Mean work functions effective for negative-ionic, electronic and positive-ionic emissions from polycrystalline surfaces

To elucidate the thermionic property of polycrystalline surfaces, a further study is made on the mean work functions (φ⁻, φ^e and φ⁺) effective for negative-ionic, electronic and positive-ionic emissions. Comparison between theoretical analyses and experimental data yields the conclusions as follows. (1) The equation of φ⁻ = φ^e holds always with both mono- and polycrystalline surfaces. (2) The relation of φ⁻ = φ^e < φ⁺ applies to polycrystalline surfaces because they bear the thermionic contrast (Δφ^* ≡ φ⁺ − φ^e > 0). (3) The value of Δφ^* ranges from ∼0.4 to 0.9 eV depending upon the surface species of polycrystalline metals (e.g., W, Re and Pt), whilst Δφ^* = 0 for monocrystalline surfaces. (4) When the degree of monocrystallization (δ_m) is less than ∼50%, the theoretical value of Δφ^* is virtually independent of δ_m and agrees well with experimental data, nearly the same within ±0.1 eV among the so-called “polycrystalline” surfaces of W. (5) As δ_m increases beyond ∼80 up to 100%, Δφ^* decreases rapidly down to 0 eV, showing again a good agreement between theory and experiment. (6) Our theoretical model is valid in evaluating the effective mean work functions, irrespective of the range of δ_m.     

Theoretical study of local crystal structure in KZnF3:Fe system

An analysis of the relationship between the EPR trigonal-field parameters and the local crystal structure of KZnF₃:Fe³⁺system is presented by diagonalizing the complete energy matrices for a d⁵ configuration ion in a trigonal crystal field. We propose a two-layer-ligand model, in which the ligands consist of six nearest-neighbor F⁻ ions in the first layer and eight next nearest-neighbor K⁺ ions in the second layer. The calculation indicates that the local structure distortion of KZnF₃:Fe³⁺system is due to the displacement of a K⁺ ion along C₃ axis towards the Fe³⁺ ion, which leads to the shift of the F⁻ ions away from C₃ axis. By simulating the EPR low-symmetry parameters D and (a−F), the distorted angles between the Fe³⁺-F⁻ bonds and C₃ axis are determined, Δθ₁=2.58°, Δθ₂=−1.4° at room temperature (300 K) and Δθ₁=2.84°, Δθ₂=−1.4° at low temperature (77 K). Those results are in good agreement with the experimental findings Δθ₁=2.8±0.3°and Δθ₂=−1.1±0.3°.     

Syntheses, crystal structures, visible and near-IR luminescent properties of ternary lanthanide (Dy, Tm) complexes containing 4,4,4-trifluoro-1-phenyl-1,3-butanedione and 1,10-phenanthroline

The ligands 4,4,4-trifluoro-1-phenyl-1,3-butanedione (Hbfa) and 1,10-phenanthroline (phen) were used to prepare ternary lanthanide (Ln) complexes [Dy(bfa)₃phen and Tm(bfa)₃phen]. Crystal data: Dy(bfa)₃phen C₄₂H₂₆F₉N₂O₆Dy, triclinic, P1¯, a=9.9450(6) Å, b=14.0944(9) Å, c=14.6043(9) Å, α=82.104(1)°, β=87.006(1)°, γ=76.490(1)°, V=1971.1(2) Å³, Z=2; Tm(bfa)₃phen C₄₂H₂₆F₉N₂O₆Tm, triclinic, P1¯, a=9.898(5) Å, b=13.918(5) Å, c=14.753(5) Å, α=83.517(5)°, β=86.899(5)°, γ=76.818(5)°, V=1965.3(14) Å³, Z=2. The coordination number of the central Ln³⁺ (Ln=Dy, Tm) ion is eight, with six oxygen atoms from three Hbfa ligands and two nitrogen atoms from the phen ligand. The photophysical properties of the two complexes were studied by absorption spectra, diffuse reflectance spectra, and emission spectra. They show the characteristic luminescence of the corresponding Ln³⁺ ion in both visible and near-IR (NIR) region. Additionally, the energy transfer mechanisms between the ligands and central Ln³⁺ ions were discussed.     

