Hemisphere Of Radius Research Articles

Ferroelectric structure and related properties of Pb5W3O9F1

Lead tungsten oxyfluoride, Pb5W3O9F10, is a new ferroelectric material that crystallizes in the tetragonal system with space group I4 and four formulas in the unit cell. The lattice constants at 295 K are a=14.583(3) and c=7.365(1) Å for λ(MoKα1,2)=0.709 30, 0.713 59 Å. The integrated intensities of all 10 757 reflections within a reciprocal hemisphere of radius (sin θ)/λ ≤0.91 Å−1 were measured on a CAD-4 diffractometer with MoKα radiation and resulted in 7080 F2m ≥2σF2m. Correction for linear anisotropic radiation damage, followed by averaging the symmetry equivalent reflections, gave 2933 independent 〈 F2m〉≥2σ〈 F2m〉 with internal agreement factor Rint〈Fm〉 =0.0289. The crystal structure was solved from the Patterson function and difference Fourier series and refined on w〈 F2m〉 by the method of least squares. The final agreement factors between measurement and model were R=0.0382 and wR=0.0329 on 〈 Fm〉. The as-grown crystal contained equal volumes of ±Z domains. The three independent tungsten atoms form WO6, WO3F3, and WO2F4 octahedra, and the three independent lead atoms form two different PbO4F5 and a PbO4F4 polyhedron. The major local dipole contribution to the spontaneous polarization is associated with a tungsten atom that is displaced 0.172(43) Å from the center of its octahedron. The corresponding predicted Curie temperature is 590(260) K, in agreement with the experimental value of 785(15) K. All six metal atoms undergo anharmonic thermal vibrations, which are fitted either to fifth or sixth order in a Gram–Charlier expansion of the probability density function. The average interatomic distances are W–F=1.798(8) Å, W–O=1.944(18) Å, Pb–F=2.68(19) Å, and Pb–O=2.72(21) Å: standard deviations are from Bessel’s relationship. The atomic rearrangement required in forming Pb5W3O9F10 from PbF2 and WO3 is discussed, and similarities and contrasts between the related families for which Pb5W3O9F10, SrAlF5, and Sr3(FeF6)2 provide the archetype structures are considered. All three families exhibit ferroelectric behavior.

ChemInform Abstract: ESR Line Shape Studies of N‐(4‐n Butyl benzilidine)4‐amino 2,2,6,6‐Tetramethyl Piperidine 1‐Oxide (BBTMPO) in Toluene.

BBTMPO was synthesized by a condensation of 4‐n butyl benzaldehyde and 4‐amino 2,2,6,6‐tetramethyl piperidine N‐oxide (4‐tempamine). The shape of the resulting spin probe, estimated using the Dreiding models, is a spherocone of length 10.7 A capped by hemispheres of radii 6.6 and 4.0 A, respectively. The magnetic parameters of BBTMPO in toluene at 77 K were determined. Variable temperature ESR line shape studies in the temperature range of 142–211 K revealed that BBTMPO exhibited anisotropic rotational diffusion in toluene along the long axis of the spherocone and in the direction parallel to the N–O bond. Rotational diffusion along the long axis was found to be 3±0.5 and 7±0.5 times faster for frozen and liquid toluene, respectively, than the other two axes. The line shapes have been simulated by using the stochastic Liouville theory of slow‐motional effects on ESR spectrum. The slow tumbling spectrum (158 K) in the model‐sensitive region showed good agreement for a Brownian model. The plot of τR vs 1/T ...

ESR line shape studies of N-(4-n butyl benzilidine) 4-amino 2,2,6,6-tetramethyl piperidine 1-oxide (BBTMPO) in toluene

