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At least 1,459 records · Page 81

Infrared spectroscopy of polycyclic aromatic hydrocarbon cations. 1: Matrix-isolated naphthalene and perdeuterated naphthalene

Ionized polycyclic aromatic hydrocarbons (PAHs) are thought to constitute an important component of the interstellar medium. Despite this fact, the infrared spectroscopic properties of ionized PAHs are almost unknown. The results we present here derive from our ongoing spectroscopic study of matrix isolated PAH ions and include the spectra of the naphthalene cation, C10H8(+), and its fully deuterated analog, C10D8(+), between 4000 and 200/cm. Ions are generated by in situ Lyman-alpha photoionization of the neutral precursor. Bands of the C10H8(+) ion are observed at 1525.7, 1518.8, 1400.9, 1218.0, 1216.9, 1214.9, 1023.2, and 758.7/cm. Positions and relative intensities of these bands agree well with those in the available literature. The 758.7/cm band has not previously been reported. C10D8(+) ion bands appear at 1466.2, 1463.8, 1379.4, 1373.8, 1077.3, 1075.4, and 1063.1/cm. Compared to the analogous modes in the neutral molecule, the intensities of the cation's CC modes are enhanced by an order of magnitude, while CH modes are depressed by this same factor. Integrated absorption intensities are calculated for the strongest bands of C10H8 and for the observed bands of C10H8(+). Absolute intensities derived for the naphthalene cation differ from earlier experimental results by a factor of approximately 50, and from theoretical predictions by a factor of approximately 300. Reasons for these discrepancies and from the astronomical implications of PAH cation spectra are discussed.

Hudgins, D. M.↗

Measurement of the CMB temperature power spectrum and constraints on cosmology from the SPT-3G 2018 T T , T E , and E E dataset

We present a sample-variance-limited measurement of the temperature power spectrum ( T T ) of the cosmic microwave background using observations of a ∼ 1500 deg 2 field made by the SPT-3G in 2018. We report multifrequency power spectrum measurements at 95, 150, and 220 GHz covering the angular multipole range 750 ≤ ℓ < 3000 . We combine this T T measurement with the published polarization power spectrum measurements from the 2018 observing season and update their associated covariance matrix to complete the SPT-3G 2018 T T / T E / E E dataset. This is the first analysis to present cosmological constraints from SPT T T , T E , and E E power spectrum measurements jointly. We blind the cosmological results and subject the dataset to a series of consistency tests at the power spectrum and parameter level. We find excellent agreement between frequencies and spectrum types and our results are robust to the modeling of astrophysical foregrounds. We report results for Λ CDM and a series of extensions, drawing on the following parameters: the amplitude of the gravitational lensing effect on primary power spectra A L , the effective number of neutrino species N eff , the primordial helium abundance Y P , and the baryon clumping factor due to primordial magnetic fields b . We find that the SPT-3G 2018 T T / T E / E E data are well fit by Λ CDM with a probability to exceed of 15%. For Λ CDM , we constrain the expansion rate today to H 0 = 68.3 ± 1.5 km s - 1 Mpc - 1 and the combined structure growth parameter to S 8 = 0.797 ± 0.042 . The SPT-based results are effectively independent of Planck, and the cosmological parameter constraints from either dataset are within < 1 σ of each other. The addition of temperature data to the SPT-3G T E / E E power spectra improves constraints by 8–27% for each of the Λ CDM cosmological parameters. When additionally fitting A L , N eff , or N eff + Y P , the posteriors of these parameters tighten by 5–24%. In the case of primordial magnetic fields, complete T T / T E / E E power spectrum measurements are necessary to break the degeneracy between b and n s , the spectral index of primordial density perturbations. We report a 95% confidence upper limit from SPT-3G data of b < 1.0 . The cosmological constraints in this work are the tightest from SPT primary power spectrum measurements to date and the analysis forms a new framework for future SPT analyses.

