angular momentum and energy quanta to orbital electrons. Energy levels depend on angular momentum squared.
rotation
Rotations can be spins or orbits. Spins have orientation, frequency, and angular momentum. Orbits have orientation, frequency, angular momentum, and spin-orbit angular-momentum interactions. Spins and orbits have no net linear momentum, because motion is in all directions equally. Rotation is around point or line. Rotation defines plane perpendicular to rotation axis. Rotation axes have orientations in space.
rotation: compared to vibration
Vibrations are oscillations or waves. Vibration is between two extremes. Vibration along length has spatial orientation. Vibration around angle is in plane. Vibrations have frequency. Waves have motion direction. Wave vibration can be transverse to, or longitudinal with, motion direction.
angular momentum
Spins and orbits have angular momentum, because motion is around rotation axis. Orbitals have axis orientation.
width
Orbit width is same as atom diameter, by uncertainty principle. Electrons move all over orbit, by uncertainty principle, but most motion is near shell radius.
independent
Orbitals are orthogonal to all others, with no overlap or interaction, because electrons are fermions and cannot be together in same place (Pauli exclusion principle).
time
Orbitals do not change with time.
speed
Electron orbital speed is 600,000 meters per second and so is not relativistic.
large atoms
For large atoms, inner electrons shield outer electrons from atomic nucleus, so outer electrons have orbits farther from nucleus and have less kinetic energy than with no shielding.
Electrostatic force between nucleus and electron causes electrons to orbit atomic nuclei in main regions {shell, atom}| {atomic shell} at specific distances. Atoms have up to seven shells, from one to seven unit distances from nucleus.
energy
Electron kinetic energy E depends on reciprocal of shell number n squared: E = 1 / n^2. For first shell, n = 1 and E = 1/1 = 1 unit. For second shell, n = 2 and E = 1/4 = 0.25 unit. For third shell, n = 3 and E = 1/9 = 0.11 unit. For fourth shell, n = 4 and E = 1/16 = 0.07 unit, For fifth shell, n = 5 and E = 1/25 = 0.04 unit. For sixth shell, n = 6 and E = 1/36 = 0.03 unit. For seventh shell, n = 7 and E = 1/49 = 0.02 unit. Energy levels are closer together at higher shells, because force depends directly on reciprocal of radius squared.
K shell is 10^4 times atomic-nucleus radius. L shell is 1.5 times farther from nucleus than K shell. M shell is 1.67 times farther from nucleus than K shell. N shell is 1.75 times farther from nucleus than K shell.
electrons
Shells farther from nucleus can hold more electrons, because they allow more quanta combinations. Shells can hold 2 * n^2 electrons, where n is shell number. First shell {K shell} can hold two electrons. Second shell {L shell} can hold eight electrons. Third shell {M shell} can hold 18 electrons. Fourth shell {N shell} can hold 32 electrons. Fifth shell {O shell} can hold 50 electrons. Sixth shell {P shell} can hold 72 electrons. Seventh shell {Q shell} can hold 98 electrons.
shell
Atomic-electron orbits have different radii and energy levels {shell, orbital}. From lowest to highest potential energy, and highest to lowest kinetic energy, radius is 1, 2, 3, 4, 5, 6, and 7 units. Orbit radii increase linearly. Units differ for different atoms. Potential energy depends on radius, so quantum energy changes between shells are equal.
wavelength
Smallest orbit has circumference equal to one wavelength. Wavelength depends on radial force and resistance to force. Smallest orbit has highest frequency. Second-smallest orbit has circumference with wavelength equal to two original wavelengths. Second-smallest orbit has half original frequency. Third-smallest orbit has circumference with wavelength equal to three original wavelengths. Third-smallest orbit has one-third original frequency, and so on.
