Corrections to Kramers Formula Describing the Decay Rate of a Metastable State

Abstract:

The work investigates the accuracy of the Kramers approximate formula describing the decay rate of a metastable state. The study was performed by comparing the rate obtained through Kramers formula with the quasistationary rate derived by dynamical modeling that was performed by solving the Langevin equation. We have demonstrated that the non-correlation between Kramers rate and dynamical rate reached 15%, while a better correlation was expected. The study allowed us to generate corrections to Kramers formula by accounting for the higher derivatives of the potential. The rate obtained by the corrected formula exhibits the correlation with the dynamic rate of better than 1%.

• Periodical: Key Engineering Materials (Volume 685)
• Pages: 128-132
• DOI: https://doi.org/10.4028/www.scientific.net/KEM.685.128
• Online since: February 2016
• Authors: Nurken E. Aktaev, K.A. Bannova
• Keywords: Dynamical Modeling, Kramers Formula, Kramers Rate, Metastable State, Quasistationary Rate

References

[1] H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940) 284-304.

DOI: 10.1016/s0031-8914(40)90098-2

[2] W.F. Brown Jr, Thermal fluctuations of a single-domain particle, Phys. Rev. 130 (1963) 1677-1686.

DOI: 10.1103/physrev.130.1677

[3] V.M. Strutinsky, The fission width of excited nuclei, Phys. Lett. B, 47 (1973) 121-123.

[4] J. Burki, C.A. Stafford, D.L. Stein, Theory of metastability in simple metal nanowires, Phys. Rev. Lett., 95 (2005) 090601.

[5] H. Hofmann, F.A. Ivanyuk, Mean first passage time for nuclear fission and the emission of light particles, Phys. Rev. Lett. 90 (2003) 132701.

DOI: 10.1103/physrevlett.90.132701

[6] С. Schmitt, P.N. Nadtochy, A. Heinz, B. Jurado, A. Kelic, K. -H. Schmidt, First experiment on fission transient in fissile spherical nuclei produced by fragmentation of radioactive beams, Phys. Rev. Lett. 99(2007) 042701.

DOI: 10.1103/physrevlett.99.042701

[7] N.E. Aktaev, I.I. Gontchar, A modified Kramers approach to describing the fission of excited atomic nuclei, Bulletin of the Russian Academy of Sciences: Physics, 75 (2011) 994-997.

DOI: 10.3103/s1062873811070045

[8] D.J. Bicout, A.M. Berezhkovskii, A. Szabo, G.H. Weiss, Kramers-Like Turnover in Activationless Rate Processes, 83 (1999) 1279-1282.

DOI: 10.1103/physrevlett.83.1279

[9] E. Pollak, Variational transition state theory for activated rate processes, Jour. Chem. Phys., 93 (1990) 1116-1124.

DOI: 10.1063/1.459175

[10] E.G. Pavlova, N.E. Aktaev, I.I. Gontchar, Modified Kramers formulas for the decay rate in agreement with dynamical modeling, Physica A 391 (2012) 6084-6100.

DOI: 10.1016/j.physa.2012.06.064

[11] I.I. Gontchar, R.A. Kuzyakin, E.G. Pavlova, N.E. Aktaev, The nuclear fission process as Brownian motion: modifying the Kramers fission rates, Journal of Physics: Conference Series 381 (2012) 012089.

DOI: 10.1088/1742-6596/381/1/012089

[12] I.I. Gontchar, M.V. Chushnyakova, N.E. Aktaev et. al., Disentangling effects of potential shape in the fission rate of heated nuclei, Phys. Rev. C, 82 (2010) 064606.

DOI: 10.1103/physrevc.82.064606

[13] P. Hanggi, P. Talkner, M. Borkovec, Reaction-rate theory: fifty years after Kramers, Reviews of Modern Physics 62 (1990) 251-341.

DOI: 10.1103/revmodphys.62.251

[14] E.G. Pavlova, N.E. Aktaev, I.I. Gontchar, Corrections to Kramers formula for the fission rate of excited nuclei, Bulletin of the Russian Academy of Sciences: Physics, 76 (2012) 1098-1102.

DOI: 10.3103/s1062873812080217

[15] I.I. Gontchar, E.G. Pavlova, A.L. Litnevsky, N.E. Aktaev, How much accurate is description of nuclear fission rate by means of Kramer's formula?, 3rd International Conference on Current Problems in Nuclear Physics and Atomic Energy, NPAE 2010 - Proceedings, (2011).

DOI: 10.1088/1742-6596/312/8/082023

[16] O. Edholm, O. Leimar, The accuracy of Kramers' theory of chemical kinetics, Physica A. 98 (1979) 313-324.

DOI: 10.1016/0378-4371(79)90182-1

[17] P. Fröbrich and G. –R. Tillack, Path-integral derivation for the rate of stationary diffusion over a multidimensional barrier, Nuclear Physics A. 540 (1991) 353 – 364.

DOI: 10.1016/0375-9474(92)90209-3

[18] P. Fröbrich and A. Ecker, Langevin description of fission of hot metallic clusters, Euro Physics Journal D. 3 (1998) 245 – 256.

DOI: 10.1007/s100530050171

[19] J. -D. Bao and Y. Jia, Determination of fission rate by mean last passage time, Physical Review C. 69 (2004) 027602.

DOI: 10.1103/physrevc.69.027602

[20] J. -D. Bao and Y. Jia, Last passage time statistics for barrier-crossing processes, Journal of Statistical Physics. 123№ 4 (2006) 861 – 869.

DOI: 10.1007/s10955-006-9082-2

[21] B. Yilmaz, S. Ayik, Y. Abe and D. Boilley, Non-Markovian diffusion over a parabolic potential barrier: influence of the friction-memory function, Physical Review E. 77 (2008) 011121.

DOI: 10.1103/physreve.77.011121

[22] P. Fröbrich and G. –R. Tillack, Path-integral derivation for the rate of stationary diffusion over a multidimensional barrier, Nuclear Physics A. 540 (1991) 353 – 364.

DOI: 10.1016/0375-9474(92)90209-3

[23] D. Boilley and Y. Lallouet, Non-markovian diffusion over a saddle with a generalized Langevin equation, Journal of Statistical Physics. 125 № 2 (2006) 477 – 493.

DOI: 10.1007/s10955-006-9197-5

[24] P.N. Nadtochy, A. Kelic and K. -H. Schmidt, Fission rate in multi-dimensional Langevin calculation, Physical Review C. 75 (2007) 064614.