Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry

We describe experiments on superfluid ³He in a cylinder of 1 mm in diameter. This geometry causes the preferred orientation of the n-vector in the superfluid B-phase to be locally different, resulting in a curved configuration across the sample. Exclusive to our experiment is the observation that...

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Datum:	2013
Hauptverfasser:	Benningshof, O.W.B., Jochemsen, R.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2013
Schriftenreihe:	Физика низких температур
Schlagworte:	Сверхтекучий ⁴He и ³He
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spelling	irk-123456789-1187452017-06-01T03:06:31Z Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry Benningshof, O.W.B. Jochemsen, R. Сверхтекучий ⁴He и ³He We describe experiments on superfluid ³He in a cylinder of 1 mm in diameter. This geometry causes the preferred orientation of the n-vector in the superfluid B-phase to be locally different, resulting in a curved configuration across the sample. Exclusive to our experiment is the observation that we succeeded in obtaining a texture which is metastable and unchanged in our pressure and temperature ranges, most likely because the experiment is performed at low pressures and low magnetic fields. As this texture can be considered as a potential for spin waves, we had the unique opportunity to study spin waves for several pressures in exactly the same texture. Our geometry causes this texture potential to be nearly quadratic, allowing an analytic solution of the theory which can be compared to our experimental results. As predicted we find the intensities of all spin wave modes more or less equal. Increasing the pressure shows a gradual increase in the number of spin wave modes in our cell. Finally we were able to cause a transition from the metastable to the predicted stable texture, concluding unexpectedly that the metastable texture is realized if the growing (or cooling) speed is sufficiently slow. 2013 Article Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry / O.W.B. Benningshof, R. Jochemsen// Физика низких температур. — 2013. — Т. 39, № 9. — С. 986–991. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 67.30.he, 67.30.hj, 67.30.ht http://dspace.nbuv.gov.ua/handle/123456789/118745 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Сверхтекучий ⁴He и ³He Сверхтекучий ⁴He и ³He
spellingShingle	Сверхтекучий ⁴He и ³He Сверхтекучий ⁴He и ³He Benningshof, O.W.B. Jochemsen, R. Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry Физика низких температур
description	We describe experiments on superfluid ³He in a cylinder of 1 mm in diameter. This geometry causes the preferred orientation of the n-vector in the superfluid B-phase to be locally different, resulting in a curved configuration across the sample. Exclusive to our experiment is the observation that we succeeded in obtaining a texture which is metastable and unchanged in our pressure and temperature ranges, most likely because the experiment is performed at low pressures and low magnetic fields. As this texture can be considered as a potential for spin waves, we had the unique opportunity to study spin waves for several pressures in exactly the same texture. Our geometry causes this texture potential to be nearly quadratic, allowing an analytic solution of the theory which can be compared to our experimental results. As predicted we find the intensities of all spin wave modes more or less equal. Increasing the pressure shows a gradual increase in the number of spin wave modes in our cell. Finally we were able to cause a transition from the metastable to the predicted stable texture, concluding unexpectedly that the metastable texture is realized if the growing (or cooling) speed is sufficiently slow.
format	Article
author	Benningshof, O.W.B. Jochemsen, R.
author_facet	Benningshof, O.W.B. Jochemsen, R.
author_sort	Benningshof, O.W.B.
