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/

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L1: Fundamentals• Nuclear angular

momentum andmagnetism

• Space quantization• Energy levels• Transitions• Populations - polarization• Precession

Literature: Hore chap 1, Hornak NMR chap 1 and 3

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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

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Nuclear magnetism and spin

!

µ = "I

magnetic moment, µmagnetogyric ratio, γangular momentum, I

µ || I

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Properties of elementaryparticles• mass: interaction with gravitational field• charge: interaction with electric field• spin: interaction with magnetic field

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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

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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

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Energy levels for I = 1/2

!

E = "µzB0

= "mh#B0

!

"E = h#B0

= h$0

!

B0|| z

!

m = "1 2

!

m = +1 2 Larmorfrequency

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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

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cloud of e-

chemical shielding

!

"B01#$( ) =%

0

B0

NMR spectroscopy

ω

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Spin-spin couplingsB0 modified by field from neighboring spins

B0

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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

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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?