Key words: Corundum; energy levels of Gd*** in Al2O3; quadrupole transitions; spin Hamiltonian; transition probabilities; ultrasonic (paramagnetic) resonance; ultrasonic transition probabilities; wave functions of Gd+++.

A computation is made of energy levels, wave functions and transition matrix elements of the Gd3+ ion in Al2O3. The crystal field parameters are taken from Geschwind and Remeika's paramagnetic resonance experiments. The transition probabilities are given for dipole radiation in three polarization directions. For ultrasonic work we give the real and imaginary parts of the five matrix elements of the quadrupole transition. From these one can easily deduce the transition probabilities in any given direction.

The magnetic field directions are described by the angles and , the polar and azimuthal angles with respect to the crystalline c axis. The values of go from 0 to π/2 in six steps and two values of π are chosen, 0 and 2 π/3 for which the variation is largest. The magnetic field strengths are from 0 to 0.6 tesla (6000 gauss); beyond this value the spin can be considered as "free." Some consideration is given to the analytical behavior of the energy versus field diagram for the direction == 0.

November-December 1971

The accuracy of measurements of viscosity of liquids, R. S. Marvin, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 535-540 (Nov.-Dec. 1971).

Key words: Absolute measurement of viscosity; accuracy of viscosity measurement; calibration of viscometers; viscosity; uncertainty of viscosity measurements. Most absolute measurements of viscosity have utilized capillary flow, and required semiempirical corrections amounting to several times their precision and estimated accuracy. The range of values found from these measurements and the possibility of unrecognized systematic errors make it impossible to base a realistic estimate of accuracy on the results of only one type of measurement. The results of two independent absolute measurements involving different types of flow, reported in the two accompanying papers, are summarized here. The estimated accuracy in each case is about 0.1 percent. The two results differ by 0.5 percent. It is suggested that we continue to base the calibration of relative viscometers on the value of 1.002 centipoise (cP) for the viscosity of water at 20 °C and one atmosphere. This provides a generally accepted base which limits comparability only by the precision of the measurements. However, whenever the true values of viscosity are required the limits of uncertainty including an estimate of systematic error should be taken as no better than ± 0.25 percent.

An absolute determination of viscosity using a torsional pendulum, H. S. White and E. A. Kearsley, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 541-551 (Nov.-Dec. 1971).

Key words: Absolute measurement; calibration; standard; torsional pendulum; viscosity.

An absolute measurement of the viscosity of di(2-ethylhexyl) sebacate with a torsional pendulum viscometer. The apparatus and calibration procedures are discussed and a detailed study of errors is made. The systematic error was estimated to be less than ± 0.07 percent and the variability was shown to be less than ±0.07 percent. There is an unexplained difference of 0.3 to 0.4 percent between these measurements and measurements with a capillary instrument.

An absolute determination of viscosity using channel flow, R. W. Penn and E. A. Kearsley, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 553-560 (Nov.-Dec. 1971).

Key words: Absolute measurement; calculation of bounds; calibration; Poiseuille flow; standard; viscosity; viscous flow.

The viscosity of a sample of di(2-ethylhexyl) sebacate has been determined by measuring the pressure at taps along a closed channel containing the flowing liquid. By means of relative viscosity measurements in conventional capillary viscometers, we are able to express our results in terms of the viscosity of water at 20 °C. We find a value of 0.010008 poise. An appendix outlines the calculation of upper and lower bounds for the geometrical flow constant.

A new determination of the atomic weight of zinc, G. Marinenko, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 561-564 (Nov.-Dec. 1971).

Key words: Atomic weight; coulometry; zinc, atomic weight.

The atomic weight of zinc determined coulometrically for 5 metallic reference samples is 65.377±0.003 where the uncertainty figure is the 95 percent confidence interval for the mean plus an allowance for the mean plus an allowance for known possible sources of systematic error.

High-speed (subsecond) measurement of heat capacity, electrical resistivity, and thermal radiation properties of niobium in the range 1500 to 2700 K, A. Cezairliyan, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 565-571 (Nov.-Dec. 1971).

Key words: Electrical resistivity; emittance; heat capacity; high-speed measurements; high temperature; niobium; thermal radiation properties; thermodynamics.

