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Z. Phys. B 102, 331–336 (1997) ZEITSCHRIFT FÜR PHYSIK B c Springer-Verlag 1997 Specimen geometry effects on the irreversible magnetization in the low field regime for specimens of bulk Nb3Sn M. Däumling1,? , W. Goldacker2 1 2 Département de Physique de la Matière Condensée, Université de Genève, CH-1211 Genève 4, Switzerland Forschungszentrum Karlsruhe, D-76021 Karlsruhe, Germany Received: 18 October 1996 Abstract. Irreversible magnetization curves in particular in self field conditions were investigated for specimens of bulk Nb3 Sn as an isotropic and non-granular alternative to high Tc materials. The specimens were cut into squares with side length to thickness ratios of about 0.8 to almost 20, leading to diamagnetic slopes |χd | ranging from slightly more than 1 (slab in parallel field) to 13 (plate in perpendicular field). The peak in the low field magnetization (caused by the field dependence of jc ) moves from negative field for specimens with small |χd | (where it should be according to the standard Bean model) to zero or even positive fields for specimens with large |χd |. Finite element self consistent calculations with field dependent and thus spacially varying jc were used to fit the measured hysteresis curves. A strong current peak in the center plane of the specimens – caused by the demagnetization field – probably causes the shift of the magnetization peak to positive field. Best agreement between the experimental and calculated magnetizations have been obtained for the extreme geometries: either thin disk in perpendicular (χd < −4) or thin slab in parallel magnetic field (χd > −1.2). PACS: 74.25.Ha; 74.60.Jg; 74.70.Ad 1. Introduction The irreversible magnetization of superconductors is usually connected with the critical current density jc by the critical state model [1, 2]. The current density values are limited by flux pinning, and determine the macroscopic magnetic field profile in the specimens according to (neglecting contributions from surface currents) jc = curlH. (1) Here jc and H are vectorial quantities. If the specimen has a cylindrical geometry with the symmetry axis z then we can write (j in circumferential direction) ? Current address: Forschungszentrum Karlsruhe, D-76021 Karlsruhe, Germany jc = dHz dHr − dz dr (2) The original critical state model was conceived for cylindrical or slab geometries without demagnetization effects. Thus the first term could be neglected. As a result of the disk or plate like geometry of single crystals of high temperature superconductors [3–11] lately much interest has been concentrated on how the critical state would behave in geometries that show considerable demagnetization fields. These magnetic fields are generated in the specimen and are usually not parallel to the applied magnetic field. Thus flux line bending occurs and the critical current is not only determined by the flux line gradient, but also the curvature of the flux lines. In this case the second term becomes important, and even dominant in the case of thin tapes or disks in a perpendicular field. However, most work has been carried out considered flux penetration for field independent jc in the zero thickness limit, applicable for thin strips or disks. Recently the field dependence of jc has been taken into account ([8] and references therein), and attempts to model flux penetration [12] and distribution [13, 14] in three dimensional objects have been made. One particular aspect that was measured in plate shaped specimens of the high Tc materials was that the peak in the irreversible magnetization that usually occurs in negative field when decreasing the external magnetic field was positioned at a positive field [15–17]. This peak is caused by the field dependence of jc or flux creep [18] and must be – in the classical critical state model – always located in a negative magnetic field. This is the case because a negative applied field is necessary to create some minimum effective field – and thus maximum shielding current – in the specimen interior. In this article we will concentrate on exactly this problem: the irreversible magnetization in the vicinity of zero magnetic field, the self-field regime. Here the magnetic field generated by the specimen itself is of the same order of magnitude or larger than the external field. Significant field variations exist across the specimen. In addition, the demagnetization field twists the applied magnetic field away from its original orientation. 332 The goal of this article is to examine this phenomenon using specimens of bulk polycrystalline Nb3 Sn. These specimens have the advantage that they are – contrary to the high Tc materials – isotropic, have rather high critical current densities and can be cut fairly easily into different shapes. In addition, flux creep effects [18] are negligible. 