Theoretical evaluation of the effective work functions for positive-ionic and electronic emissions from polycrystalline metal surfaces

Effective work functions (φ⁺ and φ^e) for positive-ionic and electronic emissions from polycrystalline metals of Nb, Mo, Ta, W and Ir are calculated according to our theoretical model by using those published data on both fractional surface area (F_i) and local work function (φ_i) of each metal surface composed of several patchy faces (1, 2, …, i). Comparison between the theoretical values thus obtained and those experimental data published to date yields the conclusions as follows. (1) With a slight error of less than ∼0.1 eV, the value of φ^e calculated with each of the metals is in fair or good agreement with that determined by experiment. (2) Such agreement is found also with φ⁺ for W. (3) In a typical case of W, where the degree of monocrystallization (δ_m) corresponding to the largest among the values of F_i is less than ∼0.5, the thermionic contrast (Δφ* ≡ φ⁺ − φ^e) is found again to be nearly equal to both theoretical and experimental values reported previously. (4) Each of the five metals shows that Δφ* at δ_m = 0.68-0.95 is smaller than Δφ* at δ_m < 0.5. (5) This result strongly supports our theoretical prediction that Δφ* decreases gradually to zero as δ_m increases beyond ∼0.5 up to ∼1. (6) Particularly, such a surface which has δ_m ≥ 0.96 exhibits Δφ* ≈ 0, apparently equivalent to the so-called “monocrystalline surface (δ_m = 1)”. These results lead to the conclusion that our theoretical model is valid for evaluating the effective work functions probably with a slight error of less than ∼0.1 eV, irrespective of both the surface species and the range of δ_m. In addition, our simple model makes it possible to analyze the mechanism of change in φ⁺ and φ^e according to the change in surface characters of both φ_i and F_i.     

Structural and luminescence properties of [Ln(ODA)(phen)·4H2O] complexes (Ln=Sm and Dy, ODA=oxydiacetate, phen=1,10-phenanthroline)

Two novel complexes of Sm(III) and Dy(III) with mixed oxydiacetate (ODA) and 1,10-phenanthroline (phen) ligands were synthesized and their structure and luminescence properties were characterized. The complexes of [Ln(ODA)(phen)·4H₂O]Cl·5H₂O [Ln=Sm and Dy] crystallize in the monoclinic space group P2₁/n with Sm: a=12.3401(14) Å, b=16.821(2), c=12.6847(11) Å, β=107.939(10)°, V=2505.0(5) Å³, Z=4 and ρ=1.841 mg/m³, and with Dy: a=12.289(7) Å, b=16.805(6) Å, c=12.705(4) Å, β=108.144(18)°, V=2493.4(19) Å³, Z=4 and ρ=1.786 mg/m³. The complexes of [Sm(ODA)(phen)·4H₂O]⁺ and [Dy(ODA)(phen)·4H₂O]⁺ excited by UV light produce orange red and lightly white emissions, respectively, via the nonradiative energy transfer from phen to the metals. The quantum yield of the sensitized luminescence of [Dy(ODA)(phen)·4H₂O]⁺ (Q=19%) is much greater than that of [Sm(ODA)(phen)·4H₂O]⁺ (Q=1.4%). The luminescence decay times of the complexes were in a few microsecond range and independent of temperature.     

The crystal structure and magnetic properties of the 1-dimensional dihalide-bridged polymers dichlorobis(thiazole)cobalt(II) and dibromobis(thiazole)-cobalt(II)

The isostructural polymeric compounds Co(thiazole)₂X₂ (X=Cl (1), Br(2)) have been synthesised by the addition of thiazole to an ethanolic solution of the corresponding anhydrous cobalt halide. Powder X-ray and neutron diffraction measurements were used for structural determination. The structures were determined using powder neutron diffraction data and Rietveld techniques: (1) C2/c, a=17.806(2) Å, b=3.6806(6) Å, c=14.807(3) Å, β=94.78(1)°, V=967.1(3) ų, Z=4; (2) C2/c, a=18.079(3) Å, b=3.8138(8) Å, c=15.022(4) Å, β=92.71(1)°, V=1034.6(4) ų, Z=4. Each linear polymer chain is composed of pseudo-octahedral, high-spin Co²⁺ centres, doubly linked by halide bridges. Magnetisation measurements of 1 and 2 at 5 K between 0 and 10 kG reveals a metamagnetic transition between antiferromagnetic and ferromagnetic states. Low temperature susceptibility data have been fitted to a one-dimensional Ising model with a mean field correction and were found to be anisotropic with ferromagnetic intrachain interactions along the b-axis and weaker antiferromagnetic interchain interactions.     

