University:
Widener University
Course:
PHYS-121 | Fundamentals of Physics I
Academic year:

2023
Views:

476

Pages:

6
Author:

bydredigz7h

Energy Levels in Molecules, contd. + - + + - 1 Absorption and emission Ьу gases 1 For transitions between two states, initial i and final f, the Einstein "Ь" coefficient is given Ьу and we need this to Ье » О for strong ("allowed") absorption or emission. This defines "selection rules". Ф (r, 0, ф) = space-dependent part of the wavefunction describlng state of molecule. 2 (о) = spin-dependent ра rt of wavefunction (total wavefunction = Ф . 2 ) µ = dipole moment operator = е г (а) Selection rules from µ (dipole) µ ., For strong absorption/emission need О (i) pure rotational transitions - permanent dipole required vibrational transitions where vibration does not produce а dipole moment (ii) permanent dipole required (iii) vibrational transitions where vibration produces а (transient) dipole moment permanent dipole not reguired. (iv) when vibrational transitions allowed then simultaneous vibrational + rotational transitions allowed electronic transitions - permanent dipole not required (always а dipole between (v) nuclei and electrons) (Ь) Selection rules from Ф; and Фr (symmetry) Since µ (or г) is an odd function in space, and we require total integrated Ф; µФ 1 to Ье ап even function to prevent significant cancellation between volume elements, then Ф; and Ф1 must have opposite parity. (see Energy Level diagram) Shape of absorption and emission lines (а) "natural" broadening due to (Heisenberg*) uncertainty in energy of upper and lower levels. Integrated line strength = Sю = Jk v dv о Natural width *NOTE Лv 1 == -- = 2лЛt 2al0 (recall 2а ю аю 1 = -- ) 4л"С;о ( лt) (Ь) "doppler" broadening due to motion toward and away from observer (use Maxwellian distribution of velocities) Doppler width 2у (с) "pressure" broadening due to collisional and long distance interactions perturblng the energy levels S;o = fkv dv = f ko о 1 + V - Vo ( ) Г/4:n; 2 dv NOTE Collision freq.(sec-1) in а 2component gas (m1, Г1, П1 and m2, Г2, П2) is: 1 _______ (п Г о: collisional frequency � + n2 )(r1 + r2) 2 (sлкт(¾ + ¾ 1 2 1 2 )) i.e. г = Г 0 (1 atm, 27ЗК) Р" т-i! (г 0 , �' � (- - 21) а= 1, � = from laboratory measurements) Pressure-broadened Width {Г/2n) ko/2 (d) Summary atural lf iпсгеаsеТ: Natural ➔ Doppler If lncreaseP: Doppler ➔ Pressure Doppler 1 Absorption bands Сап get rotational, vibrational, and electronic transitions together: (а) rotational bands (e.g. Н 2 O) - due to ЛJ = ±1 (or emission) e.g. for а llnear molea.Jle (СО, СО21 NO, е с.) �--ч---�-+'--.........l-'-�'-+-'�---1►V гоt = 2El(J+l), J=0, 1,2,---- (miaowave) plus far - IA (Ь) vibrational-rotation bands - due to ЛJ = О, :t 1 and лv = :tl k Р branch ( ааа-1) (ог emisslon) Q branch e.g. ог а llnear molecule -Q.µ1.L.:L....L.;i.....u....u..J,.L..L..f...),..l,Jo...u......J...L.�_u__L!,........L\-+-V ---P(J)--P{1') R(O)--- R(J)---Q(O') +2В V vib-rot = VО { +О -2В / {IR) rotn. rise + vibn. rise * (J + 1) J=0, 1, 2 .................... (R) * (J) ]=0, 1, 2 .................... (Q) J = 1, 2, 3 .................... (Р) rotn. fall + vibn. rise *В=� 8Jt2I (с) electronic - vibration - rotation bands vibn. bands k (or emission) s � J/rotп. 11·ne} � rontinuum (vls.+uv)