3D model calculations using magnetic field dependent – and thus spatially varying – jc values were carried out in order to fit the measured magnetization curves. z z r x H H I(r,z) I(x) φ Fig. 1a,b. Definitions of the geometry for magnetic field and current calculations 2. Experimental The Nb3 Sn specimens are polycrystalline, non-porous, and were prepared powder-metallurgically [19] in a high pressure/high temperature reaction between liquid tin and niobium. The specimens were sparc eroded first into a long square rod of cross section (3.72x3.72 mm2 ) From this rod slices with thicknesses of 4.25, 1.7, 0.71, 0.33 and 0.2 mm were eroded. The onset critical temperature (measured with a SQUID magnetometer) is 18.1K. From the data of Orlando et al. [20] we can estimate that the lower critical field Hc1 is of order 30mT (10K) or 50mT (5K). The microstructure of the specimens was investigated with optical microscopy. For this purpose a slice – broken into two pieces – was embedded into epoxy resin with the two pieces perpendicular to each other. The surfaces were then polished and investigated with the aid of an optical microscope. This way it should have been possible to detect anisotropies in the grain structure. However, none was detected; the grains and grain structure were isotropic. The grain size is approximately 10 µm. X-ray scattering shows the specimen to be essentially single phase Nb3 Sn with some traces of unreacted Nb and no texture of the A15 phase. The magnetization was measured with a vibrating sample magnetometer (VSM) in a superconducting high field solenoid. The specimen temperature was controlled by admitting helium gas into the specimen chamber through a needle valve. Thus temperatures between 2 and 120K can be achieved. The specimen stay in He gas (p < 10mbar) all the time at all temperatures. In order to obtain as many different configurations as possible the plate like specimens were mounted both perpendicular and parallel to the magnetic field. Hysteresis curves were measured at 10 and 15K. At 5K most specimens showed flux jumping, in particular the thicker ones. 3. Calculations 3.1. Cylindrical geometry The squares used in the experiment were approximated by disks of the same diameter. As typical shielding fields in our specimens are significantly larger than Hc1 we have neglected contributions from Meissner currents. We will see (below) that in a more sophisticated calculation these effects will have to be included. Calculations of the critical state were carried out for a disk of height d and radius r0 in cylindrical coordinates (see Fig. 1a). The field Happl was applied in the z direction. From the cylindrical symmetry it follows that currents only flow in the circumferential direction. The disk was divided into n x m ring segments (n along the radius and m along the height). The radial and axial fields (hr and hz , respectively) at arbitrary coordinates r and z created by a current loop of radius a with the current I flowing are given by [21] hr (r, z) = hz (r, z) = I z [−K(k) 2π r[(a + r)2 + z 2 ]1/2 a2 + r 2 + z 2 E(k)] + (a − r)2 + z 2 I 1 [K(k) 2 2π [(a + r) + z 2 ]1/2 a2 − r 2 − z 2 E(k)] + (a − r)2 + z 2 (3) (4) with k 2 = 4ar[(a + r)2 + z 2 ]−1 . The center of the loop is located at r = z = 0. K and E are complete elliptic integrals of the first and second kind. The magnetic field at the coordinates of each segment is then the sum of the fields generated by all the other segments, plus the applied field. The iterative calculation proceeds as follows: first a constant current density j=j(Hmax ) is assumed to flow in each segment. Here Hmax is the maximal magnetic field considered for a magnetization curve on the reverse sweep. The field generated by this current is determined and added to the applied field. The current direction was chosen as to cause a positive (trapped) field in zero applied field. After this first step the local j values were modified according to the critical state, fulfilling the condition j = jc (H) at p all points in the disk where H = (Happl + Σhz )2 + Σh2r is the total (radial plus axial) field at each point. Then the field was again calculated from the modified current distribution, thus changing the fields (and currents) again, and so on. The magnetic moment of the sample was computed by summing up the contributions from the individual current loops. The criterion for convergence was the magnetic moment. If it changed less than 0.1 % from one iteration to the next, the calculation proceeded to the next lower external field. Hereby the current distribution from the previous field was kept as