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X. Song (2013). Atomic structure and chemistry of dense nanoprecipitates in $M g B_{2}$ superconductor , Ceramic International, 39, 4229-4305.

HREM images and corresponding Fourier transformation of the nano-precipitates

Ceramic $M g B_{2}$ superconductor has great potential for applications in the devices. Doping $M g B_{2}$ with carbon or SiC has been successful in improving critical current density $J_{c}$ , the upper critical field $H_{c 2}$ and flux-pinning. However, the presence of nano-precipitates, and their microstructure and chemistry, upon alloying/doping $M g B_{2}$ grains are largely unexplored. In this paper, the atomic structure and chemistry of 5–10 nm dense nano-precipitates were discovered in purified $M g B_{2}$ pellets by removing $B_{2} O_{3}$ . The nano-precipitates are platelet-shaped and fully coherent with the $M g B_{2}$ grains and having the longer axis of the plate perpendicular to the c-axis of $M g B_{2}$ grains. Systematic conventional and analytical Transmission Electron Microscopy (TEM) was carried out to study the atomic structure and chemistry of those nano-precipitates. The practical experimental approach of optimizing the composition and synthesis heat treatment of $M g B_{2}$ to induce dense uniformly distributed nano-precipitates is also discussed.

X. Song (2007). (110) facets and dislocation structure of low-angle grain boundaries in $Y B a_{2} C u_{3} O_{7 - δ}$ and $Y_{0.7} C a_{0.3} B a_{2} C u_{3} O_{7 - δ}$ thin film bicrystals , J. Materials Research, 22, 950-957.

$Diffraction contrast image of 5° grain boundary of YBCO grown under the higher deposition rate and diffraction contrast image$

The facet and dislocation structure of 5° and 7° [001]-tilt grain boundaries of $Y B a_{2} C u_{3} O_{7 - δ}$ (YBCO) and $Y_{0.7} C a_{0.3} B a_{2} C u_{3} O_{7 - δ}$ (YCaBCO) thin film bicrystals were studied. A 24° [001]-tilt YBCO grain boundary was also examined to contrast with the low angle grain boundary faceting behavior. All the low-angle grain boundaries exhibit strong faceting along (100)/(010) and (110) and possess both straight symmetric segments containing equally spaced [100] unit dislocations and step asymmetric segments composed of (110) and (100)/(010) facets. Grain boundaries with a higher degree of meander acquired up to 40% (110) facets. The atomic structure of (110) facets was revealed by the atomic resolution Z-contrast imaging. The (110) facets are dissociated for both the YBCO and YCaBCO grain boundaries. We also found the Ca-doped (110) facets to be more extended along the grain boundary plane, consistent with our earlier finding of a dissociated dislocation core in Ca-doped (100) facets. These 5° and 7° misorientations that we studied are just in the range at which YBCO grain boundaries start to become obstacles to current flow. The above results will be helpful for understanding the current transport across YBCO low-angle grain boundaries.

X. Song, Z. Chen, S. Kim, M. Feldmann, D. Larbalestier, J. Reeves, Y. Xie, V. Selvamanickam (2006). Evidence for Enhanced Flux-pinning by Small, Dense nanoprecipitates in a Sm-doped $Y B a_{2} C u_{3} O_{7 - δ}$ coated conductor , Applied Physics Letters, 88, 212508.

$(a) TEM plan-view diffraction contrast showing the nanoprecipitate array within YBCO grain. (b) Complex electron diffraction taken along the [001] axis of YBCO grain. (c) Index of complex electron diffraction shown in Fig. 1(b). The solid arrows indicate the diffraction spots from the square shaped precipitates. The dashed and dotted arrows indicate the diffraction spots from two sets of rectangular shaped precipitates. (d) High resolution TEM image showing the faceting of the nanoprecipitates$

About 17vol.% of ∼10nm∼10nm sized $(Y, S m)_{2} O_{3}$ precipitates have been developed in a metal-organic chemical vapor deposition grown Sm-doped $Y B a_{2} C u_{3} O_{7 - δ}$ coated conductor. The precipitate spacing of ∼15nm suggests strong vortex-precipitate pinning interactions, which are evidenced by a shift in the peak in the flux pinning force curve, an enhanced irreversibility field exceeding 8T at 77K, and a lack of temperature scaling of the flux pinning force over the temperature range of 65–82K.

