The Differences Between the Two Forms of Semi-Analytical Nodal Methods on Solving the Third-Order Simplified Spherical Harmonics Method Equations1

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Pan Qingquan ◽  
Lu Haoliang ◽  
Li Dongsheng ◽  
Wang Kan
Keyword(s):  
Spherical Harmonics ◽  
High Accuracy ◽  
Calculation Methods ◽  
Expansion Process ◽  
Third Order ◽  
The Third ◽  
Nodal Method ◽  
Key Points ◽  
Reactor Calculation ◽  
The Difference

Solving the third-order simplified spherical harmonics method (SP3) equations is one of the key points in the development of advanced reactor calculation methods and has been widely concerned. The semi-analytical nodal method (SANM), based on transverse-integrated diffusion equation, has the advantages of high accuracy and convenience for multigroup calculation. Due to its advantages, the method is expected to be used in solving the SP3 equations. However, the traditional SANM is not rigorous since the expansion process does not take the special modality of the SP3 equations and their analytical solutions into consideration. There are two modalities of the SP3 equations, so there are two traditional SANM forms on solving the SP3 equations, and the differences between the two forms will be very important in further research on the SANM. A code is developed to solve the SP3 equations under the two different forms. After the calculation of the same benchmark, the difference between the two forms on solving the SP3 equations is found. According to the results, and in view of the special modality of the SP3 equations, points on a more rigorous SANM for solving the SP3 equations are discussed.

Study on Semi-Analytical Nodal Method for Solving SP3 Equation

2017 ◽  
Author(s):  
Pan Qingquan ◽  
Lu Haoliang ◽  
Li Dongsheng ◽  
Wang Kan
Keyword(s):  
Flux Distribution ◽  
High Accuracy ◽  
Neutron Diffusion ◽  
Expansion Process ◽  
Reactor Physics ◽  
Key Technology ◽  
Nodal Method ◽  
Order Algebraic ◽  
The Difference ◽  
Neutron Diffusion Equation

Solving the SP3 equation is the key technology of the Next Generation Reactor Physics Calculation, and has been widely concerned. The semi-analytical nodal method (SANM) based on transverse-integrated neutron diffusion equation has the advantages of high accuracy and convenience for multi-group calculation. The 0th-order flux and the 2nd-order flux being Expanded with the existing 4th-order SANM polynomials and being solved respectively, the 4th-order algebraic accuracy flux distribution is also obtained, however, this solving process is not the semi-analytical nodal method since the polynomial expansion process does not take the special modality of SP3 equation and it’s analytical solution into consideration. There are two modality SP3 equation, so there are two SANM expansion forms. A code is developed to solve the SP3 equation under the two different forms. After the calculation of the same benchmark, the difference between the two forms of SP3 equation is found. According to the results, and in view of the special modality of the SP3 equation, advices for a strict semi-analytical method for solving SP3 equation are discussed.

GENERATION OF THE TERAHERZ RADIATION USING χ(3) IN SEMICONDUCTOR

1995 ◽  
Vol 04 (01) ◽  
pp. 163-189 ◽  
Author(s):  
J. B. KHURGIN
Keyword(s):  
Energy Coupling ◽  
Difference Frequency Generation ◽  
Difference Frequency ◽  
Alternative Methods ◽  
Third Order ◽  
Photogalvanic Effect ◽  
Practical Applications ◽  
The Third ◽  
Coupling Techniques ◽  
The Difference

A rigorous theory of the difference frequency mixing of two signals, one with frequency ω and the other with frequency near 2ω, in semiconductors is presented. It is shown that a lower-frequency (DC ~ 10THz) directional photocurrent and voltage are generated as a result of this nonlinear interaction. This result conclusively links the 'directional photogalvanic effect' with the third-order nonlinearity. The magnitude of the difference frequency response is evaluated as a function of frequency and the efficiency of the method is examined for various energy coupling techniques. Comparison with alternative methods for difference frequency generation using the second order nonlinearities is made and the practical applications are considered.

scholarly journals Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation

2009 ◽  
Vol 54 (21) ◽  
pp. 6477-6493 ◽  
Author(s):  
Yujie Lu ◽  
Ali Douraghy ◽  
Hidevaldo B Machado ◽  
David Stout ◽  
Jie Tian ◽  
...  
Keyword(s):  
Spherical Harmonics ◽  
Third Order ◽  
Bioluminescence Tomography ◽  
The Third ◽  
Spectrally Resolved

A Second-Order Three-Level Difference Scheme for a Magneto-Thermo-Elasticity Model

2014 ◽  
Vol 6 (3) ◽  
pp. 281-298 ◽  
Author(s):  
Hai-Yan Cao ◽  
Zhi-Zhong Sun ◽  
Xuan Zhao
Keyword(s):  
Difference Scheme ◽  
Hyperbolic Equations ◽  
A Priori ◽  
A Priori Estimate ◽  
Second Order ◽  
Priori Estimate ◽  
Third Order ◽  
The Third ◽  
The Difference ◽  
Second Order Convergence

AbstractThis article deals with the numerical solution to the magneto-thermo-elasticity model, which is a system of the third order partial differential equations. By introducing a new function, the model is transformed into a system of the second order generalized hyperbolic equations. A priori estimate with the conservation for the problem is established. Then a three-level finite difference scheme is derived. The unique solvability, unconditional stability and second-order convergence inL∞-norm of the difference scheme are proved. One numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.

SECOND HYPERPOLARIZABILITIES OF MOLECULAR AGGREGATES: INTERMOLECULAR ORBITAL-INTERACTION AND SPIN-CONFIGURATION EFFECTS

2002 ◽  
Vol 01 (05n06) ◽  
pp. 545-549
Author(s):  
MASAYOSHI NAKANO ◽  
SATORU YAMADA ◽  
MASAHIRO TAKAHATA ◽  
KIZASHI YAMAGUCHI
Keyword(s):  
Optical Properties ◽  
Nonlinear Optical ◽  
Exchange Interactions ◽  
Orbital Interaction ◽  
Molecular Aggregates ◽  
Third Order ◽  
Spin Configuration ◽  
The Third ◽  
Weak Exchange ◽  
The Difference

Toward an understanding of intermolecular-interaction effects on the third-order nonlinear optical properties of nanostructured molecular aggregates, we investigate the longitudinal second hyperpolarizabilities (γ) of model dimers composed of neutral ( C 5 H 7) and charged [Formula: see text] monomers using ab initio molecular orbital methods. It is found that π–π orbital interaction in the stacking direction remarkably affects the γ values of dimers, while the difference in spin configuration hardly causes significant changes in γ for the present models due to weak exchange interactions between monomers.

A perturbation calculation of properties of the 1 s σ and ­2 p σ states of HeH 2+

1956 ◽  
Vol 238 (1213) ◽  
pp. 276-285 ◽  
Keyword(s):  
Dipole Moments ◽  
High Accuracy ◽  
Wave Functions ◽  
Quadrupole Moments ◽  
Perturbation Calculation ◽  
Third Order ◽  
The Third ◽  
Wide Range ◽  
Fifth Order ◽  
Kinetic And Potential Energies

A perturbation calculation, valid in the limit of large separations, of various properties of the 1 s σ and 2 p σ states of HeH 2+ is carried out. The total energy and the kinetic and potential energies are calculated to the fifth order, the dipole moments to the third order and the quadrupole moments to the second order. The results are compared with those obtained using exact and variationally determined two-centre wave functions and also with those obtained from an approximate application of perturbation theory and it is shown that perturbation calculations of molecular properties are capable of high accuracy over a wide range of nuclear separations.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

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