starting condition for the iteration procedure of the new field. A self-consistent state was usually reached within less than five iterations. However, if the field dependence of jc is made very steep numerical instabilities occur, in particular near the magnetic field at which first negative magnetic flux enters the rim of the disk. If the segment number was increased these instabilities sometimes could be avoided. Practical limits (of order 5000 elements) were, 333 1500 3 // 3⊥ 6⊥ 2 // 2⊥ 4 // 4⊥ 5⊥ WG4 10K 5 Bp ) 2 1000 m A jcB (A/mm 2 ) 0 -3 0 1( m -5 χd -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 10K 500 3.0 Field (T) Fig. 2. Irreversible magnetization curves at 10K of Nb3 Sn platelet 1.7 mm thick in parallel magnetic field Table 1. Diamagnetic slopes χd and thicknesses for all Nb3 Sn specimens used in the magnetization experiment. All samples are square shaped with a side length of 3.73 mm. A perpendicular orientation means that the field is aligned perpendicular to the square specimen WG3 WG6 WG2 WG4 WG5 thickness [mm] 0.2 0.33 0.71 1.7 4.25 field orientation k ⊥ ⊥ k ⊥ k ⊥ ⊥ 1.18 11.1 5.35 1.22 3.58 1.28 2.12 1.44 −χd however, given by the calculation speed of the computer used (DEC 3000-900S). The field dependence of the current used here was of the Kim type [22] with an additional Gaussian term H1 H2 (5) j(H) = j0 Aexp(− 2 ) + H + H1 H0 The parameters A, H0 and H1 were used to fit the measured magnetization. 3.2. Slab geometry The slab was assumed to extend into infinity for the z and y direction (see Fig. 1b). The slab width is 2W along the x direction, zero being located in the center. The external magnetic field is pointed along the z direction, all currents flow along the y axis. The equation resolved was dH(x)z = j(H(x)) (6) dx Typically this equation was solved for 1000 segments along x. The calculation now proceeded in a similar manner to the one described above. However, compared to the previous paragraph the calculation was much simpler and faster, as the current only varies in one dimension. j(x) = 4. Experimental results The magnetization curve at 10K of one specimen is shown in Fig. 2. Diamagnetic slopes were determined from the first branch of the curve after zero field cooling. Within measurement accuracy the value of the slope is independent of sample temperature up to 15K, as is expected for specimens much larger than the magnetic penetration depth. The maximum shielded field (self field) h∗ is roughly equal to the 0.1 1 Field (T) Fig. 3. Critical current density of all Nb3 Sn specimens at 10K as a function of magnetic field field at which the diamagnetic branch and the envelope curve join, thus about 0.5T for the specimen shown. The curves were measured up to high magnetic field (typically 2T), so that the specimen is always fully penetrated on the reverse leg. The diamagnetic slopes of all specimens are given in Table 1. It should be noted that – contrary to high Tc materials – flux creep plays a very small role here. Measurements of dynamic relaxation at 15K show that the magnetization curves for two different field ramp rates differing by a factor of 5 fall on top of each other within measuring accuracy. Thus flux creep effects are negligible in this case, and the shielding current flowing can be identified with the critical current denstity jc . The values of the critical current density jc as a function of magnetic field are shown in Fig. 3. These were calculated using a modification of the standard Bean formula [1, 23] −1 2 a ad b− (7) jcB = ∆m 2 3 where b is the length of the sample, a its width, d its thickness and ∆m the difference between the upper and lower branch of the magnetic moment. Evidently this expression is only correct if everywhere in the sample the same current j is flowing, which is only true if the magnetic field does not vary significantly across the specimen, an assumption which is certainly violated in low applied fields. As expected, the jcB values for all specimens converge into a single curve for magnetic fields higher than about 1T. The remaining differences at higher field stem probably from the errors in determining the geometry of the specimens. This shows that in fact bulk pinning is responsible for the measured hysteresis at high fields, irrespective of specimen shape. The slabs in parallel field and disks in perpendicular fields behave identical. The field dependence of jc in high field above about 1T can be approximated by a Kim type 2 dependence (eq. 5 with A=0) with j0 = 4300A/mm and µ0 H1 = 57mT. As the material of all specimens is identical we can now conclude that the true jc (H) behavior is identical for all specimens, even in low magnetic field. Significant differences in the jcB (= apparent jc ) values – calculated assuming constant current flow across the 334 0.05 2.5 2.0 0.00 WG3 10K peak position Bp (T) 1.5 -0.05 1.0 ) 2 -0.10 0.5 m A 0.0 -3 -0.15 0 1( m 10K 15K calc. 10K -0.20 -0.25 -0.5 -1.0 data -1.5 fit -2.0 -0.30 0 2 4 6 8 10 -2.5 12 -1.0 - diamagnetic slope -0.5 0.0 0.5 1.0 Field (T) Fig. 4. Field value at which the peak in magnetization in decreasing magnetic field occurs, The calculated values are shown as dashed lines (10K only) 0.3 fit WG3 data 10K 0.2 specimens – exist in low magnetic field. The thin specimens show much higher apparent jc values than the thick specimens. Thus the self field and thus inhomogeneous current flow must play some role in suppressing the true field dependence of jc for in particular the thicker specimens, which create a higher self field with the consequence of lower jcB . The position of the peak in the magnetization on the upper branch (field ramped down, see arrow in Fig. 2) as a function of the diamagnetic slope is shown in Fig. 4. The peak is located at low negative fields for the thin slabs in parallel magnetic fields, and moves to more negative fields for thicker slabs. For further increasing diamagnetic slope |χd | the peak moves towards zero, and for the thinnest platelets in perpendicular field is located at a positive field. This trend is similar to the one observed by us for some high Tc materials [17, 24] and others for single crystals of YBa2 Cu3 O7−δ [15, 16]. ) 0.1 2 m A 0.0 -3 0 1( -0.1 m -0.2 -0.3 -1.0 Figure 5 shows the result of the calculations of the irreversible magnetization curves for the thinnest platelet in parallel and perpendicular direction using the jc (H) dependence of (5) with the parameters A=2, j0 =4300 A/mm2 , µ0 H1 = 57mT and µ0 H0 = 75mT. We note that the general shape of the peak is reproduced well by the calculation. The differences between calculated and measured curve in Fig. 5a are small, and somewhat larger in Fig. 5b. It should be noted that the reversible magnetization is negligibly small for all our specimens (of order 10−6 Am2 for the specimens shown). However, in Fig. 5a the peak in the calculated curve is occurring in negative magnetic field, and not in positive field as in the experimental curve (for better resolution see Fig. 4). In particular the variable A has some influence on the peak position. For larger A the peak moves to the right, thus in the direction of the measured curve. This is illustrated in Fig. 6 for three values of A. More field dependence of jc shifts the peak to the right, towards more positive magnetic field, while at the same time sharpening the peak in the magnetization somewhat. An even larger A – for example A=5 – causes large zero field current densities, and thus large field gradients, rendering the calculation unstable. The 0.0 0.5 1.0 Field (T) Fig. 5a,b. Magnetic moment vs. field for decreasing branch for specimen WG3 in perpendicular a and parallel b field orientation together with calculated curve using jc (H) of (5) 10K 2.0 ) 2 5. Result of calculations -0.5 m A -3 0 1.5 (1 m A=2 A=1 A=0 1.0 -0.2 -0.1 0.0 0.1 0.2 Field (T) Fig. 6. Calculated magnetic moment vs. field for decreasing branch for 0.2mm thick platelet in perpendicular field orientation using jc (H) of (5). with A=0, 1 and 2 (see text) same happens if a large screening current at the edge of a current and field free diamagnetic zone is allowed. The position of the peak Bp of the calculated magnetization curves (with A=2) are shown in Fig. 4 together with the experimental data. The agreement with the data of Fig. 4 is good, but not everywhere quantitative. In particular for the intermediate diamagnetic slopes the slab calculation overestimates |Bp | in the slab approximation, and underestimates it in the disk approximation (1.5 < |χd | < 4). Figure 7 shows the calculated current and field distributions for the thin platelet WG3 in perpendicular magnetic field, Fig. 8 the corresponding data in parallel field using identical parameters. In the perpendicular constellation the 335 parameter: field // z (mT) -140 -120 -60.0 -20.0 1.5 -100 -80.0 -40.0 0 20.0 40.0 60.0 80.0 100 120 r (mm) 140 1.0 160 180 200 0.5 220 240 260 280 300 320 0.0 0.10 340 0.05 0.00 -0.05 -0.10 z (mm) 8000 6000 4000 jc (A/mm2) 2000 0.00 0.05 z 1.0 r (m m) (m m 0.5 ) -0.10 0 -0.05 0.0 1.5 2.0 0.10 Fig. 7a,b. Calculated axial field a and current b distributions for thin platelet (0.2mm) in perpendicular magnetic field at 10K. The field unit is mT. 250 200 150 100 5000 Field (mT) j (A/mm2) 10000 50 0 0 0.000 0.025 0.050 0.075 0.100 x (mm) Fig. 8. Calculated field and current distributions for thin platelet (0.2mm) in parallel magnetic field current maximum occurs near, but not at the edge. This is due to the demagnetization field