title	Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry
title_short	Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry
title_full	Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry
title_fullStr	Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry
title_full_unstemmed	Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry
title_sort	spin waves in the b-phase of superfluid ³he in confined cylindrical geometry
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2013
topic_facet	Сверхтекучий ⁴He и ³He
url	http://dspace.nbuv.gov.ua/handle/123456789/118745
citation_txt	Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry / O.W.B. Benningshof, R. Jochemsen// Физика низких температур. — 2013. — Т. 39, № 9. — С. 986–991. — Бібліогр.: 22 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT benningshofowb spinwavesinthebphaseofsuperfluid3heinconfinedcylindricalgeometry AT jochemsenr spinwavesinthebphaseofsuperfluid3heinconfinedcylindricalgeometry
first_indexed	2025-07-08T14:34:40Z
last_indexed	2025-07-08T14:34:40Z
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fulltext	© O.W.B. Benningshof and R. Jochemsen, 2013 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9, pp. 986–991 Spin waves in the B-phase of superfluid 3 He in confined cylindrical geometry O.W.B. Benningshof and R. Jochemsen Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, the Netherlands E-mail: Jochemsen@Physics.LeidenUniv.nl Received March 4, 2013 We describe experiments on superfluid 3 He in a cylinder of 1 mm in diameter. This geometry causes the pre- ferred orientation of the n-vector in the superfluid B-phase to be locally different, resulting in a curved configu- ration across the sample. Exclusive to our experiment is the observation that we succeeded in obtaining a texture which is metastable and unchanged in our pressure and temperature ranges, most likely because the experiment is performed at low pressures and low magnetic fields. As this texture can be considered as a potential for spin waves, we had the unique opportunity to study spin waves for several pressures in exactly the same texture. Our geometry causes this texture potential to be nearly quadratic, allowing an analytic solution of the theory which can be compared to our experimental results. As predicted we find the intensities of all spin wave modes more or less equal. Increasing the pressure shows a gradual increase in the number of spin wave modes in our cell. Final- ly we were able to cause a transition from the metastable to the predicted stable texture, concluding unexpectedly that the metastable texture is realized if the growing (or cooling) speed is sufficiently slow. PACS: 67.30.he Textures and vortices; 67.30.hj Spin dynamics; 67.30.ht Restricted geometries. Keywords: superfluid B-phase, metastable texture, 3 He. Introduction The superfluid B-phase of liquid 3 He is characterized by the relative broken symmetry of the spin angular mo- mentum (S) and orbital angular momentum (L) of the Cooper pairs, of which the order parameter is proportional with the spin-orbit rotation matrix R(n, ) [1]. The angle between the two vectors is fixed, and equals the Leggett angle L 104 . The normal to the plane formed by L and S is named the rotation axis n, which is a convenient vector quantity to describe any orientation effects in the B- phase. The bulk superfluid B-phase is isotropic, conse- quently the n-vector will not have any preferred orienta- tion. However, walls and magnetic fields will introduce a preferred orientation of the n-vector. In confined geometries and in a magnetic field the orientation of the n-vector will be locally different, mean- ing that the n-vector is bent over the sample. The bending is of the typical size of the magnetic healing length ,H and forms a potential for spin waves. These spin waves can be detected by transverse Nuclear Magnetic Resonance (NMR) experiments, and are observed as satellite peaks in the absorption spectrum. In general the differential equa- tions concerning these spin dynamics are not trivial to solve for a given geometry. However, for the parallel-plate and cylindrical geometries the differential equation, to a certain extent, can be solved [2]. In the case of slab geometry (separation of the plates )HL and 0B parallel to the plates, transverse NMR experiments did detect spin waves [3]. Here the spacing between the spin wave modes (NMR resonances) was more or less constant, and the intensity dropped as a func- tion of 1,k where k is the spin wave mode. This is in good agreement (at least to first order) with the solution of the differential equation for this geometry [4,5]. The spin wave experiments performed in cylindrical geometries (R is few times )H do show a decrease of intensity with increasing k as well, see for example Hako- nen et al. [6]. The experiments, concerning this geometry, could be explained by numerical calculations [7]. Here the configuration of the texture (which forms the effective potential) was determined by minimizing the appropriate free energy and solving the resulting Euler–Lagrange equa- tions. The magnetic healing length decreases as function of temperature, consequently the effective potential for the spin waves is temperature dependent. Normally, this is considered the dominating temperature effect for spin waves in the cylindrical geometry [8]. Spin waves in the B-phase of superfluid 3 Hе in confined cylindrical geometry Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 987 Here, we present that we have been able to grow a n- texture, which is meta-stable and does not change below a certain temperature. Not only do we create this way a tem- perature independent potential, but this potential is also close to a quadratic one, for which the corresponding diffe- rential equation can be solved analytically. 