Atom electrons are in shells with orbit types {orbital}|. Orbital can have zero, one, or two electrons.
energy level
Electron orbitals have different energy levels. From lowest to highest, they are one 1s, one 2s, three 2p, one 3s, three 3p, one 4s, five 3d, three 4p, one 5s, five 4d, three 5p, one 6s, seven 4f, five 5d, three 6p, one 7s, seven 5f, five 6d, and three 7p. Number in parentheses is number of possible orbits with that energy. Before using f orbitals, orbital hybridization causes one electron to go into a d orbital.
electronic transitions
Electrons can jump from orbital to higher or lower orbital. Both orbitals must be anti-symmetric to allow angular-momentum conservation. Angular-momentum units are the same for orbiting and spinning.
angular momentum
Same-shell electrons can have different orbital angular momenta {orbital angular momentum, atom}. Angular momentum adds centrifugal force to electrostatic force. Orbital angular momentum has units h / (2 * pi), where h is Planck constant. First shell allows only 0 units. Second shell allows 0 and 1 units. Third shell allows 0, 1, and 2 units. Fourth shell allows 0, 1, 2, and 3 units, and so on.
shape
In shells, orbit shape determines orbital angular momentum. Spherical s orbital allows zero angular momentum. Double-ellipsoid p orbital allows zero or one angular-momentum unit. Quadruple-ellipsoid or double-ellipsoid/torus d orbital allows zero, one, or two angular-momentum units. Octuple-ellipsoid f orbital allows zero, one, two, or three angular momentum units, and so on.
First shell can only have spherical orbital, because it has minimum potential energy and cannot alter. Second shell can have spherical orbital and three oriented orbitals. Shells above first shell can have spherical orbital, three oriented orbitals, and five, seven, and so on, multiply oriented orbitals.
interactions
Orbital orientation and spin orientation interaction changes angular momentum by precession. Spin-axis orientation is always along z-axis. If orbital-axis orientation is along z-axis, no interaction happens, and total angular momentum does not change. If orbital-axis orientation is perpendicular to z-axis, torque interaction {spin-orbit interaction} effects add or subtract angular momentum units. Electric coupling forces cause torque that causes orbital to precess around orbital vertical axis. Spin-orientation interaction can change angular momentum by -3, -2, -1, 0, +1, +2, or +3 units.
Atom electrons are in orbitals {electron configuration}|. Orbitals {degenerate orbital} can have same energy levels.
Electrons fill orbitals from lowest energy to highest energy {Aufbau principle}. Before using f orbitals, orbital hybridization causes one electron to go into a d orbital.
Electrons tend to enter all shell orbitals before they fill any orbital with two opposite-spin electrons {Hund's rule} {Hund rule}. Hund's rule is true for small atoms, because it takes more energy to put two electrons into one orbital than into two different orbitals.
Fermions are electrons, neutrons, protons, and the like. Because fermions have half-unit spins, when identical fermions interchange, their wavefunctions become the negative of the other. Therefore, no two fermions can have same energy quanta {Pauli exclusion principle, fermion}|.
Hund's rule is true for small atoms. Other rules {Slater's rules} {Slater rules} apply for large atoms.
Orbital shape can be spherical {s orbital}, with zero crossing points. Because spheres are radially symmetric, with electron orbits in all directions and so filling space, spherical orbits have no net orientation, so no interaction with spin makes added angular momentum 0. There can only be one kind of spherical orbital, because it must have radial symmetry.
Orbital shape can be double ellipsoidal along straight line {p orbital}, with one crossing point and one rotation axis. p orbital has two elongated lobes along line with one crossing in middle. Double-ellipsoidal orbit can orient in three spatial directions. If axis is along z-axis, aligned with spin, added angular momentum is 0. If axis is along x-axis or y-axis, perpendicular to spin, added angular momentum is -1 or +1. There can only be three kinds of double-ellipsoidal orbital, because one axis can have only three independent spatial orientations, which fill space. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Orbital shape can be quadruple-ellipsoidal four-leaf clover {d orbital}, with two crossing points and two rotation axes. d orbitals have four elongated lobes, two each along both orthogonal lines, with two crossings in middle. Four-leaf-clover quadruple ellipsoidal orbit can align with x-axis and y-axis; between xy-axis, xz-axis, or yz-axis; or with z-axis, as double ellipsoid and torus. If with x and y or between xy, added angular momentum is -2 or +2, because both axes are perpendicular to z-axis. If between xz or yz, added angular momentum is -1 or +1, because one axis is perpendicular to z-axis. If with z, added angular momentum is 0, because axis aligns with spin axis. There can only be five kinds of quadruple-ellipsoidal orbital, because axes can have only five independent spatial orientations, which fill space. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Orbital shape can be octuple-ellipsoidal eight-lobed clover {f orbital}, with three crossing points and three rotation axes. f orbitals have six elongated lobes, with two each along three orthogonal lines, with three crossings in middle. Successive and more complex clover-leaf-shaped orbits can have 7, 9, or 11 distinct orientations. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Elementary particles have intrinsic angular momentum {spin, particle}| {particle, spin} {intrinsic angular momentum}. Spin conserves energy, momentum, and angular momentum.
