Dynamics of Zeeman and dipolar states in the spin locking in a liquid entrapped in nano-cavities: Application to study of biological systems

Abstract

We analyze the application of the spin locking method to study the spin dynamics and spin-lattice relaxation of nuclear spins-1/2 in liquids or gases enclosed in a nano-cavity. Two cases are considered: when the amplitude of the radio-frequency field is much greater than the local field acting the nucleus and when the amplitude of the radio-frequency field is comparable or even less than the local field. In these cases, temperatures of two spin reservoirs, the Zeeman and dipole ones, change in different ways: in the first case, temperatures of the Zeeman and dipolar reservoirs reach the common value relatively quickly, and then turn to the lattice temperature; in the second case, at the beginning of the process, these temperatures are equal, and then turn to the lattice temperature with different relaxation times. Good agreement between the obtained theoretical results and the experimental data is achieved by fitting the parameters of the distribution of the orientation of nanocavities. The parameters of this distribution can be used to characterize the fine structure of biological samples, potentially enabling the detection of degradative changes in connective tissues.

Keywords: Nano-cavity; Spin locking; Spin-lattice relaxation; Zeeman and dipolar reservoirs.

Fig. 1

A nano-cavity containing water molecules. Here φ is the angel between the main axis Z_c of the cavity and external magnetic field H ₀, θ is the angle between the Z₀ -axis given by an averaged orientation of the nano-cavity main axes and the magnetic field, $\zeta$ and $\xi$ are the polar and azimuthal angles characterizing the deviation of the main axis Z_c of a cavity from the Z₀ axis.

Fig. 2

Time and angle dependences of the inverse dimensionless spin temperatures of (a) the Zeeman and (b) dipolar reservoirs, calculated using Eqs. (23a) and (23b), respectively.

Fig. 2

Time and angle dependences of the inverse dimensionless spin temperatures of (a) the Zeeman and (b) dipolar reservoirs, calculated using Eqs. (23a) and (23b), respectively.

Fig. 3

Time dependences of the inverse dimensionless spin temperature of the Zeeman reservoir, calculated using Eq. (23a) (red, solid curve - at θ=0.8 rad and black dotted - at θ=0.82 rad) and time dependences of the inverse dimensionless spin temperature of the dipolar reservoir calculated using Eq. (23b) (green, dot-dashed curve - at θ=0.8 rad and blue dashed - at θ=0.82 rad)

Fig. 4

(a) Time and angle dependences of the inverse dimensionless spin temperature of the Zeeman reservoir, calculated using Eq. (24a); (b) These dependences of the difference between inverse dimensionless spin temperatures of the Zeeman and the dipolar reservoirs, calculated using Eqs. (24a) and (24b).

Fig. 4

(a) Time and angle dependences of the inverse dimensionless spin temperature of the Zeeman reservoir, calculated using Eq. (24a); (b) These dependences of the difference between inverse dimensionless spin temperatures of the Zeeman and the dipolar reservoirs, calculated using Eqs. (24a) and (24b).

Fig. 5

The normalized NMR signals of tendon at different strength of spin-locking field is calculated using Eq. (26) and the experimental data from [40]: ${\omega }_1$ = 500 Hz – green solid curve presents theoretical results and green triangles are the experimental data; ${\omega }_1$ = 1000 Hz – red dashed curve (theoretical results) and red squares (the experimental data); ${\omega }_1$ = 3000 Hz – blue dotted curve (theoretical results) and blue circles (the experimental data); ${\omega }_1$ = 5000 Hz – navy dashed-dotted curve presents theoretical results and navy pentagons are the experimental data. Note that error bars for the experimental date are slowly higher than the symbol sizes.

Fig. 6

Time dependences of the Zeeman and dipolar inverse temperatures in the case of high strength of the spin-locking field (${\omega }_1\gg {\omega }_{loc}$).