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Results are reported from an experimental study investigating triply ionized holmium in 10 garnets using the point-change model to predict theoretical energy levels and temperature-dependent branching ratios for the 5I7 to 5I8 manifolds for temperatures between 50 and 400 K. Plots were made for the largest lines at 300 K. YScAG was plotted twice, once for each set of X-ray data available. Energy levels are predicted based on theoretical crystal-field parameters, and good agreement to experiment is found. It is suggested that the present set of theoretical crystal-field parameters provides good estimates of the energy levels for the other hosts on which there are no experimental optical data. X-ray and index-of-refraction data are used to evaluate the performance of 10 lasers via a quantum mechanical model to predict the position of the energy levels and the temperature-dependent branching rations of the 5I7 to 5I8 levels of holmium. The fractional population inversion required for threshold is also evaluated.
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Lu is alpha Samarium structured and crystallizes in the trigonal R-3m space group. The structure is three-dimensional. there are seven inequivalent Lu sites. In the first Lu site, Lu is bonded to twelve Lu atoms to form LuLu12 cuboctahedra that share corners with eighteen LuLu16 cuboctahedra, edges with eighteen LuLu12 cuboctahedra, and faces with twenty LuLu12 cuboctahedra. There are six shorter (3.44 Å) and six longer (3.49 Å) Lu–Lu bond lengths. In the second Lu site, Lu is bonded to twelve Lu atoms to form a mixture of edge, corner, and face-sharing LuLu12 cuboctahedra. There are a spread of Lu–Lu bond distances ranging from 3.42–3.49 Å. In the third Lu site, Lu is bonded to twelve Lu atoms to form a mixture of edge, corner, and face-sharing LuLu12 cuboctahedra. All Lu–Lu bond lengths are 3.49 Å. In the fourth Lu site, Lu is bonded to twelve Lu atoms to form LuLu12 cuboctahedra that share corners with fifteen LuLu12 cuboctahedra, edges with twenty-one LuLu16 cuboctahedra, and faces with nineteen LuLu12 cuboctahedra. There are a spread of Lu–Lu bond distances ranging from 3.42–3.49 Å. In the fifth Lu site, Lu is bonded to twelve Lu atoms to form a mixture of edge, corner, and face-sharing LuLu12 cuboctahedra. There are three shorter (3.44 Å) and six longer (3.49 Å) Lu–Lu bond lengths. In the sixth Lu site, Lu is bonded to twelve Lu atoms to form a mixture of edge, corner, and face-sharing LuLu12 cuboctahedra. There are three shorter (3.42 Å) and six longer (3.49 Å) Lu–Lu bond lengths. In the seventh Lu site, Lu is bonded to sixteen Lu atoms to form LuLu16 cuboctahedra that share corners with twenty-five LuLu12 cuboctahedra, edges with twenty-one LuLu16 cuboctahedra, and faces with thirty-five LuLu16 cuboctahedra. There are a spread of Lu–Lu bond distances ranging from 3.49–6.97 Å.
Lu(TiGa2)2 crystallizes in the tetragonal I4/mmm space group. The structure is three-dimensional. Lu is bonded in a distorted square co-planar geometry to twelve Ga atoms. There are four shorter (2.86 Å) and eight longer (3.31 Å) Lu–Ga bond lengths. Ti is bonded in a 10-coordinate geometry to two equivalent Ti and eight Ga atoms. Both Ti–Ti bond lengths are 2.73 Å. All Ti–Ga bond lengths are 2.78 Å. There are seven inequivalent Ga sites. In the first Ga site, Ga is bonded in a 11-coordinate geometry to three equivalent Lu, four equivalent Ti, and four Ga atoms. There are two shorter (2.66 Å) and two longer (2.90 Å) Ga–Ga bond lengths. In the second Ga site, Ga is bonded in a 11-coordinate geometry to three equivalent Lu, four equivalent Ti, and four Ga atoms. The Ga–Ga bond length is 2.66 Å. In the third Ga site, Ga is bonded in a 11-coordinate geometry to three equivalent Lu, four equivalent Ti, and four Ga atoms. The Ga–Ga bond length is 2.66 Å. In the fourth Ga site, Ga is bonded in a 11-coordinate geometry to three equivalent Lu, four equivalent Ti, and four Ga atoms. There are two shorter (2.66 Å) and two longer (2.90 Å) Ga–Ga bond lengths. In the fifth Ga site, Ga is bonded in a 11-coordinate geometry to three equivalent Lu, four equivalent Ti, and four Ga atoms. The Ga–Ga bond length is 2.66 Å. In the sixth Ga site, Ga is bonded in a 11-coordinate geometry to three equivalent Lu, four equivalent Ti, and four Ga atoms. Both Ga–Ga bond lengths are 2.90 Å. In the seventh Ga site, Ga is bonded in a 11-coordinate geometry to three equivalent Lu, four equivalent Ti, and four Ga atoms.
