Figure 1

(Color online) Schematic of the different VL structures, illustrated by the white shapes, observed as a function of temperature and field $\mathbf{H}$ applied parallel to the fourfold [001] crystal axis in our cylindrical sample Nb-1. The IMS is indicated by the shaded region and its boundary with the mixed state is indicated by the dashed line; this line depends on the demagnetization of the sample and lies $\approx 30\text{mT}$ lower (at temperatures $\approx 2\text{K}$) for our spherical sample Nb-2. At higher applied fields $\left(\gtrsim 200\text{mT}\right)$ away from the IMS-mixed-state boundary, the transition fields between VL phases are essentially the same for both Nb-1 and Nb-2, with minor variation due to the slightly different sample purities. Details of the position and nature of these VL structural transitions for $\mathbf{H}\parallel \left[001\right]$ can be found in Ref. 5.Reuse & Permissions

Figure 2

(Color online) Images of the diffraction pattern from the vortex lattice (VL) at 2 K: (a) at field ${\mu }_0\mathbf{H}=200\text{mT}$ applied parallel to the [001] axis in sample Nb-1, (b) at 350 mT applied parallel to the [001] axis in Nb-2, (c) at 200 mT lying within $0.1°$ of the $\left(\overline{1}10\right)$ mirror plane and $1.9°$ from [001] in Nb-2, and (d) at 350 mT lying close to the (110) mirror plane, $14°$ from [001] in Nb-2. These pictures were prepared by summing several detector measurements over a range $\pm 1°$ of rocking angles about the horizontal and vertical axes, allowing all Bragg reflections to be presented in a single image. The undiffracted neutrons at the center of the picture have been masked. The (110) mirror plane cuts these pictures horizontally. In (a) and (c) the VL half-unit cell is scalene triangular with four domains. In (a), $\mathbf{H}$ is parallel to [001] and the domains are degenerate; in (c) $\mathbf{H}\stackrel{̸}{\parallel }\left[001\right]$ breaking the degeneracy; their unequal occupancies lead to different integrated intensities of the Bragg spots. To distinguish between domains, we enumerate those observed in (a) and (c). We shall call “2” and “3,” respectively the domains indicated by the dashed red and dotted yellow hexagons. Domains “1” (“4”) are related to domains “2” (“3”) by reflection in the (100) [or (010)] planes that cut images (a) and (c) diagonally. In (b) and (d) the VL unit cells are square. In (b) the two degenerate VL domains we shall denote as “24” (“13”) are indicated by the dotted red (dashed yellow) squares while in (d) the single VL domain is shown by the dotted orange square.Reuse & Permissions

Figure 3

(Color online) Schematic representation of the rotation angles available with the cylindrical sample Nb-1 in our SANS setup. The vertical and horizontal axes of rotation of the cryomagnet are denoted by $\omega$ and $\phi$ respectively. These axes rotate the sample and applied field together, allowing VL Bragg spots to be rocked through the Ewald sphere, and diffraction images (Fig. 2) to be constructed. The single-crystal sample can be rotated relative to the field through $\zeta$ and $\chi$. $\zeta$ represents rotations of the sample insert about the vertical axis relative to the cryomagnet, while $\chi$ is provided by a special sample mount developed at Risø-DTU. For the spherical sample Nb-2, the $\chi$ rotation axis was not available.Reuse & Permissions

Figure 4

(Color online) Population of the VL domains in sample Nb-2 at (a) 200 mT, 2 K (scalene phase), (b) 200 mT, 5 K (isosceles △ phase), and (c) 350 mT, 2 K (high-field square phase), as the field direction is rotated by an angle $\zeta$ close to [001] in a plane belonging to the $\left[\overline{1}10\right]$ zone but elevated $1.9°$ from (110), as depicted in Fig. 5. In (a) the arrow indicates the angle corresponding to Fig. 2c. In that figure are given the denotations of the four scalene VL domains represented here by different colors: “1”–black, “2”–red, “3”–green, and “4”–blue. In 4(b), are shown the populations of the two corresponding isosceles domains present at a higher temperature; the circles (diamonds) denote domains “12” (“34”). In 4(c) the two square domains “13” (“24”) are represented by the circles (diamonds). The dashed lines in this figure are fits derived from Fig. 6 and described in the text.Reuse & Permissions

Figure 5

(Color online) Stereographic projection of the VL structure and orientation in real space at high fields $\ge 330\text{mT}$ applied in the vicinity of the [001] axis in bcc Nb. The dotted (dashed) lines show the planes in which the applied field was rotated for samples Nb-2 (Nb-1); the filled triangles depict our observations at $\ge 330\text{mT}$ and 2 K of the scalene triangular half-unit cells for selected rotations away from [001]. For fields exactly in the twofold {110} and {100} planes, a scalene VL domain and its image in the mirror plane are degenerate and have equal populations. Here, for rotations in the (100) plane, we show which of the two scalene domains dominates as the field is rotated out of the plane by displacing each triangle to lie on the appropriate side of (100). At fields $\ge 330\text{mT}$ applied exactly parallel to [001] the VL unit cell is square (Ref. 14) but with sides oriented $\pm 11°$ from $⟨100⟩$, and two degenerate VL domains coexist [Fig. 2b]. The open and filled squares correspond to domains “13” and “24,” respectively [Figs. 2b, 4c]; the displacements of these squares from [001] depict which domain dominates as the field is rotated slightly away from [001]. Using these data the dominant VL domain and its shape and orientation can be deduced for fields parallel to equivalent directions of the cubic crystal as exemplified here in the top left quadrant by open triangles.Reuse & Permissions