axis
Particles always travel at light speed along a space-time motion line. Spin axis is parallel to motion line and is either counter-clockwise or clockwise around that space-time momentum vector.
classical mechanics
In classical mechanics, spin has linear continuous projections onto other axes (and orthogonal axes have no spin components). For example, if object spins around z-axis, observers can measure spin around xz-axis and yz-axis, but spin around x-axis and y-axis (both orthogonal to z-axis) is zero.
Fundamental particles are points (or strings or loops with negligible radius), and some have no mass, so fundamental-particle intrinsic angular momentum is not due to mass rotating at a distance around an axis. Classical mechanics cannot account for elementary-particle spin.
quantum mechanics
Elementary-particle spin is quantum-mechanical and special relativistic. To reconcile quantum mechanics and special relativity, quantum-mechanical-wavefunction components are matrices, not just numbers. Matrices have transformations that are equivalent to spin angular momentum. Reconciling quantum mechanics and general relativity requires that momentum (energy) and position (time) affect each other, so matrices have complex-number elements.
In quantum mechanics, observers can measure spin around any axis. Measurement of elementary-particle spin around any axis finds that spin is an angular-momentum quantum unit, either clockwise or counterclockwise around axis. For example, measuring independent-electron intrinsic angular momentum finds spin equals (0.5 * h) / (2 * pi), where h is Planck constant, which is 1/2 angular-momentum quantum unit. (Electron spin cannot be zero, because electrons have mass.) Spin counterclockwise around motion axis adds 1/2 angular momentum unit, so spin is +1/2. Spin clockwise around motion axis subtracts 1/2 angular-momentum unit, so spin is -1/2.
Measuring independent-photon intrinsic angular momentum finds spin equals (0.5 * h) / pi, where h is Planck constant, which is 1 angular-momentum quantum unit. (Photon spin cannot be zero, because photons have energy.) Spin counterclockwise around motion axis adds 1 angular-momentum unit, so spin is +1. Spin clockwise around motion axis subtracts 1 angular-momentum unit, so spin is -1.
spin: vectors and spinors
Real-number vectors have magnitude, one direction (component), and one orientation (in that direction): (a). Rotating real-number vectors 360 degrees makes the same vector, because vector direction and orientation return to original direction and orientation. Spinning real-number vectors any number of degrees makes the same vector, because vectors have no extensions in perpendicular directions. For example, turning a straight line around its axis keeps the same shape.
Complex-number vectors have magnitude, one direction (in local two-dimensional space), and one orientation (in that direction): (a + b*i). Rotating complex-number vectors 360 degrees makes the same vector, because vector direction and orientation return to original direction and orientation. Spinning complex-number vectors any number of degrees makes the same vector, because vectors have no extensions in perpendicular directions.
Spinors have two complex-number (or quaternion) components: (a + b*i, c + d*i). Spinors have magnitude, two directions, and one orientation that depends on which component goes first. Rotating spinors 360 degrees makes original direction but opposite orientation, like rotating around a Möbius strip, because parity changes. Spinor rotation differs from vector rotation because spinor rotation has phase effects. Spinning spinors any number of degrees makes a different spinor, because spinors have extensions in perpendicular directions.
spin: rotation
Fermion odd-half-integer-spin particles have different statistics than boson integer-spin particles. For bosons, spin and rotation are independent and add. For fermions, spin and rotation are dependent and multiply.
spin: symmetries
Elementary-particle intrinsic angular momentum is about wavefunction symmetries.