Lu(MnAl)6 crystallizes in the orthorhombic Immm space group. The structure is three-dimensional. Lu is bonded in a 8-coordinate geometry to twelve Mn and eight Al atoms. There are four shorter (3.18 Å) and eight longer (3.30 Å) Lu–Mn bond lengths. There are a spread of Lu–Al bond distances ranging from 2.88–3.04 Å. There are two inequivalent Mn sites. In the first Mn site, Mn is bonded in a 12-coordinate geometry to two equivalent Lu, four Mn, and six Al atoms. There are two shorter (2.46 Å) and two longer (2.52 Å) Mn–Mn bond lengths. There are a spread of Mn–Al bond distances ranging from 2.52–2.62 Å. In the second Mn site, Mn is bonded to two equivalent Lu, four equivalent Mn, and six Al atoms to form a mixture of distorted edge, corner, and face-sharing MnLu2Mn4Al6 cuboctahedra. There are a spread of Mn–Al bond distances ranging from 2.61–2.64 Å. There are three inequivalent Al sites. In the first Al site, Al is bonded in a 10-coordinate geometry to one Lu, six Mn, and three Al atoms. There are one shorter (2.70 Å) and two longer (2.80 Å) Al–Al bond lengths. In the second Al site, Al is bonded in a 8-coordinate geometry to one Lu, six Mn, and one Al atom. The Al–Al bond length is 2.80 Å. In the third Al site, Al is bonded in a 12-coordinate geometry to two equivalent Lu, six Mn, and two equivalent Al atoms.
Lu(CrAl)6 crystallizes in the orthorhombic Immm space group. The structure is three-dimensional. Lu is bonded in a 8-coordinate geometry to twelve Cr and eight Al atoms. There are four shorter (3.28 Å) and eight longer (3.33 Å) Lu–Cr bond lengths. There are a spread of Lu–Al bond distances ranging from 2.91–3.03 Å. There are two inequivalent Cr sites. In the first Cr site, Cr is bonded to two equivalent Lu, four Cr, and six Al atoms to form a mixture of distorted edge, corner, and face-sharing CrLu2Al6Cr4 cuboctahedra. There are two shorter (2.51 Å) and two longer (2.52 Å) Cr–Cr bond lengths. There are a spread of Cr–Al bond distances ranging from 2.58–2.66 Å. In the second Cr site, Cr is bonded to two equivalent Lu, four equivalent Cr, and six Al atoms to form a mixture of distorted edge, corner, and face-sharing CrLu2Al6Cr4 cuboctahedra. There are a spread of Cr–Al bond distances ranging from 2.65–2.69 Å. There are three inequivalent Al sites. In the first Al site, Al is bonded in a 10-coordinate geometry to one Lu, six Cr, and three Al atoms. There are one shorter (2.67 Å) and two longer (2.84 Å) Al–Al bond lengths. In the second Al site, Al is bonded in a 7-coordinate geometry to one Lu, six Cr, and two equivalent Al atoms. Both Al–Al bond lengths are 3.03 Å. In the third Al site, Al is bonded in a 12-coordinate geometry to two equivalent Lu, six Cr, and four Al atoms.
