Nuclear magnetic resonance restricted diffusion between parallel planes in a cosine magnetic field: An exactly solvable model

Journal of Chemical Physics

Volumn 126, Issue 10, 2007, Pages

  Grebenkov, Denis S ^a  

^a LABORATOIRE DE PHYSIQUE DE LA MATIÈRE CONDENSÉE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFUSION; EIGENVALUES AND EIGENFUNCTIONS; LAPLACE TRANSFORMS; MAGNETIC FIELD EFFECTS; MATHEMATICAL OPERATORS; NUCLEAR MAGNETIC RESONANCE;

COSINE MAGNETIC FIELDS; DIFFUSING SPINS; PARALLEL PLANES; SOLVABLE MODELS;

MAGNETIC FIELDS;

EID: 33947310259     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2539073     Document Type: Article

References (55)

1. P. T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy (Clarendon, Oxford, 1991).
2. P. T. Callaghan, A. Coy, D. MacGowan, K. J. Packer, and F. O. Zelaya, Nature 0028-0836 10.1038/351467a0 351, 467 (1991).
3. P. P. Mitra, P. N. Sen, L. M. Schwartz, and P. Le Doussal, Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.68.3555 68, 3555 (1992).
4. W. P. Halperin, J. -Y. Jehng, and Y. -Q. Song, Magn. Reson. Imaging 0730-725X 10.1016/0730-725X(94)91509-1 12, 169 (1994).
5. M. D. Hürlimann, K. G. Helmer, L. L. Latour, and C. H. Sotak, J. Magn. Reson., Ser. A 111, 169 (1994).
6. M. D. King, J. Houseman, S. A. Roussel, N. van Bruggen, S. R. Williams, and D. G. Gadian, Magn. Reson. Med. 32, 707 (1994).
7. R. W. Mair, G. P. Wong, D. Hoffmann, M. D. Hürlimann, S. Patz, L. M. Schwartz, and R. L. Walsworth, Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett. 83.3324 83, 3324 (1999).
8. Y. -Q. Song, S. Ryu, and P. N. Sen, Nature 0028-0836 10.1038/35018057 406, 178 (2000).
9. V. G. Kiselev and D. S. Novikov, Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.89.278101 89, 278101 (2002).
10. H. E. Möller, X. J. Chen, B. Saam, K. D. Hagspiel, G. A. Johnson, T. A. Altes, E. E. de Lange, and H. -U. Kauczor, Magn. Reson. Med. 0740-3194 10.1002/mrm.10173 47, 1029 (2002).
11. E. J. R. van Beek, J. M. Wild, H. -U. Kauczor, W. Schreiber, J. P. Mugler III, and E. E. de Lange, J. Magn. Reson Imaging 20, 540 (2004).
12. D. Le Bihan, Nat. Rev. Neurosci. 4, 469 (2003).
13. S. Axelrod and P. N. Sen, J. Chem. Phys. 0021-9606 10.1063/1.1356010 114, 6878 (2001).
14. P. N. Sen, Concepts Magn. Reson. 1043-7347 10.1002/cmr.a.20017 23A, 1 (2004).
15. D. S. Grebenkov, Rev. Mod. Phys. (2007) (in press).
16. P. P. Mitra, P. N. Sen, and L. M. Schwartz, Phys. Rev. B 0163-1829 10.1103/PhysRevB.47.8565 47, 8565 (1993).
17. L. L. Latour, P. P. Mitra, R. L. Kleinberg, and C. H. Sotak, J. Magn. Reson., Ser. A 101, 342 (1993).
18. K. G. Helmer, M. D. Hürlimann, T. M. de Swiet, P. N. Sen, and C. H. Sotak, J. Magn. Reson., Ser. A 115, 257 (1995).
19. B. T. Saam, D. A. Yablonskiy, V. D. Kodibagkar, J. C. Leawoods, D. S. Gierada, J. D. Cooper, S. S. Lefrak, and M. S. Conradi, Magn. Reson. Med. 44, 174 (2000).
20. D. A. Yablonskiy, A. L. Sukstanskii, J. C. Leawoods, D. S. Gierada, G. L. Bretthorst, S. S. Lefrak, J. D. Cooper, and M. S. Conradi, Proc. Natl. Acad. Sci. U.S.A. 0027-8424 10.1073/pnas.052594699 99, 3111 (2002).
21. E. L. Hahn, Phys. Rev. 0031-899X 10.1103/PhysRev.80.580 80, 580 (1950).
22. H. Y. Carr and E. M. Purcell, Phys. Rev. 0031-899X 10.1103/PhysRev.94.630 94, 630 (1954).
23. D. E. Woessner, J. Chem. Phys. 0021-9606 10.1063/1.1731821 34, 2057 (1961).
24. D. E. Woessner, J. Phys. Chem. 67, 1365 (1963).
25. E. O. Stejskal and J. E. Tanner, J. Chem. Phys. 0021-9606 10.1063/1.1695690 42, 288 (1965).
26. B. Robertson, Phys. Rev. 0031-899X 10.1103/PhysRev.151.273 151, 273 (1966).
27. C. H. Neuman, J. Chem. Phys. 0021-9606 10.1063/1.1680931 60, 4508 (1974).
