Atomic Flashcards

Question

What is L-S coupling (Russell-Saunders coupling)?

Answer 1

* low Z atoms when e-e interaction dominates * individual e- spin combines to form S * individual orbital angular momenta combines to form L * L & S couple to form J

Answer 2

* When spin orbit splitting ≈ separation of the sub-levels due to weak B field * ΔE_SO ≈ g_Jμ_BB

Answer 3

* energy level splitting from coupling of net nuclear spin I and J (F = I + J) * Inclusion of net nuclear spin causes further energy perturbation besides spin-orbit coupling * interaction of net nuclear spin I with atom's internal magnetic field

Answer 4

* e-s are fermions so each e- in multi e- atom must be described by a different spin-orbital * total wavefns must be antisymmetric wrt exchanged of e-s * since no 2 fermions can occupy the same quantum state * if nlm = n'l'm' then it would not satisfy the antisymmetry of the total wavefn

Answer 5

1. Absorption 1. atom absorbs photon from field & is excited to higher E state 1. rate of absorption is proportional to pop. of ground state (N) & energy density U 2. Spontaneous emission 1. atom decays from excited state emitting a photon 1. rate of decay is proportional to excited state pop. (N₂) 3. Stimulated emission 1. photon from field causes excited atom to decay and emit photon 1. rate of emission proportional to excited state pop. N₂ & energy level U

Answer 6

* To achieve cooling, atoms must be sufficiently slowed to about 1cms^-1 * (at room temp. atoms have velocities of several hundred ms^-1) * atoms are bombarded with many photons with energies close to transition frequency * many since one has too little effect as p_photon \<\< p_atoms * atoms subjected to pair of counterpropagating (photon) laser beams * Atoms see photons as doppler shifted * beam towards (opposite direction to) atom will be blueshifted (higher frequency) * beam in (same) direction of atoms motion will be redshifted (lower frequency) * probability of scattering (absorption + re-emitting) a photon depends exponentially on how close a laser is tuned to resonance * blueshifted [v'= v_L(1-v/c)] =\> closer to resonance so probability increases (mainly absorbed) * redshifted [v' = v_L(1-v/c)] =\> further from resonance so probability decreases * Upon scattering, net momentum of p = hv_L/c transferred to atom * atom decays and emits photon causing atom to recoil * Direction of emission random so effect of recoil averages to 0 =\> net momentum transfer to atoms against atoms direction of motion * Frictional force on atoms depends on velocity (viscous damping) * faster atoms experience larger force slowing them down * =\> narrowing of momentum distribution as T decreased to around 10^-6 K * atom trap wtih 3 lasers along each x, y, z axis, and inhomogenous B field added to confine atoms * BEC (all atoms occupy & is described by same wavefn) formed at around 10^-7 ~ 10^-9 =\> requires further techniqe of evaporative cooling (letting "hotter" atoms escape)

Answer 7

* v_if = frequency of main x-ray emission line * Z = atomic number * S = screening constant (depends on series) * Z_eff = Z -S * S_K = 1 * S_L = 7.4 * C_n = empirical constant independent of Z * This law is used to find the frequencies of characteristic x-rays =\> applied in characteristic x-ray spectra

Answer 8

Transitions form series (groups of lines) which are defined by final state of transitions n_f * n_f = 1 =\> K-series * n_f= 2 =\> L-series * n_f = 3 =\> M-series Within each series, Δn = n_i - n_f is denoted by subscripts * Δn = 1 =\> α * Δn = 2 =\> β * Δn = 3 =\> γ

Answer 9

* Δv = ±1 (vibrational - from selection rules of harmonic oscillator) * |J - J'| = 1 (rotational) * Infra red frequencies

Answer 10

* distribution of final vibrational states after transitions between molecular eelctronic states are determined by overlap of vibrational wavefns between ground and excited state (vibrational energy levels) * transitions between vibrational states typically occur for states in which overlap is greater since transition is much faster than nuclei speed, therefore nuclei don't move far in this time * hence smaller difference in nuclear coordinates of these states * so states with bigger overlaps have higher transmission probability * =\> higher intensity bands * I = Franck-Condon Factor * v" = vibrational wavefn of lower state (final) * v' = vibrational wavefn of upper state (initial)

Answer 11

* state S = 0 * gives M_s= 0 * =\> singlet state * Antisymmetric w.r.t exchange * S = 1 * gives M_s = -1, 0, 1 * =\> triplet state * Symmetric w.r.t exchange. * spin-up × spin down cancels