Lu(CuAl)6 crystallizes in the orthorhombic Immm space group. The structure is three-dimensional. Lu is bonded in a 12-coordinate geometry to twelve Cu and eight Al atoms. There are four shorter (3.19 Å) and eight longer (3.29 Å) Lu–Cu bond lengths. There are a spread of Lu–Al bond distances ranging from 2.95–3.08 Å. There are two inequivalent Cu sites. In the first Cu site, Cu is bonded in a 12-coordinate geometry to two equivalent Lu, four Cu, and six Al atoms. There are two shorter (2.51 Å) and two longer (2.53 Å) Cu–Cu bond lengths. There are a spread of Cu–Al bond distances ranging from 2.52–2.63 Å. In the second Cu site, Cu is bonded in a 12-coordinate geometry to two equivalent Lu, four equivalent Cu, and six Al atoms. There are a spread of Cu–Al bond distances ranging from 2.60–2.73 Å. There are three inequivalent Al sites. In the first Al site, Al is bonded in a 10-coordinate geometry to one Lu, six Cu, and three Al atoms. There are one shorter (2.63 Å) and two longer (2.83 Å) Al–Al bond lengths. In the second Al site, Al is bonded in a 8-coordinate geometry to one Lu, six Cu, and one Al atom. The Al–Al bond length is 2.64 Å. In the third Al site, Al is bonded in a 12-coordinate geometry to two equivalent Lu, six Cu, and two equivalent Al atoms.
Lu-doped zirconia (LuDZ) was reported in the 1960s to be a poor oxygen ion conductor. However, we identified this system as worthy of reexamination based on the results of a machine learning-based screen for promising oxygen ion conductors, further supported by a physical argument: because Lu is the heaviest and smallest of the lanthanides, it should offer the possibility of high conductivity due to its high polarizability and size compatibility with the zirconia structure. Here the relationship between dopant concentration and conductivity in the LuDZ system was determined using a high throughput experimental methodology in which spatially resolved conductivity was measured across a compositionally graded film, spanning from Zr 0.93 Lu 0.07 O 1.965 to Zr 0.82 Lu 0.18 O 1.91 . The conductivity broadly peaks with composition at 15 cation % Lu, coinciding with the approximate phase boundary of the fully stabilized cubic phase. The composition-dependent conductivity measurements reveal a surprising increase in activation energy for ion transport with increasing Lu content with no change in this trend at the composition of the conductivity maximum. The pre-exponential factor increases sharply with Lu content, but displays a distinct plateau once the conductivity maximum composition is attained. At 500 °C, the conductivity of Zr 0.85 Lu 0.15 O 1.925 is 2 × 10 –3 S cm –1 , approaching that of optimally doped Sc-stabilized zirconia, which has the highest reported conductivity among zirconia compounds. In comparison to other oxide ion conductors, Lu-stabilized zirconia offers the advantage over ceria of negligible electronic conductivity and is easier to process than (La,Sr)(Ga,Mg)O 3-δ .
Lu(MnGa)6 crystallizes in the orthorhombic Fmmm space group. The structure is three-dimensional. Lu is bonded in a 12-coordinate geometry to eight Mn and twelve Ga atoms. There are four shorter (3.14 Å) and four longer (3.32 Å) Lu–Mn bond lengths. There are eight shorter (3.21 Å) and four longer (3.42 Å) Lu–Ga bond lengths. There are two inequivalent Mn sites. In the first Mn site, Mn is bonded in a 12-coordinate geometry to two equivalent Lu, four equivalent Mn, and four equivalent Ga atoms. All Mn–Mn bond lengths are 2.69 Å. All Mn–Ga bond lengths are 2.46 Å. In the second Mn site, Mn is bonded in a 10-coordinate geometry to one Lu, seven Mn, and six Ga atoms. There are a spread of Mn–Mn bond distances ranging from 2.62–3.10 Å. There are a spread of Mn–Ga bond distances ranging from 2.57–2.79 Å. There are two inequivalent Ga sites. In the first Ga site, Ga is bonded to two equivalent Lu, four equivalent Mn, and six Ga atoms to form a mixture of distorted face and corner-sharing GaLu2Mn4Ga6 cuboctahedra. There are two shorter (2.42 Å) and four longer (2.61 Å) Ga–Ga bond lengths. In the second Ga site, Ga is bonded in a 12-coordinate geometry to two equivalent Lu, six Mn, and four Ga atoms. There are one shorter (2.91 Å) and one longer (3.06 Å) Ga–Ga bond lengths.
