# Near/far-side angular decompositions of Legendre polynomials using the amplitude-phase method

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## Abstract

A decomposition of Legendre polynomials into propagating angular waves is derived with the aid of an amplitude-phase method. This decomposition is compared with the ’Nussenzveig/Fuller’ so called near/far-side decomposition of Legendre polynomials. The latter decomposition requires the Legendre function of the second kind. This is not the case with the amplitude-phase decomposition. Both representations have the same asymptotic expressions for large values of \((l+1/2)\sin \theta \), where *l* and \(\theta \) are the polynomial degree and the angle respectively. Furthermore, both components of both representations satisfy the Legendre differential equation. However, we show the two representations are not identical.

## Keywords

Scattering Legendre polynomials Amplitude-phase method Differential cross section Chemical reaction theory## 1 Introduction

Differential cross sections measure angular distributions of nuclear, atomic and molecular scattering events and are expressed in terms of angular eigenfunction expansions, typically involving several, tens or hundreds of terms [1, 2]. Various approximations, like the infinite-order-sudden approximation in rotationally inelastic molecular scattering, may express differential cross sections in terms of Legendre polynomials \(P_{l}(\cos \theta )\), where *l* and \(\theta \) are the (non-negative integer) polynomial degree and the scattering angle respectively. The analysis in terms of single Legendre polynomials may be complicated. Usually semiclassical theories involving quantum interferences of semi-classical trajectories are of great help, but they involve further approximations of the partial-wave expansion [2]. An alternative, yet exact, way is to use a near/far-side decomposition. This approach collects terms in the differential cross sections corresponding to semiclassical contributions originating from repelled and attracted classical trajectories. In this way one has a complementary computational tool explaining interference structures valid beyond the semiclassical view in terms of near(repulsive)/far(attractive) trajectories [3, 4, 5, 6, 7, 8].

The near/far-side analysis of differential cross sections of radially symmetric interactions based on partial-wave expansions uses decompositions of the Legendre polynomials \(P_{l}(\cos \theta )\) into propagating angular wave functions. The propagating angular waves are required to satisfy the Legendre differential equation and also to satisfy semiclassical expressions as \((l+1/2)\sin \theta \rightarrow \infty \) [3, 4, 5, 6, 7, 8]. Advantages of the near/far-side decomposition of a partial-wave expansion of scattering amplitudes have been pointed out by several authors in molecular scattering [3, 4, 5, 6, 7, 8], and in nuclear scattering [9, 10, 11, 12, 13, 14, 15].

An amplitude-phase method for solving second-order ordinary differential equations was recently applied to obtain Legendre functions and associated Legendre functions of the first kind [16], in particular for complex values of the degree *l*. The method provides numerically ’exact’ solutions from analytically exact boundary conditions, in particular for calculating the Legendre polynomials. It expresses any Legendre function in terms of two fundamental amplitude-phase solutions [16]. Fundamental amplitude-phase solutions allow direct exponential representations of Legendre polynomials, without requiring linear combinations with Legendre functions of the second kind. Therefore it suggests an alternative way to separate near/far-side angular contributions in differential cross sections.

The amplitude-phase representation of the Legendre polynomial turns out to be different from the typically used propagating angular waves as defined by Nussenzveig and Fuller [17, 18, 19, 20]. The differences are however not manifested, as shown in this study, under semiclassical conditions where scattering involves significant contributions from large angular momentum quantum numbers.

Section 2 deals with the real-valued fundamental amplitude-phase solutions of the Legendre differential equation and their behaviors as \((l+1/2)\sin \theta \rightarrow \infty \). The particular linear combinations of amplitude-phase solutions representing the Legendre polynomials are presented in Sect. 3, where also the semiclassical expression of the Legendre polynomials is derived. Section 4 deals with the near/far-side components of the Legendre polynomials in terms of the amplitude-phase solutions and in the Nussenzveig/Fuller approach. In Sect. 5 the propagating angular functions of the Nussenzveig/Fuller- and the amplitude-phase types corresponding to Legendre polynomials are compared numerically. Conclusions are in Sect. 6.

## 2 Fundamental amplitude-phase solutions of the Legendre differential equation

*l*is assumed to be a non-negative integer, although the amplitude-phase formulas presented are still valid without this assumption. By introducing a new angular function

*any*solution of the non-linear Milne-type differential equation [16]

### 2.1 Asymptotic behaviors of the fundamental amplitude-phase solutions

*u*by assuming

*u*is sufficiently slowly varying that \(u'\) and \(u''\) can be neglected, leading to

*l*) approximation

*l*approximations:

## 3 Amplitude-phase expressions of Legendre polynomials

*first*kind are given for \(\theta =\pi /2\) (see [22])

*first*kind’ and \(\chi _{\pi /2}^{(1)}\) is shorthand for \(\chi ^{(1)}(\pi /2)\), as for the corresponding derivative \(\chi '^{(1)}(\pi /2)\). The minus sign in Eq. (15) comes from the differential relation \({\text {d}}\cos \theta =-{\text {d}}\sin \theta \).

