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Vyacheslav
Kuzmin
*^{a},
Kajum
Safiullin
*^{a},
Gleb
Dolgorukov
^{a},
Andrey
Stanislavovas
^{a},
Egor
Alakshin
^{a},
Timur
Safin
^{a},
Boris
Yavkin
^{a},
Sergei
Orlinskii
^{a},
Airat
Kiiamov
^{a},
Mikhail
Presnyakov
^{b},
Alexander
Klochkov
^{a} and
Murat
Tagirov
^{ac}
^{a}Institute of Physics, Kazan Federal University, 420008 Kazan, Russian Federation. E-mail: slava625@yandex.ru; kajum@inbox.ru; Fax: +7 843 233 7355; Tel: +7 843 233 7306
^{b}National Research Centre “Kurchatov Institute”, 123182 Moscow, Russian Federation
^{c}Academy of Sciences of the Republic of Tatarstan, Institute of Perspective Research, 420111 Kazan, Russian Federation

Received
29th August 2017
, Accepted 5th December 2017

First published on 5th December 2017

In this article a method to assess the location of paramagnetic centers in nanodiamonds was proposed. The nuclear magnetic relaxation of adsorbed ^{3}He used as a probe in this method was studied at temperatures of 1.5–4.2 K and magnetic fields of 100–600 mT. A strong influence of the paramagnetic centers of the sample on the ^{3}He nuclear spin relaxation time T_{1} was found. Preplating the nanodiamond surface with adsorbed nitrogen layers allowed us to vary the distance from ^{3}He nuclei to paramagnetic centers in a controlled way and to determine their location using a simple model. The observed T_{1} minima in temperature dependences are well described within the frame of the suggested model and consistent with the concentration of paramagnetic centers determined by electron paramagnetic resonance. The average distance found from the paramagnetic centers to the nanodiamond surface (0.5 ± 0.1 nm) confirms the well-known statement that paramagnetic centers in this type of nanodiamond are located in the carbon shell. The proposed method can be applied to detailed studies of nano-materials at low temperatures.

The application of NV centers as nanoscale sensors of the surrounding environment is an actively developing field of research. Shallow NV centers have recently been applied as atomic-sized NMR sensors^{8} that allow the performance of nanoscale NMR with 1 ppm resolution.^{9} Experimental realization of the predefined near-surface deposition of NV centers could be done using, e.g. the δ-doping technique which allows one to form arrays of NV centers with exceptional spin coherence properties.^{10,11} The experimental technique that allows one to determine the actual depth of the NV center deposition inside the nanoparticle is in high demand.

^{3}He is a premium candidate for studies of the magnetic properties of nanosized samples at low temperatures due to the absence of a nuclear quadrupole moment and sufficiently long intrinsic T_{1} relaxation times. Therefore ^{3}He nuclear magnetic relaxation can be sensitive to the magnetism of the sample. Properties of adsorbed ^{3}He are well understood and it is commonly used as a model system of 2D solids with quantum exchange at low temperatures.^{12–14} Usually it is assumed that nuclear magnetic relaxation in this system occurs due to dipole–dipole interactions modulated by quantum exchange or thermal 2D motions.^{14,15} There are also many studies of direct dipole–dipole interactions between adsorbed ^{3}He and nuclear^{16,17} spins or indirect interactions with electronic spins inside the solids.^{18,19} Similar direct dipolar couplings were also observed in liquid ^{3}He surrounding solid samples.^{20,21}

In this article we present a new technique that provides detailed information on the magnetic properties of the surface layer of solids and the distribution of paramagnetic impurities in nanodiamonds. It is based on the measurement of relaxation times of adsorbed ^{3}He distant from the sample surface by variable adsorbed layer thickness of noble or inactive gases. The observed minima in T_{1} temperature dependences provide direct information on correlation times and local magnetic fields.^{12,22,23} The proposed method allows the variation of the distance between the ^{3}He layer and paramagnetic centers and allows one to obtain information on its distribution in nanoparticles.