Answer 12

* e-s are fermions hence total wavefn is antisymmetric * if spin = antisymmetric then spatial = symmetric; vice versa * e-s in ground state of helium have same n, l and m numbers * so the spatial part is symmetric (since the antisymmetric part is 0) * hence spin part has to be antisymmetric =\> spin singlet

Answer 13

* total (σ_T): **effective** area (normal to direction of incidence) provided by a target to an incoming projectile * differential: particle flux cattered by each target nucleus into solid angle dΩ divided by incoming intensity * flux = particles per unit time * intensity = particles per unit time per unit area = flux per unit area

Answer 14

* n = density * l = length of cell * I/I₀ = 100% - %attenuation * σ_T = total collisional cross-section

Answer 15

* Wave-functions are anti-symmetric w.r.t exchange of identical fermions * and symmetric w.r.t exchange of identical bosons.

Answer 16

* If L is odd =\> odd pairing of e-s with the same l * so there maybe an e- which doesn't have a pair e- with the same n and l value * therefore under exchange, the set up is now different * =\> spatial wavefunction is antisymmetric * forces the spin part to be symmetric. * But L+S is odd so S is even * spin wavefunction is antisymmetric * if both spin and spatial are antisymmetric * then total wavefunction is symmetric * =\> forbidden by Pauli

Answer 17

* If the splitting due to B field is bigger than the splitting due to LS coupling.

Answer 18

1. Microwave - Transitions between rotational levels only 2. Infrared - Transitions between rotational and vibrational states together 3. Ultraviolet - Transitions between rotation, vibrational and electronic states. So high energy as electronic energy level differences are much greater.

Answer 19

* 2 photons from stimulated emission have same frequency (coherent) * each photon can stimulate further transitions =\> monochromatic waves coherent on a large scale * stimulated emission to be dominant need population inversion * but cannot be produced in 2 level system where population of state 2 is increased until N₁= N₂ (saturated transition so absorption & stimulated emission are equally likely but stimulated emission cannot become dominant process) * To obtain population inversion, need at least 3-level system 1. atoms pumped with photon of frequency v₁₃ 2. atoms accumulate in level 3 which spontaneously decays to level 2 3. atoms accumulate in metastable level 2 4. so levels 2 & 1 have inverted populations 5. incoming v₁₂ photons result in laser light at frequency v₁₂

Answer 20

**Ruby laser**: * Green light (550 nm) pumps Cr3+ ions in Ruby crystal (Cr3+ is impurity in Al2O3). * Rapid (non-radiative) de-excitation to lower energy, metastable state =\> **population inversion** * This can lase in red at 694 nm. * Set-up requires \> 50 % of ground state to be pumped ie large pumping power. * Atoms in level 1 will absorb laser photons. This type of laser not very efficient.

Answer 21

Consider the a and b that label the wavefuction to be a given set of quantum numbers n, l, m and n', l', m'.

Answer 22

* Nuclei move much more slowly than electrons since they are much heavier =\> m_e/M_N \<\< 1 * Consider the nuclei as fixed and find the reduced mass of the system of nuclei. * For simplicity, consider also ideal diatomic molecules.

Answer 23

ΔJ = ±1 (for molecules with per. dip. mom. =\> can be rotated by external E field) ΔJ = 0 =\> Possible for non-linear molecules

Answer 24

1. Rotation - dictated by spherical harmonics 2. Vibration - Radial eqn

Answer 25

* Ionic molecules have a permanent electric dipole moment =\> obey ΔJ = ±1 * Purely covalent molecules, like homonuclears, have no electric dipole moment =\> ΔJ = 2 forbidden transitions which are much weaker. * Ionic molecules have much stronger rotational transition intensities

Answer 26

* Note that the spacing in the ENERGY SPECTRUM is 2B.

Answer 27

* v is the vibrational quantum number * For harmonic approximation: Δv = ±1 , |J - J'| = 1 * Δv = ±1 comes from harmonic oscillator approx., and Δv = ±2,±3,±4,.. (overtones) are much weaker transitions.

Answer 28

* v small: harmonic approximation is best near bottom of well * J small: centrifugal distortion tends to stretch molecule and lower rotational energy. * Rotational constant B depends on v. Since molecules vibrate, effective bondlength =/= R_e=\> error when only considering rotation * Potential V only approximately parabolic, for low v. For high v, corrections due to anharmonicity of potential needed.

Answer 29

* Since ΔJ = 0 is not allowed, the line at Hbar\*w₀ is missing.

Answer 30

* Electronic transitions are always accompanied by vibrational and rotational. * Selection rules: Δ Λ = 0, ±1 ΔS = 0, parity change g =\> u * Since energy diff in electronic energy levels are much larger, transitions occur in ultraviolet.