BBTMPO was synthesized by a condensation of 4-n butyl benzaldehyde and 4-amino 2,2,6,6-tetramethyl piperidine N-oxide (4-tempamine). The shape of the resulting spin probe, estimated using the Dreiding models, is a spherocone of length 10.7 Å capped by hemispheres of radii 6.6 and 4.0 Å, respectively. The magnetic parameters of BBTMPO in toluene at 77 K were determined. Variable temperature ESR line shape studies in the temperature range of 142–211 K revealed that BBTMPO exhibited anisotropic rotational diffusion in toluene along the long axis of the spherocone and in the direction parallel to the N–O bond. Rotational diffusion along the long axis was found to be 3±0.5 and 7±0.5 times faster for frozen and liquid toluene, respectively, than the other two axes. The line shapes have been simulated by using the stochastic Liouville theory of slow-motional effects on ESR spectrum. The slow tumbling spectrum (158 K) in the model-sensitive region showed good agreement for a Brownian model. The plot of τR vs 1/T over the whole temperature range yield a good linear fit with two slopes, with the activation energy for reorientation process of 4.9 kcal/mol in the liquid state, and 9.9 kcal/mol in the solid state. The higher activation energy of rotational reorientation coupled with a lower anisotropy of molecular reorientation (N) in the solid state, indicates that the rotational degrees of freedom is more restricted in the frozen state than in the liquid state. Molecular reorientation for BBTMPO in toluene was analyzed in terms of the hydrodynamic free space model for molecular relaxation in liquids. The stickiness factor S was found to be 0.6 while the anisotropic interaction parameter κ which measures the coupling of rotation to translation was found to be 0.7. For perdeuterated 2,2,6,6-tetramethyl-4-piperidine N-oxide (PD-Tempone) in toluene the stickiness factor was calculated to be 0.3, and κ was 0.4. The doubling in size and weight in going from PD-Tempone to BBTMPO increases the stickiness factor and κ from 0.3 to 0.6, and 0.4 to 0.7, respectively. The similarity in the structure of BBTMPO to that of liquid crystal methoxy benzilidine butyl aniline (MBBA) and its potential use as a spin probe for MBBA is discussed.

Selected fretting-wear-resistant coatings for Ti-6%Al-4%V alloy

A Ti-6%Al-4%V alloy (Ti-6Al-4V), in the solution-treated overaged condition, was subjected to fretting exposures against uncoated Ti-6Al-4V as a baseline and against various coatings and surface treatments applied to Ti-6Al-4V. Fretting experiments were conducted in dry air and in saturated air by using a hemisphere of radius 4.7 mm on a flat surface. Typical conditions included a 1.47 N normal load, an 80 Hz frequency and a 70 μm amplitude at room temperature. The coatings evaluated included plasmasprayed tungsten carbide with 12% Co, aluminum oxide (Al 2O 3) with 13% titanium oxide (TiO 2), chromium oxide, aluminum bronze with 10% aromatic polyester, polymer-bonded polyimide, polyimide with graphite fluoride, polyimide with molybdenum disulfide (MoS 2) and methylphenylsiliconebonded MoS 2 and the surface treatments included a pre-oxidation surface treatment, a nitride surface treatment and a sputtered MoS 2 coating. Results of wear measurements on both the coated and the uncoated surfaces after 3 × 10 5 cycles indicated that the polyimide coating was the most wear resistant and caused the least wear to the uncoated surfaces. Increased exposure, however, showed a sharp increase in fretting wear rate after 10 6 cycles. Surface-nitrided Ti-6Al-4V fretted against a polytetrafluoroethylene-impregnated Al 2O 3-TiO 2 coating on Ti-6Al-4V resulted in immeasurable wear to both surfaces.

Base Metal Contacts: An Exploratory Study of Separable Connection to Tin-Lead

A study was made of solder plated butt contacts to determine the feasibility of designing separable connectors for high reliability low energy circuits without gold or other precious metals. The chemical reactivity of tin-lead and the penetrability of films that formed on it were determined. It is little degraded by sulfur vapor, 0 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , SO2, H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> S, CO, and NH3. However, films from NO2 and Cl <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> can grow to large thickness, limited only by time and concentration of the reactant. Corrosion of tin-lead was promoted by a synthetic dust and by salt spray. The penetrability of tin-lead depends on contact pressure, and, owing to creep, on temperature and time. Except at light loads with certain geometries, 0.025 mm (0.001 inch) tin-lead is penetrated nearly to the copper substrate. Reconnection to such spots may be less reliable than to fresh areas, particularly if the contacts are aged between separation and remating. The ability of films on tin-lead to be broken is dependent on contact geometry, load, and wipe. The most effective geometries are cones and short (2.5 ram) wedges having an acute included angle (60°), followed by cones and wedges with an obtuse angle (120°), followed by hemispheres of increasing radius (0.38 to 1.5 mm). The cones and wedges had rounded tips of 0.05 mm radius. The order of merit may be different with thickly plated contacts (much in excess of 0.025 mm); but thick coatings are undesirable on contacts because of their deformability. When good contact is obtained, however, as in the mating of clean surfaces, the ranking of probe geometries in maintaining good contact in aggressive atmospheres is different, with the 1.5 mm radius hemisphere being best. It may be possible to design reliable butt connectors with tin-lead coated contacts for critical applications. However, additional studies need to be undertaken. These include: development of field experience in varied environments, study of long term behavior of mated contacts such as cold welding at elevated temperature, and rematability of contacts that have been aged in corrosive environments between initial use and subsequent matings.