Pulsed ^{3}He NMR experiments were performed at 1.5–4.2 K temperatures and in the range of resonance frequencies f_{0} = 5–19 MHz using a laboratory made NMR spectrometer^{24} with a glass cryostat. The purity of ^{3}He used in the NMR experiments was better than 99.99%. The ^{3}He spin–lattice relaxation time T_{1} was measured by a saturation-recovery technique. The ^{3}He nuclei transverse magnetization relaxation time T_{2} was obtained by a standard Hahn echo sequence.

Some experiments were performed with the sample surface covered by a certain amount of solid N_{2} layers. The presence of nitrogen nuclei (^{14}N I = 1) with a quadrupole moment does not have an influence on ^{3}He nuclei magnetization relaxation as the corresponding cross-relaxation process is expected at lower NMR frequencies (≈3.48 MHz).^{25} Careful annealing of nitrogen layers was done in order to prevent any effects of nonuniformity of the N_{2} layers. For this purpose the sample was connected to a large volume balloon where a necessary amount of N_{2} gas was introduced at room temperature and then the sample temperature was slowly lowered down to 4.2 K. In order to achieve homogenous nitrogen distribution on the sample surface and create proper nitrogen layers we fulfilled the following conditions during nitrogen adsorption. During a slow temperature decrease the nitrogen gas pressure was carefully managed to be significantly below saturated vapor pressure values^{26} that excluded capillary condensation. According to estimates these measures prevent any undesirable effects of capillary condensation in the pores of nanodiamond powder larger than a few nanometers in diameter.

Implementation of the Scherrer equation for a spherical shape of nanodiamonds allows us to estimate the average diamond core diameter d_{core}: d_{core} = 0.9λ/(βcosθ), where the X-ray wavelength λ = 0.154 nm and β is the FWHM of the peak at θ. The d_{core} estimates made for three pattern peaks give almost the same values with the average d_{core} = 4.24 ± 0.06 nm.

Fig. 2 The TEM image of the nanodiamond powder sample. The interplanar distance between the core planes is 2.1 Å and between the shell layers is 3.6–3.7 Å (both are highlighted by white lines). |

The concentration of the paramagnetic centers in the nanodiamond sample is 5 × 10^{20} spin per g as was determined in the X-band at room temperature using a Cu^{2+}(DETC)_{2} sample with a known number of spins.

Spin–spin (T^{e}_{2}) and spin–lattice (T^{e}_{1}) relaxation times of the paramagnetic centers were measured at the W-band using Hahn echo decay and inversion-recovery pulse sequences, respectively. Relaxation curves were fitted by a single exponential function. The relaxation measurements were performed in the range of temperatures from 300 down to 8 K. Spin–lattice relaxation time T^{e}_{1} slowly increases from 20 to 60 μs with a decrease in temperature, whereas T^{e}_{2} remains almost constant at a value of 300 ns. The spin–lattice relaxation is almost independent at temperatures below 50 K, possibly due to a strong spin–spin interaction between near-surface paramagnetic centers (similar to the case of NV centers in the diamond^{31}), therefore we expect T^{e}_{1} on the order of 100 microseconds at temperatures of the ^{3}He NMR experiments.

Since all NMR experiments were performed with adsorbed ^{3}He accurate measurements of ^{3}He adsorption isotherms were done at a temperature of 4.2 K for all N values. The monolayer capacities of adsorbed ^{3}He on the sample surface were obtained by point A^{33,34} technique. The found ^{3}He monolayer stp quantity for N = 0 is V_{ads} = 24.1 ± 1.6 cm^{3}. Following the surface helium capacities,^{35,36} the corresponding specific surface area of our sample is 342.0 ± 22.2 m^{2} g^{−1} (^{3}He), which agrees well with the results of N_{2} BET analysis.

With the assumption of non-porous spherical particles, the obtained specific surface area value provides the estimated average particle diameter:^{28}d = 6/(ρS) = 5.51 ± 0.21 nm. This is in good agreement with XRD and TEM results.

Similar ^{3}He adsorption isotherms were measured for N = 0.96, 2.15 and 3.69 and the following ^{3}He monolayer capacities were obtained: 18.15 ± 0.19 cm^{3}, 14.40 ± 0.18 cm^{3}, and 12.11 ± 0.14 cm^{3}, respectively. All obtained ^{3}He monolayer values were used in further NMR experiments to adjust the ^{3}He adsorbed monolayer coverage fraction and to determine the exact number of nitrogen layers.

