Skip to main content
SpringerLink
Log in

Bispiral Approach for Calculation of Electron Paramagnetic and Nuclear Magnetic Resonance Powder Spectra

  • Original Paper
  • Published:
Applied Magnetic Resonance Aims and scope Submit manuscript

Abstract

The simulation of powder spectra involves summation of spectra calculated for N reference directions of the external magnetic field. Usually, the directions are given by regularly or randomly distributed points on a sphere. Due to an excessive number of points with the same polar angle \(\theta\) but with different azimuthal angles \(\varphi\), axial distributions produce jagged spectra, especially for spin systems with a weak azimuthal anisotropy. To improve the quality of the obtained spectra, a triangulation and subsequent interpolation of resonance fields/frequencies for hundreds of additional directions between triangle vertices or an average over a range of magnetic fields/frequencies (tent) are applied. A single spiral method with graduate steps for both the \(\theta\) and \(\varphi\) angles works better for systems with weak azimuthal anisotropy but allows for only a few interpolation points along the spiral. The proposed bispiral approach combines the best features of both spiral and triangular approaches: exact calculations for \(N\) reference spiral directions, joining of neighboring points of two spirals into a triangular net, and interpolation over hundreds of additional directions or the tent average. For systems with C1 symmetry, the angular space between primary and complementary spirals is exactly equal to the phase space of the magnetic fields (hemisphere). For systems with higher symmetry, the angular space can be significantly reduced by choosing the \(\varphi\)-shift for the second spiral, on par with the space reduction for axial distributions. Spectra simulated for axial, random, and bispiral distributions with two-dimensional interpolation over triangles and for the semispiral grid with one-dimensional interpolation are compared.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. F.K. Kneubühl, Line shapes of electron paramagnetic resonance signals produced by powders, glasses, and viscous liquids. J. Chem. Phys. 33, 1074–1078 (1960). https://doi.org/10.1063/1.1731336

    Article  ADS  Google Scholar 

  2. G. van Veen, Simulation and analysis of EPR spectra of paramagnetic ions in powders. J. Magn. Reson. 30, 91–109 (1978). https://doi.org/10.1016/0022-2364(78)90228-7

    Article  ADS  Google Scholar 

  3. Y.G. Kliava, EPR spectroscopy of disordered solids (Zinante, Riga, 1988)

    Google Scholar 

  4. M. She, X. Chen, X.-S. Yu, A method for evaluation of Hamiltonian and line shape parameters from electron paramagnetic resonance powder spectra. Can. J. Chem. 67, 88–92 (1989). https://doi.org/10.1139/v89-015

    Article  Google Scholar 

  5. M. Eden, M.H. Levitt, Computation of orientational averages in solid-state NMR by gaussian spherical quadrature. J. Magn. Reson. 132, 220–239 (1998). https://doi.org/10.1006/jmre.1998.1427

    Article  ADS  Google Scholar 

  6. M.J. Duer, Solid state NMR spectroscopy: principles and applications (Wiley, Hoboken, 2008)

    Google Scholar 

  7. M. Eden, Computer simulations in solid-state NMR. III. Powder averaging. Conc. Magn. Res. 18A, 24–55 (2003). https://doi.org/10.1002/cmr.a.10065

    Article  Google Scholar 

  8. A. Ponti, Simulation of magnetic resonance static powder lineshapes: a quantitative assessment of spherical codes. J. Magn. Reson. 138, 288–297 (1999). https://doi.org/10.1006/jmre.1999.1758

    Article  ADS  Google Scholar 

  9. S. Stoll, Spectral simulations in solid-state electron paramagnetic resonance. PhD thesis (ETH Zurich, Zürich, 2003), pp. 1–141. https://doi.org/10.3929/ethz-a-004529758

    Book  Google Scholar 

  10. S. Stoll, Computational modeling and least-squares fitting of EPR spectra, in Multifrequency electron paramagnetic resonance. (Wiley, Hoboken, 2014), pp. 69–138

    Chapter  Google Scholar 

  11. J.R. Pilbrow, Transition ion electron paramagnetic resonance (Clarendon Press, Oxford, 1990)

    Google Scholar 

  12. C. Craciun, Homogeneity and EPR metrics for assessment of regular grids used in CW EPR powder simulations. J. Magn. Reson. 245, 63–78 (2014). https://doi.org/10.1016/j.jmr.2014.05.009

    Article  ADS  Google Scholar 

  13. C. Craciun, Behavior of twelve spherical codes in CW EPR powder simulations. Uniformity and EPR properties. Stud. Ubb Chem. 4, 177–188 (2016)