X. Song, G. Daniels, M. Feldmann, A. Guriech, D. Larbalestier (2005). Electromagnetic, atomic structure and chemistry changes induced by Ca-doping of low-angle $Y B a_{2} C u_{3} O_{7 - δ}$ grain boundaries , Nature Materials , 4, 470-475.

Z-contrast TEM images of [100] dislocation cores a, for pure 7° [001]-tilt YBCO bicrystal. The Y-Ba and Cu-O columns are indicated in the middle image, and b, for 0.3Ca-doped 7° [001]-tilt bicrystal. The expanded, more-disordered tensile region of the dislocation core is indicated in the middle image. The sketches are of atomic column positions at [100] dislocation cores for the pure (a) and 0.3Ca-doped (b) GBs. Dashed lines indicate the dislocation core regions for pure YBCO GB (a) and the expanded core regions for the Ca-doped GB (b). The grey marks three columns with anomalous intensity in the Ca-doped GB

Practical high-temperature superconductors must be textured to minimize the reduction of the critical current density $J_{g b}$ at misoriented grain boundaries. Partial substitution of Ca for Y in $Y B a_{2} C u_{3} O_{7 - δ}$ has shown significant improvement in $J_{g b}$ but the mechanisms are still not well understood. Here we report atomic-scale, structural and analytical electron microscopy combined with transport measurements on 7° [001]-tilt $Y_{0.7} C a_{0.3} B a_{2} C u_{3} O_{7 - δ}$ and $Y B a_{2} C u_{3} O_{7 - δ}$ grain boundaries, where the dislocation cores are well separated. We show that the enhanced carrier density, higher $J_{g b}$ and weaker superconductivity depression at the Ca-doped boundary result from a strong, non-monotonic Ca segregation and structural rearrangements on a scale of ∼1 nm near the dislocation cores. We propose a model of the formation of $C a^{2 +}$ solute atmospheres in the strain and electric fields of the grain boundary and show that Ca doping expands the dislocation cores yet enhances $J_{g b}$ by improving the screening and local hole concentration.

X. Song, V. Braccini, D. Larbalestier (2004). Inter- and intra-granular nanostructure and possible spinodal decomposition in low resistivity bulk $M g B_{2}$ with varying critical fields , J. Materials Research , 19 (8), 2245-2255.

HREM images showing the nano-precipitates from samples A and B. (a) Moiré fringes caused by the overlapping of the crystalline nano-particles and the MgB2 matrix. (b) A cluster with 4–5 nanoprecipitates in close proximity. (c) Nanoscale strain field contrast, indicating fully coherent precipitates

Three electromagnetically well-characterized bulk samples with nominal resistivities at 40 K [ρ(40 K)], varying from 1 to 18 μΩcm, were investigated by conventional and high-resolution transmission electron microscopy. Clean, coherent, or semi-coherent grain boundaries and dirty-grain boundaries wetted by amorphous phases were found in all three samples, even though the starting sample A had the very low resistivity of 1 μΩcm at 40 K, characteristic of clean-limit samples. Taking into account its porosity and wetted-grain boundary area, the true resistivity value is about 0.5 μΩcm. Additional samples B and C, prepared by exposing sample A to Mg vapor, showed enhanced ρ(40 K) values of 14 and 18 μΩcm, without noticeable change in either inter- or intra-granular microstructure. Intragranular nanoprecipitates with characteristics of a spinodal of $M g B_{7}$ , with a size of 1–5 nm, were observed in a few areas of samples A and B at high local density; however at too low an overall density to explain the increased resistivities and upper critical fields.

L. Cooley, X. Song, J. Jiang, D. Larbalestier, T. He, K. Regan, and R. Cava (2002). Core pinning by intragranular nanoprecipitates in polycrystalline $M g C N i_{3}$ , Physical Review B, 65, 214518.