All measured ^{3}He longitudinal magnetization recovery and transverse magnetization decay curves in this work are well described by a single exponential function. In the case of the sample not preplated with N_{2} (N = 0) the relaxation rate is quite fast; T_{1} is on the order of milliseconds, which is unusually short compared with the adsorbed ^{3}He spin–lattice relaxation on various diamagnetic samples.^{37–39}

A set of temperature dependences of the ^{3}He longitudinal magnetization recovery times T_{1} were measured for various ^{3}He adsorbed layer coverages of the sample with and without nitrogen preplate. Typical measured T_{1}^{−1} temperature dependences are presented in Fig. 4. All measured temperature dependences of T_{1} and T_{2} for various N and x_{3} are presented in a log scale in the ESI.† The obtained temperature dependences have a parabolic shape versus inverse temperature. For convenience T^{−1,max}_{1} and T_{max} will denote the maximum spin–lattice relaxation rate and the corresponding temperature in the mentioned dependences. T_{max} decreases as the ^{3}He coverage fraction x_{3} increases (Fig. 5). Especially, T_{max} shifts at complete ^{3}He monolayer surface coverages x_{3} = 1.0.

Fig. 5 The relation between the temperature of the ^{3}He spin–lattice relaxation maximum rate T^{−1,max}_{1} (Fig. 4) and the adsorbed ^{3}He coverage fraction x for performed experiments of adsorbed ^{3}He on the nanodiamond powder sample with the number of preplated nitrogen layers N = 0; 0.96; 2.15; 3.69. |

The maximum values T^{−1,max}_{1} of the ^{3}He spin–lattice relaxation rate are slower for the case of the sample preliminarily covered by nitrogen layers. The obtained relation between T^{−1,max}_{1} and the number of nitrogen layers on the sample surface is displayed in Fig. 6 for various x_{3} values. Clearly, the ^{3}He relaxation rate strongly depends on the number of nitrogen layers and rapidly decreases with N. T^{−1,max}_{1} is almost independent of x_{3} except when N = 0.

Fig. 6 Dependence of the ^{3}He spin–lattice relaxation rates T^{−1,max}_{1} at x_{3} = 0.45, 0.6, 0.8 and 1.0 monolayer coverage of the nanodiamond powder sample on the number of preplated nitrogen layers N (x_{4} = 0). Results of ^{3}He–^{4}He experiments are shown as supplementary data (x_{3} = 0.45) (ESI†). Solid lines represent the fits by the relaxation model (see Appendix) and the following d_{0} values are obtained: 0.43 ± 0.06; 0.49 ± 0.08; 0.54 ± 0.08; 0.61 ± 0.10 nm for x_{3} = 0.45; 0.6; 0.8; 1.0, correspondingly. |

Similar experiments were performed with a partial substitution of adsorbed ^{3}He with ^{4}He atoms for N = 0 and N = 0.96. The x_{3} = 0.45 amount of ^{3}He was adsorbed on the sample surface and then we increased the adsorbed layer density with x_{4} amounts of ^{4}He to obtain similar layer coverages x = x_{3} + x_{4} = 0.45; 0.6; 0.8; 1.0 as before. In these experiments the observed T_{max} (x = x_{3} + x_{4}) values are the same (within experimental error) as those in pure ^{3}He experiments (x_{4} = 0) with N = 0.96 (see Fig. 5 and 6), but the ^{3}He spin–lattice relaxation rate slows down within 5–20% in general compared to that for x_{4} = 0. The observed T_{max}vs. x for N = 0 does not coincide with the one of pure ^{3}He experiments (Fig. 5).

Fig. 7 shows the measured frequency dependence of the ^{3}He spin–lattice relaxation time T_{1,min} (corresponds to T^{−1,max}_{1}) for a complete ^{3}He monolayer (x_{3} = 1.0, N = 0). The observed ^{3}He relaxation time linearly increases with the Larmor frequency ω.