Measurements of heat capacity, electrical resistivity, hemispherical total emittance, and normal spectral emittance of niobium in the temperature range 1500 to 2700 K by a subsecond duration pulse heating technique are described. Results obtained on the above properties are reported and are compared with those in the literature. A sharp increase in heat capacity above 2000 K was observed. Electrical resistivity showed a negative departure from linearity in the curve of electrical resistivity against temperature. Estimated inaccuracy of measured properties is: 2 percent for heat capacity, 0.5 percent for electrical resistivity, and 3 percent for hemispherical total and normal spectral emittances. Stable radical-anions derived from glyoxal bis(phenylhydrazones), A. J. Fatiadi, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 573-577 (Nov.-Dec. 1971).

Key words: Bis(phenylhydrazone); e.s.r.; glyoxal; methyl sulfoxide; oxygen; radical-anion.

A series of resonance-stabilized, hydrazyl radical-anions has been observed for the first time; they were prepared by an electron-transfer reaction involving treatment of glyoxal bis(phenylhydrazones) in methyl sulfoxide with a base and a trace of oxygen. The stability of these radical-anions appears to be dependent on the nature of the para substituents on the phenylhydrazine moiety.

Temperature dependence of photocurrents produced by x and gamma rays in silicon radiation detectors, K. Scharf and R. K. Mohr, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 579-589 (Nov.-Dec. 1971).

Key words: Gamma rays; photocurrents; radiation; radiation detector; radiation dosimetry; silicon radiation detector; temperature dependence; x rays.

The temperature dependence of dc photocurrents produced by x and gamma rays in silicon radiation detectors of the n-p, p-i-n, and surface-barrier type was investigated in a temperature range

453-313 O - 72 - 3

between 20 and 50 °C. Photodiode photocurrents, assumed as being equal to the generated currents 1,, showed a positive temperature dependence in all detectors investigated. Their temperature coefficient at 25 °C varied between +0.004 per °C and +0.002 per °C. The temperature dependence of short-circuit currents Ise measured by a compensation method, was positive and nearly linear for n-p type detectors but nonlinear and negative for p-i-n and surface-barrier type detectors. It is shown, that this different behavior of individual detectors is due to the influence of the strongly temperature-dependent junction current I, which under the short-circuit mode is drained off the generated current 1. The junction current is a function of the internal series resistance R, and the junction resistance R; of the irradiated detector (Ij=Isc R ̧/Rj). With increasing resistance ratio R/R1, the current ratio 1/1, increases and the temperature coefficient αsc of the short-circuit current decreases. Temperature coefficients (a,c)25 measured in the different detectors at 25 °C and a current density 610-7 A/cm2 decreased with increasing resistance ratio from +0.004 per °C to -0.005 per °C. Resistance ratios R/R, of the detectors investigated ranged between 0.01 and 0.24 approximately. Thus, (asc)25 measured in an individual detector can be changed by changing its effective series resistance. A decrease of (asc)25 with increasing I was observed in detectors with larger resistance ratios. This was apparently due to the voltage dependence of R, at higher junction voltages produced by larger short-circuit currents.

Isomerization processes in ions of the empirical formula C4H8*, S. G. Lias, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 591-605 (Nov.-Dec. 1971).

Key words: Butene; cyclobutane; ion structure; isomerization; methylcyclopropane; photoionization; photolysis; radiolysis.

Ions of the formula C4H,+ have been generated with different initial energies by ionizing ethylene (C2H4 + C2H4 → C4H8*, where the C4H8+ ion is formed with an initial energy of > 11.51 eV), cyclobutane (initial energy of C4H8+,> 10.84 eV), methylcyclopropane (> 10.15 eV), 1-C4H8 (> 9.58 eV), and i-C4H8 (> 9.06 eV) with 11.6-11.8 eV photons, and in some cases also with 10 eV photons and with gamma radiation. The structures of the ions have been determined from the structures of the C4H8 products formed in charge transfer reaction between the ions and charge acceptors such as dimethylamine and nitric oxide, as well as from the structures of the butanes formed in D2 transfer reactions with methylcyclopentane-d12 (C4H8++C6D12 ́ C4H8D2+C6D10*).

