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The '''[[Dirac delta function]]''' is a function introduced in 1930 by Paul Adrien Maurice Dirac in his seminal book on quantum mechanics. A physical model that visualizes a delta function is a mass distribution of finite total mass ''M''—the integral over the mass distribution.  When the distribution becomes smaller and smaller,  while ''M'' is constant, the mass distribution shrinks to a ''point mass'', which by definition has zero extent and yet has a finite-valued integral equal to total mass ''M''. In the limit of a point mass the distribution becomes a Dirac delta function.
{{:{{FeaturedArticleTitle}}}}
 
<small>
Heuristically, the Dirac delta function can be seen as an extension of the Kronecker delta from integral indices (elements of <font style="vertical-align: 13%"> <math>\mathbb{Z}</math></font>) to real indices (elements of <font style="vertical-align: 13%"><math>\mathbb{R}</math></font>). Note that the Kronecker delta acts as a "filter" in a summation:
==Footnotes==
:<math>
{{reflist|2}}
\sum_{i=m}^n \; f_i\; \delta_{ia} =
</small>
\begin{cases}
f_a & \quad\hbox{if}\quad  a\in[m,n] \sub\mathbb{Z} \\
0  & \quad \hbox{if}\quad a \notin [m,n].
\end{cases}
</math>
 
In analogy, the Dirac delta function &delta;(''x''&minus;''a'')  is defined by (replace ''i'' by ''x'' and the summation over ''i'' by an integration over ''x''),
:<math>
\int_{a_0}^{a_1} f(x)  \delta(x-a) \mathrm{d}x =
\begin{cases}
f(a) & \quad\hbox{if}\quad  a\in[a_0,a_1] \sub\mathbb{R},  \\
0  & \quad \hbox{if}\quad a \notin [a_0,a_1].
\end{cases}
</math>
 
The Dirac delta function is ''not'' an ordinary well-behaved map  <font style="vertical-align: 12%"><math>\mathbb{R} \rightarrow \mathbb{R}</math></font>, but a distribution, also known as an ''improper'' or ''generalized function''. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand ("under the integral"). Mathematicians say that the delta function is a linear functional on a space of test functions.
 
====Properties====
Most commonly one takes the lower and the upper bound in the definition of the delta function equal to <math>-\infty</math> and <math> \infty</math>, respectively. From here on this will be done.
:<math>
\begin{align}
\int_{-\infty}^{\infty} \delta(x)\mathrm{d}x &= 1, \\
\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx} \mathrm{d}k &= \delta(x) \\
\delta(x-a) &= \delta(a-x), \\
(x-a)\delta(x-a) &= 0, \\
\delta(ax) &= |a|^{-1} \delta(x) \quad (a \ne 0), \\
f(x) \delta(x-a) &= f(a) \delta(x-a), \\
\int_{-\infty}^{\infty} \delta(x-y)\delta(y-a)\mathrm{d}y &= \delta(x-a)
\end{align}
</math>
The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus. The delta function as a Fourier transform of the unit function ''f''(''x'') = 1 (the second property) will be proved below.
The last property is the analogy of the multiplication of two identity matrices,
:<math>
\sum_{j=1}^n \;\delta_{ij}\;\delta_{jk} = \delta_{ik}, \quad i,k=1,\ldots, n.
</math>
''[[Dirac delta function|.... (read more)]]''

Latest revision as of 10:19, 11 September 2020

Nuclear weapons proliferation is one of the four big issues that have held back worldwide deployment of peaceful nuclear power. This article will address the proliferation questions raised in Nuclear power reconsidered.

As of 2022, countries with nuclear weapons have followed one or both of two paths in producing fissile materials for nuclear weapons: enrichment of uranium to very high fractions of U-235, or extraction of fissile plutonium (Pu-239) from irradiated uranium nuclear reactor fuel. The US forged the way on both paths during its World War II Manhattan Project. The fundamental aspects of both paths are well understood, but both are technically challenging. Even relatively poor countries can be successful if they have sufficient motivation, financial investment, and, in some cases, direct or illicit assistance from more technologically advanced countries.

The International Non-proliferation Regime

The International Atomic Energy Agency (IAEA) has a vigorous program to prevent additional countries from acquiring nuclear weapons. The Treaty on the Non-Proliferation of Nuclear Weapons (NPT) is the cornerstone arrangement under which strategic rivals can trust, by independent international verification, that their rivals are not developing a nuclear weapons threat. The large expense of weapons programs makes it very unlikely that a country would start its own nuclear weapons program, if it knows that its rivals are not so engaged. With some notable and worrying exceptions, this program has been largely successful.

Paths to the Bomb

It is frequently claimed that building a civil nuclear power program adds to the weapons proliferation risk. There is an overlap in the two distinct technologies, after all. To build a bomb, one needs Highly Enriched Uranium (HEU) or weapons-grade plutonium (Pu-239). Existing reactors running on Low Enriched Uranium (LEU, under 5% U-235) or advanced reactors running on High Assay LEU (HALEU,up to 20% U-235) use the same technology that can enrich uranium to very high levels, but configured differently. Enrichment levels and centrifuge configurations can be monitored using remote cameras, on-site inspections, and installed instrumentation -- hence the value of international inspections by the IAEA. Using commercial power reactors as a weapons plutonium source is an extremely ineffective, slow, expensive, and easily detectable way to produce Pu. Besides the nuclear physics issues, refueling pressurized water reactors is both time-consuming and obvious to outside observers. That is why the US and other countries developed specialized Pu production reactors and/or uranium enrichment to produce fissile cores for nuclear weapons.

Future Threats and Barriers

Minimizing the risk of future proliferation in states that want to buy nuclear reactors or fuel might require one or more barriers:
1) Insisting on full transparency for all nuclear activities in buyer states, including monitoring and inspections by the International Atomic Energy Agency (IAEA).
2) Limiting fuel processing to just a few supplier states that already have weapons or are approved by the IAEA.
3) Ensuring that fuel at any stage after initial fabrication has an isotopic composition unsuitable for weapons. "Spiking" the initial fuel with non-fissile isotopes, if necessary.
4) Limiting the types of reactors deployed to buyer states. In general, breeders are less secure than burners. Sealed reactor modules are more secure than reactors with on-site fuel processing.
5) Providing incentives and assurances for buyer states to go along with all of the above.
6) Application of diplomatic pressure, sanctions, and other economic measures to non-compliant states.
7) Agreement that any reactor declared rogue by the IAEA will be "fair game" for any state feeling threatened.

Footnotes

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