# NMR Hardware

The theory sections in this wiki have been devoted to understanding how NMR works and we have made several attempts to investigate atomic nuclei using Magnetic fields, coils, and RF fields. This section is devoted to developing and NMR spectrometer from the theory side then introducing modifications to this simple spectrometer that makes NMR spectroscopy much easier. It is the goal of this page to provide enough information about NMR hardware that one (given the proper materials) could build an NMR spectrometer in their garage!

## Basic NMR Spectrometer

Shown below is the most basic NMR spectrometer you can make. For an NMR spectrometer to be operational we first need a magnet to split the degenerate nuclear energy levels of the sample. We also need a coil to apply an oscillating RF field which will be used to drive transitions between the energy levels as well as detect the signals. Therfore we will also need and RF oscillator and a oscilloscope to generate and detect the RF waves.

## Magnet

The magnet is arguably the most important part of the NMR instrumentation. The magnetic field drives the splitting of the energy levels and allows for a change in the distribution of energy levels of the degenerate ground state. As we have seen previously, the larger the magnetic field, the larger the spin difference between the upper and lower levels allowing for a stronger signal to be observed. In addition to the splitting of energy levels, several NMR interactions are dependent on the magnetic field. Both the dipolar and CSA interactions are multiplied by an increase in magnetic field, while the Quarupolar coupling interaction is decreased with increasing magnetic field.

Insert picture

Both superconducting and electromagnets magnets are comprised of loops of wires with a current passing through, which results in the formation of a magnetic field. The "north pole" of the magnetic field (using the right hand rule) is generated in the direction perpendicular to the direction of the current flow.

show equation

### Superconducting Magnets

The magnet is typically composed of an alloy that is super conducting at low temperature. While the theory of super conductivity is beyond the scope of this page, several unique properties of super conductors are important. This alloy is made into a wire which is wrapped into a coil and has a current passed through it. This generates the magnetic field! The wires are kept cooled with liquid helium at 4K, which significantly reduces the resistance of the wire to nearly zero. Therefore the current passing through the wire onyl needs to happen once and the spectrometer does not need to be plugged in. The liquid helium Dewar is insulated by a vacuum chamber and then a liquid nitrogen dwar which itself is insulated by another vacuum chamber.

### Electromagnets

## Probe

The probe is a specialized piece of hardware that contians the coils and sample and therefore is responsible for tuning to the correct frequency, applying RF pulses, receiving the signal from the nuclei. More specialized probes can also spin the sample, vary the temperature, and change the angle of the sample and coil.

Lets first optimize the efficiency of the coil so that the maximum power is transferred to the sample from RF source and from the sample to the receiver. Since both the power transfer from the RF source and the sample to the receiver is done in the coil we can rite that the power is the

\[P=VI\]

were P is power, V is the voltage, and I is the amperage. Now an EMF is generated in the coil when the magnetization processes. ANd EMF is considered to be a pure voltage source and is measure in volts. However the coil has a finite resistance to it which is related to the voltage by

\[V=IR\]

so the total voltage measurable voltage is then

\[V_{tot}=V_{EMF}-IR\]

Now measuring a voltage is fairly straightforward and will not be discussed. However The concept of impedance needs to be addressed. Impedence is another type of resistance that stems from a phase lag between the current and the voltage. Specifically, the current lags behind the voltage. As implied by the term "phase" the impedance is a complex quantity, with the rela part being the resistance and the imaginary part being called the reactance. Reactance is the opposition of a circuit element to a change in the current or voltage.

We can then show the Impedance, Z, is then

\[Z=\abs{Z}e^{i \theta}=R+iX\]

where \(\theta\) is the thase shift in polar coordinates, and X is the reactance.

Then

\[V=IZ\]

Now our total voltage can be reordered to show

\[V_{tot}=V_{EMF}-IZ\]

Lets first start with our coil. The coil is an inductor has an induction of L given by the coil dimensions as

\[L=\frac{r^2n^2}{9r+10l}\]

where r is the radius of the coil, n is the number of turns, and l is the length of the coil. In our coil we will have an oscillating magnetic field which generates both an oscillating current and voltage which can be described as

\[V(t)=V_0 cos \omega t\]

\[I(t)=\frac{V_0}{\omega L} sin \omega (t)\]

we can then immediately see that the voltage and current are 90 degrees out of phase leading to a completely imaginary impedence

\[Z_L=i \omega L\]