The temperature dependences of spin–spin relaxation times T_{2} of the adsorbed ^{3}He were also measured at various x_{3} coverages (x_{4} = 0, see ESI†). The typical T_{2} temperature dependences are presented in Fig. 8 for the sample with nitrogen preplate N = 0.96. T_{2} values decrease as the temperature T decreases and a small T_{2} deviation is observed between x_{3} coverages.

Fig. 8 The temperature dependences of ^{3}He spin–spin relaxation times T_{2} at x_{3} = 0.45, 0.6, 0.8 and 1.0 monolayer coverage of the nanodiamond powder sample preplated with one nitrogen layer nanodiamond powder sample (N = 0.96, x_{4} = 0). The solid lines represent the fits by eqn (3) and (4) (see Discussion). |

Firstly, we shall check whether intrinsic dipole–dipole interactions in the ^{3}He film could be responsible for the observed relaxation. At the maximum relaxation rate when ωτ ≈ 1 according to a simple three dimensional picture for the BPP (Bloembergen–Purcell–Pound) model one has:

T_{1,min} ≈ ω(γ_{I}^{2}〈B_{loc}^{2}〉)^{−1}, | (1) |

Assuming that ^{3}He–^{3}He dipole–dipole interactions are responsible for relaxation the estimates of a standard deviation of a fluctuating magnetic field (eqn (1)) lie within 1–3 mT depending on the number of nitrogen layers on the ND surface. These estimates of fluctuating magnetic field are too high compared to ones previously reported for a ^{3}He ensemble (0.1 mT as found for x_{3} = 0.32 at 3.26 MHz and 1.8 K by Lusher et al.^{22}). On the other hand the temperatures T_{max} depend on ^{3}He coverage x_{3} (Fig. 5), but T^{−1,max}_{1} remains the same for a fixed N (Fig. 4). All these facts additionally point out that observed ^{3}He relaxation is not governed by ^{3}He–^{3}He dipole–dipole coupling but involves an external magnetic reservoir.

The temperatures T_{max} for a given x_{3} in pure ^{3}He experiments do not vary with the number of nitrogen layers when N ≥ 1 (Fig. 5). This means that T_{max} strongly depends on the surface diffusive motion which is linked with the ^{3}He total density x_{3}. The observed T_{max}(x) for N = 0 differs from N ≥ 1 because ^{3}He motion on a non-preplated surface is distinct. The T_{max}(x) and T^{−1,max}_{1}(N) relations in ^{3}He–^{4}He experiments (N ≥ 1) are similar to the ones of pure ^{3}He because ^{3}He atom mobility does not change as ^{4}He replaces ^{3}He (Fig. 5 and 6).

The decrease of T^{−1,max}_{1} with N (see Fig. 6) unambiguously indicates that relaxation depends on the distance from the adsorbed ^{3}He to the nanodiamond surface. The data for ^{3}He–^{4}He presented on the same figure demonstrate similar relaxation rates and reflect the same tendency. Possibly the relaxation process involves paramagnetic centers, the concentrations of which are about 5 × 10^{20} spin per g. The ^{3}He amounts in our experiments do not exceed n_{3} = 6 × 10^{20} spins at most, whereas the number of paramagnetic centers is about n_{PC} = 9 × 10^{19} (according to the sample weight), so for all experimental conditions we have:

n_{PC}γ_{S}(T^{e}_{1})^{−1} ≫ n_{3}γ_{I}T^{−1,max}_{1}. | (2) |

The condition of relaxation rate maximum (ωτ ≈ 1) gives estimates for τ ≈ 8.5 ns. This time scale is much shorter than the measured spin–lattice relaxation time of paramagnetic centers T^{e}_{1} in nanodiamonds at low temperatures (≈300 μs), and is also shorter than the spin–spin relaxation time T^{e}_{2} ≈ 300 ns. Obviously T^{e}_{2} is not a correlation time for ^{3}He nuclear relaxation because the correlation time has an activation nature at low temperature and depends on the amount of ^{3}He (see Fig. 8). The found correlation time is likely associated with the ^{3}He surface diffusion that has either an activation nature (Arrhenius-like) or quantum tunneling between adsorption sites. Thermal motional averaging is indicated by the measured ^{3}He free induction decay characteristic time T_{2}* that is proportional to the temperature: T_{2}*[μs] = (85.9 ± 1.7)·T [K] in the 1.5–4.2 K range for x_{3} = 1.0. Let us assume that correlation time is the time needed for a ^{3}He atom to diffuse from one paramagnetic center to another near the surface. Assuming D ∼ 10^{−7} cm^{2} s^{−1} in the ^{3}He layer at low temperatures^{42} and that the paramagnetic centers are located in the nanodiamond shell near the surface with a mean distance between them d ≈ 0.9 nm (corresponds to the surface mean density in assumption of surface paramagnetic centers), we get τ = d^{2}/4D ∼ 20 ns which agrees with observed correlation times.