Theoretical investigation of zero-field splitting and local structure distortion for a substitution of Fe ion in CdCl2 crystal

A theoretical method for studying the inter-relation between electron and molecule structure is proposed on the basis of the complete energy matrices of the electron-electron repulsion, the ligand-field and the spin-orbit coupling for d⁵ configuration ion in a trigonal ligand-field. As an application, the local distortion structure of (FeCl₆)^3- coordination complex for Fe³⁺ ions doped into CdCl₂ is investigated. Both the second-order zero-field splitting parameter and fourth-order zero-field splitting parameter are considered simultaneously in the structural investigation. By diagonalizing the complete energy matrices, the local structure distortion parameters ΔR=−0.24 Å, Δθ=2.137° at 26 K and ΔR=−0.203 Å, Δθ=2.515° at 225 K for Fe³⁺ ions in CdCl₂ are determined. These results elucidate a microscopic origin of various ligand-field parameters which are usually used empirically for the interpretation of electron paramagnetic resonance results. It is found that the theoretical results are in good agreement with the experimental values.     

Heats of solution/substitution in TlNO3 and CsNO3 crystals and in RbNO3 and CsNO3 crystals from heats of transition: the complete phase diagrams of TlNO3-CsNO3 and RbNO3-CsNO3 systems

From measurements of the decrease in the heat (enthalpy) of transition in the solid phase using differential scanning calorimetry, the apparent molar heats of solution, slope ΔH_t/x, the partial molar heats of solution at infinite dilution, χ, and the heats of solution, ΔH_s^°, of Tl⁺ in CsNO₃ crystal and Cs⁺ in TlNO₃ crystal and Rb⁺ in CsNO₃ crystal and Cs⁺ in RbNO₃ crystal along with their recovered lattice energies, ΔH_L^°, are reported. ΔH_s^° of Tl⁺ and Rb⁺ in CsNO₃ crystal are each found to be negligible or zero representing an ideal solid solution, i.e. ΔH_mix=0. The complete phase diagrams of the TlNO₃-CsNO₃ and RbNO₃-CsNO₃ systems with details of the sub-solidus regions are included. The properties of Tl_(1−x)Cs_xNO₃ and Rb_(1−x)Cs_xNO₃ compositions are discussed in terms of a ‘mixed crystal’ or ‘crystalline solid solution’ in relation to parallel compositions of Tl_(1−x)Rb_xNO₃.     

The elliptical dimension of space-time atmospheric stratification of passive admixtures using lidar data

State-of-the-art airborne lidar data of passive scalars have shown that the spatial stratification of the atmosphere is scaling: the vertical extent (Δz) of structures is typically ≈Δx^H_z where Δx is the horizontal extent and H_z is a stratification exponent. Assuming horizontal isotropy, the volumes of the structures therefore vary as ΔxΔxΔx^H_z=Δx^D_s where the “elliptical dimension” D_s characterizes the rate at which the volumes of typical non-intermittent structures vary with scale. Work on vertical cross-sections has shown that 2+H_z=2.55±0.02 (close to the theoretical prediction 23/9).In this paper we extend these (x, z) analyses to (z, t). In the absence of overall advection, the lifetime Δt of a structure of size Δx varies as Δx^H_t with H_t=2/3 so that the overall space-time dimension is D_st=29/9=3.22…. However, horizontal and vertical advection lead to new exponents: we argue that the temporal stratification exponent H_t≈1 or ≈0.7 depending on the relative importance of horizontal versus vertical advection velocities. We empirically test these space-time predictions using vertical-time (z, t) cross-sections using passive scalar surrogates (aerosol backscatter ratios from lidar) at ∼3 m resolution in the vertical, 0.5-30 s in time and spanning 3-4 orders of magnitude in scale as well as new analyses of vertical (x, z) cross-sections (spanning over 3 orders of magnitude in both x, z directions). In order to test the theory for density fluctuations at arbitrary displacements in (Δz, Δt) and (Δx, Δz) spaces, we developed and applied a new Anisotropic Scaling Analysis Technique (ASAT) based on nonlinear coordinate transformations. Applying this and other analyses to data spanning more than 3 orders of magnitude of space-time scales we determined the anisotropic scaling of space-time finding the empirical value D_st=3.13±0.16. The analyses also show that both cirrus clouds and aerosols had very similar space-time scaling properties. We point out that this model is compatible with (nonlinear) “turbulence” waves, hence potentially explaining the observed atmospheric structures.     

