Summary Spectroscopic Techniques

Spectroscopic Techniques


1. Theory

Nuclear Magnets

Magnetic dipole

 μ: magnitude depends on spin-quantum number
Vector ⃗ I (NMR only I = ½)
 |⃗μ|= I ∙ ( I +1 ) ∙ γ ∙ℏ ( ⃗μ in A∙m2)
√
 γ : gyromagnetic ratio
h
 ℏ: reduced plank constant ( )
2π
Orientations

 μ z=m I ∙ γ ∙ ℏ ( μ z is magnetic dipole in direction of B0)
 m I : magnetic quantum number
 Nuclear spin with spin-quantum number I can take 2I + 1 orientations ( m I )
 α-state: positive m I relative to z-axis (B0), lower energy
 β-state: negative m I relative to z-axis (B0), higher energy


Energy of Nuclei in a Magnetic Field

 E=−m I ∙ γ ∙ ℏ ∙ B0
 E : energy of a nuclear magnetic dipole in a magnetic field
 E=h ∙ v

Absorption and Emission of Radiation: Transitions I

Transitions

 Absorption/emission: can change the orientation (thus the energy) of a nuclear magnetic
dipole
 Absorption: from α-state to β-state
 Emission: from β-state to α-state
 ∆ E=γ ∙ℏ ∙ B 0 (selection rule: ∆m = ±1)
γ ∙ B0
 v L=
2∙π
 v L: Larmor frequency (of the absorbed/emitted radiation)

A Real NMR Sample: Boltzmann distribution

 Proton fractions: only depend on energy and temperature
N α 1 1 γ ∙ℏ ∙ B0
 xα= = + ∙
NH 2 4 k ∙ T
N 1 1 γ ∙ ℏ ∙ B0
 xβ = β = − ∙
N H 2 4 k ∙T

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, Absorption and Emission of Radiation: Transitions II

 Absorption: H α +h ∙ v → H β (excited → ground)
 Emission: H β → H α +h ∙ v (ground → excited)
 Eexch =E|¿|− E =( N − N ) ∙h ∙ v=N ∙ ( x − x ) ∙ h∙ v ¿
emit α β H α β




2. NMR Spectroscopy

Magnetization

 Vector ⃗M : sum of all the individual vectors of the atoms (population difference of energy
levels)
γ 2 ∙ ℏ2 ∙ B0 for spin-½ nuclear magnetic dipoles (
 M z =N ∙ M z is magnetization in direction of
4∙k ∙T
B0)
2 γ 2 ∙ℏ 2 ∙ B 0 for spin-1 nuclear magnetic dipoles (
 M z =N ∙ M z is magnetization in direction of
3∙k∙T
B0)
 N : the total number of nuclear spins
Larmor precession

 Thermal equilibrium: Boltzmann distribution → magnetization parallel to B 0
 No equilibrium: magnetization not parallel to B0 → magnetization rotates
around the direction of B0
 Larmor frequency: the frequency of the processional motion, equal to
absorption frequency of the nuclear magnetic dipoles
γ ∙ B0
 v L=
2∙π

 v L: Larmor frequency
 ω L =γ ∙ B0
 ω L: Larmor angular frequency
 NMR transition: magnitude of the magnetization vector remains constant, only the orientation
with respect to B0 changes


Absorption and Emission of Radiation: Transitions III

Magnetization vector and transitions

 A: magnetization precession around the direction of the
oscillating field B1
 B: magnetization precession around the magnetic field B0
 C: combined rotation of nuclear magnetization around
the direction of the B1 field (x-axis) and B0 field (z-axis)
 the interaction of the proton magnetic dipoles with the
oscillating magnetic field B1 and the static field B0 results in a complex precessional motion of
the magnetization vector around x- and z- axes simultaneously

Arbitrary angle



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