Thus, the relaxation occurs through surface diffusional motions of nuclear spins in the magnetic fields mainly created by paramagnetic centers. This allows one to find the average distance from paramagnetic centers to the surface by applying the relaxation model described in the Appendix. This model implies the relaxation of nuclear longitudinal magnetization through paramagnetic centers distributed near the surface and intrinsic dipolar relaxation in the ^{3}He film. The observable longitudinal magnetization recovery is a single exponential because of the ^{3}He surface diffusive motion. This is different from the reported multiexponential processes of magnetization relaxation of immobile ^{1}H and ^{19}F nuclei spins at the surface of nanodiamonds^{7,43} or ^{13}C inside the nanodiamonds.^{44} A more detailed theory of nuclear magnetic relaxation in 2D fluids due to ^{3}He–^{3}He and ^{3}He–paramagnetic center dipole–dipole interactions mediated by 2D motions is given by Satoh and Sugawara.^{45} Although we consider in details in Appendix the case of fixed nuclei and paramagnetic centers it gives the same distance dependence for longitudinal relaxation rates on the distance a from the ^{3}He film to paramagnetic centers (T_{1}^{−1} ∝ a^{−4} in 2D case). Another possible mechanism of nuclear magnetic relaxation in that system is considered by Kondo et al.^{46} and Lusher et al.^{47} The authors of these works suggested that relaxation occurs due to fast spin exchange between the bulk (liquid or gas) and solid adsorbed layer where strong local magnetic fields B_{loc} are almost constant during the ^{3}He short stay in the adsorbed layer. In that case the correlation time stands for the inversed exchange frequency and the longitudinal relaxation rate is proportional to 〈B_{loc}^{2}〉 averaged over the surface. Therefore the longitudinal relaxation rate has the same distance dependence as in our model. Thus, independent of the details of relaxation mechanisms, the distance dependence for longitudinal relaxation is similar and that allows us to determine distance from the fits.

The fits of experimental data by eqn (9) and (12) (see Fig. 6) yield the average distance from paramagnetic centers to the nanodiamond surface d_{0} = 0.5 ± 0.1 nm. Here it was assumed that the distance between adsorbed N_{2} layers d_{N2–N2} is the same between each adjacent atomic N_{2} layer and is equal to 3.1 Å, as determined in numerous studies of adsorbed N_{2} structure at low temperatures (see for instance the work of Golebiowska et al.^{48}). The distance between the first adsorbed layer and the nanodiamond surface d_{1} was also taken to be equal to 3.1 Å for the sake of simplicity (close to estimates for ^{3}He on Grafoil reported by Joly et al.^{49}). The diameter of the nanoparticle was fixed to d = 5.51 nm, but we found that the fitting parameters are almost independent of d if the latter is chosen in the range 3–10 nm. In this fit we use an N-independent relaxation rate T^{−1,dd}_{1} which is attributed to the self ^{3}He–^{3}He dipole–dipole relaxation in the adsorbed ^{3}He layer (eqn (12)) and has been under study for a long time (see all references from here regarding ^{3}He experiments). The obtained T^{dd}_{1} ≈ 14 ± 2 ms from the fits is short compared to those usually observed in ^{3}He adsorbed on various amorphous non-magnetic materials such as aerogels and Vycor (0.1–1 seconds)^{36,37} and crystal powders,^{40} but similar to those observed in FSM (Folded Sheet Mesoporous materials).^{23}

The application of eqn (7) and (9) with d_{0} = 0.5 nm provides a satisfactory agreement of the model relaxation time T^{PC}_{1}(N = 0) = 4.1 ms (the case of fixed spins is applied for the sake of simplicity) with the experimental T_{1} ≈ 1 ms at a frequency of 18.8 MHz. In this estimate we assumed the average number of paramagnetic centers in each particle N_{PC} = 139 which corresponds to that defined above the mean diameter of nanoparticles and the total number of paramagnetic centers n_{PC}. Note that the measured value of T_{1} ≈ 1 ms can be obtained within this model for d_{0} = 0.3 nm which is also close to the determined above average d_{0}.