Spin-0 particles are scalars (not vectors). Scalars have no direction and so have same physics under any rotation. Because intrinsic angular momentum is zero, clockwise and counterclockwise have no meaning. Spheres have all symmetries: any-degree rotational symmetry, mirror symmetry, radial symmetry, and inversion symmetry. Turning a sphere through any angle, reflecting it through any plane through any diameter, and spinning around any axis results in the same shape and behavior. Around any axis and orientation, observers see no net spin, so spin-rotation interaction is zero. See Figure 1.
Spin-1 particles are vectors, with one symmetry axis. Spin-rotation interaction is non-zero, so observers see opposite spin (anti-symmetry) after 180-degree rotation. Turning a clockwise spinning sphere upside down reverses its orientation and changes clockwise to counterclockwise. Vectors have same physics under 360-degree (and 720-degree, 1080-degree, and so on) rotation (360-degree rotational symmetry). Turning the sphere upside down again puts it back to original orientation and clockwise spin. See Figure 2.
Spin-2 particles are tensors, with two symmetry axes. Spin 2 particles have mirror symmetry. Spin 2 has 90-degree anti-symmetry. Turning the sphere to right angle interchanges axes, so one axis keeps clockwise motion and one axis changes from clockwise to counter-clockwise, reversing the orientation. Two spin-rotation interactions are non-zero but symmetric, so flipping plane over returns system to same spin-rotation interactions. Spin-2 particles have same physics under 180-degree (and 360-degree, 540-degree, 720-degree, and so on) rotation. Turning a sphere spinning clockwise around an axis and clockwise around a perpendicular axis upside down changes clockwise to counterclockwise around both axes but also reverses both axes, so the sphere returns to its original state. See Figure 3.
Spin-1/2 particles are vectors, with two axes sharing one symmetry. Because they share one symmetry, spin-1/2 particles have different spin-rotation interactions than vector bosons, which have no shared symmetry and so spin 1. Spin-rotation interaction is perpendicular at 180-degree rotation, reversed at 360-degree rotation, and opposite perpendicular at 540-degree rotation, and original at 720-degree rotation. Spin 1/2 particles have 360-degree anti-symmetry, like rotating around a Möbius strip, changing parity. Turning a sphere spinning clockwise around an axis, clockwise around a perpendicular axis, and clockwise around a second perpendicular axis completely around changes clockwise to counterclockwise around two axes and reverses both axes, but changes clockwise to counterclockwise around the third axis, which has the same orientation, so the sphere reverses orientation. Spin 1/2 has 720-degree rotational symmetry. Turning the sphere completely around again changes clockwise to counterclockwise around two axes and reverses both axes, but changes counterclockwise to clockwise around the third axis, which has the same orientation, so the sphere returns to original state. See Figure 4.
spin: speculation
Perhaps, elementary-particle intrinsic angular momentum is imaginary-number mass rotating at imaginary-number radius around particle axis, through imaginary-number angle with imaginary-number angular velocity, perhaps through imaginary-number time. Multiplying imaginary numbers results in positive real-number momentum and energy. Hyperbolas have imaginary-number radii, because they have negative curvature. Hyperbolic-curve angles are imaginary-number angles: cos(i*A) = cosh(A) and e^A = cosh(A) + sinh(A), where A is real-number angle. Higgs field has imaginary mass. Imaginary-number time rotations make special-relativity Lorentz transformations. Using imaginary-number time can establish absolute general-relativity space-time.
spin: bosons and fermions
At high concentration and/or low temperature, with Heisenberg uncertainty, for thermal-equilibrium non-interacting bosons, exchange of two particles does not change wavefunction (Bose-Einstein statistics), because particle wavefunction product is commutative (symmetric rank-two tensor): f(a) * f(b) - f(b) * f(a). Combining two spins returns the system to original orientation: f(a) * f(b) = ((-1)^(2*spin)) * (f(b) * f(a)), where spin = +1 or -1. Relativistically applying a rotation operator in imaginary time to integer spin particles results in no Pauli exclusion principle. Bosons are indistinguishable. Only system states matter. It is incorrect to talk about first one and second one, or particle 1 and particle 2. Many bosons can have same energy, momentum, and angular momentum.