There are several points to this. First, we now see that the current is frequency dependent, that is at high frequencies the current is very small, but at low frequencies the current is maximized. This has ramification for both application of power to the coil and detection as high frequency nuclei will then be harder to detect! Remember that we are trying to measure a voltage and that any decrease in current will directly decrease the voltage and therefore the signal that we receive! The inductor also has a finite resistance, allowing it to behave as a resistor as well. For a oscillating voltage the current is then

\[I(t)=\frac{V_0}{R}cos \omega t\]

assuming the time dependent voltage is of the form

\[V(t)=V_0cos \omega t\]

as we can see there is no phase shift so the impedance of the resistor is then

\[Z_R=R\]

and the current is directly related to the voltage. We have now reached quite a predicament in the high frequency range. That is the power scales down significantly when we go to high frequency. Therefore we need some way to "reverse" this effect. One way to do this is to use a capacitor. The current flowing across a capacitor is given by

\[-C \omega V_0 sin \omega t\]

and we can see that this scales directly with frequency. Therefore capacitors allow high frequencies to pass through them and block low frequencies. Since they are 90 degrees out of phase with the voltage, they exhibit the same reactance that inductors do

## RF Circuit

### Matching and Tuning

### Cross Diodes

## Amplifier

## Detector

A simple description of the NMR detector is given in the Detectors section. The function of the NMR detector is to detect the nuclear transitions occurring in the experiment. As the magnetization is precessing, it generates an electromotive force (EMF). The induced EMF is detected through the coil and sent to the receiver.

\[EMF=\frac{d \phi}{dt}\]

where \(d\phi\) is the magnetic flux induced from the magnetization vector processing.

Let's assume our magnetization vector has length m. The time dependence of m is then

\[m(t)=|m|[sin\psi cos(\omega_0 t +\xi_0)e_x+sin\psi sin(\omega_0 t +\xi_0)e_y +cos\psi e_z\]

for a spherical basis set. The resultant magnetization at point r from the precession is

\[B(r,t)=\frac{m_0}{4\pi r^3} [3(m(t) \cdot e_r)e_r-m(t)]\]

the magnetic flux through the coil, distance r away from magnetic vector, is given by a surface integral

\[\phi(t)=\int\int B(r,t)\cdot n r drd\theta\]

Solving this integral results in

\[\phi(t)=-\frac{m_0}{4\pi}\int_0^{r_{coil}}\frac{dr}{r^2}\int_0^{2\pi} d\theta m(t) \cdot n\]

which can be simplified if the coil is positioned such that it is in the x-z or y-z plane (n=e_{y} or e_{x}). (check math).

Interestingly,using only one coil, we are only able to detect the the magnetization is precessing, not which direction it is precessing. We circumvent this issue by employing quadrature detection.

### Quadrature Detection

To find the direction of the magnetization precession, the NMR signal is split into 2 parts and applying a phase to each part.

Relation to rotating frame

Problems with imperfect phases

reference coherence pathways

describe how its done

## Shims

The shims are used to obtain a homogenous field around the sample. They are named after the thin metal pieces NMR technicians would insert into the magnetic field in the to manually make a homogenous magnetic field. Nowadays, the shims are coils of wire which pass current to generate magnetic fields in certain directions. These shims are not just about the z, y and z directions, the vary across a variety of planes to ensure a very homogenous field is achieved.

If we want our magnetic field to be homogenous, then the space we want to be homogenous must be absent of any charges or current. Therefore, a homogenous magnetic field obeys the Laplace Equation, more commonly written as

\[\left(\frac{d^2}{d^2x}+\frac{d^2}{d^2y}+\frac{d^2}{d^2z}\right)B_0=0\]

Those familiar quantum mechanics may recognize this from the Schroedinger equation in the forms

\[\nabla^2\psi\]

which represents the potential energy in three dimensions. Solving this second order differential is indeed challenging. As with any field, it can be represented by a linear of any complete orthogonal basis set. A particularly useful basis set is the spherical harmonics basis set. B) under this basis set is described as

\[B_0=\sum_{n=0}^\infty \sum_{m=0}^{n} C_{nm} (\frac{r}{a})^n P_{nm}(\cos\theta)\cos[m(\phi-\psi_{nm})]\]

where \(r\) is a distance of some point, \(a\) is the bore radius of the magnet, and \(P_{nm}\) is the Legendre polynomial.

#### Zonal Harmonics

When m =0 the variation of B0 with (\phi\) goes away and the equation is reduced to

\\[B_0=\sum_{n=0}^\infty C_{n0} (\frac{r}{a})^n P_{nm}(cos\theta)\]

These inhomogeneities can be cancelled by application of the z shims.

#### Tesseral Harmonics

For the remainder of the m values, the off-z axis shims must be used.

We must then apply magnetic fields in all directions in order to cancel the inhomegenties in B_{0.}Since modern shims are coils of wire, the current must be adjusted to create the proper field to cancel to inhomogeneity. Commonly the shims are listed in cartesian coordinated.

Conversion equation

Show the Chart of conversion

#### Shim Mixing

Unfortunately, the shims are not independent, that is adjusting one shim changes the magnetic fields of the previously set shims.

## Problems

### Questions

- Derive the magnetic flux assuming the coil is in the y-z plane
- Derive the magnetic flux assuming the coil is in the x-z plane