Answer 31

* Linear combination of H₂⁺ orbitals * Sum gives symmetric wavefunction, diff gives antisymmetric wavefunction. * Electronic energy levels labelled by total electronic angular momentum Λ, similar to L in atoms. In the same way as with S, P, D, F,... we have for molecules Σ, Π, Δ, Φ,...

Answer 32

* Δv can take any value - vib energy levels * ΔJ = 0, ±1 - rot energy levels * Since bond length R_e can change, B (rot. constant) can also change and we don't get evenly spaced roational spectrum lines. * Also, since E_n, E_v \>\> E_J - spreads electronic-vibration frequency into band of closely spaced lines.

Answer 33

* Bombard high Z atoms ( make up the anode) with energetic electrons from the heated cathode which are accelerated across the potential difference V between the anode and cathode =\> K.E = eV

Answer 34

1. Continuous x-ray spectra 2. Characteristic x-ray spectra

Answer 35

* Fast moving electrons are deflected by the coulomb field of heavy atoms * Charged particles emit radiation when accelerated (or decelerated) * This is continuous background radiation =\> Bremsstrahlung (braking radiation) * X-rays produced * Limiting case when electrons brought to a stand still and all energy is given up =\> cut off wavelength

Answer 36

* Some of the energetic electrons are able to collide with inner shell electrons with enough energy to cause ionisation of the electron. * These electrons leave "gaps/or holes" behind which are filled by higher energy state electrons making the transition to the lower state. * This results in the emission of characteristic x-rays

Answer 37

* When population of level 1 state is smaller than level 2, N₁\< N₂ - Photon more likely to be absorbed. * If level 2 more populated than 1, N₂ \> N₁ photon likely to stimulate emission. * Atoms from emission goes on to cause further stimulated emission and cascade of photons produced - LASER * Therefore, **population inversion** where **N₂ \> N₁**is a necessary requirement of laser action. * However, population inversion is not possible in a 2 level system.

Answer 38

1. Atoms pumped with photons of frequency ν₁₃. 2. Atoms accumulate in level 3 which decays spontaneously to level 2. 3. Atoms accumulate in metastable level 2. 4. Levels 2 and 1 have inverted populations. 5. ν₁₂ photons result in laser light at frequency ν₁₂.

Answer 39

1. Covalent bonds - Sharing of electrons between atoms of similar electronegativity. H₂ as example. 2. Ionic bonds -Electron found predominantly close to one atom. Atoms have very diff. electronegativity. Normally occurs for alkali-halogen pairs, i.e. LiF, NaCl, etc.

Answer 40

* Electron affinity A - Energy required to add an electron * Ionisation I - Energy to remove an electron (binding energy of valence electron) * Calculating cost of removing electron from alkali atom and adding it to Halogen: e.g. NaCl I(Na) - A(Cl) Subtract since A is the energy gained by the Chlorine atom.

Answer 41

* change Z to Z_eff * Z_eff= nuclear charge + charge of inner e-s * since outer e- may have many e-s between it & the nucleus * change n to n_eff= n - Δ_nl * Δ_nl is quantum defect * quantifies departure from hydrogenic behvaiour to account for shielding effects of inner e-s on nuclear attraction experienced by outer e- * change R_infinity to R_M * nucleus no longer assumed to be infinitely heavier than the e- * Δ_nl increases with l and n * since less nuclear attraction experienced for outer e- at high l * so more hydrogenic like * varies slower with n for a given l

Answer 42

* spin orbit correction term H_LS * due to internal atomic magnetic fields * rest frame of e-, nucleus moves around it acting like an effective current loop =\> generates its own magnetic field * e- has own magnetic dipole moment due to its spin * magnetic field of nucleus interacts with magnetic dipole moment of e- so e- experiences a torque =\> energy cost depending on relative orientation of nucleus and e- * hence energy correction term corresponding to H_LS * Zeeman effect correction term H_B * magnetic moments due to spin & orbital angular momentum interact with external B field * =\> potential energy V_B (perturbation due to external B field) corresponding to H_B * effect: * LS causes splitting of terms to give levels * B causes further splitting of these levels

Answer 43

* H_LS much greater =\> LS remain coupled & external B field is weak * good quantum numbers: J, M_J, L, S * H_B much greater =\> strong external B field causes LS to uncouple * good quantum numbers: L, M_L?