Ferroelectric Tungsten Bronze-Type Crystal Structures. III. Potassium Lithium Niobate K(6−x−y)Li(4+x)Nb(10+y)O30

Ferroelectric K(6−x−y)Li(4+x)Nb(10+y)O30, with x ≃ 0.07, y ≃ 0.23, and a Curie temperature of 613°K, crystallizes with a tungsten bronze-type structure in the tetragonal system. The lattice constants are a = 12.5764 ± 0.0002 and c = 4.0149 ± 0.0001 Å at 298°K. The space group is P4bm. There is one formula per unit cell. The integrated intensities of 6578 structure factors, inside a reciprocal hemisphere of radius (sinθ) / λ = 1.02 Å−1, were measured with PEXRAD. There are 998 symmetry-independent Fmeas significantly above background. Isomorphism with BaxSr(5−x)Nb10O30 allowed the metal atom positions to be deduced. Oxygen atom positions were obtained by Fourier series methods. Refinement of structural parameters by the method of least squares resulted in a final agreement factor R = 0.0401. The structure is generally like that of other tungsten bronzes of formula (A1)2(A2)4C4(B1)2(B2)8O30, with the A1 site occupied by 87% K and 13% Li, the A2 site by 99% K and 1% Li, and the C site by 94% Li and 6% Nb. The B sites are fully occupied by Nb. X-ray spectrochemical analysis and the present diffraction study suggest x ≃ 0.07 and y ≃ 0.23 for the nonstoichiometric formula above. Evidence for major positional disorder, excluding occupancy disorder, similar to that observed in BaxSr(5−x)Nb10O30 and in Ba(4+x)Na(2−2x)Nb10O30 is not present for K(6−x−y)Li(4+x)Nb(10+y)O30. The K/Li atom at the A1 site is 12 coordinated with average K/Li–O distance of 2.833 Å. The K/Li atom at the A2 site is nine coordinated with average K/Li–O distance of 3.024 Å. The Li/Nb atom at the C site has three nearest oxygen neighbors at an average distance of 2.153 Å and six additional next-nearest neighbors at an average distance of 2.542 Å. The two independent irregular NbO6 octahedra have an average Nb–O separation of 1.973 Å. The Nb atom displacements, of 0.167 and 0.160 Å from the nearest mean oxygen plane, may be compared with the value of 0.175 Å predicted from the parabolic relation with Curie temperature. All metal atoms are displaced from the mean planes of oxygen in the same sense, making the positive polarization direction parallel to the sense of this displacement.