79 ASTRONOMY AND ASTROPHYSICS↗

Investigation of Aerodynamic Matching Sensitivities for Conceptual Design

A methodology for aerodynamic matching was implemented in the Higher fidelity Conceptual Design and structural optimization (HCDstruct) tool to support transonic flutter simulations. The Enhanced Factor Correction Technique was implemented in HCDstruct to correct the Nastran doublet lattice method using data from corresponding computational fluid dynamics simulations. In this paper, sensitivities associated with this aerodynamic matching technique were investigated using HCDstruct and a Boeing 737-200-like aircraft configuration. Specifically, sensitivities of the aerodynamic matching method were investigated with respect to changes in flight conditions, such as changes in Mach number and angle of attack, and also in key design variables for the main wing, such as changes in aspect ratio and sweep angle. The variations in flight conditions and design parameters were mapped to estimated prediction error in sectional lift for the main wing, which helps to demonstrate the extent to which a user may expect a given aerodynamic correction matrix to provide suitable aerodynamic predictions for a realistic aircraft wing.

Jesse R. Quinlan↗

Investigation of Aerodynamic Matching Sensitivities for Conceptual Design

A methodology for aerodynamic matching was implemented in the Higher fidelity Conceptual Design and structural optimization (HCDstruct) tool to support transonic flutter simulations. The Enhanced Factor Correction Technique was implemented in HCDstruct to correct the Nastran doublet lattice method using data from corresponding computational fluid dynamics simulations. In this paper, sensitivities associated with this aerodynamic matching technique were investigated using HCDstruct and a Boeing 737-200-like aircraft configuration. Specifically, sensitivities of the aerodynamic matching method were investigated with respect to changes in flight conditions, such as changes in Mach number and angle of attack, and also in key design variables for the main wing, such as changes in aspect ratio and sweep angle. The variations in flight conditions and design parameters were mapped to estimated prediction error in sectional lift for the main wing, which helps to demonstrate the extent to which a user may expect a given aerodynamic correction matrix to provide suitable aerodynamic predictions for a realistic aircraft wing.

Jesse Ray Quinlan↗

Valence Disproportionation of GeS in the PbS Matrix Forms Pb 5 Ge 5 S 12 Inclusions with Conduction Band Alignment Leading to High n-Type Thermoelectric Performance

Converting waste heat into useful electricity using solid-state thermoelectrics has a potential for enormous global energy savings. Lead chalcogenides are among the most prominent thermoelectric materials, whose performance decreases with an increase in chalcogen amounts (e.g., PbTe>PbSe>PbS). Herein, we demonstrate the simultaneous optimization of the electrical and thermal transport properties of PbS-based compounds by alloying with GeS. The addition of GeS triggers a complex cascade of beneficial events as follows: Ge 2+ substitution in Pb 2+ and discordant off-center behavior; formation of Pb 5 Ge 5 S 12 as stable second phase inclusions through valence disproportionation of Ge 2+ to Ge0 and Ge 4+ . PbS and Pb 5 Ge 5 S 12 exhibit good conduction band energy alignment that preserves the high electron mobility; the formation of Pb 5 Ge 5 S 12 increases the electron carrier concentration by introducing S vacancies. Sb doping as the electron donor produces a large power factor and low lattice thermal conductivity (κ lat ) of ~0.61 Wm -1 K -1 . The highest performance was obtained for the 14% GeS-alloyed samples, which exhibited an increased room temperature electron mobility of ~121 cm 2 V -1 s -1 for 3 × 10 19 cm -3 carrier density, and a ZT, of 1.32 at 923 K. This is ~ 55% greater that the corresponding Sb-doped PbS sample and is one of the highest reported for the n-type PbS system. Moreover, the average ZT (ZT avg ) of ~0.76 from 400 to 923 K is the highest for PbS-based systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

MATRIX-VBS (v1.0): Implementing an Evolving Organic Aerosol Volatility in an Aerosol Microphysics Model

