Molecular Spectra and Molecular Structure - 1985 PS2 Sol

Problem Set #2 ANSWERS 1. The number of possible spin eigenfunctions for a single particle of spin I is 2I + 1. (a) How many linearly independent spin eigenfunctions are possible for two equivalent particles of spin I? ANSWER (2I + 1)(2I + 1) spin eigenfunctions (b) For a particle with I = 1, denote the three spin eigenfunctions by α, β, and γ, corresponding to the eigen­ values Mz = +, 0, −. How many linearly independent symmetric and how many linearly independent antisymmetric spin states are there for two equivalent particles with I = 1? ANSWER: 6 symmetric eigenfunctions, 3 antisymmetric eigenfunctions symmetric: |αα� , |ββ� , |γγ� , √1 (|αβ� + |βα�) , 2 √1 (|αγ� + |γα�) , 2 √1 (|βγ� + |γβ�) 2 anti-symmetric: √1 2 √1 2 √1 2 (|αβ� − |βα�) , (|αγ� − |γα�) , (|βγ� − |γβ�) Page 2 2. Atomic eigenfunctions contain a factor exp(iMφ). When the atom is a magnetic field B, the quantum number M represents the projection of the J–vector on B (−J ≤ M ≤ +J). The usual selection rules for L, S , and J still hold for moderate B, and in addition a selection rule governing the values of ΔM becomes important. The dipole-moment operators for transitions involving M are ce�� and c� e�⊥ cos φ. The coefficients c and c� are non-zero constants (for the purposes of this problem); e is the charge on the electron; and �� and �⊥ are the components of the electric field of the radiation parallel and perpendicular to B. Derive the selection rules for ΔM for radiation polarized parallel and perpendicular to B. ANSWER: Electric field of radiation � to B field dipole moment operator = ce�� (c is a constant) � 2π � 2π � � ∴ e−iM φ ce�� eiMφ dφ = ce�� ei(M−M )φ dφ 0 0 = ce�� δ M,M� ∴ ΔM = 0 Electric field ⊥ to B field applied: ∴ dipole moment operator = c� e� cos φ �� 2π � � 2π � 2π 1 � −iM � φ � iMφ i(M−M � +1)φ i(M−M � −1)φ e c e�⊥ cos φe dφ = c e�⊥ e dφ + e dφ 2 0 0 0 � � 1 = c� e�⊥ δ M� ,M+1 + δ M� ,M−1 2 ∴ transition allowed for M � = M ± 1 (or ΔM = ±1) Page 3 3. Calculate the Zeeman pattern to be expected for the sodium D–lines at 10,000 Gauss (G). You may neglect nuclear hyperfine interactions. Indicate the polarization of each Zeeman line, that is, whether the electric vector of the emitted radiation is parallel to the applied magnetic field (π–component) or perpendicular to it (σ–component). (a) Show qualitatively the Stark effect to be expected for the sodium D–lines. The splittings are proportional to what power of the electric field strength? ANSWER: B = 10kG � � (1) (S +1)−L(L+1) e� EZeeman = 2mc BM J 1 + J(J+1)+S2J(J+1) Na D lines → 2 S 1/2 − 2 P1/2 2 (1) (2 S 1/2 ) EZeeman S 1/2 − 2 P3/2 � �� 11 1 � � → |JS LM J � = � 0 ± 22 2 1 (1) → EZeeman (2 S 1/2 , M J = +1/2) = (4.76 × 10−5 cm−1 /Gauss)(10kG)(1/2)(2) = 0.467 cm−1 2 1 (1) M J = − → EZeeman (2 S 1/2 , M J = −1/2) = −0.467 cm−1 2 MJ = 2 P1/2 � �� 11 1 � → �� 1 ± 22 2 (1) (2 P1/2 , M J = +1/2) = (0.467 cm−1 )(1/3) = +0.156 cm−1 EZeeman (1) EZeeman (2 P1/2 , M J = −1/2) = −0.156 cm−1 2 P3/2 � �� 11 3 1 � � →� 1± ± 22 2 2 (1) (2 P3/2 , M J = +3/2) = (0.467 cm−1 )(4/3)(3/2) = +0.934 cm−1 EZeeman (1) EZeeman (2 P3/2 , M J = −3/2) = −0.934 cm−1 (1) EZeeman (2 P3/2 , M J = +1/2) = (0.467 cm−1 )(4/3)(1/2) = +0.311 cm−1 (1) EZeeman (2 P3/2 , M J = −1/2) = −0.311 cm−1 Allowed Transitions: ΔM J = 0, ±1 � applied ΔM J = 0 for Radiation E field � B � applied ΔM J = ±1 for Radiation E field ⊥ B Page 4 ANSWER: 3(a), continued 2P 3/2 2P 1/2 2S 1/2 � � � � � � 3/2 1/2 −1/2 −3/2 1/2 −1/2 � � �� 1/2 −1/2 Unperturbed: 2 S 1/2 − 2 P3/2 ⇒ λ = 5889.963Å ⇒ ν = 16978.035 cm−1 2S 2 −1 1/2 − P1/2 ⇒ λ = 5895.930Å ⇒ ν = 16960.853 cm With 10 kG Field .311 2 2 ������������������������������ → 16960.853 ± (0.467 − 0.156) = 16960.542, 16961.164 cm−1 2 ������������������������������ : Perpendicular Polarization → 16960.853 ± (0.467 + 0.156) = 16960.230, 16961.476 cm−1 2 ������������������������������ → 16978.035 ± (0.467 + 0.311) = 16977.257, 16978.813 cm−1 2 ������������������������������ : Perpendicular Polarization → 16978.035 ± (0.467 − 0.311) = 16977.879, 16978.191 cm−1 S 1/2 − P1/2 : Parallel Polarization .623 2 S 1/2 − P1/2 .778 2 S 1/2 − P3/2 : Parallel Polarization .156 2 S 1/2 − P3/2 1.401 ������������������������������ also 16978.035 ± (0.467 + 0.934) = 16976.634, 16979.436 cm−1 2S − 2 P1/2 2S Spectrum unperturbed → − 2 P3/2 with B–field → polarization → ⊥ � � ⊥ E ⊥ � � ⊥ ⊥ � ⊥ Page 5 Answer 3a, continued 2S 1/2 − 2 P1/2 2S Stark effect ∝ � 2 unperturbed → 1/2 − 2 P3/2 with �–field → polarization → � or ⊥ � � ⊥ ⊥ E � (b) What do you think might happen to a beam of ground-state sodium atoms passing through a strong inhomogeneous magnetic field? a strong inhomogeneous electric field? Answer: Ground State Na Atoms ⇒ 2 S 1/2 Strong Inhomogeneous Magnetic Field ⇒ Splits Atoms into 2 Beams: M J = ± 12 Strong Inhomogeneous Electric Field ⇒ 1 Beam that is bent by field. 4. An atom is in a (3d)2 3 P0 state. (a) List all L−S−J terms to which an electric dipole allowed transition might occur. Answer: (3d)2 3 P0 state → ? Selection rules: even↔odd, ΔS = 0, ΔL = 0, ±1, ΔJ = 0, ±1 (0 � 0) ∴ State must be of opposite parity ⇒ must have ◦ superscript (3dnp or 3dn f ), must be triplet state ⇒ 3 P◦1 , 3 D◦1 (b) List all two-electron configurations into which electric dipole allowed transitions can occur from (3d)2 3P . 0 Answer: n>3 (3d)2 → (3d)(np) 2 (3d) → (3d)(n f ) n≥4 Page 6 5. The “transition moment,” or the probability of transition, between two rotational levels in a linear molecule may be assumed to depend only on the permanent electric dipole moment of the molecule and thus to be the same for all allowed pure-rotational transitions. In the pure-rotational emission spectrum of H35 Cl gas, lines at 106.0 cm−1 and 233.2 cm−1 are observed to have equal intensities. What is the temperature of the gas? The rotational constant B for H35 Cl is known to be 10.6 cm−1 , and the ratio hc/k has the value 1.44 cm·K. Answer: For emission: ν = B(J � (J � + 1) − J � (J � − 1)] = 2BJ � ∴ from ν = 106 cm−1 and 233.2 cm−1 we have J � = 5 and J � = 11 I5 = (2J � + 1)e−1.44(30B)/T I11 = (2J � + 1)e−1.44(132B)/T 11e−1.44(30B)/T = 23e−1.44(132B)/T ∴ T = 2110 K 6. What would happen to the Birge-Sponer extrapolation scheme for a molecular potential that correlates with ionic states of the separated atoms? Answer: The Birge-Sponer extrapolation would underestimate D0 because the electrostatic interaction between the two atoms would cause the actual curve to approach zero much more slowly than the linear Birge-Sponer extrapolation.