Possibly, the applied technique for distance measurement can be implemented for the determination of types of paramagnetic centers and their location in nanodiamonds by selective ^{3}He relaxation measurements using the fact that paramagnetic centers of a different nature have different dynamic parameters. As the interaction between ^{3}He and paramagnetic centers is mutual, a similar idea can be applied for the same purpose by means of selective T^{e}_{1} measurements with a variable thickness of solid layers of nitrogen (or noble gas atoms) on the nanodiamond surface isolating nuclear and paramagnetic center systems. The measured single line of the EPR spectra in our sample is accumulated for a relatively broad nanodiamond size distribution. It is known that for particles smaller than 80 nm the concentration of paramagnetic centers strongly depends on the particle size.^{50} In addition, the number of paramagnetic spins is very high and the paramagnetic spin system is strongly coupled. Therefore it makes it impossible to distinguish different types of paramagnetic centers by means of EPR and ^{3}He NMR. Such a type of experiment can be carried out in nanodiamonds with a lower concentration of paramagnetic centers and a narrow size distribution, which is not the case for our sample.

Temperature dependences of ^{3}He transverse relaxation clearly show an exponential decrease of relaxation times T_{2} with inverse temperature 1/T (Fig. 8). The observed relaxation time values and temperature behaviour are similar to that of reported 2D films of ^{3}He.^{23,45,46} Additionally we found a weaker dependence of T_{2} than of T_{1} on the number of nitrogen layers (see ESI†). This points out that the influence of intrinsic dipole–dipole relaxation in the ^{3}He film on transverse ^{3}He relaxation is stronger (eqn (13)). Moreover we note that relaxation time T_{2} in 2D is always much shorter than T_{1} in dipolar coupled systems (nuclear–nuclear and/or nuclear–electron) due to peculiarities of dipolar correlation functions in reduced dimensions.^{13,45}

At low temperatures (below 2 K) T_{2} is independent on temperature and indicates crossover from thermally activated motion to quantum tunneling as is known, for instance, from studies of ^{3}He in Vycor.^{36} A commonly used rule-of-thumb for motionally narrowed 3D systems reads:

T_{2}^{−1} = τM_{2}, | (3) |

τ^{−1} ∝ exp(−E_{a}/kT). | (4) |

Γ_{1} ∝ 〈B_{loc}^{2}〉, | (5) |

(6) |

Fig. 9 An illustration of the developed relaxation model of ^{3}He adsorbed on the nanodiamond surface. |

Note that for the fixed spins:^{51}

(7) |

Let us assume that the paramagnetic center is located at a distance d_{0} from the surface, ^{3}He is at a distance d_{1} from the surface, R_{0} is the radius of nanoparticle and R = R_{0} − d_{0} and a = d_{0} + d_{1}. Then one can get:

(8) |

(9) |

(10) |

The result given by eqn (9) can be applied to a nanoparticle preplated with nitrogen layers. For that case d_{1} should be substituted in eqn (10) by:

d_{2} = d_{1} + Nd_{N2–N2}, | (11) |

Note that in the case of large nanoparticles (ξ ≪ 1, in another words a ≪ R) one has T^{−1,PC}_{1} ∝ a^{−4} dependence of relaxation rate on the distance from paramagnetic centers to the ^{3}He layer as it follows from eqn (9).

In addition to this relaxation mechanism described above one has to take into account intrinsic dipolar relaxation in ^{3}He films (T^{dd}_{1}). Thus, the observable longitudinal relaxation time of adsorbed ^{3}He can be described by the following equation:

(12) |

(13) |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp05898e |

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