Crystal Structure of Pyroelectric Paramagnetic Barium Manganese Fluoride, BaMnF4

BaMnF4, pyroelectric at room temperature and paramagnetic above 30°K, crystallizes in the orthorhombic system with space group A21am and lattice constants a = 5.9845 ± 0.0003, b = 15.098 ± 0.002, and c = 4.2216 ± 0.0003 Å at 298°K. The integrated intensities of 5085 reflections within a reciprocal lattice hemisphere of radius (sinθ) / λ = 1.02 Å−1 were measured using pexrad. The crystal structure has been solved and refined by a combination of Patterson and Fourier series and the method of least squares, using 455 independent and significantly nonzero F(meas). The final agreement factor R, for atoms with anisotropic temperature coefficients, is 0.0466. The structure consists of MnF6 octahedra sharing corners to form puckered sheets parallel to (010). These sheets are linked by Ba2+ ions only, giving rise to excellent (010) cleavage. Within the sheets there are nearly linear –Mn–F–Mn–F– chains parallel to c. Parallel to a there is a second set of planar zigzag –Mn–F–Mn–F– chains, formed by adjacent shared fluorine atoms with Mn–F–Mn angles of 138.6°. The F–Mn–F angles in this chain are 98.3°. The remaining two fluorine atoms in each octahedron are adjacent. These atoms, on the outside of the sheets, are bonded to one Mn atom only. The octahedra are distorted with Mn–F distances ranging from 2.037 to 2.151 Å, with an average of 2.098 Å. Each barium ion is surrounded by a distorted trigonal prism of fluorine atoms, with two more located in the equatorial plane through rectangular prism faces, all Ba–F distances being less than 2.88 Å. A model is presented to account for the observed ferroelectricity in BaZnF4 and the absence of ferroelectricity in isomorphous BaMnF4, based on the separation of the Zn or Mn atoms in the puckered sheets. The experimental absolute polarization sense is in agreement with that calculated. The theoretical polarization for BaMnF4 of 11.5 ± 1.5 μC cm−2 may be compared with the average value of 8.0 μC cm−2 for the ferro-electric members of the BaMF4 series.

Ferroelectric Tungsten Bronze-Type Crystal Structures. II. Barium Sodium Niobate Ba(4 + x)Na(2 − 2x)Nb10O30

Ferroelectric Ba(4+x)Na(2 − 2x)Nb10O30, with a Curie temperature of 833°K, has a tungsten bronze-type structure and crystallizes in the orthorhombic system, with subcell lattice constants a = 17.59182 ± 0.00001, b = 17.62560 ± 0.00005, and c = 3.994915 ± 0.000004 Å at 298°K. The space group is Cmm2, and there are two formulas in the subcell. The c axis of the true cell is double that of the subcell. The integrated intensities of 6911 reflections within a reciprocal hemisphere of radius (sinθ)/λ = 1.02 Å−1 were measured with PEXRAD, 1873 symmetry-independent structure factors being significantly above background. The metal atom positions were determined from the three-dimensional Patterson function and the oxygen atoms from metal-phased Fourier series. The final agreement index between measured and calculated structure factors is 0.0579. The structure differs only in detail from previously determined tetragonal tungsten bronze structures. In the general formula (A1)2(A2)4C4(B1)2(B2)8O30, the B1 and B2 sites are fully occupied by Nb, the A2 sites by Ba and the A1 site by 87.0% Na and 6.5% Ba. Evidence from chemical analysis, x-ray density calculations and the present determination suggests that the best approximation to the formula of the crystal studied is Ba4.13Na1.74Nb10O30. The Ba and O atoms at z ≃ 12 are disordered in a manner similar to the O atoms in the Ba layer in barium strontium niobate. The four crystallographically independent Nb atoms, each in octahedral coordination, are linked to O atoms by distances ranging from 1.765 ± 0.021 to 2.270 ± 0.021 Å, with a mean value of 1.989 Å. Ba is 10 coordinated, with Ba–O distances ranging upward from 2.671 ± 0.013 Å. Na is 12 coordinated, with Na–O distances ranging from 2.660 ± 0.014 to 2.990 ± 0.015 Å, with a mean of 2.788 Å. The Nb-atom displacements from the mean oxygen planes lie in the range 0.171–0.205 Å; the parabolic relation with Curie temperature predicts a displacement of 0.204 Å. The measured value of Ps at room temperature is 40 μC cm−2: the linear relation between displacement and polarization predicts a saturation value of 44–53 μC cm−2. All metal atoms are displaced from the oxygen planes in the sense given by the macroscopic positive polarity.

Prebreakdown studies using sphere-hemisphere electrode system

Measurements have been made of corona-threshold and breakdown voltages in dry air in nonuniform fields for a sphere-hemisphere electrode configuration. For a particular size of outer hemisphere, the electrostatic field was changed from quasiuniform to nonuniform either by using various sizes of inner spheres for concentric arrangements, or, for eccentric arrangements, by varying the eccentricity of the inner sphere. Two outer hemispheres of radius 7.25 cm and 3.81 cm, respectively, were used in the present investigation. For direct voltage, with concentric arrangements, the present results are in general agreement with those previously obtained by Hermstein, and photographic evidence is given of the transition from streamer to glow. Measurements have been extended to eccentric gaps and power-frequency voltages. Finally, some power-frequency-breakdown results are given for concentric conditions, for pressures up to 8atm. The significance of these new results is discussed.

