Biosc1980/Chem1590
- NMR
Solving the Bloch Equations Using Mathematica
Overview
The purpose of this exercise is to become introduced
to gain familiarity with the Bloch equations because they help explain
some major concepts in the detection of NMR signals: the phenomema of precession
of magnetizatin in an applied field, resonance between the RF irradiaiton
and the spin system's precession, "90-degree" or "p/2"
excitation pulses, selective and nonselective irradiation, the Free
Induction Decay ( FID) and Fouirier Transform methods, and the lineshape
of an NMR signal. You will learn to use Mathematica and get some
familiarity with solving matrix equations, which could be very useful in
your research later on for other mathematical modelling and graphing problems.
Outline of the Lesson
Introduction to Mathematica
and Starting the Program
The Bloch Equations
The Fourier Transform
Relaxation as a Function of
Molecular Weight and Field Strength
Starting the Program
This program is a symbolic calculator which also
does numerical calculations ion a very user-friendly way. One feature which
is especially emphasized is matrix manipulations, and our NMR exercises
will illustrate this type of calculation. Create a directory called "Mathematica"
where you will store all the Mathematica notebooks. To launch the program,
simply type "Mathematica" at the prompt. In Mathematica, select help, and
you will see an interface with a lot of nice features, such as a text search
for help and also some tutorials ("getting started/ demos"). Select "The
Mathematica Book" and look for: (1) Contents and then the short section
called "Suggestions about Learning Mathematica" (2) then look through the
tour of Mathematica -- this tour will help you to have a good overview
of the functionalities especially the short section called "Mathematica
as a Calculator" to get some very basic instructions. "A Practical
Introduction to Mathematica" is a very useful resource to look at
later. Note that to perform the calculations in a cell, you should
use the shift and the enter key simultaneously.
A simple matrix treatment for a unimolecular
chemical kinetic problem illustrates some basic math analogous to that
involved in the Bloch equations.
The following sections are Mathematica notebooks
that lead you through Bloch equations calculations.
Solving the Bloch Equations
This
Mathematica notebook concerns the Bloch equations; we will compute
solutions to the following differential equation describing evolution of
net nuclear magnetization under the influence of
-
precession due to the Larmor frequency in the presence
of the static applied field, at an offset (Omega) relative to the rotating
frame,
-
RF irradiation (strength given by gamma*B1) whose
frequency is assumed to be the rotating frame frequency,
-
and relaxation (both T2 and T1).
The derivation of this matrix form of the Bloch
equations is discussed in your reading (chapter 1). The notebook also has
a brief introduction about the derivation of the Bloch equations, if you
would like an on-line refresher... The Bloch equation written in the form
indicated above is formally (or mathematically) analogous to a lot of other
kinetic problems, including systems of coupled first-order chemical kinetics.
Thus this Mathematica notebook might be useful for you in other applications.
We solve the equation by diagonalizing the matrix numerically, a general,
if sometimes slow, approach.
In the top portion of the notebook for calculation
FIDs we define many operations. If you are intereted, use the
comments and the Mathematica HELP tools to understand the syntax for these
definations.
In the bottom portion you can enter your choice
of spectroscopic parameters and evaaluate the FID and plot it.
Apply this computation to illustrate the following
NMR phenomena, by adjusting the values for R1, R2, offset and B1 strength.
Describe your calculations and the spin trajectories in a brief informal
report.
-
On- and Off- resonance Radio Frequency Pulses of
arbitrary tip angles: calculate the Cartesian Components of the net
magnetization vector (Mx, My and Mz) after the pulse. Use this calculation
to answer the question how far off resonance the pulse can be in order
to effectively rotate the magnetization, and what is the effect of a pulse
where gamma*B1 is comparable to Omega?
-
The Free Induction Decay with Various Offsets and
Relaxation Rates: you can calculate the Cartesian components of the magnetization
as a function of time. Use this calculation to answer the question what
effects R1 and R2 have on the evolution of magnetization and the appearance
of the FID and spectrum, and what effect the frequency offset has on the
FID and spectrum.
-
What calculation could you do to illustrate the
effect of saturation of a spin?
Complex Fourier Transform of the Free Induction
Decay
The
Complex Fourier Transform as it is used in NMR spectroscopy is discussed
and illustrated; i.e. the values of Mx and My as a function of time, can
be analyzed with a complex Fourier transform to yield a properly phased
spectrum. Calculate the FIDs for a slow vs. fast relaxation times (based
on the previous exercise), and for big vs. small frequency offsets, and
look at the resulting spectra. Does T1 contribute to the linewidth of the
peak, or does only T2 determine the linewidth?
Relaxation of Molecules in Solution: T1, NOE
and Linewidths
Rates of relaxation due to proton-proton couplings
can be predicted knowing the interproton distance, the frequency
of the transition and the rotational correlation time (which in turn depends
upon the molecular weight of the molecule, temperature and viscosity of
the solution). Write your own notebook to plot the following expressions
for the NOE and thespin-lattice relaxation rate R1 rates as a function
of molecular weight and field strength. Evaluate R1 and the NOE for
molecules of weight range 0.1 to 1000 kDa and field strengths ranging from
200 to 800 MHz assuming an interproton distancce of 1.5 and 4.0 Angstroms.
