1.b Bloch equations, T 1, T 2

Transcription

1 1.b Bloch equations, T 1, T Magnetic resonance eperiments are usually conducted with a large number of spins (at least 1 8, more typically 1 1 to 1 18 spins for electrons and 1 18 or more nuclear spins). Magnetic resonance eperiments thus measure the behaviour of an ensemble of identical spins, and the directly measurable quantities are ensemble averages in the strict sense of quantum mechanics. One of the important ensemble averages in magnetic resonance (and in magnetism) is the magnetisation M. Magnetisation M = magnetic moment Volume The magnetisation M is a vector in ordinary 3-D space For individual, uncoupled spins, density n, this magnetisation can be calculated easily. The assumption of individual spins is valid to a very good approimation for nuclear spins, even at very high densities n. For electronic spins, this assumption is only valid for low density, since electrons couple strongly by echange if they are to close together. µ = g µ I = γ I I I K µ e e µ γ B e = g S = S nuclei electrons The quantity γ is called the mageto-gyric or the gyromagnetic ratio. GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 1

2 In the high temperature approimation, the magnetic energy of the spin is small compared to the thermal energy: ( γ ) B kt In this approimation, which is practically always fulfilled for nuclear spins even at high fields and low T, the magnetisation M is: M ( γ ) ( γ B ) I ( I + 1) 3kT = n M is along B in thermal equilibrium Density of the magnetic moments (1/m 3 ) This is the magnetisation in thermal equilibrium at temperature T. As the magnetisation M is proportional to B, one defines the magnetic susceptibility χ m : M B = χm H χ m µ χ m = M µ B ( γ ) I ( I + 1) χ m = µ 3kT n 1 This proportionality χ m T is called: Curie-behaviour GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt

3 Typical values for nuclear spin systems can easily be calculated. Water has a very high density of protons, I =1/. From the density of water, ρ = 1 3 kg/m 3, one calculates the density of protons, n 1H : 1H m 3 n = χ (95K) m The magnetic moment of 1 H is: γ = J 9 = Susceptibility of water at room temperature (T = 95K). All nuclear susceptibilities, even at low temperatures, are very small compared to 1! For electrons, the susceptibilities are usually much larger, due to the magnetic moment of the electron. Sodium-Electro-Sodalite has a very high density of unpaired electrons, n e = m -3 The electronic susceptibility at room temperature T = 95 K is: χ m (95K) = This is approimately 1 times as large as the nuclear susceptibility of water. Electronic susceptibilities are usually fairly complicated to calculate, since at high concentrations the electronic spins are coupled by the rather strong echange interaction. In metals, the susceptibility e.g. is temperature independent: Pauli susceptibility. In doped semiconductors, the susceptibility is beween Curie- and Pauli- type. T GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 3

4 Bloch equations In macroscopic samples, the eperiment directly measures the quantum mechanical ensemble epectation values of the observables. The signal in a MR eperiment is directly proportional to the magnetisation M or M y of the sample. This magnetisation is again related to the epectation values of the spin operators <I > and <I y > or <S > and <S y >. ˆ µ = γiˆ = γ Iˆ ˆ µ = γiˆ = γ Iˆ y ˆ µ = γiˆ = γ Iˆ y z z z y Instead of looking at the epectations values of ˆµ ˆµ y and ˆµ, one z can directly calculate the magnetisations M,M y and M z. M µ z z etc. Since the static field B is along the z-direction, the spin system is anisotropic with respect to,y and z. (By the way: Any two level system is anisotropic with a distinct quantisation ais!) z: longitudinal direction ( along B ).,y,: transverse directions GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 4