At low pressures the C4H8+ ions initially formed in ethylene, cyclobutane, and methylcyclopropane isomerize to the thermodynamically most stable configurations, i-C4H8+ and 2C4H8. The 2-C1H,+ structure predominates in all the experiments. As the pressure is raised, the i-C4H+ ion yield diminishes as that of 2-C4H8+ increases, indicating that when the precursor of the i-C4H+ ion is collisionally deactivated, it ends up as 2C4H8. At high pressures, 1-C4H8+ ions are intercepted; their yield increases with increasing pressure, indicating that 1-C4H8* is an intermediate which isomerizes further unless it is colli

sionally deactivated. The 1-C4H+ ion formed in methylcyclopropane (initial energy > 10.15 eV) is more easily deactivated than that formed in cyclobutane (initial energy > 10.84 eV). That the isomerization of the 1-C4H+ ion to lower energy structures such as i-C,H,* and 2-C4H8+ requires excess internal energy is demonstrated by the fact that in the photolysis with 10 eV photons, a negligible amount of isomerization is observed, but with 11.6-11.8 eV photons, more than half of the 1-C4H8* ions isomerize to the 2-C4H8+ structure at a pressure of 2 torr. Isomerization of the low energy i-C4H+ ions formed in the photolysis of i-C4H8 to other structures is relatively unimportant at 11.6-11.8 eV.

Taking the ratio i-C4H8+/2-C4H8+ as an indicator of the amount of energy removed by collisions from the intermediate C4H species under conditions where only i- and 2-C4H+ ions are intercepted, it is shown that the efficiency of energy transfer from the ions to helium, hydrogen, neon, krypton, xenon, nitrogen, and carbon dioxide is related to the polarizability of the added deactivator.

The photochemistry of propane at high photon energies (8.4-21.2 eV), R. E. Rebbert, S. G. Lias, and P. Ausloos, J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 6, 607-612 (Nov.-Dec. 1971).

Key words: Ion-molecule reaction; photoionization; propane; quantum yields; rare-gas resonance radiation; unimolecular dissociation.

Neon and helium resonance lamps, which deliver photons of 16.7-16.8 eV and 21.2 eV energy, respectively, have been used to photolyze C3H8, C3D8, C3H8-C3D8 (1:1) mixtures, and CD3CH2CD, and the results obtained at the two energies are compared. In particular, it is noted that although the quantum yield of ionization in propane is unity at 16.7-16.8 eV, when the energy is raised still further to 21.2 eV, the probability of ionization apparently diminishes to 0.93, an observation which suggests that at 21.2 eV, superexcited states may be reached whose dissociation into neutral fragments competes with ionization.

The quantum yields of the lower hydrocarbon products formed in the presence of a radical scavenger in CзH, and CзD are reported, and are compared with quantum yields of products formed in the vacuum ultraviolet photolysis at lower energies. (Quantum yields of products formed at 8.4 eV and 10.0 eV are reported here for the first time.) Acetylene is formed as a product in the decomposition of the neutral excited propane molecule, and its yield increases in importance with increasing energy; at 16.7-16.8 eV, where all product formation can be traced to ionic processes, acetylene is formed in negligible yields. It is concluded that ionic processes in propane do not lead to the formation of acetylene, and the observation of this product in radiolytic systems may be a reliable indicator of the relative importance of neutral excited molecule decomposition processes.

From the results obtained with the C3H8-C3D8 (1:1) mixture, and with CD3CH2CD3, details of the ion-molecule reaction mechanisms and the unimolecular decomposition of the propane ion are derived.

January-June 1971

Bounds to truncation errors in biorthogonal polynomial approximations, with illustrative applications to gamma-ray transport distributions, L. V. Spencer, J. Res. Nat. Bur. Stand. (U.S1), 75B (Math. Sci.), Nos. 1 and 2, 1-16 (Jan.-June 1971).

Key words: Biorthogonal polynomials; gamma-ray transport; moment method; neutron transport; polynomial truncation errors; Sievert function.

Problems of gamma-ray and neutron penetration in infinite, homogeneous media can be solved in terms of infinite sums of a certain type of biorthogonal polynomial system. Many calculations have used a few terms, perhaps 4 to 8, successfully due to good apparent convergence. This paper develops expressions for bounds to the truncation error of these finite sums. Examples and applications are given of the functions used to obtain the error bounds.

A theorem on matrix commutators, J. M. Smith, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 17-21 (Jan.-June 1971).

Key words: Commutator; matrix; orthogonal; skew-symmetric.

Let P=1+(-1), the direct sum of the pXp identity matrix and the negative of the qq identity matrix. The following theorem is proved.

THEOREM: If the matrices A and B are P-orthogonal (orthogonal with respect to P) and P-skew-symmetric (skewsymmetric with respect to P) then [A,B] = AB - BA may be expressed as the difference, CD, of matrices which are Porthogonal, P-skew-symmetric.

Explicit expressions for the matrices C and D are given.

A special permutation matrix decomposition for combinatorial design incidence matrices, E. C. Johnsen and T. F. Storer, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 23-30 (Jan.-June 1971).