Resonance-enhanced multiphoton ionization spectroscopy of CH3 and CD3. Two-photon absorption selection rules and rotational line strengths of the ν3- and ν4-active vibronic transitions

To explore the possibility of K-level resolved, 2+1 resonance-enhanced multiphoton ionization (REMPI) processes of the methyl radical, the two-photon absorption selection rules and rotational line strengths of the 3₀¹ and 4₀¹ vibronic bands of the transition (n=3 or 4) were reported. Stringent selection rules, which were imposed upon these two-photon transitions, are the initial K″=3p (p=0,1,2,…), ΔK=±2, ΔU=±3, and ΔN=0,±1,±2 (O, P, Q, R, and S branches). The previously assigned 2₂² vibronic band of the methyl radical should be studied by the REMPI with a better spectral resolution and analyzed by the newly derived two-photon absorption selection rules and rotational line strength formulas.     

Torsional analysis of 2-butynol

The microwave spectrum of 2-butynol, CH₃CCCH₂OH, has been investigated in the frequency range of 6-26.5 GHz. The spectra of the A and E torsional states were observed using Fourier transform microwave spectroscopy. Due to the presence of the cylindrically symmetric CC “spacer” between the methyl group and the rest of the molecule, the barrier to internal rotation, V₃, is only 7 cm⁻¹. One conformer of 2-butynol was observed and assigned. The spectrum was analyzed with the ρ-axis method using a very flexible Hamiltonian which gives a fit an order of magnitude better than that obtained with more standard code for internal rotation. The spectroscopic constants are: 23744.(18), 2093.429(1), 1966.358(1), and −400.34(2) MHz for A, B, C, and D_ab; 0.48(1) kHz, −30.3(4) kHz, and 4.5(5) MHz for Δ_J, Δ_JK, and Δ_K; and 0.1406(6), 6.93(9) cm⁻¹, −33.4(7) kHz, and 192.0(9) GHz for ρ, V₃, L_V, and F, respectively. The root mean squared error of the fit is 14 kHz. The angles between the internal rotor axis and principal axes are θ_a=12°, θ_b=78°, with θ_c held fixed at 90°.     

Low-temperature property investigation of the lead indium-niobate-lead nickel-niobate solid solution

The complex perovskite solid solution (1−x) Pb(In_1/2Nb_1/2)O₃-(x) Pb(Ni_1/3Nb_2/3)O₃ has been successfully prepared by the Columbite precursor method. The temperature dependencies of the dielectric constant and pyroelectric coefficient were measured between −261 and 200 °C. Relaxor ferroelectric behavior has been noticed in all compositions across the solid solution. The room-temperature electrostrictive coefficient, Q₃₃, was 1.83×10⁻² C²/m⁴ for x=0.10. No room-temperature piezoelectric activity was detected; however, upon cooling to −261 °C the maximum coupling coefficients k_p=29%, k_t=11%, and k₃₃=31% were observed for the composition x=1.00.     

FTIR Study of the ν1/ν4 Bands of D3SiCl near 1600 cm

The ν₁ (A₁, 1583.22 cm⁻¹) and ν₄ (E, 1615.33 cm⁻¹) Si-D stretching bands of monoisotopic D₃Si³⁵Cl have been studied by FTIR spectroscopy with a resolution of 3.3×10⁻³ cm⁻¹. We have assigned 2341 rovibrational lines for ν₁ (J_max=70, K_max=19) and 6207 for ν₄ (J_max=75, K_max=27). Both (ΔK=±1, Δ?=±1) and (ΔK=±2, Δ?=?1) interactions connect the v₁=1 and v₄=1 levels, the latter exerting moreover a weak ?(2, 2) interaction. These interactions were taken into account in a nonlinear least-squares fit, refining 29 free parameters with a standard deviation of 0.257×10⁻³ cm⁻¹ over 6722 nonzero-weighted data. Blended lines and about 250 of the 330 lines belonging to the K=11 subband of ν₁ and the KΔK=−6 subband of ν₄ were zero-weighted because they are locally perturbed respectively by the neighboring upper states of the 2ν₃+ν₆ (E, 1561.95 cm⁻¹) and 3ν₃ (A₁, 1604.81 cm⁻¹) bands. Equivalent fits were obtained for altogether three different models obeying constraints according to the theory of unitary equivalent reductions of the rovibrational Hamiltonian. By means of a band contour simulation both the transition moment ratio |M₁:M₄|=0.67 and a positive sign of the Coriolis intensity perturbation were determined.     