*l*the amplitude-phase expression (16) is

*l*

### 3.1 Asymptotic expression of Legendre polynomials

*l*it is possible to compare formulas of the present amplitude-phase approach with standard asymptotic expressions for the Legendre polynomials (see e.g. [18, 19] and/or McCabe and Connor [3, 4, 5, 6, 7, 8]), given by

*l*. The two amplitude-phase expressions to compare with are

*l*, and

*l*.

*l*based on the amplitude-phase approach.

## 4 Near/far-side decompositions

### 4.1 The amplitude-phase near/far-side decomposition

*l*with Re \(l>-1/2\). From (16) there is an obvious decomposition of \(P_{l}(\cos \theta )\) in terms of Jost-type propagating angular functions

*l*the amplitude-phase expressions (34) simplifiy to

*l*expressions (34) simplifiy to

*l*. Equation (38) can be written

### 4.2 Nussenzveig/Fuller near/far-side decomposition

## 5 Numerical comparison

The amplitude-phase method provides ways of calculating Legendre functions of the first or second kind in terms of fundamental solutions of Legendre’s differential equation. It is then possible to compare the two near/far-side representations presented in Sect. 4. Hence, Eqs. (39) and (41) are the two (exact) near/far-side representations compared numerically in the present section.

Differences in \(Q^{{(\pm )}}_{l}(\cos \theta )\) and \( P^{{(\pm )}}_{l}(\cos \theta )\) occur only in the imaginary parts. In a series of tables the differences of the imaginary parts can be seen for small, medium and large integer values of *l*, and a sequence of \(\theta \). MatLab (version 10a) was used for the computations and the relative and absolute errors are controlled by the ’tol’ parameter, set to \(2.3\times 10^{-14}\).

Tables 1, 2 and 3 show values of the imaginary parts of \(Q^{{(+)}}_{l}(\cos \theta )\) and \( P^{{(+)}}_{l}(\cos \theta )\). Values of \(Q^{{(-)}}_{l}(\cos \theta )\) and \( P^{{(-)}}_{l}(\cos \theta )\) are obtained by complex conjugate symmetry. The real parts agree to all significant figures.

Table 1 shows results for \(l=3\) and a restricted sequence of angles in the range \(5\le \theta /^{\circ } \le 85\). The polynomials exhibit exact symmetries about \(\theta =90^{\circ }\). The entries for a given angle agree to about 3 decimal positions. Low values of the angular momentum quantum number are relevant in electron- and in other sub-atomic scattering problems. For such values the semiclassical condition \((l+1/2)\sin \theta>>1\) may not be satisfied.

*l*represents typical angular momenta in atomic orbiting and rainbow scattering of hydrogen.

Numerical comparison of the imaginary parts of \(Q^{{(+)}}_{3}(\cos \theta )\) and \( P^{{(+)}}_{3}(\cos \theta )\) for selected values of the angle \(\theta \)

\(\theta /^{\circ }\) | Im \(Q^{{(+)}}_{3}(\cos \theta )\) | Im \(P^{{(+)}}_{3}(\cos \theta )\) |
---|---|---|

5 | \(-\)0.396551939796008 | \(-\)0.396412615667989 |

15 | 0.011252516825386 | 0.011248563378341 |

25 | 0.200897859221705 | 0.200827275985947 |

35 | 0.268338287533512 | 0.268244009850886 |

45 | 0.235275455754521 | 0.235192794331319 |

55 | 0.130350634271783 | 0.130304837021450 |

65 | \(-\)0.006178921957558 | \(-\)0.006176751061825 |

75 | \(-\)0.129825363966732 | \(-\)0.129779751264299 |

85 | \(-\)0.202571714339455 | \(-\)0.202500543013258 |

Numerical comparison of the imaginary parts of \(Q^{{(+)}}_{30}(\cos \theta )\) and \( P^{{(+)}}_{30}(\cos \theta )\) for selected values of the angle \(\theta \)

\(\theta /^{\circ }\) | Im \(Q^{{(+)}}_{30}(\cos \theta )\) | Im \(P^{{(+)}}_{30}(\cos \theta )\) |
---|---|---|

5 | 0.234615351459853 | 0.234615368359349 |

15 | 0.111215921672006 | 0.111215929682961 |

25 | \(-\)0.005817186004943 | \(-\)0.005817186423959 |

35 | \(-\)0.080725653595215 | \(-\)0.080725659409937 |

45 | \(-\)0.079219865834323 | \(-\)0.079219865834323 |

55 | \(-\)0.017049671863151 | \(-\)0.017049673091250 |

65 | 0.051365737103799 | 0.051365740803706 |

75 | 0.072855536558358 | 0.072855541806189 |

85 | 0.033393747988477 | 0.033393750393851 |

Numerical comparison of the imaginary parts of \(Q^{{(+)}}_{300}(\cos \theta )\) and \( P^{{(+)}}_{300}(\cos \theta )\) for selected values of the angle \(\theta \)