5 In thermal equilibrium the magnetisations must be: ( γ ) I( I + 1) M = M = B n z 3kT M = My = The spin system is polarized in the longitudinal direction. There are no transverse components in thermal equilibrium. The equations of motion for the magnetisation are derived from the torque on the magnetic moment. ˆ µ = γjˆ = γ Iˆ The torque N on the spin magnetic moment is: N = µ B The time derivative of the angular momentum J is equal to the torque N. djˆ = ˆ µ B as ˆ µ = γ Jˆ this leads to the relations: djˆ d ˆ µ = ( γ Jˆ ) B = ( γµ ˆ) B These are the basic equations of motion for the angular momentum or the magnetic moment. GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 5

6 dm dm dm dm y z = γ M B Since M µ With B in the z-direction: B = γ M B y = γ M B = Bloch equations These are the equations of motion for the magnetisation M without dissipation. = B Feli Bloch introduced in a phenomenological way two relaation rates 1/T 1 and 1/T. T 1 is called longitudinal relaation time and T transverse relaation time. The introduction of these relaation terms assures, that the magnetisation M returns to the thermal equilibrium case in eponential decays. dm dm dm y z = γ M B y = γ M B = M T 1 M M T M T GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 6 z y Transverse relaation T Longitudinal relaation T 1

7 Rotating coordinate system: y Solution of Bloch equations Transverse magnetisation M = M + M y dm dm dm y z = γ M B y = γ M B M () t = M cos( γ B t ) y M () t = M sin( γ B t) = M ( t ) = const. z This is a solution for the transverse components. The magnetisation M rotates with angular velocity ω= -γb in the -y plane. GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 7

8 It is convenient at this point to introduce the rotating frame of reference,, y, z. y y The reference frame, y rotates with -ω around the z=z ais. The equation of motion in the rotating frame of reference lead to an effective magnetic field: b ω = B + γ Of course the magnetic field B is not changed by looking at it from a different coordinate system! Only parts of the interaction are transformed away. With ω = -γb, the effective magnetic field b =. Thus, in the rotating frame of reference rotating with ω= -γb, there is no effective magnetic field. The magnetisation M in the rotating frame of reference is constant! GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 8

9 B () t = B cos( ωt)- ˆ B sin( ωt)y ˆ 1 We apply an alternating magnetic field 1 1 This alternating field is stationary along the direction in the rotating frame of reference. The effective field is: B eff B1 = b b ω = B + γ dm dm dm y z M = γ Mb y T M = γ M b + γ M B z 1 T = γ M B + y 1 M T 1 M z y These are the Bloch equations with an alternating magnetic field (t) and the relaation. This is a driven, damped oscillating system. The solutions for the magnetisation will show a resonance phenomenon. It is assumed, that the relaation times T 1 and T are not changed by the -field. GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 9

10 By defining the quantities ω, ω and ω 1, the Bloch equations can be cast into a particularly convenient form: ω = γ Larmor frequency B ω = γ B Rabi frequency 1 1 ω = ω ω Frequency difference between Larmor frequency and frame of reference. dm dm y M = + ω T M M = ω M - ω M dm M M y T 1 z z z = ω M 1 y - T1 y Bloch equations in the rotating frame of reference. GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 1

11 f off = MHz =. mt =.1 mt On Resonance T 1 = T = 5 ns Magnetisation (rel. units) M z M M y =.1 mt = 1 Gauß Time (ns) bloch_equation_1.opj 1..8 f off = MHz =. mt =. mt.6 =. mt = Gauß Magnetisation (rel. units) M z M M y bloch_equation_1.opj Time (ns) GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 11

12 f off = MHz =. mt =.4 mt On Resonance T 1 = T = 5 ns Magnetisation (rel. units) M z M M y =.4 mt = 4 Gauß Time (ns) =.8 mt = 8 Gauß bloch_equation_1.opj Magnetisation (rel. units) f off = MHz =. mt =.8 mt T 1 = 5ns T = 5ns M y M z M Time (ns) bloch_equation_1.opj GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 1

13 Magnetisation (rel. units) f off = MHz =. mt = 1.6 mt T 1 = 5ns T = 5ns M z M y M Time (ns) bloch_equation_1.opj B On Resonance 1 = 1.6 mt = 16 Gauß T 1 = T = 5 ns GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 13

