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10.4 Continuous WaveNMR Instrumentation

• coherent detection• bulk magnetization• the rotating frame, and effective magnetic field• generating a rotating frame, and precession in the laboratory frame• spin-lattice relaxation and T1• spin-spin relaxation, dephasing and T2• the 60 MHz cw NMR spectrometer

Coherent Detection

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Direct detection of absorption of NMR "photons" from an incoherent source is difficult because of saturation. Instead, the excess spins (Nα - Nβ) are coherently excited.

The precession of the resultant coherent magnetic moment will induce an electric current in a wire coil. This current is at the precession frequency and easily detected.

The unexcited spins do not produce a detectable current because the individual nuclei are precessing with random phase and average to zero.

A coherent source is defined as one where every photon has an identical phase. In the UV visible a single-mode laser is required to make a coherent electric field. In the radio frequency region, a simple wire coil can be used to generate a coherent magnetic field.

Bulk Magnetization

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z, B

y

x

Nα

Nβ

In the absence of radiation, the individual precession moments are randomly oriented along the precession circle. The vector sum of all of the α spins is a vector Nα in magnitude pointing in the positive z-direction. The vector sum of all the β spins is a vector Nβ pointing in the negative z-direction.

The resulting sum-vector is pointing along the positive z-axis with a magnitude of Nα - Nβ. It is this bulk magnetization vector that needs to acted upon by the applied rf source. Note that the bulk magnetization is proportional to N.

The Rotating Frame

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z, By

x

uv

If the sinusoidal magnetic field (10-7 T) is to produce coherent precession, it must do so by moving the nuclei against the strong fixed field, B0 (7 T), being used to split the spin states. How is this possible?

A rotating coordinate system (u,v) can be used. • If νrot = 0, the nuclear magnetic moments appear to rotate at the Larmor frequency, given by γB0. • As νrot is increased the nuclei appear to precess at a slower rate, e.g. as if they were in a field weaker than B0. • If νrot = γB0, the axes are rotating at the Larmor frequency. From the perspective of the rotating axes, the nuclear magnetic moments no longer seem to be precessing. That is, to anything in the rotating frame the nuclei behave as if they were in a zero strength external field.

Effective Magnetic Field

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The differing rates of precession are accounted for by the concept of an effective field.

If a magnetic field, B1 , can be made to rotate at the Larmor frequency, B0 will effectively go away and the nuclear magnetic moments can be made to rotate about B1 .

If the Larmor frequency is ~300 MHz for a 7 T magnetic, the nuclear magnetic moment will precess around a 10-7 T rf magnetic field with a frequency of 4.26 Hz.

10effB B ν= −

γ

Generating a Rotating Frame

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x

y

z, B0

B1

coils in the y,z-plane

Coils in the z,y-plane carry an electrical current oscillating at the Larmor frequency. The current induces a magnetic field, B1 , in the x-direction which is building and collapsing at the Larmor frequency.

This linearly polarized magnetic field can be decomposed into two counter propagating circularly polarized fields. One of these is rotating in the same direction as precession around B0 and can effect the NMR transition. The other is rotating counter to the precession and can't contribute to the signal (the magnetic field appears twice as strong, i.e. 2B0 ).

Precession in the Laboratory Frame

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z, B0

y

x

µN

detector coils

As the net, bulk nuclear magneticmoment precesses around the rf field, itmoves away from the z-axis. In thelaboratory frame, this movement is seenas a precession along the surface of asphere. For the example we have beenusing, the precession would be 300 MHzin the x,y-plane and 4.26 Hz in the z,x-plane.

To detect the induced precession, coilsare placed in the x,z-plane with their center along the y-axis. As theprecession moves down the spherical surface, the magnetic momentforms larger circles parallel to the x,y-plane. This circular component induces a 300 MHz current in the detector coils. The maximum detected current appears when the bulk magnetic moment is tipped90°. If the rf magnetic field continues to be applied, the vector will continuing working its way down the sphere until it points to -B0 . When this is achieved, all of the excess spins are in the β energy state and an inverted population is achieved.

Spin Lattice Relaxation

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Consider the situation where the rf was applied long enough to make Nα = Nβ, and then the rf was turned off. There is a thermodynamic drive for Nβ/Nα to return to the value predicted by the Boltzmann distribution.

In the UV/visible portion of the spectrum, both spontaneous emission and dark processes play a role in thermalizing the two states. In the NMR region spontaneous emission is very unlikelyand does not play a significant role.

Spin-lattice relaxation is a dark process that transfers energy from the excited nucleus to other magnetic nuclei on the same molecule or in adjacent molecules. The field of the two spin systems must have equal ∆E, but moving magnets have a wide distribution of frequencies (Doppler effect, ~ 1-3 ppm).

This process converts β levels into α levels. It is called longitudinal relaxation since the change is along B0. The relaxation time is called T1 and has values from 10-2 to 102 seconds.

Spin-Spin Relaxation

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Spin exchange can occur between chemically equivalent nuclei.

A(mI = +1/2) + B(mI = -1/2) ---> A(mI = -1/2) + B(mI = +1/2)

This process yields no net change in energy, but the phases havebeen changed depending upon when the exchange takes place during the precession.

Since nuclei with random phases are in a huge excess, it is mostprobable that spin exchange will occur with them. Thus, the coherent, excess spins will gradually assume the random phases of the bulk nuclei. This process is called dephasing.

Dephasing of Coherent Precession

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When coherent absorption occurs, all of the β nuclei are precessing about the external field with the same phase. As thespins dephase they spread out along the precession circle. Thisis called transverse relaxation and has the symbol, T2. Although transverse relaxation doesn't change the value of Nα - Nβ, it destroys the ability to detect the excess spins.

For samples in fluid solution, transverse relaxation is faster than spin-lattice relaxation. In a CW instrument the power is adjusted to just balance the loss of the coherent signal.

60 MHz NMR

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For proton resonance at 60 MHz the magnetic field is 60×106 = 42.6×106 B0

B0 = 1.4084507 T

The difference in γ values among elements is so large that only one type of nucleus is examined at a time. The permanent magnetic field is fixed so a different nucleus requires a different rf frequency. Thus, a 1.4 T field requires radiation at 56.5 MHz for 19F. This requires a different probe and electronics.

The exact energy of the NMR transition depends upon the electronic environment around the nucleus. The electrons shield the nucleus, in a structurally dependent way, so that the magnitude of the field at the nucleus is less than expected,

Beff = B0 (1 - σ)

where σ is the shielding constant. An approximate maximum proton chemical shift is about 17 ppm, which corresponds to an effective magnetic field of 1.4084272 T.

The Varian A-60 CW NMR

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sample

magnet magnet

sweep magnetic coils

transmittercoils

receivercoils

In the cw instrument, the transmitter is fixed at 60 MHz and the magnetic field varied by the sweep coils. For a chemical shift range of 0-17 ppm, the sweep coils would need to add from 0 to 0.00002325 T to the fixed magnetic field. This range will bring all chemically shifted protons into resonance with 60 MHz.

The receiver power is measured as the coil field is swept acrossall the NMR resonances. To maximize the signal, the transmitterpower is adjusted so that the absorption and relaxation rates are the same at the strongest absorption.

The Varian A-60 spectrometer was introduced in 1961 and was sold until ~1970.