2. Theory The orientation of the n-vector in superfluid 3 He–B is determined by the magnetic field in the bulk liquid and by the proximity to the wall of the cell. The strength of the magnetic field, which is the static magnetic field to per- form NMR, is around 15 mT, for which the superfluid is in the high field limit (B >> 3 mT). In this case the most im- portant orientation energies are: the bulk-field free energy ,BHF the bulk-bending free energy BBF and the surface- field free energy .SHF The total free energy to be mini- mized is = .BH BB SHF F F F (1) The bulk-field energy is minimized when the n-vector is aligned with the magnetic field [4]. In the case that 0B is perpendicular to the normal to the wall ˆ( ),s as in the case of our experiments, it is shown [4] that SHF is mini- mized when n makes an angle = arccos(1/ 5) 64.5 with respect to both ŝ and 0.B However, the change in the local orientation can only be continuous, since fast changes (discontinuities) of the order parameter will be energetical- ly unfavorable. One accomplishes a smoothing of the spa- tial variations over a finite distance, if BBF is expressed as an invariant combination of the gradients of the order pa- rameter [4,9]. The superfluid takes the configuration which minimizes the total free energy over the whole sample. This results in a continuous change of the orientation of the n-vector over the whole sample, and this configuration is called the n-texture [9]. For the case of an axial magnetic field and a cylindrical geometry with radius ,HR one can calculate that the energetically most favorable configuration of the n-texture is the flare-out configuration, as calculated by Smith, Brink- man and Engelsberg [4]. Here we expect that the n-vector has an angle with 0B and ŝ of arccos(1 5) at the wall (r = R) of the cylinder and that it will align itself over a typical length H parallel to the z-axis (and the magnetic field) in the center of the cylinder. These two boundary conditions change the n-vector orientation in a spiral-like configuration, which can be parameterized by ˆˆ ˆ ˆ= sin cos sin sin cos ,n r j z (2) in cylindrical coordinates. The angles and are func- tions of r only, and their behavior is intensely studied [7,10,11]. The angle is hardly r dependent and is in prac- tice close to /3. The angle between the n-vector and magnetic field varies smoothly over the sample be- tween values fixed by the boundary conditions. Both radial dependencies are experimentally confirmed by Spencer et al. [12,13]. The magnetic healing length is proportional to 1 / /cT T B [8,11,13], meaning that it increases during cooling, and the flare-out texture gradually grows in a spir- al way to the center. The growing should be able to contin- ue as long as the boundary conditions can be fulfilled, meaning that the flare-out texture will grow as long as it is the most energetically favorable configuration. Simulations [11] till ratios of / > 0.5H R support this idea of growing. Quantitative calculations for higher ratios seem to be tough. One may assume that the most extreme form of the flare-out configuration would be the gradual change of the n-vector connecting the two boundary conditions. Not clear is how this energetically compares to other textures. Actually, because of complexity of the calculations, it is not clear at all how transitions to other textures should oc- cur when .H R The qualitative answer for the limit H R is clear. The texture should be completely uni- form [7], and should have an angle of arccos(1/ 5), as directed by the boundary condition at the wall. To study any textural transitions we consider a cylindric- al geometry with a diameter of 1 mm. Converting back the magnetic healing length as found in literature [8,11,13] to our pressures (0–6 bar) and magnetic field (15 mT), it is 5 to 10 times longer than the radius of the cylinder at 0.7 T/Tc. If one starts cooling all textures initially started to grow in the flare-our configuration. The growing continues till tempera- tures around 0.7 T/Tc. Lower temperatures leave the texture unchanged, so a textural transition is not observed. It is be- lieved that this flare-out configuration is formed where the two boundary condition are directly connected, or with other words, the n-vector is gradually changed over the whole range of the sample. This texture is stable, but is not ex- pected on energetic grounds, for which we