    Google Scholar 

  14. R.T. Weber, WIN-EPR SimFonia manual (EPR Division, Bruker Instruments Inc., Billerica, 1995)

    Google Scholar 

  15. M.J. Nilges, Electron paramagnetic resonance studies of low symmetry Nickel(I) and Molybdenum(V) complexes. PhD thesis (University of Illinois, Urbana, 1979), pp. 1–195

    Google Scholar 

  16. G.R. Hanson, K.E. Gates, C.J. Noble, M. Griffin, A. Mitchell, S. Benson, XSophe-Sophe-XeprView. A computer simulation software suite (v. 1.1.3) for the analysis of continuous wave EPR spectra. J. Inorg. Biochem. 98, 903–916 (2004). https://doi.org/10.1016/j.jinorgbio.2004.02.003

    Article  Google Scholar 

  17. S. Stoll, A. Schweiger, EasySpin, a comprehensive software package for spectral simulation and analysis in EPR. J. Magn. Reson. 178, 42–55 (2006). https://doi.org/10.1016/j.jmr.2005.08.013

    Article  ADS  Google Scholar 

  18. S. Stoll, A. Schweiger, EasySpin: simulating cw ESR spectra. Biol. Magn. Reson. 27, 299–321 (2007)

    Google Scholar 

  19. V.I. Lebedev, Quadratures on a sphere. Comput. Math. Math. Phys. 16, 10–24 (1976). https://doi.org/10.1016/0041-5553(76)90100-2

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Stevensson, M. Eden, Efficient orientational averaging by the extension of Lebedev grids via regularized octahedral symmetry expansion. J. Magn. Reson. 181, 162–176 (2006). https://doi.org/10.1016/j.jmr.2006.04.008

    Article  ADS  Google Scholar 

  21. D.W. Alderman, M.S. Solum, D.M. Grant, Methods for analyzing spectroscopic line shapes NMR solid powder patterns. J. Chem. Phys. 84, 3717–3725 (1986). https://doi.org/10.1063/1.450211

    Article  ADS  Google Scholar 

  22. A. Kreiter, J. Huettermann, Simultaneous EPR and ENDOR powder-spectra synthesis by direct Hamiltonian diagonalization. J. Magn. Reson. 93, 12–26 (1991). https://doi.org/10.1016/0022-2364(91)90026-P

    Article  ADS  Google Scholar 

  23. T. Michaels, Equidistributed icosahedral configurations on the sphere. Comput. Math. Appl. 74, 605–612 (2017). https://doi.org/10.1016/j.camwa.2017.04.007

    Article  MathSciNet  MATH  Google Scholar 

  24. M.K. Arthur, Point picking and distributing on the disc and sphere (ARL-TR-7333, Newark, 2015), pp. 1–48

    Book  Google Scholar 

  25. S. Galindo, L. Gonzales-Tovany, Monte Carlo simulation of EPR spectra of polycrystalline samples. J. Magn. Reson. 44, 250–254 (1981). https://doi.org/10.1016/0022-2364(81)90166-9

    Article  ADS  Google Scholar 

  26. H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Am. Math. Soc. 84, 957–1041 (1978). https://doi.org/10.1090/S0002-9904-1978-14532-7

    Article  MathSciNet  MATH  Google Scholar 

  27. C. Craciun, Application of the SCVT orientation grid to the simulation of CW EPR powder spectra. Appl. Magn. Reson. 38, 279–293 (2010). https://doi.org/10.1007/s00723-010-0129-9

    Article  Google Scholar 

  28. M. Bak, N.C. Nielsen, REPULSION, a novel approach to efficient powder averaging in solid-state NMR. J. Magn. Reson. 125, 132–139 (1997). https://doi.org/10.1006/jmre.1996.1087

    Article  ADS  Google Scholar 

  29. I.M. Sobol, Uniformly distributed sequences with an additional uniform property. U.S.S.R. Comput. Maths. Math. Phys. 16, 236–242 (1976). https://doi.org/10.1016/0041-5553(76)90154-3

    Article  MATH  Google Scholar 

  30. I.A. Antonov, V.M. Saleev, An economic method of computing LPτ-sequences. U.S.S.R. Comput. Maths. Math. Phys. 19, 252–256 (1979). https://doi.org/10.1016/0041-5553(79)90085-5

    Article  MATH  Google Scholar 

  31. M.J. Mombourquette, J.A. Weil, Simulation of magnetic resonance powder spectra. J. Magn. Reson. 99, 37–44 (1992). https://doi.org/10.1016/0022-2364(92)90153-X

    Article  ADS  Google Scholar 

  32. E.B. Saff, A.B.J. Kuijlaars, Distributing many points on a sphere. Math. Intell. 19, 5–11 (1997)

    Article  MathSciNet  Google Scholar 

  33. V. Grachev, Double spiral and fractal approaches for calculations of EPR and NMR spectra in amorphous solids and powders (Specialized Colloque AMPERE “ESR and Solid State NMR in High Magnetic Fields,” Stuttgart, 2001), p. 50

