Download Introduction to NMR Spectroscopy and Imaging

Introduction to NMR Spectroscopy and Imaging
Assignments for Chapter 01 Elementary Magnetic Resonance
0. Choose the correct one(s) from the following statements or explain briefly your
supporting reason if you decide that one is incorrect:
a. Nuclear paramagnetism means the nuclear spins align along the magnetic
field and generate a macroscopic magnetization.
b. The nuclear magnetization of a sample disappears when the applied magnetic
field is removed (turned off), or the sample is removed from the magnet.
c. Normally, an NMR sample is diamagnetic, i.e., all electrons in the molecule
are paired up, or, there is no net magnetization from the electrons.
d. A quadrupole spin means it has very large magnetic moment.
e. The majority of NMR sensitive nuclei are quadrupolar.
f. Both NMR and MRI use the nuclear spins as reporters of structure and
dynamics of a system under study.
g. The total magnetization of a sample at equilibrium always points to the
direction of the applied magnetic field.
h. It may take very long time (seconds or longer) for a nuclear spin to make a
flip (change of spin direction from, e.g., along the magnetic field to
anti-parallel direction).
i. Basically an NMR signal records the history of an ensemble of nuclear spins
reaching their equilibrium state by making transitions between different
energy levels.
j. Wherever there is a magnetic field B, a spin there responds with a precession
at the frequency determined by the product of the magnitude of B and the
gyromagnetic ratio of that spin, γ.
1. Calculate the Larmor frequency of the following nuclei in a magnetic field of 4.7
Tesla:
Hydrogen-1, Deuterium-2, Carbon-13, Nitrogen-15,
Silicon-29, Phosporous-31, Rubidium-87, Platinum-195.
Oxygen-17,
Sodium-23,
2. Given that the NMR signal of a sample is 0.1 mV in a spectrometer with a
magnetic field of 4 T, what is the NMR signal of the sample in a magnetic field of 16
T, assuming that all other conditions are the same?
3. Derive the equation describing the net magnetization of a sample with N nuclear
spins with gyromagnetic ratio γ and spin quantum number I in magnetic field of B0
and at temperature T:
Nγ 2 2 I ( I  1) B0
 M 0 
3k BT
4. The strongest field commercial NMR magnet corresponds to 900 MHz for proton
Larmor frequency. Determine:
a) the magnetic field of the superconducting magnet (in Tesla)
b) the transition energy for proton relative to kBT at 300K
c) the resonance frequency for 13C and the ESR frequency for a hydrogen atom (in
Hz)
d) the wavelength of the photon associated with an ESR and an NMR transition in this
spectrometer.
5. a). For a spin I=1/2 nucleus in a magnetic field (B0 ) such that ω0= 200 MHz,
calculate the equilibrium population ratio (NA/NB) of the two spin states (α and β) at
20 oC. b). Assuming that there are 6.02x1023 magnetically identical nuclei (from "a"
above) in your sample, exactly how many of these spins make up the excess in the
α
-called magnetization, or M0)? c). Repeat the
calculation in "β" for ω0 = 600 MHz. d). Assuming that the Signal-to-Noise ratio is
solely determined by the magnitude of M0, and given that 100 scans must be averaged
to achieve a satisfactory S:N at 200 MHz, calculate the number of scans required to
achieve the same S:N ratio at 600 MHz.
6. Two 1H spins are 1, 2, 5 Angstroms apart from each other. Find the magnetic field
generated at each spin by the other. Compare this value with the surface magnetic
field produced by a typical magnet on fridge.
7. Two NMR spectrometers operate at the magnetic field of 11.74 T and 4.7 T,
respectively. Which signal(s) would most probably affect the NMR experiments on
these spectrometers: long wave radio, shortwave radio, TV, mobile phone transmitters,
Radar?
8. The diameter and length of the selenoid coil of an NMR probe are 5 mm and 2 cm
respectively. The sample is water. Calculate the 1H signal (voltage) received by the
preamplifier of the NMR spectrometer. Assume the magnetic field is 11.74 T and
temperature 300 K. Compare this value with the signal of a typical mobile phone.
9. A coil of length l, cross-sectional area A, and n turns is wound in the form of a
solenoid. The axis of the coil is in the x-direction, and a static field B0 is in the
z-direction.
a) Assuming the nuclear moments are in thermal equilibrium, what is the nuclear
magnetization per unit volume M0 produced by B0, in terms of B0 and the static
nuclear susceptibility χ0?
b) Compute a numerical value of χ0 for protons in water at room temperature, using
the formula
0 
Nγ 2 2 I ( I  1)
3kBT
where N is the number of spins per unit volume. (For protons, γ can be found from
the fact that the resonance occurs at 42 MHz for B0=104 Gauss.)
(Refer to problem 3)
c) The magnetization M0 is turned by a 90o pulse. Derive an expression for amplitude
V0 of the voltage induced in the coil by the precessing M0.
d) Make a numerical estimate of V0 for protons, assuming a 10-turn coil 2 cm long, 1
cm in diameter, and a B0 of 5000 Gauss.
10. Write an essay to explain the applications of NMR spectroscopy and imaging in
chemistry and/or materials science.