NMR shifts in 3 He in aerogel induced by demagnetizing fields
aa r X i v : . [ c ond - m a t . o t h e r] N ov NMR shifts in He in aerogel induced by demagnetizing fields
V. V. Dmitriev +1) , M. S. Kutuzov ∗ , A. A. Soldatov + , × , A. N. Yudin + , ◦ + P. L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia ∗ Metallurg Engineering Ltd., 11415 Tallinn, Estonia × Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia ◦ National Research University Higher School of Economics, 101000 Moscow, Russia
Submitted November 21, 2018
Magnetic materials generate demagnetizing field that depends on geometry of the sample and results in ashift of magnetic resonance frequency. This phenomenon should occur in porous nanostructures as well, e.g.,in globally anisotropic aerogels. Here we report results of nuclear magnetic resonance (NMR) experimentswith liquid He confined in anisotropic aerogels with different types of anisotropy (nematic and planar aero-gels). Strands of aerogels in pure He are covered by a few atomic layers of paramagnetic solid He whichmagnetization follows the Curie-Weiss law. We have found that in our samples the NMR shift in solid Heis clearly seen at ultralow temperatures and depends on value and orientation of the magnetic field. Theobtained results are well described by a model of a system of non-interacting paramagnetic cylinders. Theshift is proportional to the magnetization of solid He and may complicate NMR experiments with superfluid He in aerogel.
1. INTRODUCTION
As it was shown by C. Kittel [1], demagnetizingfields may result in an additional frequency shift in mag-netic resonance experiments at large values of the sam-ple magnetization. The spin susceptibility of liquid Heis small, and in this case Kittel shifts in bulk sampleswere observed in experiments with spin polarized He[2, 3] or with thin He films [4, 5]. In the latter case theshift is due to the presence of few ( ∼
2) atomic layersof solid paramagnetic He adsorbed on the surface. Inresult, the overall nuclear magnetic resonance (NMR)signal from He, containing both liquid and solid com-ponents, is observed as a single NMR line (due to fastspin exchange mechanism [5]) with the frequency shift asweighted average of those in liquid and solid fractions of He. Solid layers follow the Curie-Weiss law [6, 7, 8] andtheir magnetization (as well as the Kittel shift) at lowtemperatures may be large. The Kittel shift also maybe observable in normal He confined in different nanos-tructures, e.g., in aerogels consisting of nanostrands. Inglobally isotropic aerogel the average shift is zero, butit may appear in the anisotropic sample.Here we study the Kittel effect in pure liquid He intwo different globally anisotropic nanostructures calledbelow nematic and planar aerogels. Nematic aerogelconsists of nearly parallel strands, while in planar aero-gel the strands are uniformly distributed in a plane per-pendicular to the symmetry axis. The solid He ad-sorbed on the strands can be considered as a system ofindependent cylindrical surfaces oriented either along e-mail: [email protected] one direction (in nematic aerogel) or chaotically dis-tributed in the plane (in planar aerogel). We use thismodel to calculate the Kittel frequency shifts in He forboth cases and to interpret our experimental results.
2. THEORETICAL MODEL
The total magnetic susceptibility of He in aerogelis a sum of the susceptibilities of liquid ( χ l ) and solid( χ s ) He: χ = χ l + χ s = χ l + C s T − Θ , (1)where C s is the Curie constant, Θ is the Curie tem-perature of solid He, χ i ≡ M aeroi /H , H is an externalmagnetic field, M aeroi is a total magnetic moment of liq-uid ( i = l ) or solid ( i = s ) He per unit volume of the aerogel sample. χ l is temperature independent in nor-mal liquid He and may only decrease with temperaturein superfluid He [9], so at low temperatures the NMRsignal from solid He can prevail over that from liquid He.In experiments with pure He in aerogel the commonNMR line has the following frequency shift [5]:∆ ω ′ = χ l ∆ ω l + χ s ∆ ω s χ l + χ s , (2)where ∆ ω l is a frequency shift in liquid He, ∆ ω s is aKittel frequency shift in solid He. Here all the shifts aremeasured from the Larmor frequency ω L = γH , where γ is the gyromagnetic ratio of He.1
V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, A. N. Yudin
In the first approximation the solid He adsorbed onthe strands of nematic and planar aerogels is a combina-tion of cylindrical surfaces. The frequency shift in solid He at a separate strand [3] is∆ ω s = πγM cyls (cid:0) − ϕ (cid:1) , (3)where M cyls is a magnetization of the solid He on acylinder surface, ϕ is an angle between H and the cylin-der axis. In nematic aerogel strands are almost parallelto one another, so the mean frequency shift in He ad-sorbed on nematic aerogel is given by∆ ω s = πγ χ s s V δ H (cid:0) − ϕ (cid:1) , (4)where s V is an effective surface area per unit volumeof the aerogel, δ is a thickness of solid He layers. Theshift is positive for ϕ = 0, negative for ϕ = π/
2, whilethe ratio of the corresponding absolute values is 2.In planar aerogel strands are parallel to the distin-guished plane. After averaging over angular distributionof the non-interacting strands in the plane we get thefollowing value of the frequency shift:∆ ω s = πγ χ s s V δ H (cid:18)
32 sin ϕ − (cid:19) , (5)where ϕ is an angle between H and the normal to theplane. In contrast to the case of nematic aerogel theshift is negative for ϕ = 0, positive for ϕ = π/
2, whilethe ratio of the corresponding absolute values is still 2.