The gas-particle partitioning and chemical aging of semi-volatile organic aerosol are presented in a newly developed box model scheme, where its effect on the growth, composition, and mixing state of particles is examined. The volatility-basis set (VBS) framework is implemented into the aerosol microphysical scheme MATRIX (Multiconfiguration Aerosol TRacker of mIXing state), which resolves mass and number aerosol concentrations and in multiple mixing-state classes. The new scheme, MATRIX-VBS, has the potential to significantly advance the representation of organic aerosols in Earth system models by improving upon the conventional representation as non-volatile particulate organic matter, often also with an assumed fixed size distribution. We present results from idealized cases representing Beijing, Mexico City, a Finnish forest, and a southeastern US forest, and investigate the evolution of mass concentrations and volatility distributions for organic species across the gas and particle phases, as well as assessing their mixing state among aerosol populations. Emitted semi-volatile primary organic aerosols evaporate almost completely in the intermediate-volatility range, while they remain in the particle phase in the low-volatility range. Their volatility distribution at any point in time depends on the applied emission factors, oxidation by OH radicals, and temperature. We also compare against parallel simulations with the original scheme, which represented only the particulate and non-volatile component of the organic aerosol, examining how differently the condensed-phase organic matter is distributed across the mixing states in the model. The results demonstrate the importance of representing organic aerosol as a semi-volatile aerosol, and explicitly calculating the partitioning of organic species between the gas and particulate phases.

volatility-basis set↗

Human motion planning based on recursive dynamics and optimal control techniques

This paper presents an efficient optimal control and recursive dynamics-based computer animation system for simulating and controlling the motion of articulated figures. A quasi-Newton nonlinear programming technique (super-linear convergence) is implemented to solve minimum torque-based human motion-planning problems. The explicit analytical gradients needed in the dynamics are derived using a matrix exponential formulation and Lie algebra. Cubic spline functions are used to make the search space for an optimal solution finite. Based on our formulations, our method is well conditioned and robust, in addition to being computationally efficient. To better illustrate the efficiency of our method, we present results of natural looking and physically correct human motions for a variety of human motion tasks involving open and closed loop kinematic chains.

Non-NASA Center↗

Unveiling the Reactivity of Fluoropolymers with Sodium Metal: Mechanistic Insights and Battery Implications

Poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF-HFP) has broad applications across various metal-ion battery systems, such as a binder for electrodes, a supporting matrix for electrolytes, and a separator material. Due to its excellent mechanical properties, PVDF-HFP has become an excellent candidate for fabricating gel and solid-state electrolytes in sodium-based batteries. However, in this study, we noticed notable side reactions occurring at the interface of PVDF-HFP membranes and Na metal. These reactions not only alter chemical compositions but also further affect the surface morphology and adhesive properties of membranes. Similar phenomena are observed in other polyfluoroalkyl-based membranes (PVDF and PTFE). Therefore, we systematically studied the reaction mechanisms between the Na metal and these polymers. The influence of different functional groups (−F, −CF 3 , −H) and their arrangement on the reaction extent has also been discussed. Finally, we concluded with the key factors driving these side reactions and provided new perspectives for designing polymers tailored for sodium-based batteries.

Interfacial side reactions↗

Egr1 is a 3D matrix–specific mediator of mechanosensitive stem cell lineage commitment

While extracellular matrix (ECM) mechanics strongly regulate stem cell commitment, the field’s mechanistic understanding of this phenomenon largely derives from simplified two-dimensional (2D) culture substrates. Here, we found a 3D matrix–specific mechanoresponsive mechanism for neural stem cell (NSC) differentiation. NSC lineage commitment in 3D is maximally stiffness sensitive in the range of 0.1 to 1.2 kPa, a narrower and more brain-mimetic range than we had previously identified in 2D (0.75 to 75 kPa). Transcriptomics revealed stiffness-dependent up-regulation of early growth response 1 (Egr1) in 3D but not in 2D. Egr1 knockdown enhanced neurogenesis in stiff ECMs by driving β-catenin nuclear localization and activity in 3D, but not in 2D. Mechanical modeling and experimental studies under osmotic pressure indicate that stiff 3D ECMs are likely to stimulate Egr1 via increases in confining stress during cell volumetric growth. To our knowledge, Egr1 represents the first 3D-specific stem cell mechanoregulatory factor.