At high concentration and/or low temperature, with Heisenberg uncertainty, for thermal-equilibrium non-interacting fermions, exchange of two particles changes wavefunction (Fermi-Dirac statistics), because particle wavefunction product is anti-commutative (anti-symmetric rank-two tensor): f(a) * f(b) + f(b) * f(a). Combining two spins takes the system to opposite orientation: f(a) * f(b) = ((-1)^(2*spin)) * (f(b) * f(a)), where spin = +1/2 or -1/2. Relativistically applying a rotation operator in imaginary time to half-integer spin particles results in Pauli exclusion principle. Fermions are distinguishable. Only system states matter. It is correct to talk about first one and second one, or particle 1 and particle 2. Two particles can have same energy but must have different momentum and/or angular momentum.
Note: At low concentration and/or high temperature, without Heisenberg uncertainty, thermal-equilibrium non-interacting particles have Maxwell-Boltzmann statistics. Exchange of two particles does not matter, because wavefunction has no effect. Particles can have same energy and same or different momentum and angular momentum.
spin: measurement
To measure spin, experimenters must establish a spatial axis, and then measure angular momentum around that axis. (Experimenters cannot know electron trajectories, because electrons have wavefunctions.) Around any chosen axis, instruments measure spin as exactly +1/2 unit or exactly -1/2 unit. By uncertainty principle, instruments measuring spin simultaneously around axes perpendicular to that axis get +1/2 unit or -1/2 unit with equal probability, meaning that those spin measurements have 100% uncertainty.
Instruments cannot measure spin when two electrons are interacting, because system then includes measuring apparatus. Instruments measure after particle creation or interaction. After particle creation or interaction, instruments decohere wavefunction and so destroy particle system and make particles independent.
spin: measurement angle
For electrons (spin 1/2), if measuring axis is at angle A to a clockwise spin-vector (spin -1/2), the probability that the measurement will be spin -1/2 is (cos(A/2))^2. Perhaps, because spin-vector has two axes but shares one symmetry, it is like the spin-vector projects onto an angle A/2 axis as cos(A/2), and the angle A/2 axis vector projects onto the angle A measuring axis as cos(A/2), so the net projection is (cos(A/2))^2.
If a zero-spin state emits entangled electrons in opposite directions (conserving momentum and angular momentum), and one direction is measured at angle A and the other at angle B (with angle difference C), the both-same-spin probability is (sin(C/2))^2, and the each-opposite-spin probability is (cos(C/2))^2.
For photons (spin 1), if measuring axis is at angle A to a clockwise spin-vector (spin -1), the probability that the measurement will be spin -1 is (cos(A))^2. Perhaps, because spin-vector has one axis, it is like the spin-vector projects onto an angle A axis as cos(A) twice, so the net projection is (cos(A))^2.
If a zero-spin state emits entangled photons in opposite directions (conserving momentum and angular momentum), and one direction is measured at angle A and the other at angle B (with angle difference C), the both-same-spin probability is (sin(C))^2, and the each-opposite-spin probability is (cos(C))^2.
orbitals
Orbitals with two electrons typically have one electron with positive spin and one electron with negative spin {anti-symmetric spin state}, so net spin angular momentum is zero, and ground-state orbital is symmetric. In orbitals, paired electron spins {spin pair} cancel magnetic fields.
Outside energy can add spin angular momentum. The first excited orbital state has two electrons with positive spin or two electrons with negative spin {symmetric spin state}. Net spin angular momentum is 1, and excited-state orbital is anti-symmetric.
In orbitals, two electrons have probability 0.25 to have total spin 0 and 0.75 to have total spin 1.
In different orbitals, electrons can have same lower-energy spins. Two electrons enter two different orbitals before going into same orbital, because electrostatic repulsions are greater in energy than magnetic interactions, energy differences between orbitals are small, and repulsions between electrons in different orbitals are smaller than repulsions in same orbital.
Electron has spin and can precess {spin dragging}| or move in electric fields.
Low-temperature materials can behave like ice {spin ice}|. Magnetic poles can become unaligned.
Atom electrons have coupling {spin-orbit coupling} {Russell-Sanders coupling} {jj coupling} between orbit and spin magnetic fields.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225