28. J. Stepišnik, Physica B & C 104, 350 (1981).
29. J. Stepišnik, Prog. Nucl. Magn. Reson. Spectrosc. 17, 187 (1985).
30. J. C. Tarczon and W. P. Halperin, Phys. Rev. B 0163-1829 10.1103/PhysRevB.32.2798 32, 2798 (1985).
31. J. E. Tanner and E. O. Stejskal, J. Chem. Phys. 0021-9606 10.1063/1.1670306 49, 1768 (1968).
32. A. Coy and P. T. Callaghan, J. Chem. Phys. 0021-9606 10.1063/1.467447 101, 4599 (1994).
33. S. D. Stoller, W. Happer, and F. J. Dyson, Phys. Rev. A 1050-2947 10.1103/PhysRevA.44.7459 44, 7459 (1991).
34. T. M. de Swiet and P. N. Sen, J. Chem. Phys. 0021-9606 10.1063/1.467127 100, 5597 (1994).
35. M. D. Hürlimann, K. G. Helmer, T. M. de Swiet, P. N. Sen, and C. H. Sotak, J. Magn. Reson., Ser. A 113, 260 (1995).
36. D. J. Bergman and K. -J. Dunn, Phys. Rev. E 1063-651X 10.1103/PhysRevE.52.6516 52, 6516 (1995).
37. D. S. Grebenkov, in Proceedings of the 8th International Bologna Conference on Magnetic Resonance in Porous Media., p. 54 (Bologna, Italy, 2006).
38. L. J. Zielinski and P. N. Sen, J. Magn. Reson. 1090-7807 10.1006/jmre.2000.2185 147, 95 (2000).
39. H. C. Torrey, Phys. Rev. 0031-899X 10.1103/PhysRev.104.563 104, 563 (1956).
40. A. Caprihan, L. Z. Wang, and E. Fukushima, J. Magn. Reson., Ser. A 118, 94 (1996).
41. P. T. Callaghan, J. Magn. Reson. 1090-7807 10.1006/jmre.1997.1233 129, 74 (1997).
42. S. L. Codd and P. T. Callaghan, J. Magn. Reson. 1090-7807 10.1006/jmre.1998.1679 137, 358 (1999).
43. A. V. Barzykin, Phys. Rev. B 0163-1829 10.1103/PhysRevB.58.14171 58, 14171 (1998).
44. A. V. Barzykin, J. Magn. Reson. 1090-7807 10.1006/jmre.1999.1778 139, 342 (1999).
45. This dimensionless parameter q can be thought of as an extension of the classical q value in a PGSE experiment, (2π) -1 γgδ, to nonlinear and non-narrow magnetic fields.
46. A more general Fourier (or mixed, or relaxing) boundary condition was considered in Ref..
47. We stress that this random walk is merely a useful analogy to manipulate the Kronecker symbols in. It has no relation to the physical diffusion of spins in a slab geometry.
48. The efficiency and accuracy of this computation will, of course, depend on the temporal profile f (t). For instance, if the content of f (t) is too narrow (as in the case of Stejskal-Tanner narrow pulses), one needs to use specific distributions of moments t1, tn to perform trials.
49. The integral representation for 1 can be rigorously derived from its definition. In contrast, this definition is in general not applicable to with <1. The coefficient 32 is formally defined by series expansion of the second moment as p→0; see Ref. for details.
50. The numerical implementation of the MCF approach to compute the signal attenuation in a linear magnetic field gradient was thoroughly discussed in Ref.. In this particular case, the MCF numerical technique is similar to the stepwise gradient approximation by Barzykin (Refs.).
51. Although difficult to demonstrate rigorously, it is natural to assume that the next significant contribution after the second moment would be given by the fourth moment (n=2).
52. We stress again that this non-analytic dependence of the signal on q should be understood as asymptotic behavior.
53. To better fit the signal as a function of q, we slightly modified the relation by introducing an adjustable coefficient a0, Eexp [- a0 (π 2) (pq) 12]. We took a0 equal to 1.13, 1.00, and 0.60 for p=0.1, p=1, and p=10, respectively. This minor modification may be related to some p -dependent corrections to the asymptotic behavior.
54. A. L. Sukstanskii and D. A. Yablonskiy, J. Magn. Reson. 1090-7807 10.1006/jmre.2002.2582 157, 92 (2002).
55. D. S. Grebenkov, J. Magn. Reson. 179, 393 (2006).