Crystal Structure of the Transition-Metal Molybdates and Tungstates. V. Paramagnetic Alpha-Nh2(MoO4)3

Paramagnetic α-Nd2(MoO4)3, prepared below the transition temperature of 960°C, crystallizes in a subtle variant of the scheelite structure. The lattice constants of the basic tetragonal unit cell are a = 5.298 ± 0.007 and c = 11.705 ± 0.023 Å, with space group I41/a and 43 of a Nd2(MoO4)3 formula per unit cell. The ordered distribution of two Nd atoms over every three available sites results in additional weak scattering corresponding to a monoclinic unit cell with a = 27.02, b = 11.705, and c = 11.85 Å, with β = 105.3°, space group C2. The integrated intensities of 2490 scheelitelike reflections within a reciprocal hemisphere of radius (sinθ) / λ = 1.02 Å−1 were measured with PEXRAD, of which 198 were independent and above background. Refinement of the atomic positions, assuming a statistical distribution of 23 Nd on each scheelite 4b site, resulted in an agreement factor between measured and calculated structure factors of 0.0375. Each Nd atom is dodecahedrally coordinated to eight oxygen atoms, with four Nd–O distances of 2.492 ± 0.010 Å and four others of 2.509 ± 0.011 Å. Each Mo atom occupies a tetrahedron of oxygen atoms, with four identical Mo−O distances of 1.800 ± 0.011 Å. An arrangement of Nd atoms and vacancies in the monoclinic cell giving a predicted scattering pattern in satisfactory agreement with observation is presented.

Crystal Structure of the Transition-Metal Molybdates and Tungstates. IV. Paramagnetic CuMoO4

Paramagnetic CuMoO4 is triclinic, space group P1̄, with lattice constants a = 9.903 ± 0.028, b = 6.783 ± 0.015, c = 8.359 ± 0.0019 Å, α = 101.08° ± 0.20°, β = 96.88° ± 0.20°, and γ = 107.05° ± 0.20° at 298°K, with six formula weights in the unit cell. The integrated intensities of 6310 reflections within a reciprocal hemisphere of radius (sinθ) / λ = 1.02 Å−1 were measured with PEXRAD; of these, 3081 were independent and significantly above background. The atomic arrangement is similar to that in ZnMoO4 but with corresponding atoms displaced by as much as 0.4 Å. Refinement by the method of least squares gave a final agreement factor between measured and calculated structure factors of 0.064. Two of the independent copper atoms occupy axially distorted octahedra of oxygen atoms; the four short Cu–O bond distances in each octahedron range between 1.933 ± 0.010 and 1.980 ± 0.012 Å, with an average value of 1.953 Å. The axial Cu–O bonds range in length between 2.213 ± 0.012 and 2.619 ± 0.011 Å, with an average value of 2.364 Å. The third copper atom occupies a distorted tetragonal pyramidal arrangement of oxygen atoms, with basal Cu–O distances ranging from 1.894 ± 0.013 to 1.985 ± 0.010 Å, the average being 1.928 Å; the apical Cu–O bond length is 2.341 ± 0.009 Å. The three independent molybdenum atoms are surrounded by somewhat distorted oxygen tetrahedra with Mo–O distances varying between 1.701 ± 0.010 and 1.862 ± 0.008 Å, the average being 1.772 Å. All atoms in CuMoO4 vibrate with significant thermal anisotropy.