59 BASIC BIOLOGICAL SCIENCES↗

Improving the chi-squared approximation for bivariate normal tolerance regions

Let X be a two-dimensional random variable distributed according to N2(mu,Sigma) and let bar-X and S be the respective sample mean and covariance matrix calculated from N observations of X. Given a containment probability beta and a level of confidence gamma, we seek a number c, depending only on N, beta, and gamma such that the ellipsoid R = (x: (x - bar-X)'S(exp -1) (x - bar-X) less than or = c) is a tolerance region of content beta and level gamma; i.e., R has probability gamma of containing at least 100 beta percent of the distribution of X. Various approximations for c exist in the literature, but one of the simplest to compute -- a multiple of the ratio of certain chi-squared percentage points -- is badly biased for small N. For the bivariate normal case, most of the bias can be removed by simple adjustment using a factor A which depends on beta and gamma. This paper provides values of A for various beta and gamma so that the simple approximation for c can be made viable for any reasonable sample size. The methodology provides an illustrative example of how a combination of Monte-Carlo simulation and simple regression modelling can be used to improve an existing approximation.

Feiveson, Alan H.↗

Lattice QCD calculation of 𝑥-dependent meson distribution amplitudes at physical pion mass with threshold logarithm resummation

We present a lattice quantum chromodynamics (QCD) calculation of the 𝑥-dependent pion and kaon distribution amplitudes (DA) in the framework of large momentum effective theory. This calculation is performed on a fine lattice of 𝑎 = 0.076 fm at physical pion mass, with the pion boosted to 1.8 GeV and kaon boosted to 2.3 GeV. We renormalize the matrix elements in the hybrid scheme and match to Math output error with a subtraction of the leading renormalon in the Wilson-line mass. The perturbative matching is improved by resumming the large logarithms related to the small quark and gluon momenta in the soft-gluon limit. After resummation, we demonstrate that we are able to calculate a range of 𝑥 ∈[𝑥 0 ,1 − 𝑥 0 ] with 𝑥 0 = 0.25 for pion and 𝑥 0 = 0.2 for kaon with theoretical systematic errors under control. The kaon DA is shown to be slighted skewed, and narrower than pion DA. Although the 𝑥-dependence cannot be direct calculated beyond these ranges, we estimate higher moments of the pion and kaon DAs by complementing our calculation with short-distance factorization.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Maximum Cs-137 Curie Loading onto Crystalline Silicotitanate for the Documented Safety Analysis of the Tank Side Cesium Removal Platform

The Tank Side Cesium Removal (TSCR) system is currently being constructed to process Hanford tank waste supernates for vitrification. TSCR incorporates a filtration system and cesium (Cs) removal system using columns filled with crystalline silicotitanate (CST) ion exchanger, produced by Honeywell UOP, LLC. The documented safety analysis (DSA) developed for TSCR limits a single column curie loading to 141,600 Ci; given a 137 Cs isotopic mass fraction of 20% and the planned CST bed size of a TSCR column, this equates to 0.10 mmole Cs per g CST. Factors that influence 137Cs loading onto the CST include, but are not limited to, CST production lot (different production lots behave differently), contact temperature, contact duration, 137Cs mass fraction, and competitors in the tank waste feed. Seventeen tank waste feeds (compositions) were identified by Washington River Protection Solutions to be processed through TSCR. These feed compositions were used to develop a simulant (referred to herein as Stage 1) that would provide an upper bound to the Cs loading onto CST based on maximizing the Cs/Na activity coefficient ratios in solution while maintaining Na at no less than 5.0 M. Building upon this Stage 1 simulant, a series of four additional simulants were developed based on the cationic/anionic species that impact Cs exchange, with each successive formulation relaxing one or more matrix component concentration constraints as show in Table S.1

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Mechanistic Insights into Visible-Light-Induced ATRA Reactions Powered by the Symbiotic Relationship between Cu(II)/Cu(I) Phenanthroline Complexes