Figure 6

(Color online) Polar schematic of the population of the four scalene VL domains in sample Nb-2 at 200 mT and 2 K. The radial coordinate represents the relative VL domain population while the angular coordinate represents the azimuthal angle $\eta$ as the applied field precesses around and close to the [001] axis. The different domains are represented by different colors to correspond with Fig. 4a. The crosses mark our data. The solid lines are fits as described in the text; these are mapped back to our experimental rotation angle $\zeta$ to generate the dashed lines in Fig. 4.Reuse & Permissions

Figure 7

(Color online) Schematic in real space showing the nomenclature used for the interior angles and orientation of the VL half-unit cell as the field direction is rotated: (a) in a plane close to the (110) mirror plane, (b) exactly in the (100) mirror plane. These rotation planes are illustrated in Fig. 5. Note that descriptions (a) and (b) are equivalent when $\mathbf{H}$ lies parallel to the [001] axis. For fields exactly in a twofold plane, if either the VL shape or orientation are inconsistent with the underlying crystal mirror symmetry then two VL domains coexist, as illustrated.Reuse & Permissions

Figure 8

(Color online) The transformation of the VL structure in the mixed state of sample Nb-2 as the field direction is rotated away from [001] in the plane close to (110) described by the dotted horizontal line in Fig. 5. The nomenclature of the interior angles and orientation are described in Fig. 7a. We depict the interior angles in (a) and orientation in (b) of the half-unit cell of the dominant VL domain at low fields and high temperatures $\gtrsim 4\text{K}$ including previous observations (blue symbols) (Ref. 6) as well as our own (black symbols). Overlayed are our data at ${\mu }_{0}H=200\text{mT}$ and low temperature $T=2\text{K}$ (red symbols). The solid lines in (a) are guides to the eye. In (c) the structure of the dominant VL domain at 2 K and high fields ${\mu }_{0}H=350\text{mT}$ is shown; the interior angles are depicted in the main figure while the orientation is described in the inset.Reuse & Permissions

Figure 9

(Color online) The field-dependence of the VL structure for fields applied $\simeq 16°$ away from [001] close to the (110) plane in Nb-2 at 2 K. The nomenclature for the orientation angle $\nu$ and interior angles $\alpha ,\beta ,\gamma$ is described in Fig. 7a. The filled triangles depict the half-unit cell schematically.Reuse & Permissions

Figure 10

(Color online) The VL structure as the field is rotated in the (100) mirror plane. The interior angles and orientation of the half-unit cell are denoted in Fig. 7b. The different colors indicate how the different VL phases observed at $\mathbf{H}\parallel \left[001\right]$ transform to the single isosceles VL domain at $\mathbf{H}\parallel \left[011\right]$ ($45°$ from [001]): black follows the high-temperature phase at 200 mT 4.5 K in Nb-1; blue follows the same phase from previous observations (Ref. 6) at 4.3 K for fields in the IMS using Nb-2; red follows the low-temperature, low-field phase at 200 mT 2 K in Nb-1; green follows the high-field phase at 330 mT 2 K in Nb-1. The lines are guides to the eye indicating the evolution of each phase.Reuse & Permissions

Figure 11

(Color online) Stereographic projections showing the real-space half-unit cell of the dominant VL domain for fields applied along various directions within an irreducible segment of solid angle. In (a), the open triangles depict our results in sample Nb-2 at 200 mT, 5 K, and the filled triangles represent previous observations at 4.3 K for lower fields in the intermediate mixed state (Ref. 6). For fields exactly in the twofold {110} and {100} planes, a scalene VL domain and its image in the mirror plane are degenerate and have equal populations. The domain which dominates as the field is rotated out of a mirror plane is shown by displacing the triangle appropriately. The inset of (a) shows in a similar manner the dominance of the two isosceles domains for fields applied close to [001]; the rapid switch in dominance as the field crosses a {110} mirror plane is illustrated in Fig. 4b where domains “34” (“12”) correspond to the shaded (open) triangles. In (b) are summarized our results at 350 mT, 2 K on the same sample; the VL structures and orientations for high fields applied within $10°$ of [001] are detailed in Fig. 5.Reuse & Permissions

Figure 12

Calculated Fermi surface of Nb from Ref. 28: (a) the distorted octahedron centered on $\Gamma$ of the closed second Brillouin zone; (b) the third Brillouin zone includes six distorted ellipsoids centered at N and an open, multiply connected “jungle gym” surface.Reuse & Permissions