Consider the practical question of whether macromolecules will relax efficeintly
in the highest field strengths available and what their NOE strengths
are likely to be like in comparison with experiments done in weaker magnets.
tc =
hV/(3kT) = MW (0.4ns/kDa)
NOE = A [-tc
+ 6tc/(1+4w2tc2)]
R1 = A [tc
+ 3tc/(1+w2tc2)
+ 6tc/(1+4w2tc2)
]
where w represents the Larmor frequency and
A=(m0/4p)2(1/10)(g4h2/4p2)(r
-6) = 5.7 10-8 sec-2 for two protons
at 1Angstrom. The expression needs to be scaled by the inverse sixth
power of the distance, and by the value of the functions in ns. Both
the NOE and R1 will have units of inverse seconds (rate constanst) and
will be of the order of 0.01 to 100 per second.
Plot the NOE and the spin-lattice relaxation rate, R1, as a function
of thte Larmor frequency, w,
assuming a particular molecular weight, and assuming the interproton distance
is 1.5 Angstrons (roughly for a geminal pair in a methylene group) and
using the approximation tc
= MW(0.4ns/kDa) for the correlation time.
Plot the functions for three molecular weights (0.1, 10, 1000 kDa) on top
of one another. Make sure that the plot ranges include the frequencies
for a 200 and an 800 MHz spectrometer, and remember that your axis will
be angular frequency not Hz (so you should look at 2 p
200 MHz and 2 p
800 MHz).
We
show related and somewhat more detailed calculations in this example.
Have a look at this notebook when you have finished trying to write your
own one.
Other Material of Possible Interest
The following sections are taken from a more advanced
course emphasizing a quantum picture for resonance, response to pulse sequences,
and the FID. They are NOT required for molecular biophysics students.....
Basics of Density Matrix Theory and Anisotropic
CSA Interactions
Examples Involving the AK and AB Spin System:
Confirming the P.O. Rules, INEPT (J) Transfer and Dipolar Order
Density Matrix for a Spin-1
Product Operator Calculations in Mathematica
Other Software on the Web
Basics of Density Matrix Theory
This
Mathematica notebook concerns density matrix calculations for a single
spin-1/2, focussing on the solutions to the time-dependent Schroedinger
equation (below) for a time independent Hamiltonian. 
The Hamiltonians of interest for NMR are almost always related to the
effect of an applied magnetic field (static or radio frequency) on nuclear
spin, which are treated as rotations. We illustrate the prediction of observables
from the density matrix, such as Mx and My, which we usually measure in
a FID. We illustrate the use of this formalism by simulating a simple experiment,
the chemical shift echo, and the use of the Stochastic Liouville equation
to calculate an FID.
An
additional notebook discusses the anisotropic chemical shift and how
this anisotropy manifests itself in measurements.
Examples: The Echo, Confirming the P.O. Rules, INEPT Transfer
This
notebook concerns an introduction to direct - product bases and illustrates
calculations for A-K spin systems. After you have done the calculations
in the notebook, use your background with Mathematica and with Density
Matrix Calculations to confirm the Product Operator Rules given in class,
and to simulate a COSY spectrum. For the AK spin system, once you have
confirmed all the P.O. rules, you can save a lot of time by simply using
P.O. calculations. However, please keep in mind that for larger spin systems,
you will need to derive entirely different bookeeping rules, because the
physical laws have a different functional form, or the spin space is much
larger and will be represented with entirely different "coherences". Examples
of cases that will require entirely new P.O. rules include I=1 (e.g. Deuterons)
or I=5/2 (e.g. 17O) or more complicated coupled spin networks such as the
AB spin system (e.g. the protons in a methylene group, for example on a
beta carbon in an amino acid) or the ABX spin system (e.g. the protons
on an alpha and a beta carbon in a typical amino acid). The notebook below
shows density matrix calculations for a spin-1, the next most simple case.
For extra credit, consider whether AK spin systems coupled by the Dipolar
Coupling rather than J couplings need to be treated differently.
This
notebook concerns an introduction to the Dipolar Interaction for AB
spin systems.
Density Matrix Calculations for a Spin-1
This
notebook concerns density matrix calculations for a spin-1. Based on
this material you could derive alternative Product Operator Rules for spin-1.
Some Product Operator Calculators in Mathematica
This
Notebook concerns product operator calculations in Mathematica; it
was prepared by J. Shriver at Southern Illinois University in the Department
of Chemistry and Biochemistry; see the
link about these notebooks for more information . Some other
P.O. notebooks are also available over the web ("POMA") prepared
by Doug Morris. As we learn more advanced pulse sequences for biophysics
in class, you should try to do the corresponding P.O. calculations.
Some Freeware Tailored to this Purpose
The programs "PENCIL"
from the University of Washington is a highly visual density matrix calculator,
and "GAMMA" from Ernst's
laboratory at the ETH in Zurich, and maintained at the Florida High Magnetic
Field Laboratory at Florida State University, is a very sophisticated NMR
simulation tool.
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