LuFe10Si2 crystallizes in the orthorhombic Immm space group. The structure is three-dimensional. Lu is bonded in a 12-coordinate geometry to sixteen Fe and four equivalent Si atoms. There are a spread of Lu–Fe bond distances ranging from 2.92–3.16 Å. All Lu–Si bond lengths are 3.06 Å. There are four inequivalent Fe sites. In the first Fe site, Fe is bonded in a 10-coordinate geometry to one Lu, eleven Fe, and two equivalent Si atoms. There are a spread of Fe–Fe bond distances ranging from 2.32–2.90 Å. Both Fe–Si bond lengths are 2.60 Å. In the second Fe site, Fe is bonded in a 10-coordinate geometry to one Lu, eleven Fe, and two equivalent Si atoms. There are a spread of Fe–Fe bond distances ranging from 2.41–2.67 Å. Both Fe–Si bond lengths are 2.53 Å. In the third Fe site, Fe is bonded in a 12-coordinate geometry to two equivalent Lu, eight Fe, and two equivalent Si atoms. All Fe–Fe bond lengths are 2.42 Å. Both Fe–Si bond lengths are 2.57 Å. In the fourth Fe site, Fe is bonded to two equivalent Lu, eight Fe, and two equivalent Si atoms to form distorted FeLu2Fe8Si2 cuboctahedra that share corners with four equivalent SiLu2Fe10 cuboctahedra, corners with ten equivalent FeLu2Fe8Si2 cuboctahedra, edges with two equivalent SiLu2Fe10 cuboctahedra, edges with four equivalent FeLu2Fe8Si2 cuboctahedra, faces with four equivalent SiLu2Fe10 cuboctahedra, and faces with six equivalent FeLu2Fe8Si2 cuboctahedra. Both Fe–Fe bond lengths are 2.37 Å. Both Fe–Si bond lengths are 2.38 Å. Si is bonded to two equivalent Lu and ten Fe atoms to form distorted SiLu2Fe10 cuboctahedra that share corners with six equivalent SiLu2Fe10 cuboctahedra, corners with eight equivalent FeLu2Fe8Si2 cuboctahedra, edges with three equivalent SiLu2Fe10 cuboctahedra, edges with four equivalent FeLu2Fe8Si2 cuboctahedra, a faceface with one SiLu2Fe10 cuboctahedra, and faces with eight equivalent FeLu2Fe8Si2 cuboctahedra.
Lu(CuSn)2 crystallizes in the tetragonal P4/nmm space group. The structure is three-dimensional. Lu is bonded in a 12-coordinate geometry to eight Cu and eight Sn atoms. All Lu–Cu bond lengths are 3.34 Å. There are four shorter (3.24 Å) and four longer (3.64 Å) Lu–Sn bond lengths. There are two inequivalent Cu sites. In the first Cu site, Cu is bonded in a 12-coordinate geometry to four equivalent Lu, four equivalent Cu, and four equivalent Sn atoms. All Cu–Cu bond lengths are 3.04 Å. All Cu–Sn bond lengths are 2.60 Å. In the second Cu site, Cu is bonded in a 9-coordinate geometry to four equivalent Lu and five Sn atoms. There are one shorter (2.47 Å) and four longer (2.66 Å) Cu–Sn bond lengths. There are two inequivalent Sn sites. In the first Sn site, Sn is bonded to four equivalent Lu and four equivalent Cu atoms to form a mixture of distorted edge and face-sharing SnLu4Cu4 tetrahedra. In the second Sn site, Sn is bonded in a 9-coordinate geometry to four equivalent Lu and five Cu atoms.
The systems of equations governing chemically nonequilibrium flows are solved numerically by two lower-upper(LU) algorithms, namely, LU-SSOR (Yoon and Jameson, 1987) and LU-SW (Steger and Warming, 1981). The methods of flux-Jacobian splitting differentiate the two LU algorithms. The LU-SSOR gives a simpler implicit operator with less temporal damping, while the LU-SW converges faster but requires more computation for each iteration. Relative efficiencies and mutual merits of these two LU algorithms are compared. The results show that LU-SW with upwind-differenced right-hand-side gives the best convergence.