consider this a metastable texture. The reason may be that a transition cor- responds with sharp jumps in the n-texture, which will defi- nitely be discouraged by the bulk-bending free energy. However, the bending of the n-vector will act as an attrac- tive potential for spin waves, resulting in a temperature independent potential for various pressures for tempera- tures below 0.7 T/Tc. For the flare-out texture in a long circular cylinder with the axis aligned with the magnetic field one can formulate the differential equation concerning the spin dynamics. In the transverse case, relevant for our transverse NMR expe- riments, it is given by [2,7,10,11,14] 2 2 , 1 ( \| \| )L r D k d d K K R r r dr dr 2 , , , 1 (1 ) = .sin 2 k k kE (3) O.W.B. Benningshof and R. Jochemsen 988 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 The first term can be seen as the kinetic part. Here K and K are dimensionless parameters which equal 8/5 and 4/5, respectively, if calculated in the weak-coupling Ginz- burg–Landau [15] approximation. L rR are the transverse components of the spin-orbit rotation matrix, and D is the dipole coherence length. The second part in Eq. (3) is the potential part, which is formed by the relevant trans- verse components of Eq. (2). The eigenvalues are given by the following equation 2 2 2 , , ,= ,k k L B k BE (4) where ,k corresponds with the frequency of the kth spin wave mode, L is the Larmor frequency and B the lon- gitudinal NMR frequency of the B-phase. In general the differential Eq. (3) is not trivial to solve, especially in combination with a temperature dependent potential. However, in our case we have a temperature in- dependent potential, and moreover, our potential only dif- fers maximally 6.5% if the term 21 sin 2 is approximated by a quadratic dependency in r. As the potentials are very alike, it is convenient to compare the data with the results from a quadratic potential, for which the differential equa- tion is analytically solvable. It reduces to the Schrödinger equation for the two dimensional harmonic oscillator, for which the eigenvalues are given by , 1 = 1 2 ( 1), 2 D kE K K k (5) where is the measure of curvature of the potential. Only s-wave states couple to the homogeneous rf field, or with other words: only the = 0l modes have nonvanishing in- tensities in experiments using uniform rf fields [7]. Conse- quently, only the even k modes couple to the correspond- ing NMR frequencies. The eigenvalues have constant separation and, as this is a two dimensional system in a quadratic potential (density of states is constant), the inten- sities should all be equal. Interesting is that in the Ginz- burg–Landau regime [2], the eigenvalues are pressure de- pendent, as D changes as function of pressure, which tunes it between 32 m (zero pressure) and 7 m (melting pressure). As the energy landscape (potential energy) for all pressures is the same, while the level spacing of the eigenvalues decreases with increasing pressure, the amount of spin wave modes in the cylinder should grow. In reality D does also change with temperature [8], but this dependency is weak, certainly compared to the tempera- ture dependency in B [16], as it is proportional with 2( ) / .BT The temperature behavior of the energy gap ( )T and the susceptibility B for the superfluid B-phase are both experimentally and theoretically known [17–19], meaning that our system, including the growing of spin wave modes, is in theory completely known. 3. Cell A schematic drawing of the final cell is shown in Fig. 1. The cell is constructed on a block of copper, which fits on the experimental space of the nuclear stage and is 10 mm thick. In this way the cell with all the various components could be assembled outside the cryostat, including the magnet, and could easily be screwed on the nuclear stage. The cell itself mainly consists out of silver and poly- etherimide (PEI). The silver piece sliced into the copper block, and was ri- gidly fixed with additional bolts. One should be sure that copper and silver pieces are squeezed sufficiently against each other, to prevent extra impedance for the thermal con- duction. The silver piece itself, which has good thermal con- ducting properties at low temperatures [20], should cool the 3 He to the temperatures of the nuclear stage. Silver is pre- ferred over copper, despite the fact that it has a lower ther- mal conductance, because its properties are more favorable in the presence of a magnetic field. However, the interfacial thermal resistance, better known as the Kapitza resistance ,KR between the liquid 3 He and the silver becomes rather high at low temperatures. The amount of vibrations (pho- nons) is strongly decreased at low temperatures and in com- bination with the mismatch at the interface (low scatter probability) this gives a high Kapitza resistance. The tem- perature difference across such interface is given by = ,KR Q T A (6) where Q is the heat flow and A the surface area of the interface. In the case of an interface between 3 He and sil- ver the Kapitza resistance at 2 mK is =KR 10 5 m 2 K W –1 [20]. To prevent