$Left figure: Electron micrographs of a polished sample. (b) Transmission electron micrograph shows that the MgCNi3 colonies are separated by a graphite layer. (c) Within each colony, MgCNi3 grains are 100–300 nm in size. The appearance of grain-boundary dislocations suggests low angles of misorientation between some grains. Right figure: Selected area diffraction patterns from the [101] and [211] zone axes, respectively. Arcs centered around the central electron beam (solid lines in the sketches below) are due to primary diffraction of graphite. Arcs centered around MgCNi3 diffraction spots (dashed lines in the sketches) are due to double diffraction$ $Selected area diffraction patterns from the @101# and @211# zone axes, respectively.$

The magnetic properties and nanostructure of polycrystalline $M g C N i_{3}$ prepared from a batch with overall composition $M g C_{1.5} N i_{3}$ , were studied by vibrating sample magnetometry and electron microscopy. Very high critical current density, e.g., 1.8 $m A / c m^{2}$ at 1 T and 4.2 K, is deduced from the magnetic hysteresis and evident subdivision of the sample into 10μm clusters of $M g C N i_{3}$ grains by excess graphite. The bulk pinning force $F_{p} (H)$ is comparable to that of other strong flux-pinning superconductors, such as NbN, Nb-Ti, and $N b_{3} S n$ , all of which have higher critical temperatures. While $F_{p} (H)$ indicates the expected grain-boundary pinning mechanism just below Tc≈7.2K, a systematic change to a core-pinning mechanism is indicated by a shift of the $F_{p} (H)$ curve peak to higher (reduced) field with decreasing temperature. The lack of temperature scaling of $F_{p} (H)$ suggests the presence of pinning sites at a nanometer scale inside the grains, which are smaller than the diameter of fluxon cores 2ξ(T) at high temperature and become effective when the coherence length ξ(T) approaches the nanostructural scale with decreasing temperature. High-resolution transmission electron microscopy imaging and electron diffraction revealed a substantial volume fraction of cubic and graphite nanoprecipitates comparable to ξ(0)≈5nm in size, consistent with the hypothesis above. Dirty-limit behavior seen in previous studies may thus be tied to electron scattering by the precipitates. To our knowledge, no other fine-grained bulk intermetallic superconductor exhibits a similar change from grain boundary to core pinning with decreasing temperature, suggesting that the arrangement of pinning sites in $M g C N i_{3}$ is unique. These results also indicate that strong flux pinning might be combined with a technologically useful upper critical field if variants of $M g C N i_{3}$ with higher Tc can be found.

C. Eom, M. Lee, J. Choi, L. Belenky, X. Song, L. Cooley, M. Naus, S. Patnaik, J. Jiang, M. Rikel, A. Polyanskii, A. Gurevich, X. Cai, S. Bu, S. Babcock, E. Hellstrom, D. Larbalestier, N. Rogado, K. Regan, M. Hayward, T. He, J. Slusky, K. Inumaru, M. Haas, and R. Cava (2001). High critical current density and enhanced irreversibility field in superconducting $M g B_{2}$ thin films , Nature, 411, 558-560.

$Transmission electron microscopy. a, Dark-field, plan-view image of film 2 in which only a fraction of the grains show bright intensity, clearly revealing the 10 nm grain size of the material. b, c, Selected area diffraction patterns from about 1um square areas of film 2. All of the rings can be attributed to either MgB2 or MgO. The pattern in b was collected with the incident electron beam aligned approximately parallel to the film normal. It shows only complete rings and those rings that correspond to the MgB2 phase are all of the (hk0) type. The pattern in c was collected with the incident beam oriented at an angle to the film normal. It shows an uneven distribution of intensity along all MgB2 rings. Considered together, these diffraction patterns indicate that the ®lm possesses a (00l ) (that is, c axis) ®bre texture oriented parallel to the ®lm normal with little or no texture in the plane of the film$

The discovery of superconductivity at 39 K in magnesium diboride offers the possibility of a new class of low-cost, high-performance superconducting materials for magnets and electronic applications. This compound has twice the transition temperature of $N b_{3} S n$ and four times that of Nb-Ti alloy, and the vital prerequisite of strongly linked current flow has already been demonstrated. One possible drawback, however, is that the magnetic field at which superconductivity is destroyed is modest. Furthermore, the field which limits the range of practical applications-the irreversibility field H*(T)-is approximately 7 T at liquid helium temperature (4.2 K), significantly lower than about 10 T for Nb-Ti and approximately 20 T for $N b_{3} S n$ (ref. 7). Here we show that $M g B_{2}$ thin films that are alloyed with oxygen can exhibit a much steeper temperature dependence of H*(T) than is observed in bulk materials, yielding an H* value at 4.2 K greater than 14 T. In addition, very high critical current densities at 4.2 K are achieved: 1 $M A c m^{- 2}$ at 1 T and 105 $A c m^{- 2}$ at 10 T. These results demonstrate that $M g B_{2}$ has potential for high-field superconducting applications.