Answer 44

* find J values * ΔE_max given by Lande interval rule with sum of all the separations of components J₁, J₂, ..., J_n (higher Js of separations where ΔJ = 1) * ΔE_max = A(L,S) J₁ + A(L,S) J₂ + ... + A(L,S) J_n * =\> ΔEmax = A(L,S) Σ_i J_i * obtain A(L,S) * then separations of each individual component is * ΔE_i= A(L,S) J_i

Answer 45

* when energy level splitting due to weak field = that due to LS * ΔE_{weak field} = g_JM_Jμ_BB \> smallest splitting (separations of components) due to LS coupling * ignore M_J since only considering energy separation between levels

Answer 46

* heteronuclear molecules * hence have some electronic distortion and so have electric dipole moment * compared to linear molecules like H₂ whose displacement is 0 * e.g. HCl and CO

Answer 47

* R_e = equilibrium bond length * D_e = potential minimum (dissociation energy) = V(R_e) * What's D₀?

Answer 48

* if V(R) is approximately parabolic =\> motion is approximately harmonic * Taylor series expansion of V(R) about R=R_e * can neglect terms higher than quadratic * at minimum dV/dR (at R=R_e) = 0 * giving the eqn (to 2nd order)

Answer 49

* ω² = k/m usually * for bonds, replace m with μ * hence ω = sqrt (k/μ) * remember mass of hydrogen = mass of proton * and divide by 2π for answer in Hz unless stated to give in s^-1

Answer 50

1. R_M correction term for R_infinity depends on μ 1. E proportional to μ 2. B depends on μ 1. E proportional to 1/μ 3. ω depends on μ 1. E proportional to 1/sqrt(μ) So rotating most sensitive to isotopic changes since E decreases rapidly for increasing μ

Answer 51

* 1st = sum of KE of each nuclei * 2nd = KE of single e- * 3rd = attraction between e- and nucleus A * 4th = attraction between e- and nucleus B * 5th = internuclear repulsion * 2nd to 5th terms make up electronic Hamiltonian

Answer 52

* terms with ∇_Rψ(r_i) do not feature in electronic Schrodinger eqn since electronic wavefn is assumed to be indt of changes in position of nuclei (nuclei positions fixed)

Answer 53

* E_rot = BJ(J+1) * calculate B for each transition * and calculate the average B from those Bs

Answer 54

* Divide through by hc to get cm^-1

Answer 55

* n=1 of H (only has s state) has atoms with no intrinsic dipole moment * since H ground state is a non-degenerate state (has definite parity) & cannot possess an intrinsic electric dipole moment * induced dipole moment by E_ext * V_E = (-1/2)α(E_ext)² hence quadratic Stark effect * ΔE = -E²_ext(A+BM²_J) * n\>1 have atoms with intrinsic dipole moment * atoms have l-degeneracy i.e. all l levels for a given n have same energy * ΔE = ∓3eE_exta₀ * new eigenstates of new H are also eigenstates of unperturbed H₀ so no WD in mixing wavefns so energy is linear in E_ext

Answer 56

* The stark energy change is equal to the change in kientic energy. Equate them to find the magnitude of the electric field, dF.

Answer 57

* most probable radius from maximum of radial prob. distribution i.e. dPnl(r)/dr=0 * Include 4π in Pnl(r)=r²|Rnl|² if wavefn is not normalised * average radius * integrate from 0 to infinity of rP(r)dr

Answer 58

* degeneracy wrt m is a feature of central potential (i.e. V = V(r)) which depends on |r| * hence potential has spherical symmetry * degeneracy wrt l is characteristic of 1/r potential

Answer 59

* most alkali atoms * one loosely bound outer e- which is optically active

Answer 60

* spin singlet e-s are closer together and experience stronger repulsion * spin triplet keeps electrons apart (tend to avoid each other and Coulomb repulsion reduced) * spatial separation is a result of spin orientation

Answer 61

* behaviour of sign dependency on orientation of e- spins * consequence of Pauli Principle which couples spin and space states (one depends on the other)

Answer 62

* Orbital electron motion causes magnetic field within atom whose direction depends on L; * Internal magnetic field causes torque on electron’s spin magnetic moment with direction determined by S; * Torque doesn’t change magnitude of S, and reaction torque doesn’t change magnitude of L; * Torques cause precessional motion of S and L with orientation dependent on each other; * S and L precess about their sum J =L+S; * Sz and Lz are not constants of motion.

Answer 63

Total inelastic cross-section σ_ij gives probability (effective area) for scattering changing internal state of target from level i to level j. Change accompanied by corresponding loss in (gain of) energy by the projectile. σ_ij can be calculated from the Schr¨odinger equation.

Answer 64

LOL what card, that is the card