Lu(FeGe)6 crystallizes in the hexagonal P6/mmm space group. The structure is three-dimensional. Lu is bonded to twelve equivalent Fe and eight Ge atoms to form distorted face-sharing LuFe12Ge8 hexagonal bipyramids. All Lu–Fe bond lengths are 3.26 Å. There are two shorter (2.81 Å) and six longer (2.94 Å) Lu–Ge bond lengths. Fe is bonded in a 12-coordinate geometry to two equivalent Lu, four equivalent Fe, and six Ge atoms. All Fe–Fe bond lengths are 2.55 Å. There are four shorter (2.51 Å) and two longer (2.66 Å) Fe–Ge bond lengths. There are three inequivalent Ge sites. In the first Ge site, Ge is bonded in a 8-coordinate geometry to one Lu, six equivalent Fe, and one Ge atom. The Ge–Ge bond length is 2.52 Å. In the second Ge site, Ge is bonded in a 6-coordinate geometry to six equivalent Fe atoms. In the third Ge site, Ge is bonded in a 12-coordinate geometry to three equivalent Lu and six equivalent Fe atoms.
Background: Mainly because of its long half-life and despite its scientific relevance, spectroscopic measurements of 176 Lu forbidden β decays are very limited and lack formulation of shape factors. A direct precise measurement of its Q value is also presently unreported. In addition, the description of forbidden decays provides interesting challenges for nuclear theory. The comparison of precise experimental results with theoretical calculations for these decays can help to test underlying models and can aid the interpretation of data from other experiments. Purpose: Perform the first precision measurements of 176 Lu β-decay spectra and attempt the observation of its electron capture decays, as well as perform the first precision direct measurement of the 176 Lu β-decay Q value. Compare the shape of the precisely determined experimental β spectra to theoretical calculations, and compare the end point energy to that obtained from an independent Q value measurement. Method: Here, the 176 Lu β-decay spectra measurements and the search for electron capture decays were performed with an experimental setup that employed lutetium-containing scintillator crystals and a NaI(Tl) spectrometer for coincidence counting. The β decay Q value was determined via high-precision Penning trap mass spectrometry (PTMS) with the LEBIT facility at the National Superconducting Cyclotron Laboratory. The β-spectrum calculations were performed within the Fermi theory formalism with nuclear structure effects calculated using a shell model approach. Results: Both β transitions of 176 Lu were experimentally observed and corresponding shape factors formulated in their entire energy ranges. The search for electron capture decay branches led to an experimental upper limit of 6.3×10 –6 relative to its β decays. The 176 Lu β-decay and electron capture Q values were measured using PTMS to be 1193.0(6) and 108.9(8) keV, respectively. This enabled precise β end point energies of 596.2(6) and 195.3(6) keV to be determined for the primary and secondary β decays, respectively. The conserved vector current hypothesis was applied to calculate the relativistic vector matrix elements. The β-spectrum shape was shown to significantly depend on the Coulomb displacement energy and on the value of the axial vector coupling constant gA, which was extracted according to different assumptions. Conclusion: The implemented self-scintillation method has provided unmatched observations of 176 Lu, independently validated by the first direct measurements of its β-decay Q value by Penning trap mass spectrometry. Theoretical study of the main β transition led to the extraction of very different effective gA and log10f values, showing that a high-precision description of this transition would require a realistic nuclear structure with nucleus deformation.
Lu(BC)2 crystallizes in the tetragonal P4/mbm space group. The structure is three-dimensional. Lu is bonded in a 8-coordinate geometry to eight equivalent B and eight equivalent C atoms. All Lu–B bond lengths are 2.69 Å. All Lu–C bond lengths are 2.63 Å. B is bonded in a distorted trigonal planar geometry to four equivalent Lu and three equivalent C atoms. There is one shorter (1.52 Å) and two longer (1.59 Å) B–C bond length. C is bonded in a 3-coordinate geometry to four equivalent Lu and three equivalent B atoms.