I. J. Thompson, R. Feenstra, D. Christen, A. Gapud, and X. Song (2006). Critical currents of ex situ $Y B a_{2} C u_{3} O_{7 - δ}$ thin films on rolling assisted biaxially textured substrates: Thickness, field, and temperature dependencies , Physical Review B 73, 134502.

Angular dependence of the critical current density for a 700 nm YBCO film at 77 K. The transport measurements were conducted in applied fields of 0.5, 1, and 2 T, with the field always perpendicular to the current direction maximum force configuration

The critical current density Jc flowing in thin $Y B a_{2} C u_{3} O_{7 - δ}$ (YBCO) films of various thicknesses d has been studied magnetometrically, both as a function of applied field H and temperature T, with a central objective to determine the dominant source of vortex pinning in these materials. The films, grown by a $B a F_{2}$ ex situ process and deposited on buffered rolling assisted biaxially textured substrates (“RABiTS”) substrates of Ni−5%W, have thicknesses d ranging from 28nm to 1.5μm. Isothermal magnetization loops M(H;T) and remanent magnetization $M_{r e m} (T)$ in $H = 0$ were measured with H∥c-axis (i.e., normal to film plane). The resulting $J_{c} (d)$ values (obtained from a modified critical state model) increase with thickness d, peak near d∼120nm, and thereafter decrease as the films get thicker. For a wide range of temperatures and intermediate fields, we find $J_{c} \propto H^{- α}$ with α∼(0.56–0.69) for all materials. This feature can be attributed to pinning by large random defects, which theoretically has power-law exponent $α = 5 ∕ 8$ . Calculated values for the size and density of defects are comparable with those observed by TEM in the films. As a function of temperature, we find $J_{c} (T, s f) ~ [1 - (T ∕ T_{c})^{2}]^{n}$ with n∼1.2–1.4. This points to “ $δ T_{c}$ pinning” (pinning that suppresses $T_{c}$ locally) in these YBCO materials.

Z. Chen, M. Feldamnn, X. Song, D. Larbalestier (2007). Three-dimensional vortex pinning behavior in a Sm-doped $Y B a_{2} C u_{3} O_{7 - x}$ coated conductor with a high density of nano-precipitates , Superconductor Science & Technology, 20, S205.

Cross-sectional SEM image shows ∼20 vol% of non-superconducting Ba–Cu–O amorphous phase indicated with white arrows. The red rectangle highlights the region where the amorphous phase spans up to half the film thickness. Material degradation can be seen near the top surface of the YBCO, as indicated by red arrows

We report on the thickness and angular dependence of the critical current density $J_{c} (H, θ)$ , the irreversibility field $H^{i r r}$ , and the bulk pinning force $F_{p} (H)$ of a metal-organic chemical vapour deposition (MOCVD) grown $Y B a_{2} C u_{3} O_{7 - x}$ (YBCO) coated conductor, which contains ~17 vol% of ~10 nm sized $(Y, S m)_{2} O_{3}$ precipitates with an average spacing of ~10–15 nm. Some surface porosity and amorphous second-phase particles on the scale of ~0.5–1 µm appear to reduce the current-carrying cross-section, which controls the magnitude of $J_{c}$ but not the vortex pinning. We observed an enhanced $H^{i r r} ~ 9 T$ at 77 K along the c-axis which, like the shape of $J_{c} (H)$ and $F_{p} (H)$ , was independent of thickness as the sample was milled down to ~0.16 µm. Angular-dependent measurements of $J_{c}$ showed the usual excess vortex pinning along the c-axis and along the ab-plane, but with a background that could only be fitted with an unusually small anisotropy parameter of 3, which, like the high $H^{i r r}$ and the thickness-independent shape of $F_{p} (H)$ , we ascribe to strong vortex pinning centre interactions. Together, these measurements show very different behaviour from most pulsed-laser-deposited films, which exhibit strong thickness-dependent properties. We ascribe the present different results to the dense array of small, insulating precipitates, which act as strong pinning centres and produce strong three-dimensional (3D) vortex pinning, because their separation of 10–15 nm is always much smaller than the film thickness.