As photoredox catalysis continues to yield promising chemical transformations, there is an increased need to understand how specific photocatalysts function to improve reaction efficiencies while expanding their scope. Copper-phenanthroline-based photocatalysts such as Cu II (dap)Cl 2 (dap = 2,9-di(p-anisyl)-1,10-phenanthroline) and [Cu I (dap) 2 ]Cl were both found to be equally capable of olefin activation through electrophilic atom transfer radical addition (ATRA) reactions. Although these molecular catalysts have proven successful, many intermediates suggested in the proposed catalytic cycles have never been detected. One undetermined aspect in this chemistry is related to how one equivalent of Cu II (dap)Cl 2 generates half of an equivalent of [Cu I (dap) 2 ] + during the photocatalytic sequence. To this end, we initially used more synthetically accessible model systems, namely, [Cu I (dpp) 2 ]Cl and Cu II (dpp)Cl 2 (dpp = 2,9-diphenyl-1,10-phenanthroline), to glean detailed mechanistic insights into this unusual symbiotic relationship. We directly detected several intermediates involved in the ATRA photocatalytic cycle using these model chromophores in conjunction with electronic spectroscopy, infrared matrix-assisted laser desorption electrospray ionization (IR-MALDESI) mass spectrometry, electronic structure calculations, EPR spin-trap experiments, and 1 H NMR spectroscopy. We found that the unique ligand lability and coordinating properties of acetonitrile enable both the in situ oxidation of [Cu I (dpp) 2 ] + by tosyl chloride into Cu II (dpp)Cl 2 and the visible-light-induced homolysis of the Cu II –Cl bond, which initiates the conversion to the Cu I species [Cu I (dpp) 2 ][Cu I Cl 2 ]. Here, the combined findings from the present study of the catalytic cycle demonstrate that the symbiotic relationship between Cu II (dpp)Cl 2 and [Cu I (dpp) 2 ] + , as well as between Cu II (dap)Cl 2 and [Cu I (dap) 2 ] + , is the critical factor enabling the ATRA photoreaction by departing from either photocatalyst.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Echo mapping of active galactic nuclei broad-line regions: Fundamental algorithms

We formulate and test a series of algorithms for echo mapping the emission-line regions near active galactic nuclei from measurements of correlated variability in their line and continuum light curves. The linear regularization method (LRM) employs a direct inversion of evenly spaced light-curve data, with a regularization parameter that can be used to control the trade-off between noise and resolution. Matrix formulas express the formal solution as well as its variance and covariance in terms of uncertainties in the measurements. Unlike the maximum-entropy method (MEM), LRM applies to kernels with both positive and negative values, but the results are somewhat limited by ringing effects. A positivity constraint proves effective in controlling the ringing. MEM combines regularization and positivity in a natural way, but similar results are also found using positivity constraints with nonentropic regularization functions. Direct inversions of unevenly sampled light curves require interpolating the noisy data. In this case better results are found by solving for both the continuum light curve and kernel function in a simultaneous fit to the data. Our conclusion is that while echo mapping currently gives ambiguous results, the algorithms are not the limiting factor. Progress depends on efforts to increase the accuracy and completeness of sampling of the observed light curves.

Vio, Roberto↗

Convergence Acceleration of Runge-Kutta Schemes for Solving the Navier-Stokes Equations

The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 can be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. This RK/implicit scheme is used as a smoother for multigrid. Fourier analysis is applied to determine damping properties. Numerical dissipation operators based on the Roe scheme, a matrix dissipation, and the CUSP scheme are considered in evaluating the RK/implicit scheme. In addition, the effect of the number of RK stages is examined. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. Turbulent flows over an airfoil and wing at subsonic and transonic conditions are computed. The effects of the cell aspect ratio on convergence are investigated for Reynolds numbers between 5:7 x 10(exp 6) and 100 x 10(exp 6). It is demonstrated that the implicit preconditioner can reduce the computational time of a well-tuned standard RK scheme by a factor between four and ten.

Swanson, Roy C., Jr.↗

Assimilation of satellite microwave observations over the rainbands of tropical cyclones

