KEMM17 VT-10, L1 1
KEMM17, Magnetic resonance:spectroscopy and imaging
Daniel Topgaard
KEMM17 VT-10, L1 2
KEMM17 VT-10, L1 3
Lecture notes can be downloaded from:
http://www.omm.lu.se/people/senior_scientists/topgaard/teach/kemm17/
KEMM17 VT-10, L1 4
L1: Fundamentals• Nuclear angular
momentum andmagnetism
• Space quantization• Energy levels• Transitions• Populations - polarization• Precession
Literature: Hore chap 1, Hornak NMR chap 1 and 3
KEMM17 VT-10, L1 5
Basis of nuclear magneticresonance (NMR)
mag
netic
fiel
d, B
0energy difference, ΔEPlanck constant, hresonance frequency, ν
1H: ν = 500 MHz @ B0 = 11.7 T
nuclearmagnet
!
"E = h#
spins - magnet - RF radiation
KEMM17 VT-10, L1 6
Nuclear magnetism and spin
!
µ = "I
magnetic moment, µmagnetogyric ratio, γangular momentum, I
µ || I
KEMM17 VT-10, L1 7
Properties of elementaryparticles• mass: interaction with gravitational field• charge: interaction with electric field• spin: interaction with magnetic field
KEMM17 VT-10, L1 8
Spin quantum number, I
• I: integer or half-integer• Examples
• I = 1/2: 1H, 31P, 13C “spin half”• I = 1: 2H “spin one”• I = 3/2: 23Na
!
I = h I I +1( )[ ]1 2
reduced Planck constant,ћ = h/2π = 1.055×10-34 Js
KEMM17 VT-10, L1 9
Space quantization
!
Iz
= mh
z-component of I, Izmagnetic quantumnumber, mm = -I, -I+1, …, I-1, I
Ix, Iy undetermined!
m = "1 2
!
m = +1 2
!
I =1 2
spin up
spin down
!
z
KEMM17 VT-10, L1 10
Energy levels for I = 1/2
!
E = "µzB0
= "mh#B0
!
"E = h#B0
= h$0
!
B0|| z
!
m = "1 2
!
m = +1 2 Larmorfrequency
KEMM17 VT-10, L1 11
Angular and cyclic frequency
!
" = 2#$
ω: angular frequency [rad·s-1]ν: cyclic frequency [s-1], [Hz]
ω often used to get rid of factor 2π
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Transitions between levels
!
"0
= #B0
B0
• Transitions with Δm = ± 1 allowed• Transitions induced by electromagnetic
radiation with frequency
!
m = "1 2
!
m = +1 2
KEMM17 VT-10, L1 13
cloud of e-
chemical shielding
!
"B01#$( ) =%
0
B0
NMR spectroscopy
ω
KEMM17 VT-10, L1 14
Spin-spin couplingsB0 modified by field from neighboring spins
B0
KEMM17 VT-10, L1 15
Magnetic resonance imaging(MRI)
!
"B0r( ) =#
0r( )
position
Fig from Nobel website
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Resonance condition
!
"0
= #B0
ω0 depends on:• external field• chemical surroundings• neighboring spins• position
spectroscopy
imaging
KEMM17 VT-10, L1 17
Draw energy level diagrams for 1H, 2H, 17O,and 23Na nuclei in a magnetic field. Label eachspin state with the associated magnetic quantumnumber and indicate the allowed transitions.
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Calculate the resonance frequencies (in MHz)for 1H, 12C, 13C and 14N in magnetic fields of4.70 and 11.7 T.
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PopulationsBoltzmann distribution at thermal eq.
!
n"
n#
= exp $%E
kT
&
' (
)
* +
!
"E
number of spins up, n↑number of spins down, n↓Boltzmann constant, k = 1.381·10-23 J/Kabsolute temperature, T
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Polarization, P
!
P =n" # n$
n" + n$=1# exp #%E kT( )1+ exp #%E kT( )
&%E
2kT
high-T approx.!
"E
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Macroscopic magnetization, M• M vector sum of µ• NMR signal ∝ M0
• Behaves classically!
!
M0
= n" # n$( )µz=n" + n$( )% 2h2B0
kT
B0
M0
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Precessioncf. spinning top
!
dI
dt= µ "B
0
solution: rotation of µaround B0 with frequencyω0 = -γB0
µ || I
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Consider an ensemble of protons at a magneticfield of 21.0 T and the temperature 25.0 °C.a) Calculate the energy difference between thetwo spin states and compare it with the thermalenergy kT.b) Calculate the polarization at thermalequilibrium.c) What is the population difference betweenthe two levels (n↑-n↓) if there are 106 spins intotal?