Exact Diagonalization of the Quadrupole Hamiltonian in Nuclear Magnetic Resonance Spectroscopy

Suchandan Pal1

1Florida State University

Lattice Sum Calculations

Dipole Field:

Introduction:

What is NMR?

NMR Spectroscopy is a technique for probing the internal structure of molecules using unique magnetic properties of nuclei. Under normal conditions the nuclear spins precess randomly due to their thermal energy. When an external magnetic field is applied the nuclear spins allign and precess about the magnetic field. The potential energy due to the interaction of the nuclei with the magnetic field is known as the Zeeman energy. The precession frequency, called the Larmor frequency, is unique for each nuclei.

The Zeeman interaction splits the Quantum mechanical energy levels of the nucleus. Transition between these levels are induced by an RF pulse. When a RF pulse is sent out nuclei will only respond by to frequencies that match their Larmor frequency, a phenomenon called resonance. After the absorption of the energy the nuclei return to equilibrium through a relaxation process by exchanging energy with the environment. This energy is detected as the NMR signal.

Magnetic moments from neighboring atoms affect the resonance frequency of the nuclei. The Dipole contribution is calculated by the sum below.

The Zeeman interaction is given by magnetic Hamiltonian, where K is the Knight shift, an extra shift due to the extra magnetic field contributed by the spin of Conduction electrons in metals, I the spin and H the magnetic field.

However the total Hamiltonian, HT is given by HM and the Quadrupole Hamiltonian, HQ. The calculation is complicated by the fact that the Quantization axes of the two Hamiltonians are not necessarily collinear, so one has to transform two Hilbert Spaces with Euler Angles.

The problem is actually difficult, the answer becomes the sum of the expected values over a region, however high degrees of precision are unnecessary, and a lattice sum with simplifications suffices. We neglected the wave functions associated with electrons comprising the atom assuming charges to be point like and calculated the dipole field as follows.

2Dipole Field

With Equipotential lines

Euler Angles:

Euler angles provide a method of decomposing a rotation in a three dimensional space into three elementary rotations. The first rotation is about the z axis by an angle phi, the second an angle theta about the rotated x axis and then an angle of psi about the rotated z axis. This can be written as a product of three rotation matrices.

Notice that the above equation falls away as 1/r3 like a classical dipole field. Here r1 is the inner Lorenz radius and r2 is the outer Lorenz radius. m is the magnetic moment of the nucleus.

Electric Field Gradient:

The purpose of this program was to calculate the Electric Field Gradient at a particular field site, this is important because it contributes to Quadrupole energy. The Electric Field Gradient matrix is given by:

Quadrupole Hamiltonian:

The Hamiltonian is an expression for the observable corresponding to the total energy of a system. The electric potential due to the charge distribution of the nucleus can be written as an expansion, a weighted sum of the dipole, Quadrupole, octupole, etc. The dipole and octupole terms happens to be zero. Quadrupole effects are significant. The next non-zero term, the hexadecapole term is of order of (rn/re)2~ (10-12 cm/10-8 cm) ~ 10-8 times the Quadrupole term, which is beyond current equipment and as a result can be neglected.

In a crystal the Electric Field Gradient (EFG) interacts with the Nuclear Quadrupole Moment which modifies the nuclear energy levels. The Quadrupole Hamiltonian describes this extra energy. I wrote programs to diagonalize the Quadrupole Hamiltonian of nuclei using a method described by Cohen and Reif1. The EFG results from the interaction of the nucleus with different charges surrounding it. It turns out that the effects of electron shielding, though significant are difficult to calculate. One interesting problem is however the effect of electrons in the orbitals of the atom, these, although possibly the most significant are very difficult to calculate due to the irregular shape of orbitals. If bonds of significant covalent character exist than the “shape” of the orbitals is not well understood.

The final rotation, A, is a product of three rotations, B, C and D.

Where an element in a particular row, column = (a,b) is given by

There are many conventions for describing Euler angles we use the most common, the “x convention”. The rotation is described by a triplet of angles described above.

Where is the ath element of rij,

Acknowledgements

1. Quadrupole Effects in Nuclear Magnetic Resonance Studies of Solids; Cohen, H. M.; Reif, F., Solid State Physics v5, Academic Press, 1957

2. Wikipedia. July 24, 2007. Dipole. http://en.wikipedia.org/wiki/Dipole

3 Weisstein, Eric W. "Euler Angles." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EulerAngles.html

Dr. Arneil P. Reyes; Dr. Phil Kuhns

Energy levels

The Eigenvalues of the Hamiltonian correspond to the Energy levels of the nuclei. Various plots of the energy levels are shown below.

The graph on the left shows the splitting of the energy levels at different conditions of field and Quadrupole effects, for a nucleus of spin 3/2. When both the Quadrupole energy and the applied magnetic field are zero the energy levels are degenerate and there is no resonance. When Hm is not zero the lines are equally split leading to a single resonance frequency. For a finite HQ << HM the Zeeman levels are shifted leading to three resonance peaks. The bottom panel shows the situation where HM is a perturbation to HQ.

We want to find Where V(x) is the potential arising from the nuclear charge distributed with density p(x) over the nuclear volume S.

It turns out that the formula for calculating the Quadrupole Hamiltonian is:

Where I+and I- are given by