Key words: Block design; combinatorial analysis; configurations; König's theorem; matrices; matrix equations; permutation matrix decompositions; 0, 1 matrices.

By a well-known theorem of König, every 0,1 matrix A of order v with all row and column sums equal to k>0 can be decomposed into a sum of k permutation matrices of order v. Here we consider whether A, trace A = 0 or v, can have a special permutation matrix decomposition called a special König decomposition. The question is answered more or less satisfactorily using certain graph factorization theorems of Petersen and Tutte. Symmetric counterexamples are given to show that not all such matrices have special König decompositions; however, it is shown that symmetric incidence matrices for (v, k, λ) designs always have special König decompositions.

The Casimir check for the algebraic matrices of the configuration (d+s)"p, C. Roth, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 31-47 (Jan.-June 1971).

Key words: Casimir operator; checking parameters; (d+ s)"p, SU3.

By using the eigenvalues of the Casimir operator for the group SU3, checking parameters were obtained for 12 electrostatic interaction matrices of the configuration (d+s)"p.

Convex homotopy, W. A. Horn, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 49-61 (Jan.-June 1971).

Key words: Convex functions; convex sets; homotopy. Given any two continuous, convex functions ƒ and g on a compact, convex set with certain properties ("regular" convexity), it is shown that an infinite number of homotopies of ƒ and g exist which are also convex. One such homotopy in particular (the "basic" homotopy) is shown to have nice monotonicity properties and can be used as a basis for constructing other homotopies. A method of constructing the basic homotopy is given for the case where the domain of ƒ and g is a line segment. (Theorems are for normed linear spaces only.)

The powers of a connected graph are highly hamiltonian, V. N. Bhat and S. F. Kapoor, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 63-66 (Jan.-June 1971).

Key words: Graph; hamiltonian; power of a graph.

Let G be a connected graph on p points. It is proved that if any k points are deleted from the graph Gm+2, 1 ≤ m≤ p − 3 and 0 ≤ km. then the resulting graph is hamiltonian.

Exact solutions of linear equations with rational coefficients, A. S. Fraenkel and D. Loewenthal, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 69-75 (Jan.-June 1971).

Key words: Congruences; exact solutions; linear equations; modular arithmetic.

Improvements of a congruential method for finding the exact solutions of systems of linear equations with rational coefficients are described. Typical execution times on the CDC 1604-A are given, as well as the Fortran program.

An improvement of the Fischer inequality, R. Merris, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 77-80 (Jan.June 1971).

Key words: Fischer inequality; Hadamard determinant theorem; permanent; positive definite hermitian matrix.

An improvement of the classical Fischer inequality for the determinant of a positive definite hermitian matrix is proved. It is used to analyze the Hadamard determinant theorem.

On the Smith normal form, M. Newman, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 81-84 (Jan.-June 1971). Key words: Invariant factors; principal ideal rings; Smith normal form.

An elementary proof is given of the fact that if A, B are nonsingular n×n matrices over a principal ideal ring R, then the kth invariant factor of AB is divisible by the kth invariant factor of A and by the kth invariant factor of B, 1 ≤ k ≤n. Some consequences are drawn.

On certain optimal quadrature formulas, S. Haber, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 1 and 2, 85-88 (Jan.June 1971).

Key words: Analytical functions; approximation; Gaussian quadrature; integration; optimal quadrature; quadrature.

A class of optimal quadrature formulas defined by V. L. N. Sarma in a probabilistic context is shown to be identical with a class of formulas defined previously, in a different manner, by P. J. Davis. As a result a contrast is drawn between the maximum of the error of a quadrature formula over the unit sphere in a certain function space, and its average-in Sarma's sense-over that sphere.

July-December 1971

Some elementary formulas in "matrix calculus" and their applications, J. T. Fong, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 97-107 (July-Dec. 1971).

Key words: Chain rule; continuum mechanics; gradient; matrices; matrix calculus; partial differentiation; product rule; tensor function; trace.

A collection of elementary formulas for calculating the gradients of scalar- and matrix-valued functions of one matrix argument is presented. Using some of the well-known properties of the operator "trace" on square matrices, alternative definitions of gradients and simple examples of calculating them using the product rule and the chain rule for differentiation are treated in an expository fashion in both component and matrix notations with emphasis on the latter. Two examples in continuum mechanics are presented to illustrate the application of the socalled "matrix calculus" of differentiable functions.

Linear transformations on matrices, M. Marcus, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 109-115 (July-Dec. 1971).

Key words: Determinant; generalized matrix function; invariants; matrices; rank.