3. SAMPLES AND METHODS
In the experiments as a nematic aerogel we haveused a nanomaterial called nafen [10] produced by ANFTechnology Ltd. It consists of Al O strands whichare oriented along the same direction, have diameters d ≈ ∼ ρ = 243 mg/cm ,porosity p = 93 . ℓ ≈
40 nm and nafen-910 with ρ = 910 mg/cm , p =78%, ℓ ≈
20 nm. The sample of nafen-910 was obtainedfrom nafen with density of 72 mg/cm by a techniquedescribed in Ref. [12].The sample of planar aerogel was produced from analuminum silicate (mullite) nematic aerogel consistingof strands with d ≈
10 nm (see Ref. [13]). It is a fibrousnetwork in the plane with p = 88%, ρ = 350 mg/cm ,and with characteristic lengths of separate strands of ∼ µ m which is much bigger than their diameters.The spin diffusion measurements in normal He con-fined by these nanostructures [13, 14] confirm theirstrong anisotropy. / l T (mK)
Fig. 1. The total magnetic susceptibility of pure Hein nafen-243 normalized to χ l in normal He. ϕ = 0, P = 7 . ω L / (2 π ) = 880 kHz. The solid curveis a fit to Eq. (1). Θ = 0 .
37 mK, χ s /χ l ≈ . T = T c = 1 .
643 mK / ( ) ( H z ) T/T c T ca Fig. 2. The original frequency shifts of pure He innafen-243 in lower ( ω L / (2 π ) = 361 kHz, circles, left y -axis) and higher ( ω L / (2 π ) = 880 kHz, triangles, right y -axis) magnetic fields. ϕ = 0, P = 7 . He in aerogel T ca ≈ . T c . Solid curves are fits toCurie-Weiss law at T > T ca Samples of nafen had a form of cuboid with a side4 mm, the sample of planar aerogel was a stack of threeplates with thickness ≈ × ∼ MR shifts in He in aerogel induced by demagnetizing fields He was about0.01% in experiments with nafen and 0.07% in experi-ments with planar aerogel, which for the surfaces insidethe experimental chamber dominated by the heat ex-changer with area of ≈
40 m corresponds to preplatingof aerogel strands with ∼ . ∼ . He respectively. We were able to rotate H by any angle ϕ defined in the previous section. Additional gradientcoils were used to compensate the magnetic field inho-mogeneity. The necessary temperatures were obtainedby a nuclear demagnetization cryostat and measured bya quartz tuning fork. Below the superfluid transitiontemperature ( T c ) of bulk He the fork was calibrated byLeggett frequency measurements in bulk He-B. Above T c the temperature was determined in assumption thatthe resonance linewidth of the fork in normal He isinversely proportional to the temperature [16].