the interface of becoming the highest im- pedance of the heat transport, the area should be made suf- ficiently large. This is accomplished by a silver sinter pressed on the silver piece. Here the sinter works as a sponge in the 3 He liquid and has an enormous surface area. The silver sinter was 0.5 mm thick and had an effective area Fig. 1. (Color online) Cross section of the experimental cell. Spin waves in the B-phase of superfluid 3 Hе in confined cylindrical geometry Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 989 of 25 m 2 , which gives a /Q T ratio of 2.5 10 –4 W K –1 . This is one order of magnitude better than the heat transport through helium in the cylinder, as we will see later, and made this interface not a limitation in the cool down process. The part of the cell which fits into the magnet should be made of a non-metallic material. Otherwise it would lead to significant losses like eddy currents, which is bad for temperature stability and NMR measurements. For this purpose this part of the cell is mainly made out of polye- therimide. PEI is a plastic which is easily to machine and has proven to be suitable to work with at low temperatures, meaning it does not crack after multiple cool downs. The disadvantage of working with PEI, whose molecular struc- ture per unit polymer is given by C37H24O6N2, is the quan- tity of hydrogen atoms in it. The gyromagnetic ratio of hydrogen is relatively close to the one of 3 He (they differ approximately by a factor of ~ 1.31) and because the 2T of hydrogen in the polymer is short, the tail of the NMR ab- sorption is visible at the resonance frequency of 3 He. From this point of view it is more desirable to work with quartz glass (SiO2), it is also a nonmetal and has not a net nuclear spin. However, the fabrication and machining of quartz glass cylinders is much more complicated. Together with the fact that several cell’s needed to be constructed to find the optimal results for cooling the liquid, SNR of the NMR experiments, fiber gluing, etc., made it more convenient to construct the cell out of PEI. The experiment is performed in a circular cylinder made of PEI, which axis is aligned with the static magnetic field of the NMR. The total length is 70 mm and 1 mm in diameter. The read out is performed with a weakly coupled transformer technique [22], where the rf-coil is positioned in the middle of the cylinder. 4. Results Measurements are performed between zero and 6 bar, and till temperatures below 300 K. A typical sequence of NMR spectra at different temperatures is plotted in Fig. 2. Here the measurement is performed at 6 bar, and the temperature sweep is between 0.57 mK and 1.56 mK (transition temperature). Above cT the resonance fre- quency occurs at the Larmor frequency. Directly below cT , the liquid has undergone the phase transition to the superfluid B-phase, and the growing of the flare-out con- figuration is observed. The absorption spectrum has be- come wider, caused by the radial changes in the orienta- tion of the n-vector in the cylinder. The texture gets stuck around 1 mK, and from this point it forms a constant potential. Also around this temperature the spin waves modes become visible, as the separation between the modes is enough to distinguish each mode. The spin wave absorption lines are more or less equally separated, but more importantly they have approximately the same intensities. In total 5 spin wave modes become clearly visible at the lowest temperatures, which is ex- pected at this pressure according to Eq. (5). The separation between the modes does increase as B increases by de- creasing temperature [3]. Below 0.3 T/Tc hardly any tem- perature dependency is observed, as the susceptibility and energy gap are almost saturated. This texture seems stable for the whole temperature range (at least till 100 K), and no textural transition is noticed. The NMR spectrum of the spin wave modes at several different pressures obtained at the lowest achievable tem- peratures (T < 0.3 Tc) are shown in Fig. 3. No noticeable temperature effects are expected below 0.3 Tc, so the data can be compared with the results of the theory at zero tem- perature (Eqs. (4) and (5)). The zero temperature theory predicts the frequencies of the spin wave with only the pressure as a variable. The longitudinal NMR frequency B of the B-phase is very well known as a function of pressure [3,16]. The most important approximation is the replacement of the real texture potential in Eq. (3) by a quadratic one, which allowed the analytical solution of the spin wave frequencies as given in Eq. (5). The theoretical frequencies for the various spin wave modes are shown in Fig. 3 as colored dotted-solid curves. The black (1) and red (2) curve represents the k = 0 and k = 2 mode, respectively. Both exists at all pressures. The sequence green (3), blue (4) and cyan (5) represent the k = 4, 6 and 8 modes, which only exist at increasing pressure, allowing 5 modes at a pressure of 6 bar. Fig. 2. (Color online) NMR absorption scans of helium in a cy- linder with a radius of 0.5 mm at 6 bar for various temperatures. The absorption