14 Magnetisation (rel. units) π/-puls t = 5.7 ns π-puls t = 11.3 ns f off = MHz =. mt = 1.6 mt T 1 = 5ns T = 5ns Time (ns) = 1.6 mt, f 1 = 44.8 MHz t π/ = 1/(4*f 1 ) = 5.58 ns tπ = 1/(*f1) = ns GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 14 M z M y M bloch_equation_1.opj On Resonance T 1 = T = 5 ns

15 Magnetisation (rel. units) Time (ns) M z M y M f off = 5.6 MHz =. mt =.1 mt T 1 = 5ns T = 5ns bloch_equation_1.opj 1. Off Resonance B off =.mt T 1 = T = 5 ns =.1 mt = 1 Gauß.8 =. mt = Gauß Magnetisation (rel. units) f off = 5.6 MHz =. mt =. mt T 1 = 5ns T = 5ns Time (ns) GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 15 M z M y bloch_equation_1.opj M

16 Magnetisation (rel. units) Time (ns) f off = 5.6 MHz =. mt =.4 mt T 1 = 5ns T = 5ns bloch_equation_1.opj M z M y M 1. Off Resonance B off =.mt T 1 = T = 5 ns =.4 mt = 4 Gauß.8 =.8 mt = 8 Gauß Magnetisation (rel. units) f off = 5.6 MHz =. mt =.8 mt T 1 = 5ns T = 5ns bloch_equation_1.opj Time (ns) GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 16 M z M y M

17 Particularly revealing are the steady state solutions of the Bloch equations: dm dm y dm z These solutions are obtained, when one waits long enough. Long enough is a time scale several times the relaation times T 1, T. M M M ω ( γb ) T 1 = M 1 + ( ωt) + ( γb1) T1 T ( γ B ) T = M 1 y 1 + ( ωt) + ( γb1) T1 T 1 + ( ωt ) = M z 1 + ( ωt) + ( γb1) T1 T Steady state solutions of the Bloch equations in the rotating frame of reference. These solutions directly give the frequency response of the magnetisation. In cw spectroscopy, the detected signal is proportional to the components M and M y of the magnetisation. cw: continuous wave Note: if a spin system is characterized by relaation times T 1 and T (and thus is described by the Bloch equations), the measured signal can be calculated quantitatively. GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 17

18 The term ( γ B ) T T is proportional to the microwave (or radio frequency) power P B For low power P, this term is small: ( γ B ) T T This condition is called: no saturation. The approimate solutions under these conditions are: M M ω ( γb ) T 1 M 1 + ( ωt) y ( γ B1) T 1 + ( ωt ) M ~ M is in phase with the eciting alternating field (t). ~ M is called: Dispersion ~ M y is 9 out of phase with the eciting alternating field (t). ~ M y is called: Absorption Both signals are proportional to M and proportional to. The proportionalities M B 1 M B y 1 leads to M, M y Power Some ESR-spectrometers (and most NMR spectrometers) allow to measure both absorption and dispersion. Very often, the measured signal in CW-ESR is a miture between absorption and dispersion, depending on the phase of the detection with respect to the ecitation phase. GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 18

19 Absorption Dispersion T = 56.8 ns B / T γ = /s Calculated with: bloch 1.mws = T GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 19

20 First derivative spectra Dispersion Absorption B / T T = 56.8 ns γ = /s Calculated with: bloch 1.mws = T GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt

21 First derivative spectra 8% dispersion Pure absorption % dispersion % dispersion 4% dispersion 6% dispersion 8% dispersion B / T T = 56.8 ns γ = /s Calculated with: bloch 1.mws = T GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 1

22 Dispersion solution of the Bloch equations. = T T 1 =T =56.8ns γ = /s = 1-4 T = T = T B / T = 1-5 T = T Calculated with: bloch_1.mws GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt

23 = T = 1-5 T = T = T = 1-4 T = T Absorption solution of the Bloch equations. T 1 =T =56.8ns γ = /s Calculated with: bloch_.mws B / T GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 3

24 First derivative of the absorption solution. T 1 =T =56.8ns γ = /s = T = 1-4 T = T = T = 1-5 T = T B / T Calculated with: bloch_4.mws GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 4

25 T 1 =T =56.8ns γ = /(Ts) Absorption Signal B = 1 γt Calculated with: bloch_4.mws B / T GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 5

26 Absorption Signal 1 st derivative B pp B pp T 1 =T =56.8ns γ = /(Ts) = B 3 S pp B = 1 γt B / T Calculated with: bloch_4.mws GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 6

27 1 Ma: at =.77 mt Line-wih B pp (mt), S pp (rel. units) 1.1 Bpp= B=.115mT 3 S pp.1 1E Field (mt) B pp signal_strength.opj GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 7

28 GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 8

29 Rectangular cavity: a metal enclosed bo of volume V cav = a b d y TE 1 - mode d b a Magnetic field lines z Electromagnetic fields can eist in a variety of different configurations called modes. If the electric field E 1 is perpendicular to the z- direction, these modes are denominated TEmodes. TE mnp : m,n,p are the number of half-cycles along the,y,z, direction respectively. Electric field E 1 lines GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 9

30 Magnetic field lines, TE 1 mode a k m π a n π b = ky = kz = p π d d For the empty cavity, the resonance frequency for a general TE mnp mode is: c k c f = = k + k + k π π y z c m n p a b d TE mnp f = + + TE 1 f c = + a d Typical resonance frequencies of the Bruker-rectangular cavities are 9.4 GHz. This can be achieved by: a = 4 mm d = 4.7 mm a = 4 mm Calculation by: rechteckresonator.mws GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 3

31 Modepicture: Reflected microwave-power versus frequency In the tune-mode of ESR-spectrometers, the microwave frequency is swept over a range of appr. 1 MHz around the nominal centre frequency. The reflected power is depicted versus the frequency on an oscilloscope display. 5 MHz GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 31

32 .8 Rechteck-Resonator rectangular cavity.6 Reflected Power (rel.units) Datafile: 'resonator.dat' Formel1= lor(,p,p1,p)+p3+p4* P= R= -8 P1=.1883 R1= 3 P= R= 1 P3=.6679 R3= P4= R4=.1 Fehlerquadratsumme= wih: ν =.1883 MHz data: 1..3 Q = 94 MHz/.1883 MHz = Frequency Offset (MHz) rechteck_resonator.opj GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 3

33 The quality factor Q cc is the critically coupled cavity. By the definition of critical coupling: Q cc = 1/ Q u Q u is the quality factor of the uncoupled cavity The quality factor Q u can be calculated, if the dimensions a,b,d and the resisitvity ρ of the metal is known. a = 4 mm, d = 4.5 mm, b = 1 mm ρ = Ωm (Cu at T = 95K) Q u = 9663 The measured value of Q cc = 496 compares to an epected value of Q u / = 4831 GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 33

34 ESR-Signal: Sodium-Electro-Sodalite (SES), T = 95 K 1 5 db ESR-SIGNAL (1 st der.) db 35 db 4 db 45 db MAGNETIC FIELD (Gauß) ses_power_.opj GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 34

35 ESR-Signal: Sodium-Electro-Sodalite (SES), T = 95 K 15 db 5 db ESR-SIGNAL (1 st der.) db 15 db db MAGNETIC FIELD (Gauß) ses_power_1.opj GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 35

36 45 Sodium electro sodalite (SES) ESR-signal, T = 95 K 4 att = db att = 5 db ESR-signal intensity (db) att = 5 db att = 3 db att = 35 db att = 4 db att = 1 db att = 15 db att = db slope: ESR-signal = const. power.5174 att = 45 db Microwave power (db) ses_power_o3.opj GKMR ESR lecture WS5/6 Denninger Bloch-equation_T1_T.ppt 36