A novel Bayesian Monte Carlo integration (BMCI) technique was developed to retrieve geophysical variables from satellite microwave radiometer data in the presence of tropical cyclones. The BMCI technique includes three steps: generating a stochastic database, simulating satellite brightness temperatures using a radiative transfer model, and retrieving geophysical variables such as profiles of temperature, relative humidity, and cloud liquid and ice water content from real observations. The technique also provides uncertainty estimates for each retrieval and can output the error covariance matrix of selected parameters. The measurements from the Advanced Technology Microwave Sounder (ATMS) on board Suomi National Polar-Orbiting Partnership (Suomi NPP) and the Global Precipitation Measurement (GPM) Microwave Imager (GMI) were used as input. A new technique was developed to correct the ATMS and GMI observations for the beam-filling effect, which is due to small-scale variability of precipitation and clouds when compared with the instrument footprint and also the nonlinear relation between the brightness temperature and precipitation. In addition, the assimilation of the BMCI retrievals into the NASA GEOS model is discussed for Hurricane Maria. The results show that assimilating the BMCI retrievals can influence the dynamical features of the cyclone, including a stronger warm core, a symmetric eye, and vertically aligned wind columns. Two possible factors that may limit the impact of the BMCI retrievals include 1) the resolution of the model (about 25 km), which was too coarse to show the potential of the BMCI data in improving the representation of tropical storms in the model forecast, and 2) the data assimilation system not being able to consider vertically correlated observation errors.

Isaac Moradi↗

Detection of Br and I as atomic anions using liquid sampling – atmospheric pressure glow discharge/Orbitrap mass spectrometry

The quantification and determination of halogen isotope composition is essential in many fields including geochemistry and nuclear forensics. Detection of the halogen elements is commonly attempted with inductively coupled plasma mass spectrometry (ICP-MS) however, there are multiple challenges faced. Most significantly this includes very poor ionization efficiency and the potential for isobaric interferences from either matrix or plasma species. Thus, sensitivity and accuracy are often limiting issues. Here a liquid sampling-atmospheric pressure glow discharge (LS-APGD) ionization source is coupled with an ultrahigh resolution Orbitrap MS to perform halogen detection of bromine and iodine as initial analytes. This facilitates simple and sensitive detection of these elements as atomic anions from simple aqueous salt solutions. Collision-induced dissociation (CID) and higher-energy collisional dissociation (HCD) energies were optimized to produce the maximum response of Br − and I − . The LS-APGD conditions were optimized using a design of experiments (DOE) approach for Br − and I − concurrently. Response curves for Br − and I − solutions determined limit of detection (LOD) values of 50 pg and 5 pg, respectively, in 20 μL aliquots. The curves indicated response factors of 0.67 for 79 Br − and 0.90 for 127 I − . Br isotope ratios were determined with precision between 0.7 and 7.3% RSD, with the isotope ratios determined with precision 0.8% RSD at concentrations above the limit of quantification. Preliminary tests were conducted to evaluate the effect of different cations (Na + , K + , and Mg 2+ ) on Br − responses, with no discernible impacts observed on the halogen signal responses. This study demonstrates the potential use of the LS-APGD-Orbitrap-MS for the detection of multiple halogens while avoiding interferences, minimizing sample preparation, and overcoming ionization barriers.

Lang, Dehlia A. [Clemson University, SC (United St↗

The effects of gravity darkening on the ultraviolet continuum polarization produced by circumstellar disks

We investigate the effects of gravity darkening on the UV continuum polarization produced by an axisymmetric disk that surrounds a rapidly rotating star. Although the model is a single scattering approximation, we do include the effects of attenuation (electron scattering plus hydrogen bound-free absorption) by the disk, using an approach similar to that of Sobolev (1963). Because of the gravity darkening of the star and the attenuation within the disk, the radiation field is not axially symmetric about the radius vector. This implies that the polarization source functions are no longer provided by the finite disk depolarization factors of Cassinelli, Nordsieck, & Murison (1987), which are functions of the intensity moments in a spherically symmetric atmosphere. We reformulate the polarization source functions using generalized intensity moment tensors (J, H(sub i), K(sub ij)) that are valid for an arbitrary radiation field and envelope geometry. We find that the polarization source functions are simplest when using intensity moments in the observer's reference frame. On the other hand, the intensity moments are most easily evaluated in the stellar reference frame. Using the rotation transformation properties of the generalized intensity moments, we relate the observer's moments to those evaluated in the stellar reference frame. Our procedure for determining the polarization source functions thus merely involves choosing a set of Euler angles for the coordinate rotations, and then evaluating the associated rotation matrix. The geometrical complications of polarization transfer are thus reduced to obtaining a coordinate rotation matrix.

Bjorkman, J. E.↗