solid state nmr section 2.1-2.2

What components make up solid state nmr interactions?

Vector I = spin angular momentum of the nucleus under study

Vector S = spin angluar momentum of another nucleus

OR

Vector Bo = applied magnetic field

Depends on the interaction with the nucleus

The coupling can be described mathematically with tensors

What is the Hamiltonian for the quantum mechanical description of nmr?

H = k x I x G x P

K = constant

I = spin angular momentum of the observed nucleus

G = tensor describing the 3D anisotropy of the interaction

P = another type of vector (e.g. S or Bo)

What are G and P for each type of nmr interaction?

Shielding (chemical shift) - P = Bo, G = sigma (shielding tensor) or delta (chemical shift tensor)

J-coupling (indirect) - P = S (spin angular momentum of second nucleus) G = J (indirect dipolar coupling tensor)

Direct dipole-dipole interaction - P = S, G = D (dipolar coupling tensor)

Quadrupolar interaction - P = I 9spin angular momentum of observed nucleus) G = q (electric field gradient tensor)

How can second rank tensor matrices link the properties of vectors?

Each component of a vector depends on all three components of another vector

Their coefficients come from the tensor

E.g. Q is one vector, P another and G a tensor

Ax = Gxx Px + Gxy Py + Gxz Pz

The coefficients can then be written into a matrix

These tensors are symmetric

What is the principle axis system?

From the second rank tensor matrix, it can be simplified because Gxy = Gyx, Gxz = Gxz and Gyz = Gzy

This leaves just Gxx Gyy and Gzz as the only non-zero components, so they are called the principle components

What is the trace of a tensor?

Tr G = Gxx + Gyy + Gzz

The trace is the sum of the principle components

1/3 Tr G is the isotropic average of the tensor

What are axially symmetric tensors?

Where two principle components are equal (Gxx = Gyy)

Gxx and Gyy are perpendicular components and Gzz is a parallel component

Why does shielding depend on?

Nuclei are shielded from Bo by the surrounding electrons, giving ‘chemical shift’

The electrons aren’t usually distributed spherically around the nucleus so the shielding is anisotropic

So shielding depends on orientation of the molecule relative to the applied magnetic field

How can you calculate the value of shielding for an angle theta of molecular motion?

1/3 (sigma parallel + 2sigma perpendicular) + 1/3 (3cos2(theta) - 1)(sigma parallel - sigma perpendicular)

The first term is independent of molecular orientation and is the isotropic shielding so is equal to the isotropic value of the tensor (1/3 Tr G = 1/3 (sigma parallel + 2sigma perpendicular))

The second term depends on the molecular motion. With rapid isotropic motion, this averages to 0. (Average over all angles makes the term 0)

What is the CSA powder pattern?

A powder spectrum where all crystal orientations are equally populated, giving a spectrum of the sum of the individual spectra for different values of theta

With axial symmetry, it has a large peak that decreases exponentially on one side

The large peak is sigma perpendicular (90) and where the exponential part ends is sigma parallel (0)

Intensity reflects number of molecules at that orientation

For non-axial symmetry, there are three independent principle components rather than just 2

What makes a CSA powder pattern more broad?

The shielding aniosotropy increases as the magnitude of the applied magnetic field increases

This makes the powder pattern broader

When do you get a single line of the isotropic chemical shift?

Liquid state nmr, in solid state nmr when the nucleus is in a site with cubic, octahedral or tetrahedral symmetry

This is because they have high symmetry