Crystal Structure of Piezoelectric Bismuth Germanium Oxide Bi12GeO20

Piezoelectric Bi12GeO20 crystallizes in the cubic system with space group I23 and lattice constant a = 10.1455±0.0008 Å at 298°K. The complete x-ray scattering pattern within a reciprocal lattice hemisphere of radius (sinθ)/λ = 1.02 Å−1 was determined with PEXRAD. A total of 4812 reflections were measured, resulting in 631 independent and significantly nonzero Fmeas. The crystal structure was solved using three-dimensional Patterson and Fourier series, and refined by the method of least squares. The final agreement factor R is 0.062. The germanium atoms occupy geometrically regular tetrahedra, with Ge–O=1.717±0.028 Å. The bismuth atoms are heptacoordinated: five oxygen atoms form an incomplete octahedral arrangement, with Bi–O distances ranging from 2.076±0.027 to 2.640±0.028 Å. The remaining two oxygen atoms are electrostatically coordinated, on either side of the 6s2 inert electron pair in Bi3+, at distances of 3.082 and 3.170 Å from bismuth. The Bi atom vibrates anisotropically. The absolute configuration of the atomic arrangement has been determined. The positive piezoelectric polarization induced by compressive stress applied along 〈111〉 is given by the normal to a triangular GeO4 tetrahedral face in the direction from the Ge atom to the face.

Crystal Structure of the Transition-Metal Molybdates and Tungstates. II. Diamagnetic Sc2(WO4)3

Sc2(WO4)3, diamagnetic above 30°K, crystallizes in the orthorhombic system, Space Group Pnca, with lattice constants a=9.596±0.004, b=13.330±0.003, and c=9.512±0.004 Å at 298°K. The complete x-ray scattering pattern within a reciprocal lattice hemisphere of radius (sinθ)/λ=1.02 Å−1 was measured with PEXRAD. The crystal structure was solved by use of three-dimen sional Patterson and Fourier series and refined by the method of least squares, using 1731 independent structure factors. The final agreement factor R is 0.0622. Scandium atoms occupy slightly distorted octahedra, with average Sc–O=2.063 Å and Sc–O distances ranging from 2.026±0.015 to 2.124±0.010 Å. Two crystallographically independent W atoms are surrounded by somewhat distorted tetrahedra: the W–O distances vary from 1.695±0.009 to 1.829±0.016 Å, the average being 1.761 Å. The thermal vibrations are significantly anisotropic. Sc2(WO4)3 forms the structure type for 23 trivalent metal tungstates and molybdates, including the nine smaller rare-earth tungstates. The larger rare-earth tungstates, crystallizing in the Eu2(WO4)3 structure type, have 8 coordination about the rare-earth ion; the smaller have 6 coordination. A simple correlation is found between the variation in radius ratio due to the lanthanide contraction and the change in coordination.

Crystal Structure of Paramagnetic Ludlamite, Fe3(PO4)2·4H2O, at 298°K

Paramagnetic Fe3(PO4)2·4H2O is monoclinic, with lattice constants a=10.541±0.005, b=4.646±0.004, c=9.324±0.005 Å and β=100°25.8′±1.0′ at 298°K, with two formula weights in the unit cell. The space group is P21/a. The positions of the iron, phosphorus, and oxygen atoms, determined by Ito and Mori in 1951, together with those of the hydrogen atoms, have been refined by the method of least squares. The integrated intensities of 6212 reflections in a reciprocal lattice hemisphere of radius (sinθ)/λ=1.02 Å−1 were measured with PEXRAD; 1919 symmetry-independent F(hkl) were significantly above background. The final agreement factor between measured and calculated structure factors was 0.0336. There are two crystallographically unrelated iron atoms per unit cell, each occupying a distorted oxygen octahedron; the iron—oxygen distances range from 2.016±0.002 through 2.362±0.002 Å. The average distance is 2.156 Å. The phosphate group is very nearly a regular tetrahedron, the range in P–O distance being from 1.536±0.002 through 1.549±0.002 Å. The average O–H distance is 0.87 Å. The shortest O–H···O bond is 2.541±0.003 Å. One phosphate—oxygen atom forms no hydrogen bond, one forms two, and the remaining two oxygen atoms form one hydrogen bond each. The thermal vibrations are significantly anisotropic.

429. [R. 4. a.] Three equal smooth spheres of radius r rest in a hollow hemisphere of radius R with their centres in the same horizontal plane; a cone whose weight is equal to the weight of a sphere and whose semivertical angle is a, is inserted symmetrically between the spheres with its vertex downwards; prove that the spheres will

429. [R. 4. a.] Three equal smooth spheres of radius r rest in a hollow hemisphere of radius R with their centres in the same horizontal plane; a cone whose weight is equal to the weight of a sphere and whose semivertical angle is a, is inserted symmetrically between the spheres with its vertex downwards; prove that the spheres will