which is already negative at the specimen edge. The current is flowing mainly in the center plane, diminishing strongly towards the top and bottom surfaces as a consequence of the increasing radial field. In the slab case the maximum current is flowing directly at the specimen edges where the magnetic field is zero. for specimens with diamagnetic slopes smaller than about -4 to -5. The slab model leads to best results if used for slabs with |χd | < 1.2. If the slab model is used for higher values of |χd | the measured |Bp | occurs at smaller values than the calculated ones, indicating penetration along the specimen edges. Similarly the thick disks behave a bit more like slabs than the calculation assumes. Thus for those specimens |Bp | is underestimated in the disk approximation. However, even for the very thin disks a peak in a positive field could not be obtained in the calculation. There may be several reasons for that. Firstly, an even steeper jc (H) dependence causes a shift of the peak in the correct direction. However, due to convergence problems in the calculation we cannot really verify this. Secondly, we have used a disk approximation to specimens which in fact have a square shape. This means that the current can fill out the edges of the specimens, which changes current patterns. Thirdly, in principle a field free annihilation zone should be present at the H=0 line. In such a zone the magnetic field would have to be below Hc1 , while at the edges of it a large shielding current flows. We were unable to model this zone due to problems with the numerical stability of the calculation caused by large field gradients. However, in order to accurately model the shielding behavior this numerical problem must be overcome, as the current approach is insufficient. A solution to the problem has been outlined by Brandt [12], however at the moment not yet incorporating the field dependence of jc . The observed crossover of the virgin and the envelope curves (see for example Fig. 5b) may be reproduced when calculating complete magnetization loops that use Meissner currents as well as a field dependent jc in the mixed state. This comment also applies with respect to other materials, in particular tapes of the (Bi,Pb)2 Sr2 Ca2 Cu3 Ox material where a magnetization peak at a positive magnetic field has been also observed [24]. However, the peak in this case is much more narrow and sharp than the one found here in Nb3 Sn. It is located at Bp ≈ +h∗ . By contrast, in the thinnest Nb3 Sn platelet measured here the peak occurs at about + h∗ / 10, and is much broader. In the calculations we had difficulties to even reproduce the quite broad peak in the magnetization of Nb3 Sn, and we can only speculate that an even stronger field dependence of jc (in addition to a low field zone) is necessary in order to generate these magnetization curves. A jc anisotropy may also play an additional role, as well as possibly changing length scales for the shielding currents due to field dependent granularity in the tapes. Another mechanism leading to a peak in the magnetization was pointed out by Schnack et al. [18]. Here the influence of the magnetic field on the flux creep rate is considered, with the result that a peak in the irreversible magnetic moment is generated even for jc = const due to a field variable flux creep rate. This mechanism will certainly be important in particular for high Tc materials like that (Bi,Pb)2 Sr2 Ca2 Cu3 Ox tape, but not for Nb3 Sn. 6. Discussion Best agreement between the experimental curves and the calculations have been obtained for the extreme geometries: either thin disk in perpendicular or thin slab in parallel magnetic field. In order to obtain quantitative values for flux penetration and shielding the thin disk model should be used 7. Conclusions We have measured the irreversible magnetization with particular emphesis on the self field regimeof specimens of Nb3 Sn with various aspect ratios, giving diamagnetic slopes 336 of about −1 to −13. The peak in the magnetization shifts from negative to positive magnetic field as the diamagnetic slope increases, just like in high Tc superconductors. Model calculations are used to fit the measured magnetizations, and derive magnetic field and current distributions. One requirement for the positive peak field appears to be a strong field dependence of the critical current density in zero magnetic field. This causes strongly inhomogeneous current flow, qualitativly explaining the measured magnetizations. Practical limits for the usability of slab or disk approximations for the critical state model are given. This work has been partly supported by the Swiss National Funds (NFP30). 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