^{1,a)}, K. A. Earle

^{2}, A. Mielczarek

^{3}, A. Kubica

^{4}, A. Milewska

^{4}and J. Moscicki

^{4}

### Abstract

A general theory of lineshapes in nuclear quadrupole resonance (NQR), based on the stochastic Liouville equation, is presented. The description is valid for arbitrary motional conditions (particularly beyond the valid range of perturbation approaches) and interaction strengths. It can be applied to the computation of NQR spectra for any spin quantum number and for any applied magnetic field. The treatment presented here is an adaptation of the “Swedish slow motion theory,” [T. Nilsson and J. Kowalewski, J. Magn. Reson.146, 345 (2000)10.1006/jmre.2000.2125] originally formulated for paramagnetic systems, to NQR spectral analysis. The description is formulated for simple (Brownian) diffusion, free diffusion, and jump diffusion models. The two latter models account for molecular cooperativity effects in dense systems (such as liquids of high viscosity or molecular glasses). The sensitivity of NQR slow motion spectra to the mechanism of the motional processes modulating the nuclear quadrupole interaction is discussed.

This work was supported by funds for science in years 2009–2012 as research Project No. N N202 105936 (Polish Ministry of Science and Education). K.A.E. thanks the University at Albany for partial support of this work through a Faculty Research Award Program grant.

I. INTRODUCTION

II. INTERMEDIATE MOTIONAL RANGE

III. SLOW MOTION APPROACH TO NQR SPECTRA

IV. NQR SPECTRA UNDER DIFFERENT MOTIONAL CONDITIONS

V. MORE COMPLEX MODELS OF MOLECULAR TUMBLING

VI. CONCLUSIONS

### Key Topics

- Quadrupoles
- 48.0
- Diffusion
- 24.0
- Electron paramagnetic resonance spectroscopy
- 18.0
- Zeeman effect
- 9.0
- Magnetic fields
- 8.0

## Figures

^{2}H spectra at *B* _{0} = 0.1*T* for *a* _{ Q } = 220 kHz, η = 0 for different correlation times τ_{ R }.

^{2}H spectra at *B* _{0} = 0.1*T* for *a* _{ Q } = 220 kHz, η = 0 for different correlation times τ_{ R }.

(a) ^{14}N spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3.6 MHz, η = 0 for different correlation times τ_{ R }. (b) ^{14}N spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3.6 MHz, τ_{ R } = 0.5 μs for different asymmetry parameters η.

(a) ^{14}N spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3.6 MHz, η = 0 for different correlation times τ_{ R }. (b) ^{14}N spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3.6 MHz, τ_{ R } = 0.5 μs for different asymmetry parameters η.

^{35}Cl spectra at *B* _{0} = 1*T* for *a* _{ Q } = 30 MHz, η = 0 for different correlation times τ_{ R }.

^{35}Cl spectra at *B* _{0} = 1*T* for *a* _{ Q } = 30 MHz, η = 0 for different correlation times τ_{ R }.

^{79}Br (solid lines) and ^{35}Cl (dashed lines) spectra at *B* _{0} = 1*T* for *a* _{ Q } = 30 MHz, τ_{ c } = 50 *ns* for different quadrupole asymmetry parameters η.

^{79}Br (solid lines) and ^{35}Cl (dashed lines) spectra at *B* _{0} = 1*T* for *a* _{ Q } = 30 MHz, τ_{ c } = 50 *ns* for different quadrupole asymmetry parameters η.

(a) ^{2}H spectra at *B* _{0} = 0.1*T* for *a* _{ Q } = 220 kHz, η = 0, τ_{ R } = 0.5 μs for the free diffusion model of Eq. (7) for different correlation times τ. (b) ^{2}H spectra at *B* _{0} = 0.1*T* for *a* _{ Q } = 220 kHz, η = 0, τ_{ R } = 0.5 μs for the jump diffusion model of Eq. (8) for different correlation times τ.

(a) ^{2}H spectra at *B* _{0} = 0.1*T* for *a* _{ Q } = 220 kHz, η = 0, τ_{ R } = 0.5 μs for the free diffusion model of Eq. (7) for different correlation times τ. (b) ^{2}H spectra at *B* _{0} = 0.1*T* for *a* _{ Q } = 220 kHz, η = 0, τ_{ R } = 0.5 μs for the jump diffusion model of Eq. (8) for different correlation times τ.

^{14}N spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3.6 MHz, η = 0.4, τ_{ R } = 50 *ns* for different correlation times τ; solid lines—free diffusion model of Eq. (7) and dashed lines—jump diffusion model of Eq. (8).

