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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

After decades of failure to slow the rising global consumption of coal, oil and gas,[1] many countries have proceeded as of 2024 to reconsider nuclear power in order to lower the demand for fossil fuels.[2] Wind and solar power alone, without large-scale storage for these intermittent sources, are unlikely to meet the world's needs for reliable energy.[3][4][5] See Figures 1 and 2 on the magnitude of the world energy challenge.

Nuclear power plants that use nuclear reactors to create electricity could provide the abundant, zero-carbon, dispatchable[6] energy needed for a low-carbon future, but not by simply building more of what we already have. New innovative designs for nuclear reactors are needed to avoid the problems of the past.

(CC) Image: Geoff Russell
Fig.1 Electricity consumption may soon double, mostly from coal-fired power plants in the developing world.[7]

Issues Confronting the Nuclear Industry

New reactor designers have sought to address issues that have prevented the acceptance of nuclear power, including safety, waste management, weapons proliferation, and cost. This article will summarize the questions that have been raised and the criteria that have been established for evaluating these designs. Answers to these questions will be provided by the designers of these reactors in the articles on their designs. Further debate will be provided in the Discussion and the Debate Guide pages of those articles.

Footnotes
  1. ↑ Global Energy Growth by Our World In Data
  2. ↑ Public figures who have reconsidered their stance on nuclear power are listed on the External Links tab of this article.
  3. ↑ Pumped storage is currently the most economical way to store electricity, but it requires a large reservoir on a nearby hill or in an abandoned mine. Li-ion battery systems at $500 per KWh are not practical for utility-scale storage. See Energy Storage for a summary of other alternatives.
  4. ↑ Utilities that include wind and solar power in their grid must have non-intermittent generating capacity (typically fossil fuels) to handle maximum demand for several days. They can save on fuel, but the cost of the plant is the same with or without intermittent sources.
  5. ↑ Mark Jacobson believes that long-distance transmission lines can provide an alternative to costly storage. See the bibliography for more on this proposal and the critique by Christopher Clack.
  6. ↑ "Load following" is the term used by utilities, and is important when there is a lot of wind and solar on the grid. Some reactors are not able to do this.
  7. ↑ Fig.1.3 in Devanney "Why Nuclear Power has been a Flop"

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