is expressed in voltage V, which is the signal measured with the pick-up coil. The temperature range is between 0.57 mK and cT (1.56 mK). At and above the transition tempera- ture the absorption peak is at the Larmor frequency. At lower temperatures the resonance frequency shows a shift due to texture effects. At even lower temperatures, / < 0.7,cT T several spin waves modes become visible. O.W.B. Benningshof and R. Jochemsen 990 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 The measurements are in good qualitative agreement with the theory, both the frequencies of the modes and the number of existing spin modes at a certain pressure. The theory predicts a somewhat smaller spacing between the frequencies of the spin modes, which can be explained by the fact that the real potential is a bit steeper than the qua- dratic potential, except close to the cell wall [21]. The number of observed spin modes at each pressure is exactly as predicted by theory, with one exception: for zero pres- sure one would expect 2 spin wave modes, while 3 modes can be distinguished in the NMR absorption line. 5. Discussion The flare-out configuration for the texture is not the ex- pected minimum-energy state for the ratios of / HR in our experiment. However, it seems to be a stable state and once the liquid has been cooled to the superfluid state in this configuration no transition to an other textural confi- guration is observed. We guess that the forming of this meta-stable texture occurs at sufficiently slow cooling. Once grown far enough in the flare-out configuration this state is meta-stable, as the threshold to jump to an other textural configuration is too high. If one would cool down more rapidly, the magnetic healing length H would grow and oversize the radial dimension of the system much faster than in normal proce- dure. At the moment the superfluid is formed, the n̂ -vector is not given enough the time to grow smoothly into the flare-out configuration. This condition will favor the crea- tion of the forming of the uniform-texture. To cool as fast as possible with our setup, the liquid was pre-cooled to the lowest temperatures, and than locally heated to the normal state by a rf-pulse. While the rest of the system is still cold the heat only needs to be removed locally, which should make the cooling much faster than in any other method available. Enough heat could be produced with a pulse in the rf-coil, the absorption was enough to overcome locally the energy gap, and it turns out the superfluid cools into another textural state. This is depicted in Fig. 4. The black (1) absorption line shows the usual stable situation at 6 bar (around 300 K) with the 5 spin wave modes. After the heat pulse, the resonance frequency is at the Larmor fre- quency, as shown by the green (3) absorption line, indicat- ing that the liquid is in the normal state. The locally heated liquid then starts to cool and the absorption peak moves uniformly to the left and becomes wider as the liquid be- comes superfluid. The red (2) curve in Fig. 4 shows the equilibrium result, when the liquid is cooled back to 300 K. The spin wave modes have disappeared, and the NMR absorption signal only shows a single peak, indicat- Fig. 3. (Color online) Spin wave absorption peaks as a function of magnetic field at pressures of 0, 1.5, 4 and 6 bar. Each absorp- tion line is obtained at a temperature below 0.3 T/Tc. The absorp- tion is expressed in arbitrary units. The longest peak corresponds with the mode k = 0 in Eq. (5), the neighboring peak corresponds to the mode k = 2, and so on. The (more or less vertical) curves represent the theoretically predicted spin wave frequencies for the various modes at zero temperature. The black (1), red (2), green (3), blue (4) and cyan (5) dotted/solid curves correspond with the mode 0, 2, 4, 6 and 8, respectively. All curves are plotted for increasing pressures (started from 0 pressure), but the curve is dashed when the mode is predicted not to exist, and it becomes a solid curve when theory predicts the mode to exist. Fig. 4. (Color online) Local heating of the superfluid with NMR. The black (1) and red (2) curves show the absorption spectrum of 6 bar at 300 K before and after heating the sample, respectively. The corresponding values of both absorption curves are put on the left y-axis. The green (3) curve corresponds with the absorp- tion curve while heating the sample, absorption values are put on the right y-axis. Here the values are 2 orders of magnitude higher, so enough energy is dissipated to locally warm up the liquid to the normal state, as is indicated by a jump of the peak to the Lar- mor frequency. After the local heating, the texture is changed from flare-out to uniform configuration. Spin waves in the B-phase of superfluid 3 Hе in confined cylindrical geometry Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 991 ing that the superfluid is in an uniform texture. 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Spin waves in the B-phase of superfluid ³He in confined cylindrical geometry

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