Variable-temperature FT-IR studies on the thermodynamics of carbon dioxide adsorption on a faujasite-type H-Y zeolite

Adsorption of carbon dioxide on a faujasite-type H-Y zeolite (Si:Al = 2.6:1) was studied by variable-temperature (200-290 K range) infrared spectroscopy. Adsorbed CO₂ molecules interact with the Brønsted acid Si(OH)Al groups located inside the zeolite supercage, bringing about a characteristic bathochromic shift of the O-H stretching mode from 3645 cm⁻¹ (free OH group) to 3540 cm⁻¹ (hydrogen-bonded CO₂ adsorption complex). Simultaneously, the asymmetric (ν₃) mode of adsorbed CO₂ is observed at 2353 cm⁻¹. From the observed variation of the integrated intensity of the 3645 and 2353 cm⁻¹ IR absorption bands upon changing temperature, corresponding values of standard adsorption enthalpy and entropy were found to be ΔH° = −28.5(±1) kJ mol⁻¹ and ΔS° = −129(±10) J mol⁻¹ K⁻¹. Comparison with the reported values of ΔH° for CO₂ adsorption on other zeolites is briefly discussed.     

Analysis of BPSK homodyne receivers based on modified balanced optical phase-locked loop in the presence of non-negligible loop propagation delay

Optical homodyne receivers based on modified balanced optical phase-locked loop is analyzed taking into account loop propagation delay. This modified loop contains all the components of a standard BOPLL in conjunction of an additional phase modulator. This modified loop offers a much improved tracking performance and also shows good improvements in the reduction performance over previously reported methods, like, relaxed line-width requirement and improved lock-in range. This study for the first time shows the imperfect-phase-recovery-induced power penalty as a function of laser line-width with the optimum phase deviations in the presence of non-negligible loop propagation delay. It is found that in order to maintain a 10⁻¹⁰ BER system performance with ξ = 1, R = 1 A/W, P_R = −53 dBm, Δυ = 1 MHz, phase modulator sensitivity K_PM = 10 rad/V and 10° phase deviation between the two transmitted bits, the loop delay must be kept below 3 ns. Further, the required line-width with the non-negligible loop delay time is evaluated and found to be (2.1 × 10⁻³)/τ, where τ (s) is the loop delay. This number corresponds to BER = 10⁻¹⁰, imperfect phase recovery power penalty of 1 dB and phase modulator sensitivity K_PM = 20 rad/V.     

Atomic beam magnetic resonance investigations in the3 F 2 Ground State of177Hf,179Hf and180Hf

The 5d ²6s ^{2 3} F ₂ ground state of¹⁷⁷Hf,¹⁷⁹Hf and¹⁸⁰Hf has been studied using the atomic beam magnetic resonance method. The atomic beam was produced by an universal evaporation technique described in a previous paper. The results are¹⁸⁰Hfg _j (³ F ₂)=0.695812 (10)¹⁷⁷Hf Δv(³ F ₂;F=11/2?F=9/2)=991.7917 (10) MHz Δv(³ F ₂;F=9/2?F=7/2)=477.0081 (10) MHz Δv(³ F ₂;F=7/2?F=5/2)=162.8890 (10) MHz¹⁷⁹HfΔv(³ F ₂;F=13/2?F=11/2)=82.1320 (10) MHz Δv(³ F ₂;F=11/2?F=9/2)=392.8498 (10) MHz. The magnetic dipole and electric quadrupole moments of the¹⁷⁷Hf and¹⁷⁹Hf nuclear ground states as calculated from these hyperfine structure measurements are the following: μ(177)=0.75(8)μ _k , Q(177)=4.34 (65) barns μ(179)=?0.61 (6)μ _k , Q(179)=4.90 (75) barns.     

