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Nuclear Quadrupole Resonance (NQR)
Nuclear Quadrupole Resonance (NQR) spectroscopy is the study of the electric field gradient in the vicinity of the quadrupolar nucleus whose nuclear spin number bigger than ½. The electrostatic potential due to the electrons and other nuclei at the immediate neighborhood of the nucleus introduces an asymmetry to the shape of charge distribution of the nucleus [Fig 1]. The asymmetry causes a small splitting in the nuclear ground state. Study of quadrupolar interaction is a method to determine the electric interaction of spin > 1/2 nuclei with the surrounding electric fields. The study of NQR is a method to determine the local environment of the nucleus and the bonding condition of which it participates. In contrast with NMR where the predominant energy splitting is caused by the nuclear interaction with an external static magnetic field, NQR experiments can be done in the static magnetic field, so called “pure NQR”, since the electric field gradient (EFG) within the material will cause the splitting itself.

Quantum mechanical description
The hamiltonian is written in the principal axis system. The Principal Axis System (PAS) is defined in a way that the Nuclear Quadrupole (NQ) hamiltonian takes its simplest form and the EFG ($$\overset{\longleftrightarrow}V$$ ) is a diagonal trace-less tensor. In other words, the PAS is chosen so that z-axis is along the direction that EFG's absolute magnitude is largest and x-axis is along the smallest absolute magnitude of EFG. The principal values of the electric field gradient tensor are:

$$V_{xx}= \frac {-1}{2}eQ(1-\eta)$$ $$V_{yy}= \frac {-1}{2}eQ(1+\eta)$$

$$V_{zz}= eQ$$

The corresponding hamiltonian can be written as:

$$  H_Q=  \frac{e^2 qQ}{4 I (2I-1)} [3I_z^2 -I ^2 + \eta ( I_x^2 - I_y^2)] $$

with the interaction energy of:

$$  E_Q=  \frac{1}{6} (\sum_{\alpha,\beta} V_{\alpha,\beta} \overset{\longleftrightarrow}Q_{\alpha,\beta} ) $$

where $$e^2qQ$$ is the quadrupole coupling constant (QCC), $$I$$ is the nuclear spin number and η is the asymmetry parameter. The quadrupole coupling constant is a product of two quantities, the magnitude of the electric field gradient along the z-axis of the principal axis system ($$eq=V_{zz}$$) and the electric quadrupole moment of the nucleus (eQ) [Fig 2], giving us a measure of the strength of the interaction. Since the nucleus is in a state with definite angular momentum (which is equivalent to the classical statement that the charge has cylindrical symmetry), only one nuclear constant is enough for all nine components of the hamiltonian.

Asymmetry parameter (η) is defined as:

$$\eta =\frac {V_{xx}-V_{yy}}{V_{zz}}$$

and takes values between 0 and 1. In case of η=0, the EFGs have an axis of cylindrical symmetry along the z axis of the EFG PAS frame. One of the goals of an NQR measurement is to determine the quadrupole coupling constant $$e^2qQ$$ and the asymmetry parameter η, which contain information about the surrounding environment of the nucleus. For spin-1 systems, the eigenvalues of $$I_z$$ are $$m_s$$ = -1, 0, and 1, and they define a linear vector space. One can write the hamiltonian using angular momentum in Zeeman basis energy transitions for spin-1 NQR:

$$H_Q =\Biggl(\begin{matrix} 1 & 0 &  \eta \\ 0 & -2 & 0 \\  \eta &0 & 1 \end{matrix} \Biggr)$$

In some literature the hamiltonian is written in terms of $$w_Q$$ which is the first order quadrupole coupling constant and has the form: $$w_Q = \frac {3eQ}{2I(2I-1)\hbar}V_{zz}$$

For nuclei with spin $$ I \ge 2$$, there are additional electric and magnetic interactions (like the ‘electric hexadecapole moment’) with the fourth derivative of the electric potential. There are also high-order magnetic interactions. In practice, all higher-order electromagnetic interactions are weak and negligible.

Strong quadrupole coupling constant in halogen nuclei
When a halogen atom is in a state with a mixture of a pure covalent bond (P-state) and an ionic bond (closed shell), the electronic wave function is a mixture of s and p-state. The quadrupole coupling to the ionic bonding vanishes which is to say the s-state does not contribute to the quadrupole coupling. According to the following table, the frequency of pure quadrupole transitions for covalently bonded halogens are much higher than that of the NMR transitions for typical achievable magnetic fields in the laboratory. Thus, quadrupole couplings can be used to study bond hybridization, degree of covalency, double bonding and so on.

