Lecture 5: Bloch equation and detection of magnetic resonance

Transcription

1 Lecture 5: Bloch equation and detection of magnetic resonance Lecture aims to eplain:. Bloch equations, transverse spin relaation time and *. Detection of agnetic Resonance: Free Induction Deca

2 Bloch equations

3 Bloch equation in a vector form Bloch equation describes the evolution of sample magnetisation in magnetic field (with a large static -component) taking into account spin relaation: Important: d (i) deca of the transverse and longitudinal spin components is assumed to be eponential γ H + ( ) k In thermal equilibrium magnetisation will tend to align along H. χ is the static magnetic susceptibilit χ H (ii) Deca of the and, components is described b different time constants and Note, in contrast to longitudinal deca, transverse deca conserves energ in the static field

4 Bloch equations (eplicit epressions for all components) d d d γ( γ( H H ) ) + γ( H ) the transverse spin relaation time

5 Static field solutions for Bloch equations Solutions for Bloch equations in case HH k are given b: (t) (t) e e (t) t/ t/ [ [ ()e t/ () cosω t () cosω t + ( ) ( ) ( ) (- + e -t/ () sin ω t] () sin ω t] he equilibrium or stead-state solutions are found from t )

6 Evolution of magnetiation according to Bloch equations

7 Rough estimation of in solids See also eamples 3. and 3. Each nucleus eperiences a local magnetic field from its neighbours given b (in SI units): H loc if we use data for GaAs crstal: γ for 69 Ga rad s - - µ.56-6 V s/(a m) r.5 nm μ μ μγ 3 3 r r 58μ Use: V s esla m N A m Random precession of different nuclei in this magnetic field will lead to transverse spin relaation with time of the order γh loc 3μs

8 How does transverse relaation (or dephasing) work Sine functions agnetisation (arb. units) ransverse spin components of different nuclei precess with different periods according to sin(ωt) law ime (s) -7 Sine functions he resultant magnetisation (the sum of all sine functions) quickl decas as described b Bloch equations using the relaation constant agnetisation (arb. units) ime (s) -7

9 versus * here is an additional dephasing of the magnetiation introduced b eternal field inhomogeneities, and also b inhomogeneities of the spin ensemble (for eample due to the chemical or Knight shifts). his reduction in an initial deca of can be characterised b a separate deca time. hus the total deca rate will be defined: * + ' Note, that the deca due to field or ensemble inhomogeneities is reversable (phase relationship between spins is recovarable) in spin-echo eperiments. Deca due to is not reversible.

10 pical magnitudes of transverse spin relaation time aterial/issue (ms) (ms) Gra matter 95 White matter 6 8 uscle 9 5 Cerebrospinal fluid 45 Fat 5 6 Blood - GaAs crstal ~ ~. Self-assembled semiconductor quantum dot > 6 ~

11 Detection of agnetic Resonance: Free Induction Deca

12 Farada s law of induction he induced electromotive force (EF) in an closed circuit is equal to the time rate of change of the magnetic flu through the circuit EF dφ he magnetic flu through the circuit is defined as: Φ B ds coil area

13 Free induction deca Eample 5. Describe the evolution of nuclear spins after a π/-pulse. otion of spins will be independent of the oscillating field H and will onl be defined b the static eternal field H he angle of rotation in the plane normal to H is given b: θ γh t Free refers to free of the oscillating field H In a standard RI eperiment, the field associated with a precessing magnetiation sweeps past fied receiving coils

14 Detection of free induction deca Once the magnetisation has a transverse component an electromotive force (emf) will be created in a coil, a consequence of Farada s law. he time-dependent form of this current carries the information that is eventuall transformed into an image of the sample. Advantage of FID, voltage needed to create H is onl applied for a short time. Note, FID signal decas with time H

15 SUARY Bloch equations describe the evolution of sample magnetisation in magnetic field. wo spin relaation times are eplicitl introduced for longitudinal ( ) and transverse ( ) spin relaation d d d γ( γ( H H ) ) + γ( H ) here is an additional dephasing of the magnetiation introduced b eternal field inhomogeneities, and also b inhomogeneities of the spin ensemble (for eample due to the chemical or Knight shifts). hus the total deca rate will be defined: Free induction deca provides the simplest wa for R detection using a coil where the varing magnetic flu will produce emf * + '