Monotonic properties of the shift and penetration factors

Brune, Carl Richard; Hale, Gerald M.; Paris, Mark W.

doi:10.1103/PhysRevC.97.024603

Title: Monotonic properties of the shift and penetration factors

Abstract

Here, we study derivatives of the shift and penetration factors of collision theory with respect to energy, angular momentum, and charge. Definitive results for the signs of these derivatives are found for the repulsive Coulomb case. In particular, we find that the derivative of the shift factor with respect to energy is positive for the repulsive Coulomb case, a long anticipated but heretofore unproven result. In conclusion, these results are closely connected to the properties of the sum of squares of the regular and irregular Coulomb functions; we also present investigations of this quantity.

Authors:

Brune, Carl Richard ^[1]; Hale, Gerald M. ^[2];

^[2]

Ohio Univ., Athens, OH (United States)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Mon Feb 05 00:00:00 EST 2018

Research Org.:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1479931

Alternate Identifier(s):: OSTI ID: 1419605

Report Number(s):: LA-UR-17-29016
Journal ID: ISSN 2469-9985; PRVCAN

Grant/Contract Number:: AC52-06NA25396; FG02-88ER40387; NA0002905

Resource Type:: Accepted Manuscript

Journal Name:: Physical Review C

Additional Journal Information:: Journal Volume: 97; Journal Issue: 2; Journal ID: ISSN 2469-9985

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Atomic and Nuclear Physics; R-matrix theory

Citation Formats


                    Brune, Carl Richard, Hale, Gerald M., and Paris, Mark W. Monotonic properties of the shift and penetration factors.  United States: N. p., 2018. 
Web.  doi:10.1103/PhysRevC.97.024603.

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                    Brune, Carl Richard, Hale, Gerald M., & Paris, Mark W. Monotonic properties of the shift and penetration factors.  United States.  https://doi.org/10.1103/PhysRevC.97.024603

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                    Brune, Carl Richard, Hale, Gerald M., and Paris, Mark W. Mon .  
"Monotonic properties of the shift and penetration factors".  United States.  https://doi.org/10.1103/PhysRevC.97.024603.  https://www.osti.gov/servlets/purl/1479931.

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@article{osti_1479931,

  title        = {Monotonic properties of the shift and penetration factors},

  author       = {Brune, Carl Richard and Hale, Gerald M. and Paris, Mark W.},

  abstractNote = {Here, we study derivatives of the shift and penetration factors of collision theory with respect to energy, angular momentum, and charge. Definitive results for the signs of these derivatives are found for the repulsive Coulomb case. In particular, we find that the derivative of the shift factor with respect to energy is positive for the repulsive Coulomb case, a long anticipated but heretofore unproven result. In conclusion, these results are closely connected to the properties of the sum of squares of the regular and irregular Coulomb functions; we also present investigations of this quantity.},

  doi          = {10.1103/PhysRevC.97.024603},

  journal      = {Physical Review C},

  number       = 2,

  volume       = 97,

  place        = {United States},

  year         = {Mon Feb 05 00:00:00 EST 2018},

  month        = {Mon Feb 05 00:00:00 EST 2018}

}

Copy to clipboard

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https://doi.org/10.1103/PhysRevC.97.024603

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Cited by: 2 works

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Figures / Tables:

TABLE I.: Summary of the result of Secs. III and IV for the variation of S and P with E, $\ell$, and η. The second column is deduced from the first using Eqs. (11) and (12).

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Works referenced in this record:

Alternative parametrization of R -matrix theory
journal, October 2002

Brune, C. R.
Physical Review C, Vol. 66, Issue 4
DOI: 10.1103/PhysRevC.66.044611

On Positivity of Fourier Transforms
journal, January 2006

Tuck, E. O.
Bulletin of the Australian Mathematical Society, Vol. 74, Issue 1
DOI: 10.1017/S0004972700047511

R-Matrix Theory of Nuclear Reactions
journal, April 1958

Lane, A. M.; Thomas, R. G.
Reviews of Modern Physics, Vol. 30, Issue 2
DOI: 10.1103/RevModPhys.30.257

LXXV. Integral representations for products of Whittaker functions
journal, November 1938

Erdélyi, A.
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 26, Issue 178
DOI: 10.1080/14786443808562178

Some Properties of the Shift and Penetration Factors in Nuclear Reactions
journal, January 1958

Prosser, F. W.; Biedenharn, L. C.
Physical Review, Vol. 109, Issue 2
DOI: 10.1103/PhysRev.109.413

Theory of the Effective Range in Nuclear Scattering
journal, July 1949

Bethe, H. A.
Physical Review, Vol. 76, Issue 1
DOI: 10.1103/PhysRev.76.38

Product Formulas and Nicholson-Type Integrals for Jacobi Functions. I: Summary of Results
journal, February 1978

Durand, Loyal
SIAM Journal on Mathematical Analysis, Vol. 9, Issue 1
DOI: 10.1137/0509007

The R -matrix theory
journal, February 2010

Descouvemont, P.; Baye, D.
Reports on Progress in Physics, Vol. 73, Issue 3
DOI: 10.1088/0034-4885/73/3/036301

Regular and Irregular Coulomb Wave Functions Expressed in Terms of Bessel-Clifford Functions
journal, April 1954

