NMR Magnetic Moment: Understanding Bloch Equations and Relaxation Terms

In summary: Finally, for the off resonance pulse consider T1 to be infinite and H0 to be purely along the y axis. That just causes M to rotate about y.
  • #1
the_kid
116
0
Let M be the magnetic moment of a system. Below are the Bloch equations, including the relaxation terms.

[tex] dM_x/dt=({\bf M} \times \gamma {\bf H_0})_x-M_x/T_2 [/tex]
[tex] dM_y/dt=({\bf M} \times \gamma {\bf H_0})_y-M_y/T_2 [/tex]
[tex] dM_z/dt=({\bf M} \times \gamma {\bf H_0})_z+(M_{\infty}-M_z)/T1 [/tex]

At t=0, [tex] {\bf M}=(0,0,M_{\infty}) [/tex].

[tex] {\bf H_0}=H_0 {\bf k'} [/tex] where primed coordinates are in the lab frame.

Now suppose an on resonance pulse is applied along the i direction of the rotating frame for [tex] T_{pi/2} [/tex]=0.005 milliseconds, then it is turned off to watch the free induction decay. T_2=5 milliseconds, T_1=5000 milliseconds.

So, naturally we will have nutation due to the pulse, T_2 decay of the transverse magnetization, and T_1 recovery of the longitudinal magnetization. Due to the timescales, they will proceed sequentially.

I'm trying to sketch the time evolution of the above three components of the magnetic moment in both the rotating frame and lab frame. I'm supposed to zoom in on the interesting regions; i.e. where the aforementioned behavior occurs. I'm having some trouble understanding how these processes are affected by the parameters. Could anyone assist me with this?
 
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  • #3
the_kid said:
Could anyone assist me with this?
Do you have Mathematica? I have a notebook for calculating the Bloch equations.
 
  • #4
DaleSpam said:
Do you have Mathematica? I have a notebook for calculating the Bloch equations.

I do have Mathematica!
 
  • #5
the_kid said:
I do have Mathematica!
Here you go. The notebook has two methods of calculating the Bloch equations for a single isochromat. Both methods are in the rotating frame, not the lab frame. You will either need to modify them or treat the lab frame as a really big off resonance.

The first assumes that the RF pulses are very short compared to the relaxation times and off-resonance frequency. I.e. they can be represented by simple tips. It is based on matrix multiplication. So in your case you would make one matrix representing the RF pulse followed by one representing the free precession then you would calculate the final magnetization by: [itex]M_F = FP \cdot RF \cdot M_I[/itex]. This can give you closed-form analytical solutions. Note that it has a strange approach. The magnetization vectors have 4 dimensions: {Mx, My, Mz, M0} where M0 is the fully-relaxed longitudinal magnetization. So a fully relaxed isochromat of unit magnetization would be {0,0,1,1}.

The second uses the numerical differential equation solver and doesn't make any assumptions. With that one you express things like RF and gradients in terms of functions that are passed as options to the solver. I have several examples. My approach would be to do the rotating frame using the numerical solver, show that there is no appreciable relaxation during the RF pulse, and then use the matrix solver to get an analytical solution for the lab frame. The numerical solver will have trouble with all of the cycles for the lab frame.

Enjoy.
 

Attachments

  • BlochSolvers2.nb
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  • #6
I'm still having some trouble visualizing these processes...
 
  • #7
To visualize the best thing is to consider only the rotating frame and to make some simplifying assumptions.

First, for an RF pulse consider T1 and T2 to be infinite and consider H0 to be purely along the x axis. That just causes M to rotate about x.

Then, for T2 decay consider T1 to be infinite and H0 to be zero. That just causes any transverse magnetization to decay.

Then, for T1 regrowth consider T2 to be really short and H0 to be zero. That just causes longitudinal magnetization to relax.
 

1. What is NMR Magnetic Moment?

NMR Magnetic Moment refers to the magnetic moment of a subatomic particle or nucleus, which is the measure of its ability to interact with an external magnetic field. In NMR (Nuclear Magnetic Resonance) spectroscopy, the magnetic moment of a nucleus is used to study the structure and dynamics of molecules.

2. What are Bloch Equations?

Bloch Equations are a set of mathematical equations that describe the behavior of nuclear spins in a magnetic field. They are used to understand the precession and relaxation of nuclear spins, which are essential for NMR spectroscopy.

3. What is the significance of Relaxation Terms in NMR?

Relaxation terms in NMR refer to the processes of spin-lattice relaxation (T1) and spin-spin relaxation (T2), which are responsible for the decay of the NMR signal. These terms provide valuable information about the molecular structure and dynamics of a sample.

4. How are Bloch Equations and Relaxation Terms related?

Bloch Equations describe the behavior of nuclear spins in a magnetic field, including the effects of relaxation. The relaxation terms in the Bloch Equations govern the rate of decay of the NMR signal, which is crucial for interpreting NMR spectra and obtaining information about the sample.

5. How can I use NMR Magnetic Moment and Bloch Equations in my research?

NMR Magnetic Moment and Bloch Equations are widely used in various fields of research, including chemistry, biochemistry, and medicine. They can provide valuable insights into the structure, dynamics, and interactions of molecules, making them powerful tools for studying biological systems, drug discovery, and material science.

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