S. Kim, A. Gurevich, X. Song, X. Li, W. Zhang, T. Kodenkandath, M. Rupich, T. Holesinger and D. Larbalestier (2006). Mechanisms of weak thickness dependence of the critical current density in strong-pinning ex situ metal–organic-deposition-route $Y B a_{2} C u_{3} O_{7 - δ}$ coated conductors , Superconductor Science & Technology, 19, 968.

Cross-sectional TEM images of the CC270 sample: (a) a broad overview of the porosity; (b), (c) in higher resolution. Porosity with a scale in the range 0.2–0.5 µm increases towards the top of the YBCO. The rather uneven top YBCO surface where it contacts the silver cap layer has a height variation of up to 0.2 µm. The film also shows many planar stacking faults parallel to the YBCO ab-planes, which are typically 30–50 nm apart along the c-axis, of high density, and distributed rather evenly through the film thickness. The laminar and porous microstructure is in strong contrast to the columnar and dense microstructure of PLD conductors

We report on the thickness dependence of the superconducting characteristics including critical current $I_{c}$ , critical current density $J_{c}$ , transition temperature $T_{c}$ , irreversibility field $H_{i r r}$ , bulk pinning force plot $F_{p} (H)$ , and normal state resistivity curve $ρ (T)$ measured after successive ion milling of ~1 µm thick high- $I_{c}$ $Y B a_{2} C u_{3} O_{7 - x}$ films made by an ex situ metal–organic deposition process on Ni–W rolling-assisted biaxially textured substrates (RABiTSTM). In contrast to many recent data, mostly on in situ pulsed laser deposition (PLD) films, which show strong depression of $J_{c}$ with increasing film thickness t, our films exhibit only a weak dependence of $J_{c}$ on t. The two better textured samples had full cross-section average $J_{c, a v g}$ (77 K, 0 T) ~4 $M A c m^{- 2}$ near the buffer layer interface and ~3 $M A c m^{- 2}$ at full thickness, despite significant current blocking due to ~30% porosity in the film. Taking account of the thickness dependence of the porosity, we estimate that the local, vortex-pinning current density is essentially independent of thickness, while accounting for the additional current-blocking effects of grain boundaries leads to local, vortex-pinning $J_{c}$ values well above 5 $M A c m^{- 2}$ . Such high local $J_{c}$ values are produced by strong three-dimensional vortex pinning which subdivides vortex lines into weakly coupled segments much shorter than the film thickness.

Gurevich, S. Patnaik, V. Braccini, K. H. Kim, C. Meilke, X. Song, L. D. Cooley S. D. Bu, D. M. Kim, J. H. Choi, L. J. Belenky, J. Giencke, M. K. Lee, W. Tian, X.Q. Pan, A. Siri, E. E. Hellstrom, C. B. Eom, D.C. Larbalestier (2004). Very high upper critical fields in $M g B_{2}$ produced by selective tuning of impurity scattering , Superconductor Science & Technology, 17: 278-286.

The upper critical fields Hc2(T) (full squares) and irreversibility fields H*(T) (open squares) for a high resistivity 220 µΩ cm film, and Hc2(T) for a low resistivity 7 µΩ cm film (full triangles). Full curves show Hc2(T) calculated from equation (2) with the coupling constants λσσ = 0.81, λππ = 0.28, λσπ = 0.115 and λπσ = 0.09 [38]. (a) Corresponds to , for which the theoretical curve is plotted for Dπ = 0.12Dσ. (b) Corresponds to , for which the theoretical curve is plotted for [Dπ(c)Dπ(ab)]1/2 = 0.2[Dσ(c)Dσ(ab)]1/2. The broken curves show Hc2(T) plotted for MgB2 single crystals