Abramowitz, Milton
Journal of Mathematics and Physics, Vol. 33, Issue 1-4
DOI: 10.1002/sapm1954331111

Bessel function expansions of Coulomb wave functions
journal, April 1985

Humblet, J.
Journal of Mathematical Physics, Vol. 26, Issue 4
DOI: 10.1063/1.526602

Wave Functions for Large Arguments by the Amplitude-Phase Method
journal, December 1937

Wheeler, John A.
Physical Review, Vol. 52, Issue 11
DOI: 10.1103/PhysRev.52.1123

Completely monotonic functions
journal, December 2001

Miller, K. S.; Samko, S. G.
Integral Transforms and Special Functions, Vol. 12, Issue 4
DOI: 10.1080/10652460108819360

A note on the calculation of Coulomb penetration and Coulomb wave functions
journal, April 1957

Thaler, R. M.; Biedenharn, L. C.
Nuclear Physics, Vol. 3, Issue 2
DOI: 10.1016/0029-5582(57)90107-4

T matrix and effective range function for Coulomb plus rational separable potentials especially for I =1
journal, May 1977

van Haeringen, H.
Journal of Mathematical Physics, Vol. 18, Issue 5
DOI: 10.1063/1.523373

Higher monotonicity properties of certain Sturm-Liouville functions
journal, January 1963

Lorch, Lee; Szego, Peter
Acta Mathematica, Vol. 109, Issue 0
DOI: 10.1007/BF02391809

On Differential Equations and the Function $J_\{mu}^2 + Y_\{mu}^2$
journal, January 1961

Hartman, Philip
American Journal of Mathematics, Vol. 83, Issue 1
DOI: 10.2307/2372726

On the Determination of Reduced Widths from the One-Level Dispersion Formula
journal, January 1951

Thomas, R. G.
Physical Review, Vol. 81, Issue 1
DOI: 10.1103/PhysRev.81.148

On Differential Equations, Volterra Equations and the Function J 2 μ + Y 2 μ
journal, October 1973

Hartman, Phillip
American Journal of Mathematics, Vol. 95, Issue 3
DOI: 10.2307/2373730

Pole structure of the $J^{π}$ ${= 3 / 2}^{+}$ resonance in ${}^{5}{He}$
journal, August 1987

Hale, G. M.; Brown, Ronald E.; Jarmie, Nelson
Physical Review Letters, Vol. 59, Issue 7
DOI: 10.1103/PhysRevLett.59.763

Bessel functions for large arguments
journal, January 1958

Goldstein, M.; Thaler, R. M.
Mathematics of Computation, Vol. 12, Issue 61
DOI: 10.1090/S0025-5718-1958-0102906-3

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Similar Records

Title: Monotonic properties of the shift and penetration factors

Abstract

Citation Formats

Figures / Tables:

Alternative parametrization of R -matrix theory journal, October 2002

On Positivity of Fourier Transforms journal, January 2006

R-Matrix Theory of Nuclear Reactions journal, April 1958

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Theory of the Effective Range in Nuclear Scattering journal, July 1949

Product Formulas and Nicholson-Type Integrals for Jacobi Functions. I: Summary of Results journal, February 1978

The R -matrix theory journal, February 2010

Regular and Irregular Coulomb Wave Functions Expressed in Terms of Bessel-Clifford Functions journal, April 1954

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Completely monotonic functions journal, December 2001

A note on the calculation of Coulomb penetration and Coulomb wave functions journal, April 1957

T matrix and effective range function for Coulomb plus rational separable potentials especially for I =1 journal, May 1977

Higher monotonicity properties of certain Sturm-Liouville functions journal, January 1963

On Differential Equations and the Function $J_\{mu}^2 + Y_\{mu}^2$ journal, January 1961

On the Determination of Reduced Widths from the One-Level Dispersion Formula journal, January 1951

On Differential Equations, Volterra Equations and the Function J 2 μ + Y 2 μ journal, October 1973

Pole structure of the J π = 3 / 2 + resonance in He 5 journal, August 1987

Bessel functions for large arguments journal, January 1958

Alternative parametrization of R -matrix theory
journal, October 2002

On Positivity of Fourier Transforms
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R-Matrix Theory of Nuclear Reactions
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LXXV. Integral representations for products of Whittaker functions
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Some Properties of the Shift and Penetration Factors in Nuclear Reactions
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Theory of the Effective Range in Nuclear Scattering
journal, July 1949

Product Formulas and Nicholson-Type Integrals for Jacobi Functions. I: Summary of Results
journal, February 1978

The R -matrix theory
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Regular and Irregular Coulomb Wave Functions Expressed in Terms of Bessel-Clifford Functions
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Bessel function expansions of Coulomb wave functions
journal, April 1985

Wave Functions for Large Arguments by the Amplitude-Phase Method
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Completely monotonic functions
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A note on the calculation of Coulomb penetration and Coulomb wave functions
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T matrix and effective range function for Coulomb plus rational separable potentials especially for I =1
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Higher monotonicity properties of certain Sturm-Liouville functions
journal, January 1963

On Differential Equations and the Function $J_\{mu}^2 + Y_\{mu}^2$
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On the Determination of Reduced Widths from the One-Level Dispersion Formula
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