EPR study of Cu(II) dopant ions in single crystals of bis(l-asparaginato)Zn(II)

We report an electron paramagnetic resonance (EPR) study at 33.9 GHz and room temperature of oriented single crystal samples of bis(l-asparaginato)Zn(II) doped with Cu(II). The variation of the spectra with magnetic field orientation was measured in three crystal planes (a*b, bc and a*c, with a*=b×c). These spectra display two groups of four peaks arising from the hyperfine interaction with the I_Cu=3/2 nuclear spins of copper. They were assigned to Cu(II) ions in two lattice sites related by a 180° rotation around the b-crystal axis. The g and hyperfine coupling (A) tensors of the Cu(II) ions were evaluated from the single crystal data. Some indeterminacy in the assignment of the signals was avoided measuring the EPR spectrum of a powder sample. Their principal values are g₁=2.060(1), g₂=2.068(2), g₃=2.283(2), and A₁≈0.1×10⁻⁴, A₂=13×10⁻⁴ and A₃=165×10⁻⁴ cm⁻¹. The eigenvectors corresponding to g₃ and A₃ are coincident within the experimental error; the other eigenvectors are rotated 5.6° in the perpendicular plane. Considering the crystal structure of bis(l-asparaginato)Zn(II), our EPR results indicate that the Cu(II) impurities replace Zn(II) ions in the host crystal. We propose a molecular model based on the EPR data and the structural information, and analyse the results comparing the measured values with those obtained in similar systems.     

EPR theoretical study of local lattice structure in Al2O3:Fe system

By analyzing the EPR spectrum of transition-metal ion Fe³⁺ in Al₂O₃:Fe³⁺ system, the local lattice structure around impurity Fe³⁺ ion in the crystal has been studied by means of the diagonalization of the energy matrices of the electron-electron repulsion, the ligand-field and the spin-orbit coupling for a d⁵ configuration ion in a trigonal ligand-field. Both the second-order and fourth-order EPR parameters D and (a−F) are taken simultaneously in the structural investigation. The results indicate that the two three-edge-pyramids elongated obviously along C₃ axis. The two distortion angles Δθ₁=−1.1±0.1°,Δθ₂=−1.8° as well as the two Fe-O bond lengths R₁=2.016 A, R₂=1.907 A are determined, respectively.     

The Ground State Spectroscopic Parameters and Equilibrium Structure of PD3

Infrared spectra of PD₃ have been measured in the 20-320 cm⁻¹ range and in the region of the ν₂/ν₄ and ν₁/ν₃ fundamental bands near 750 and 1690 cm⁻¹, respectively, with a resolution of ca. 0.0025 cm⁻¹. Furthermore, submillimeter-wave spectra covering the J=4-3, 13-12, and 14-13 clusters in the vibrational ground state were recorded. The observed ΔJ=+1 rotational lines were augmented by about 5500 ground state combination differences formed from transitions belonging to the fundamental bands. Of these, 1300 involved perturbation-allowed lines with ΔK≠0. These data and observations taken from the literature were appropriately weighted and fitted to 14 ground state molecular constants. The A and B reductions of the rotational Hamiltonian were found to be equivalent. Improved effective ground state and equilibrium structures were determined for both PH₃ and PD₃; the equilibrium structures, r_e (PH)=141.1607(83) pm and α_e (HPH)=93.4184(95)° and r_e (PD)=141.1785(57) pm and α_e (DPD)=93.4252(68)°, are in good agreement.     

Step-shape angular spin distribution in layered systems by Fe Mössbauer spectroscopy: A general treatment

In the so-called ‘step-shape’ angular spin distribution model for layered systems, the non-collinear directions of the atomic magnetic moments are confined to the film plane and form a homogeneous fan spanning inside an (in-plane) angular interval Δφ centered at an angle φ₀. A general approach for deriving the two parameters φ₀ and Δφ via ⁵⁷Fe Mössbauer spectroscopy measurements is discussed. The analysis extends our previously reported treatment, which assumed that the angular aperture Δφ develops symmetrically versus a fixed direction φ₀ (e.g., the in-plane easy axis of magnetization) oriented either along or perpendicular to the in-plane projection of the Mössbauer γ-ray direction. The proposed approach is also applicable for those cases when not only the spin aperture Δφ is changing but also the aperture center φ₀ is rotating under the influence of different external parameters, such as applied field, temperature, stress, etc. The method is suitable for applications to nanoscale layered heterostructures with in-plane uniaxial or unidirectional magnetic anisotropy. The method is applied to experimental data obtained on a 2-nm thick defected Fe layer with in-plane magnetic texture.