Nuclear Quadrupole Resonance

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Nuclear Quadrupole Resonance

Nuclear Quadrupole Resonance is a general term given to descibe a small number of related techniques, all of which are aimed at detecting nuclear magnetic transitions at very low or zero field.

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The Nuclear Quadrupole Coupling Hamiltonian

Definitions and units

The electric quadrupole moment of the nucleus is generally represented in the magnetic resonance literature by the product eQ. The SI units for a quadrupole moment are Coulomb.meter² (C.m²); however, it is common to dimension out the charge component, leaving the quadrupole moment with units of area. In standard tables, the quadrupole moment of the nucleus is given in fm², or, in the older literature, in the non SI units of barns: 1 barn = 10^-28m². e is of course the charge of the electron: e = 1.60217653(14) x 10^-19C (2002 CODATA recommended value).

For example, the accepted experimental value of the deuteron quadrupole moment is 0.2859(3) fm² (1). In SI units, the quadrupole moment eQ then becomes 0.2859 X 10^-30 m² X 1.60217653 x 10^-19C = 4.581(5) x 10^-50 C.m²

The electric field gradient (EFG) at the nuclear site is a traceless symmetric second-rank tensor V = grad E. Its magnitude is generally expressed by the most distinct element V_zz (in the older literature this quantity is expressed as eq). The SI units of EFG are Volts.meter^-2(V.m^-2). This quantity can be calculated in ionic lattices by standard methods (it is, however, amplified at the nuclear site by the so called Sternheimer antishielding); it is also a property directly calculable by ab initio programs such as GAMESS and Gaussian. The latter programs give as output V_zz/e in atomic units, which are Hartree.Bohr^-2.

The electric quadrupole coupling constant is simply the product of the nuclear quadrupole moment and the EFG, converted to frequency units by dividing by Planck's constant. It is sometimes denoted by C_Q=eQV_zz/h. If Q is expressed in fm², and V_zz/e in atomic units, then a useful proportionailty factor is K_Q = 2349647.81(26) Hartree^-1 .Bohr² fm^-2 s^-1 (2), which allows us to obtain C_Q(Hz) = -V_zz/e (a.u.) X K_Q X Q (fm²). Thus, for a deuteron, a 1 a.u. EFG gives a quadrupole coupling constant of 1 X 0.2859 X 2349647.91 = 671.8(7) kHz.

The nuclear quadrupole coupling Hamiltonian, in frequency units in the principal axis frame of the EFG tensor, is now:

H = [eQV_zz/(4I(2I - 1)h)][3I_z² - I(I + 1) + η(I_x² - I_y²)]

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NQR Frequencies at zero field

If there is three fold or higher symmetry about an axis, the value of the asymmetry parameter η is zero. In this case, the Hamiltonian is diagonal in the frame of reference of the EFG, and the eigenvalues can be obtained by simple inspection. The frequencies are obtained using the standard NMR magnetic dipole selection rule $\Delta m = \pm 1$ . The frequency difference between level m and level m – 1 then becomes

$\Delta\nu = \left ( \frac{3C_Q}{4I} \right) \left (\frac{1-2m}{1-2I} \right)$

...with $I \ge m \ge -I+1$ . Some basic properties:

(1) The equation leads to pairs of positive and negative frequencies, corresponding to pairs of coherences of opposite sign. Since these pairs cannot in general be separated, the counter-rotating coherences lead to cancellation of magnetization perpendicular to the sample coil. Thus, one experimental difference between NMR and NQR is that a cross-coil arrangement of transmitter and detector in NMR, which can be used to determine the sign of the gyromagnetic ratio, leads to no signal in NQR; and as a corrolary the sign of C_Q cannot directly be determined by NQR.

(2) The 1/2 - -1/2 transition is at zero frequency and invisible.

(3) For spin 3/2, there is a single observable degenerate pair of frequencies at zero field; for spin 5/2, two degenerate pairs, and so on.

If the system is less symmetric, the asymmetry parameter will be non-zero, and the expression for the frequencies becomes complex. Specific solutions are as follows:

Spin 3/2: a single pair of frequencies at $\Delta\nu = \pm \left ( \frac{C_Q}{2} \right) \left (1+\frac{\eta^2}{3} \right)^\frac{1}{2}$ . Because there is one measurable frequency and two parameters, neither C_Q nor η can be independently determined by simple NQR, although the NQR frequency can be no more than 15.5% larger than C_Q/2

Spin 5/2: two pairs of frequencies. The zero field energies of the system are the roots of the secular equation $\left ( \frac{C_Q}{40} \right) (160 + 84\lambda-\lambda^3+4\eta^2(7\lambda-40))$ . This has analytical roots, but in practice it is probably easier to solve the roots numerically.

References

(1) Deuteron properties and the nucleon-nucleon interaction. S Klarsfeld, J Martorell and D W L Sprung. J. Phys. G. 10, 165-180 (1984).

(2)¹⁴N Quadrupolar, ¹⁴N and ¹⁵N Chemical Shift and ¹⁴N-¹H Dipolar Tensors of Sulfamic Acid. G. S. Harbison , Y.-S. Kye, G. H. Penner, M. Grandin, M.Monette. J. Phys. Chem. B 106 10285-10291 2002. Note that this reference confusingly uses the notation C_Q for K_Q. Mea maxima culpa.

--Gerard.Harbison 14:24, 16 August 2006 (CDT)

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Nuclear Quadrupole Resonance

From ISMARpedia