[Paper Review] Formal derivation of an exact series expansion for the Principal Field Emission Elliptic Function v

Jonathan H. B. Deane, Richard G. Forbes+1 more|ArXiv.org|Aug 7, 2007

Vacuum and Plasma Arcs9 references3 citations

TL;DR

This paper presents a formal mathematical derivation of an exact series expansion for the Principal Field Emission Elliptic Function $v(l')$, showing it satisfies a novel second-order ordinary differential equation (ODE) with index $n = 3/16$. Using the method of Frobenius and boundary conditions from Cayley's 1876 result, the authors derive a convergent series in powers of $l'$ and $ less l'$, confirming that logarithmic terms are essential while fractional powers are not, providing a rigorous foundation for high-accuracy approximations in cold field electron emission theory.

ABSTRACT

An exact series expansion is now known for the Principal Field Emission Elliptic Function v, in terms of a complementary elliptic variable l' equal to y*y, where y is the Nordheim parameter. This expansion was originally found by using the algebraic manipulation package MAPLE. This paper presents a formal mathematical derivation. It has been discovered that v(l') is a particular solution of the ordinary differential equation (ODE) l'(1-l')d^2v/dl'^2=nv, when the index n = 3/16. This ODE appears to be new in mathematical physics and elliptic-function theory. The paper first uses an 1876 result from Cayley to establish the boundary condition that dv/dl' satisfies as l' tends to zero. It then uses the method of Frobenius to obtain two linearly independent series solutions for the ODE, and hence derives the series expansion for v(l'). It is shown that terms in ln{l'} are required in a mathematically correct solution, but fractional powers of l' are not. The form of the ODE also implies that it is mathematically impossible for simple Taylor expansion methods to generate good approximation formulae valid over the whole range 0 =< l' =< 1; this conclusion may also apply to barriers of other shapes. It is hoped that this derivation might serve as a paradigm for the treatment of other tunnelling barrier models for cold field electron emission, if in any particular case an ODE can be found for which the tunnelling-exponent correction function is a particular solution.

Motivation & Objective

• To provide a formal mathematical derivation of the exact series expansion for the Principal Field Emission Elliptic Function $v(l')$, which was previously obtained numerically using MAPLE.
• To establish that $v(l')$ is a particular solution of a new second-order ODE: $l'(1-l') \frac{d^2v}{dl'^2} = \frac{3}{16}v$, which is novel in mathematical physics and elliptic-function theory.
• To resolve the mathematical correctness of the series by confirming the necessity of $\ln l'$ terms and the absence of fractional powers of $l'$, countering earlier assumptions.
• To demonstrate that standard Taylor expansion methods fail to produce good approximations over the full range $0 \leq l' \leq 1$, due to the ODE's singular nature at $l' = 0$.
• To propose this derivation as a paradigm for analyzing other tunnelling barrier models, provided a governing ODE for the correction function can be identified.

Proposed method

• The authors identify that $v(l')$ satisfies the ODE $l'(1-l') \frac{d^2v}{dl'^2} = \frac{3}{16}v$, which is shown to be a new equation in mathematical physics.
• They use Cayley’s 1876 result to derive the boundary condition for $\frac{dv}{dl'}$ as $l' \to 0$, essential for solving the ODE uniquely.
• The method of Frobenius is applied to solve the ODE, yielding two linearly independent series solutions, from which the specific solution for $v(l')$ is extracted.
• The derivation confirms that the solution contains terms in $\ln l'$, which are mathematically necessary, but no terms in $l'^{1/2}$ or other fractional powers.
• The series is expressed in the form $v(l') = (1-l') \sum a_n l'^n + l'\ln l' \sum b_n l'^n$, ensuring exactness at both $l'=0$ and $l'=1$ when truncated.
• The resulting series is validated by matching known high-precision values and confirming convergence properties for numerical approximation.

Experimental results

Research questions

• RQ1 What is the exact mathematical structure of the Principal Field Emission Elliptic Function $v(l')$ in terms of $l'$, and why is it not expressible as a simple Taylor series?
• RQ2 Does the function $v(l')$ satisfy a second-order linear ODE, and if so, what is its form and significance in mathematical physics?
• RQ3 Why are logarithmic terms $\ln l'$ essential in the series expansion, and why are fractional powers like $l'^{1/2}$ mathematically unjustified?
• RQ4 Can the method of Frobenius be successfully applied to derive the series expansion for $v(l')$ from first principles, given the ODE and boundary condition?
• RQ5 Can this derivation serve as a general framework for analyzing other tunnelling barrier models in cold field electron emission?

Key findings

• The function $v(l')$ is a particular solution of the ODE $l'(1-l') \frac{d^2v}{dl'^2} = \frac{3}{16}v$, which is a novel equation in mathematical physics and elliptic-function theory.
• The series expansion for $v(l')$ contains logarithmic terms $\ln l'$, which are mathematically required for correctness, but does not contain any fractional powers of $l'$ such as $l'^{1/2}$.
• The coefficients of the series were computed explicitly, yielding $v(l') = 1 - 0.96729l' - 0.02330l'^2 - 0.00505l'^3 + \cdots + l'\ln l'(0.18750 + 0.01758l' + 0.00641l'^2 + \cdots)$, with high numerical accuracy.
• The form $v(l') = (1-l')(1 + 0.03271l' + \cdots) + l'\ln l'(0.18750 + \cdots)$ ensures exactness at both $l'=0$ and $l'=1$, enabling highly accurate numerical approximations.
• The derivation confirms that simple Taylor expansion methods cannot yield good approximations over the full interval $0 \leq l' \leq 1$, due to the singular nature of the ODE at $l'=0$, a limitation likely applicable to other barrier shapes.
• The work establishes a formal paradigm for analyzing other tunnelling barrier models, provided a governing ODE for the correction function can be derived and boundary conditions are known.