Let K be a field and let Mn(K) denote the vector space of all n Xn matrices over K. Suppose I(X) is an invariant defined on a subset of Mn(K). This paper surveys certain results concerning the following problem. Describe the set (1, U) of all linear transformations T: U→ U that hold the invariant / fixed:

Eigenvalues of sums of Hermitian matrices. III., R. C. Thompson and L. F. Garbanati, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 117-124 (July-Dec. 1971).

Key words: Eigenvalues; linear inequalities; singular values. Two classes of nonlinear inequalities for the eigenvalues of sums of Hermitian matrices are obtained. These nonlinear inequalities are shown to follow from linear inequalities established in parts I and II of this series. A new inequality for the singular values of matrix products is also obtained.

A new type of boundary value coupling for second order SturmLiouville systems, J. Geist, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 125-138 (July-Dec. 1971).

Key words: Boundary value problems; coupled equations; differential equations; self-adjoint equations; Sturm-Liouville systems.

A natural generalization of the familiar second order SturmLiouville system is presented. This generalization consists of considering a number of differential equations defined on different intervals, instead of just one equation on one interval. The self-adjoint character of the differential equations is retained in the generalization, but the boundary conditions are relaxed considerably. The most general boundary conditions which can be accommodated by this sort of generalization of Sturm-Liouville theory are discussed. The existence of eigenvalues is proved,

and a generalized orthogonality and a weak eigenfunction expansion theorem are derived.

An ergodic source which is not a function of any finite Markov process, R. D. Traub, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 139-148 (July-Dec. 1971).

Key words: Information theory; Markov process; probability theory; ring theory; stochastic process.

The problem of deciding when a stationary stochastic process is a function of a finite Markov process has been considered by several authors, leading to an elegant necessary and sufficient condition. Taking a different approach, this note uses elementary ring theory to prove that a certain explicitly constructed stationary ergodic process is not a function of any finite Markov process. On entire functions of exponential type, S. M. Shah and W. C. Sisarcick, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 149-156 (July-Dec. 1971).

Key words: Bounded index; entire function; exponential type; maximum modulus.

Let f be an entire function and let p > 1 and

for all sufficiently large r, then ƒ is of exponential type not exceeding {2 log (1(1/c)) + 1 + log (2N)!}. If this condition is replaced by related conditions, then also is of exponential type.

A table of integrals of the error function. II. Additions and corrections, M. Geller and E. W. Ng, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math Sci.), Nos. 3 and 4, 157-178 (July-Dec. 1971).

Key words: Error functions; indefinite integrals; special functions.

This is an extension of a compendium of indefinite and definite integrals of products of the error function with elementary or transcendental functions recently published by the authors.

A note on the time dependence of the effective axis and angle of a rotation, H. Gelman, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 179-186 (July-Dec. 1971).

Key words: Angular velocity; effective axis and angle of rotation; intrinsic vector; kinematics of a rigid body; rotation group; spatial rotation.

The time dependent rotation of one orthogonal coordinate system with respect to a fixed one is considered in the parametrization based on the effective axis and angle of the rotation, a parametrization which has recently been used to discuss the irreducible representations of the rotation group. The method of the intrinsic vector is used to derive the equations of motion for the instantaneous effective axis and angle. A new representation of the angular velocity is obtained in a rotating orthogonal coordinate system generated by a unit vector along the effective axis, and a new geometrical interpretation of the effective angle is given.

A numerical solution of integral equation of the first kind applied to slit correction in small-angle x-ray scattering, J. Mazur, J. Res. Nat. Bur. Stand. (U.S.), 75B (Math. Sci.), Nos. 3 and 4, 187-207 (July-Dec. 1971).

Key words: Conditions for convergence of solutions; integral equation of first kind; shifting operators; slit correction; slit weighting-function; small-angle x-ray scattering; strong numerical instability.

A novel approach to the problem of the slit correction in smallangle x-ray scattering is presented, based on a matrix inversion method. The integral equation for the slit correction is written as a Volterra equation of the first kind. This equation is reduced to a system of simultaneous equations, expressed in matrix form.

The order of the matrix is equal to the number of experimentally determined points. To obtain these equations, one has to expand the unknown functions in Taylor series around each point to be subsequently determined. There is, however, a difficulty inherent in this method due to the fact that most of the series expansions of the function to be determined lead to a strong numerical instability. However, a general method is developed, which enables us to find shifting operators leading to numerically stable systems of equations. The "unsmearing" of the experimental data is then performed by standard matrix inversion procedures.