We report a significant enhancement of the upper critical field $H_{c 2}$ of different $M g B_{2}$ samples alloyed with nonmagnetic impurities. By studying films and bulk polycrystals with different resistivities ρ, we show a clear trend of an increase in $H_{c 2}$ as ρ increases. One particular high resistivity film had a zero-temperature $H_{c 2} (0)$ well above the $H_{c 2}$ values of competing non-cuprate superconductors such as $N b_{3} S n$ and Nb–Ti. Our high-field transport measurements give record values T and T for high resistivity films and T for untextured bulk polycrystals. The highest $H_{c 2}$ film also exhibits a significant upward curvature of $H_{c 2} (T)$ and a temperature dependence of the anisotropy parameter opposite to that of single crystals: $γ (T)$ decreases as the temperature decreases, from to . This remarkable $H_{c 2}$ enhancement and its anomalous temperature dependence are a consequence of the two-gap superconductivity in $M g B_{2}$ , which offers special opportunities for further $H_{c 2}$ increases by tuning of the impurity scattering by selective alloying on Mg and B sites. Our experimental results can be explained by a theory of two-gap superconductivity in the dirty limit. The very high values of $H_{c 2} (T)$ observed suggest that $M g B_{2}$ can be made into a versatile, competitive high-field superconductor.

L. Cooley, X. Song, D. Larbalestier (2003). Improving flux pinning at high fields in intermetallic compounds: Clues from $M g B_{2}$ and $M g C N i_{3}$ , IEEE T APPL SUPERCON 13 (2): 3280-3283.

J. Wang, Y. Bugoslavsky, L. Cowey, A. Berenov, A. D. Caplin, L. F. Cohen, J. L. MacManus Driscoll, L. Cooley, X. Song, D. C. Larbalestier (2002). High In-Field Jc's in Bulk $M g B_{2}$ By A Scaleable, Nano-particle Addition Route , Applied Physics Letters, 81: 2026-2028.

X. Song, S. E. Babcock, C. B. Eom, D. C. Larbalestier, K. A. Regan, R. J. Cava, S. L. Bud’Ko, P. C. Canfield, and D. K. Finnemore (2002). Anisotropic Grain Morphology, Crystallographic Texture and Their Implications for Flux Pinning Mechanisms in $M g B_{2}$ Pellets, Filaments and Thin Films , Superconductor Science & Technology, 15: 511-518.

S. Patnaik, L. D. Cooley, A. Gurevich, A. A. Polyanskii, J. Jiang, X. Y. Cai, A. A. Squitieri, M. T. Naus, M. K. Lee, J. H. Choi, L. Belenky, S. D. Bu, J. Letteri, X. Song, D. G. Schlom, S. E. Babcock, C. B. Eom, E. E. Hellstrom, and D. C. Larbalestier (2001). Electronic Anisotropy, Magnetic Field-Temperature Phase Diagram and Their Dependence on Resistivity in c-Axis Oriented $M g B_{2}$ Thin Films , Superconductor Science & Technology, 14: 315-319.

$. (a) Low magnification image of the cross-sectional TEM sample showing bowing of the MgB2 film as it separated from the substrate during thinning. (b) Image of the thin film nearer the substrate in the region where fracture and delamination occurred. The amorphous material left on the substrate after delamination contains Mg and B$

Grain morphology and crystallographic texture were investigated by electron microscopy in four different polycrystalline forms of superconducting $M g B_{2}$ . The materials included a hot-pressed sintered $M g B_{2}$ pellet, a pellet reacted in situ from Mg and B, an in situ reacted $M g B_{2}$ filament and a pulsed-laser-deposited thin film grown on a single crystalline [111] oriented $S r T i O_{3}$ substrate. Thick plate-shaped grains with an aspect ratio of ~3 and large faces parallel to (0001) planes dominated the microstructure in all four types of sample. The intermediate-sized plate-shaped grains (0.1 μm × 0.3 μm on average) in the electromagnetically most homogeneous parts of the hot-pressed pellets were strongly facetted, but not textured. Large (3–5 μm) plate-shaped grains were seen in the pellet reacted directly from stoichiometric Mg and B. A tendency for parallel alignment of the [0001] axes of the considerably larger grains (~0.25 μm × 1 μm) in the filament was observed near its W core, but degradation of this texture away from the core was apparent. The very small grains (~10 nm) of the thin film possessed a well-defined fibre texture with [0001] parallel to the film normal and no preferred orientation in the plane of the film. Electrical resistivity of the finest grain samples was some 103 times higher than the largest grain sample and their critical current density about one order of magnitude higher. We conclude that, in contrast to the cuprate-based high- $T_{c}$ superconductors, grain boundaries do not limit the critical current density of polycrystalline $M g B_{2}$ and indeed act as flux-pinning centres, which enhance the critical current density.