4. RESULTS AND DISCUSSIONS
In all samples the measured magnetic susceptibility,determined from the intensity of the NMR absorptionline, has a clear Curie-Weiss behavior (Fig. 1). Due tothe fast exchange between liquid and solid He atomswe observe a single NMR line. In Fig. 2 examples oftemperature dependencies of the frequency shift in Hein nafen-243 are shown. In high magnetic field the shift(triangles) is mostly determined by the Kittel shift fromthe surface solid He, while at lower field the kink is ob-served on the data (circles) indicating a transition tosuperfluid He. At
T < T c the magnetic susceptibilityof solid He in the sample χ s ≫ χ l , so when liquid Hein aerogel is normal (∆ ω l = 0) from Eqs. (1,2,4) it fol-lows that ∆ ω ′ ≈ ∆ ω s ∝ / ( T − Θ) (solid lines in Fig. 2).In superfluid He the frequency shift is usually inverselyproportional to the magnetic field ∆ ω l ∝ /H [9], whilein the solid He on the aerogel strands ∆ ω s ∝ H ac-cording to Eqs. (4,5). Therefore, low magnetic fields inNMR experiments allow to get rid of the Kittel effectoriginating from anisotropy of the aerogel and to investi-gate purely superfluid properties of He in aerogel. Onthe other hand, in high magnetic fields the superfluidfrequency shift can be significantly suppressed with re-spect to the Kittel shift.The Kittel effect is more clearly manifested in nafen-910 which is denser than nafen-243. Superfluidity of Hein presence of solid He on the aerogel strands is com-pletely suppressed in nafen-910 [17], so using Eq. (2) wecan determine ∆ ω s from measurements of ∆ ω ′ down tothe lowest attained temperatures (see Fig. 3). The shiftis positive in the magnetic field parallel to the nafen s / ( ) ( H z ) s / l s / ( ) ( H z ) s / l Fig. 3. The Kittel shift in solid He in nafen-910 versus χ s /χ l at P = 7 . ϕ = 0 (circles) and ϕ = π/ P = 29 . ϕ ≈ ◦ in the inset). ω L / (2 π ) = 361 . ω L / (2 π ) = 78 . T , e.g.,triangles correspond to temperatures from 23 mK downto ∼ . ≈ .
9. Dashed lines are theoretical predictions(see text) s / ( ) ( H z ) s / l Fig. 4. The Kittel shift in solid He in planar aerogelversus χ s /χ l at s.v.p. ( P ≈ ϕ = 0 (cir-cles), ϕ = π/ ϕ ≈ ◦ (squares).Triangles and squares ( ω L / (2 π ) = 588 . ω L / (2 π ) = 1303 . T , e.g., circles correspond to temperaturesfrom 14 mK down to ∼ . ≈ .
1. Dashed lines aretheoretical predictions (see text)
V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, A. N. Yudin anisotropy axis ( ϕ = 0) and negative in the transversedirection of the field ( ϕ = π/ ≈ . ϕ = 2 / ω s using Eq. (4),we need to know values of χ s , s V , and δ . First, χ s canbe found from measurements of the total magnetizationof the sample that allows to determine χ s /χ l . We notethat χ l = pχ bulkl ≈ . · − emu, where p = 0 .
78 isthe porosity of nafen-910 and χ bulkl = 5 . · − emuis the magnetic susceptibility in bulk normal He at P = 7 . s V can be estimated in the as-sumption that the nafen strands are ideal cylinders. Inthis case s V = d ρρ ≈
102 m /cm , where d ≈ ρ = 910 mg/cm is nafen-910 den-sity, and ρ = 3 .
95 g/cm is Al O density. Third, thesolid He on the surfaces at P = 7 . ≈ . . Hecrystal at low temperatures) that gives δ ≈ . χ s , s V , and δ ( ± He inplanar aerogel (see Fig. 4) are also in agreement withthe theory (Eq. (5)). The shift for ϕ = 0 is negativeand ≈ . ϕ = π/
2) which is positive. At sin ϕ = 2 / χ l = pχ bulkl ≈ . · − emu (where p = 0 . χ bulkl = 3 . · − emu at s.v.p.), s V = d ρρ ≈
47 m /cm (where d ≈
10 nm, ρ = 350 mg/cm is pla-nar aerogel density, and ρ ≈ is mullite den-sity), and δ ≈ . He [18] (here we assume that ∼ . He is replaced by He [19] due to a rather“dirty” He with 0.07% of He used in the experiment).
5. CONCLUSIONS
We have observed NMR shifts due to the Kittel effectin He confined in aerogel-like nanostructures with dif-ferent types of the global anisotropy and demonstratedthat values of the shift well agree with the theoreticalexpectations. At ultralow temperatures even in moder-ate magnetic fields these shifts may be large enough to mask the He superfluid transition but can be avoidedby using lower magnetic fields or by choosing the properangle between the axis of the anisotropy and the mag-netic field.This work was supported by grant of the RussianScience Foundation (project
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