    Google Scholar 

  34. R. Swinbank, R.J. Purser, Fibonacci grids: a novel approach to global modeling. Quart. J. Roy. Meteorol. Soc. 132, 1769–1793 (2006). https://doi.org/10.1256/qj.05.227

    Article  ADS  Google Scholar 

  35. N.S. Bakhvalov, On the approximate computation of multiple integrals. Vestn. Mosk. Univ. Ser. Mat. Meh. Astr. Fiz. Chim. 4, 3–18 (1959). https://doi.org/10.1016/j.jco.2014.12.003

    Article  MathSciNet  Google Scholar 

  36. N.M. Korobov, The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959)

    MathSciNet  MATH  Google Scholar 

  37. A. Gonzales, Measurement of areas on a sphere using fibonacci and latitude-longitude lattices. Math. Geosci. 42, 49–64 (2010). https://doi.org/10.1007/s11004-009-9257-x

    Article  MathSciNet  Google Scholar 

  38. S.K. Khamas, Moment method analysis of an archimedean spiral printed on a layered dielectric sphere. IEEE Trans. Antennas Propag. 56, 345–352 (2008). https://doi.org/10.1109/TAP.2007.915466

    Article  ADS  Google Scholar 

  39. H. Ebert, J. Abart, J. Voitlander, Simulation of quadrupole disturbed NMR field spectra by using perturbation theory and the triangle integration method. J. Chem. Phys. 79, 4719–4723 (1983). https://doi.org/10.1063/1.445613

    Article  ADS  Google Scholar 

  40. C.G. Koay, Analytically exact spiral scheme for generating uniformly distributed points on the unit sphere. J. Comput. Sci. 2, 88–91 (2011). https://doi.org/10.1016/j.jocs.2010.12.003

    Article  Google Scholar 

  41. S.T. Wong, M.S. Roos, A strategy for sampling on a sphere applied to 3D selective RF pulse design. Magn. Reson. Med. 32, 778–784 (1994). https://doi.org/10.1002/mrm.1910320614

    Article  Google Scholar 

  42. R. Bauer, Distribution of points on a sphere with application to star catalogs. J. Guidance Control Dyn. 23, 130–137 (2000). https://doi.org/10.2514/2.4497

    Article  ADS  Google Scholar 

  43. D.P. Hardin, T. Michaels, E.B. Saff, A comparison of popular point configuration on S2. Dolomites Res. Notes Approx. 9, 16–49 (2016)

    MathSciNet  MATH  Google Scholar 

  44. E.A. Rakhmanov, E.B. Saff, Y.M. Zhou, Minimal discrete energy on the sphere. Math. Res. Lett. 1, 647–662 (1994). https://doi.org/10.4310/MRL.1994.v1.n6.a3

    Article  MathSciNet  MATH  Google Scholar 

  45. D. Wang, G.R. Hanson, A new method for simulating randomly oriented powder spectra in magnetic resonance: The Sydney Opera House (SOPHE) method. J. Magn. Reson. A117, 1–8 (1995). https://doi.org/10.1006/jmra.1995.9978

    Article  ADS  Google Scholar 

  46. V.G. Grachev, G.I. Malovichko, Structures of impurity defects in lithium niobate and tantalate derived from electron paramagnetic and electron nuclear double resonance data. Crystals 11, 339 (2021). https://doi.org/10.3390/cryst11040339

    Article  Google Scholar 

  47. S.A. Altshuler, B.M. Kozyrev, Electron paramagnetic resonance in compounds of transition elements (Nauka, Moscow, 1972)

    Google Scholar 

  48. F.E. Mabbs, D. Collison, Electron paramagnetic resonance of D transition metal compounds (Elsevier, Amsterdam, 1992)

    Google Scholar 

  49. M.V. Vlasova, N.G. Kakazei, A.M. Kalinichenko, A.S. Litovchenko, Radiospectroscopic properties of inorganic materials. A handbook (Naukova Dumka, Kiev, 1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SimuMag, 6030 Sir George Lane, Bozeman, MT, 59718, USA

    Valentin G. Grachev

Authors
  1. Valentin G. Grachev
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

I, Valentin G. Grachev,am solely author of this manuscript

Corresponding author

Correspondence to Valentin G. Grachev.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grachev, V.G. Bispiral Approach for Calculation of Electron Paramagnetic and Nuclear Magnetic Resonance Powder Spectra. Appl Magn Reson 53, 1481–1503 (2022). https://doi.org/10.1007/s00723-022-01484-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00723-022-01484-w

Search

Navigation