\(\theta /^{\circ }\) | Im \(Q^{{(+)}}_{300}(\cos \theta )\) | Im \(P^{{(+)}}_{300}(\cos \theta )\) |
---|---|---|

5 | 0.023085715194327 | 0.023085715194494 |

15 | 0.027593673426331 | 0.027593673426530 |

25 | \(-\)0.035365628346761 | \(-\)0.035365628347017 |

35 | 0.016311824755513 | 0.016311824755631 |

45 | 0.010483837095221 | \(-\)0.235275455754521 |

55 | \(-\)0.024826451939561 | \(-\)0.024826451939741 |

65 | 0.017819818789656 | 0.017819818789785 |

75 | 0.003059009369289 | 0.003059009369311 |

85 | \(-\)0.020452790176916 | \(-\)0.020452790177063 |

Numerical comparison of the imaginary parts of \(Q^{{(+)}}_{l}(\cos \theta )\) and \( P^{{(+)}}_{l}(\cos \theta )\) for the angle \(\theta =10^{-7}\) and two values of *l*

| Im \(Q^{{(+)}}_{l}(\cos \theta )\) | Im \(P^{{(+)}}_{l}(\cos \theta )\) |
---|---|---|

1 | \(-\)6.32146 | \(-\)6.28012 |

10 | \(-\)5.70746 | \(-\)5.70748 |

Table 3 shows results for \(l=300\) and the same angles. The entries for a given angle agree to about 12 decimal positions. Large values of *l* are typical for atomic collisions with large reduced masses compared to the case of hydrogen scattering. Further results (not shown) indicate that the agreements are independent of the parity of *l*.

At forward angles close to \(\theta =0\), and backward angles close to \(\theta =180^{\circ }\), both angular functions diverge with similar numerical agreements depending mainly on the magnitude of *l*. Table 4 shows the numerical values of the imaginary parts of \( Q^{{(+)}}_{l}(\cos \theta )\) and \( P^{{(+)}}_{l}(\cos \theta )\), for \(\theta =10^{-7}{}^{\circ }\) and \(l=1, 10\). At larger values of *l* the agreement improves.

There are problems with the separation of a Legendre polynomial into its natural propagating angular functions at such small values of *l*. Since the asymptotic condition (\((l+1/2)\sin \theta>>1\)) is not satisfied for small values of *l*; how can two near/far-side methods with the same asymptotic condition satisfied be seen as ’exact methods’? Explicitly, for \(l=0\) one has \(P_{0}(\cos \theta )= 1\) and it is not clear how to separate unity into two near/far-side angular waves other than in a symmetric way; for \(l=1\) one has \(P_{1}(\cos \theta )=\cos \theta = \frac{1}{2}\left( {\text {e}}^{{\text {i}}\theta }+{\text {e}}^{-{\text {i}}\theta } \right) \), where the *natural* propagating angular functions (\({\text {e}}^{\pm {\text {i}}\theta }\)) do not satisfy the Legendre equation separately. At present the near/far-side analysis of scattering seems to be an analysis slightly more useful than the semiclassical trajectory method mentioned in the introduction [2].

## 6 Conclusion

A near/far-side decomposition of Legendre polynomials is derived with the aid of an amplitude-phase method. For large values of \((l+1/2)\sin \theta \) expressions of the Legendre polynomial and its near/far-side components are shown to agree with the well-known semiclassical expressions. The ’Nussenzveig/Fuller’ near/far-side components of Legendre polynomials are also known to satisfy the semiclassical expressions. On account of similar numerical and asymptotic (semiclassical) properties, the Legendre polynomials can be represented by both types of complex conjugate pairs of propagating angular functions, (39) and (41), compared in this paper.

However, it is shown that the amplitude-phase and the Nussenzveig/Fuller representations are not identical. The amplitude-phase decomposition uses only functional properties of the Legendre polynomial itself, while the Nussenzveig/Fuller decomposition also uses properties of the corresponding Legendre function of the second kind. Non-negligible numerical differences occur for Legendre polynomials of degrees \(l= 1\) (and to a lesser extent for \(l =\) 2 and 3). Such a difference would possibly affect near/far-side analysis of (sub-)nuclear scattering at low energies, but not near/far-side analysis of non-resonant very low energy heavy-particle scattering.

## Notes

### Acknowledgements

The authors thank Professor JNL Connor for inspiring discussions, and the referees for their careful reading of the original manuscript.

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