Applied RF pulse in the case of spin-1
The system is excited by an oscillating magnetic dipole field chosen along the laboratory $$z'$$ axis:

$$  H_Q=  w_1 \cos (wt + \phi ) I_{z'} $$

with $$w_1 = \gamma H_1 $$and $$ H_1 $$is the RF magnetic field amplitude. The hamiltonian in the interaction frame becomes:

$$H (t) ={\displaystyle w_{1}\cos(wt+\phi )} \Biggl(\begin{matrix} 0 & -ice^{iw_zt} & ibe^{-iw_yt} \\ ice^{-iw_zt} &0&  -iae^{iw_xt} \\   -ibe^{iw_yt} & iae^{-iw_at} & 0 \end{matrix} \Biggr)$$

with a,b, and c as orientation parameters in terms of Euler angles:

$$a= \cos (\alpha)\sin (\beta) $$

$$ b=\sin(\alpha )\sin(\beta )

$$

$$ c=\cos(\beta )

$$.

The effect of the asymmetry parameter on the energy transition for the spin-1 NQR systems is shown in Fig 3. In a non-axial symmetric condition of EFGs, the $$I_{\pm}^2$$operators result in off-diagonal elements which make this basis inappropriate for NQR due to the mixing of $$\mid 1,1 >$$ and $$\mid 1,-1 >$$ states. Diagonalization of the $$  H(t) $$ is equivalent to coordinate transformation from the spherical to Cartesian basis.

By applying the interaction frame transformation, the main hamiltonian term is removed at the cost of adding additional time dependencies to the RF hamiltonian term. However, this will lead to introduction of additional oscillations at three different NQR transition frequencies. This interaction frame transformation is a rotating frame transformation around the z' axis, it induces a sinusoidal time dependence at the rotation frequency of the transformation on the other angular momentum operators. Due to the fact that the transformation of the spin components of a spin-1 system is identical to the transformation of a three-component vector in ordinary coordinate space, the excitation process can be understood through a simple physical picture.

The eigenvalues of $$H_Q$$ are the relative energies required to orient the spin-1 vector along each of the eigenvector axes of $$H_Q$$. By radiating the spin system at resonance along one of the $$H_Q$$ eigenvector axes, a quantum mechanical version of vector rotation about that axis is effected. For instance, the resonant radiation along the $$\mid z >$$ eigen-axis of $$H_Q$$ drives the transitions between the $$\mid x >$$ and $$\mid y >$$ spin states and not affecting the population of the $$\mid z >$$ state. The only resonant frequency that may be excited with the RF field along $$\mid z >$$ corresponds to the energy difference between aligning the spin state along the $$\mid x>$$ and $$\mid y >$$ directions.

NQR experimental methods
Field cycling spectrometers is used to improve the sensitivity of NQR spectrometry that contains alternative exposure of the sample to a large magnetic field and a small (or zero) magnetic field. This method is useful particularly for low frequency NQR measurements and also in cases of low natural abundance. Since large magnetic fields are difficult to turn on and off rapidly, this is commonly achieved by physically moving the sample.

Pulsed-SQUID spectrometer is used to capture maximum signal by using a coaxial pickup coil and pulse coil, but this method is limited to frequencies up to 5 MHz.

Another unique technique is acquiring NQR spectrum by means of Nitrogen-Vacancy (NV) centers using optically detected magnetic resonance (ODMR) spectroscopy. An advantage of NV-centers to other techniques is that the transition frequencies are within visible range of optical spectrum which reduces the complexity of experimental setup.

Qualitative comparison of NQR and NMR

 * 1) NQR signals will cancel out in liquid due to random tumbling motions of molecules/atoms. Thus, NQR experiments are mainly done on solids.
 * 2) The efficiency of excitation in NMR depends on the orientation of the external magnetic field and the RF field (the highest efficiency is achieved when these two are perpendicular to each other). Whereas in NQR the relative orientation of the RF field and the molecular crystal frame determines the efficiency. As a matter of fact it is not possible to excite all the crystallites in a powder crystalline sample since the crystal frame is randomly distributed in each solid crystal.
 * 3) The large NQ hamiltonian generally leads to greater NQR sensitivity caused by the higher transition frequency. But large NQ hamiltonian will cause shifts in the resonance frequency in NMR making measurements difficult. Thus, accessible systems by NQR are not necessarily accessible by NMR.
 * 4) NQR resonance frequencies have high dependencies on both temperature and crystal packing environment. Identical molecules/atoms in different environments/temperatures will have different NQR transition frequencies due to variations of electric field gradients introduced to the molecule/atom.
 * 5) For nuclei with spin less than one the NQR cannot be observed but it does not necessarily manifest as a perfect spherical-shaped nuclei.