^{14}N spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3.6 MHz, η = 0.4, τ_{ R } = 50 *ns* for different correlation times τ; solid lines—free diffusion model of Eq. (7) and dashed lines—jump diffusion model of Eq. (8).

(a) ^{2}H spectrum at *B* _{0} = 0.1*T*, *a* _{ Q } = 220 kHz, η = 0 τ_{ R } = 0.5 μs, τ = 2 μs for the free diffusion model of Eq. (7) (red line) compared with corresponding ^{2}H spectra for the simple diffusion model for (a) τ_{ R } = 1.5 μs, (b) τ_{ R } = 2.0 μs, (c) τ_{ R } = 2.5 μs, (d) τ_{ R } = 3.0 μs. (b) ^{2}H spectrum at *B* _{0} = 0.1*T*, *a* _{ Q } = 220 kHz, η = 0 τ_{ R } = 0.5 μs, τ = 3 μs for the jump diffusion model of Eq. (8) compared with corresponding ^{2}H spectra for the simple diffusion model for (a) τ_{ R } = 1.5 μs, (b) τ_{ R } = 2.0 μs, and (c) τ_{ R } = 4.0 μs.

(a) ^{2}H spectrum at *B* _{0} = 0.1*T*, *a* _{ Q } = 220 kHz, η = 0 τ_{ R } = 0.5 μs, τ = 2 μs for the free diffusion model of Eq. (7) (red line) compared with corresponding ^{2}H spectra for the simple diffusion model for (a) τ_{ R } = 1.5 μs, (b) τ_{ R } = 2.0 μs, (c) τ_{ R } = 2.5 μs, (d) τ_{ R } = 3.0 μs. (b) ^{2}H spectrum at *B* _{0} = 0.1*T*, *a* _{ Q } = 220 kHz, η = 0 τ_{ R } = 0.5 μs, τ = 3 μs for the jump diffusion model of Eq. (8) compared with corresponding ^{2}H spectra for the simple diffusion model for (a) τ_{ R } = 1.5 μs, (b) τ_{ R } = 2.0 μs, and (c) τ_{ R } = 4.0 μs.

(a) Experimental ^{2}H NMR spectra for glycerol-h_{5} taken from Ref. 56—solid black lines; fits by means of the simple diffusion model: *a* _{ Q } = 167 kHz, τ_{ R } = 0.63 μs (241 K), and τ_{ R } = 1.2 μs (236 K)—dashed red lines. (b) Experimental ^{2}H NMR spectrum for glycerol-h_{5} taken from Ref. 56—solid black line; calculations by means of the simple diffusion model: *a* _{ Q } = 167 kHz, τ_{ R } = 5.0 μs (from Ref. 56)—dotted red line, τ_{ R } = 3.6 μs–-solid red line, free diffusion model: τ_{ R } = 4.1 μs, τ = 2.0 μs—blue dashed line and jump diffusion model: τ_{ R } = 4.6 μs, τ = 0.5 μs—green dashed line.

(a) Experimental ^{2}H NMR spectra for glycerol-h_{5} taken from Ref. 56—solid black lines; fits by means of the simple diffusion model: *a* _{ Q } = 167 kHz, τ_{ R } = 0.63 μs (241 K), and τ_{ R } = 1.2 μs (236 K)—dashed red lines. (b) Experimental ^{2}H NMR spectrum for glycerol-h_{5} taken from Ref. 56—solid black line; calculations by means of the simple diffusion model: *a* _{ Q } = 167 kHz, τ_{ R } = 5.0 μs (from Ref. 56)—dotted red line, τ_{ R } = 3.6 μs–-solid red line, free diffusion model: τ_{ R } = 4.1 μs, τ = 2.0 μs—blue dashed line and jump diffusion model: τ_{ R } = 4.6 μs, τ = 0.5 μs—green dashed line.

^{35}Cl spectra at *B* _{0} = 1*T* for *a* _{ Q } = 60 MHz, η = 0 for different correlation times τ_{ R }.

^{35}Cl spectra at *B* _{0} = 1*T* for *a* _{ Q } = 60 MHz, η = 0 for different correlation times τ_{ R }.

^{79}Br spectra at *B* _{0} = 1*T* for *a* _{ Q } = 30 MHz, η = 0 for different correlation times τ_{ R }.

^{79}Br spectra at *B* _{0} = 1*T* for *a* _{ Q } = 30 MHz, η = 0 for different correlation times τ_{ R }.

^{17}O spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3 MHz, η = 0 for different correlation times τ_{ R }.

^{17}O spectra at *B* _{0} = 1*T* for *a* _{ Q } = 3 MHz, η = 0 for different correlation times τ_{ R }.

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