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CANDU reactor

The CANDU (Canada Deuterium Uranium) is a Canadian pressurized heavy-water reactor design used to generate electric power. The acronym refers to its deuterium oxide (heavy water) moderator and its use of (originally, natural) uranium fuel. CANDU reactors were first developed in the late 1950s and 1960s by a partnership between Atomic Energy of Canada Limited (AECL), the Hydro-Electric Power Commission of Ontario, Canadian General Electric, and other companies.

There have been two major types of CANDU reactors, the original design of around 500 MWe that was intended to be used in multi-reactor installations in large plants, and the rationalized CANDU 6 in the 600 MWe class that is designed to be used in single stand-alone units or in small multi-unit plants. CANDU 6 units were built in Quebec and New Brunswick, as well as Pakistan, Argentina, South Korea, Romania, and China. A single example of a non-CANDU 6 design was sold to India. The multi-unit design was used only in Ontario, Canada, and grew in size and power as more units were installed in the province, reaching ~880 MWe in the units installed at the Darlington Nuclear Generating Station. An effort to rationalize the larger units in a fashion similar to CANDU 6 led to the CANDU 9.

By the early 2000s, sales prospects for the original CANDU designs were dwindling due to the introduction of newer designs from other companies. AECL responded by cancelling CANDU 9 development and moving to the Advanced CANDU reactor (ACR) design. ACR failed to find any buyers; its last potential sale was for an expansion at Darlington, but this was cancelled in 2009. In October 2011, the Canadian Federal Government licensed the CANDU design to Candu Energy (a wholly owned subsidiary of SNC-Lavalin, now the AtkinsRéalis Group Inc.), which also acquired the former reactor development and marketing division of AECL at that time. Candu Energy offers support services for existing sites and is completing formerly stalled installations in Romania and Argentina through a partnership with China National Nuclear Corporation. SNC Lavalin, the successor to AECL, is pursuing new CANDU 6 reactor sales in Argentina (Atucha 3), as well as China and Britain. Sales effort for the ACR reactor has ended.

In 2017, a consultation with industry led Natural Resources Canada to establish a "SMR Roadmap" targeting the development of small modular reactors. In response, SNC-Lavalin developed a 300 MWe SMR version of the CANDU, the CANDU SMR, which it began to highlight on its website. In 2020, the CANDU SMR was not selected for further design work for a Canadian demonstration project. SNC-Lavalin is still looking at marketing a 300 MW SMR in part due to projected demand due to climate change mitigation.

Design and operation
CANDU Reactor Schematic.svg are known). • 1. Fuel bundle

• 2. Calandria (reactor core)

• 3. Adjuster rods

• 4. Pressurizer

• 5. Steam generator

• 6. Light-water pump

• 7. Heavy-water pump

• 8. Fueling machines

• 9. Heavy-water moderator

• 10. Pressure tube

• 11. Steam going to steam turbine

• 12. Cold water returning from turbine

• 13. Containment building made of reinforced concrete ]]

The basic operation of the CANDU design is similar to other nuclear reactors. Fission reactions in the reactor core heat pressurized water in a primary cooling loop. A heat exchanger, also known as a steam generator, transfers the heat to a secondary cooling loop, which powers a steam turbine with an electric generator attached to it (for a typical Rankine thermodynamic cycle). The exhaust steam from the turbines is then cooled, condensed and returned as feedwater to the steam generator. The final cooling often uses cooling water from a nearby source, such as a lake, river, or ocean. Newer CANDU plants, such as the Darlington Nuclear Generating Station near Toronto, Ontario, use a diffuser to spread the warm outlet water over a larger volume and limit the effects on the environment. Although all CANDU plants to date have used open-cycle cooling, modern CANDU designs are capable of using cooling towers instead.

Where the CANDU design differs from most other designs is in the details of the fissile core and the primary cooling loop. Natural uranium consists of a mix of mostly uranium-238 with small amounts of uranium-235 and trace amounts of other isotopes. Fission in these elements releases high-energy neutrons, which can cause other 235U atoms in the fuel to undergo fission as well. This process is much more effective when the neutron energies are much lower than what the reactions release naturally. Most reactors use some form of neutron moderator to lower the energy of the neutrons, or "thermalize" them, which makes the reaction more efficient. The energy lost by the neutrons during this moderation process heats the moderator, and this heat is extracted for power.

Most commercial reactor designs use normal water as the moderator. Water absorbs some of the neutrons, enough that it is not possible to keep the reaction going in natural uranium. CANDU replaces this "light" water with heavy water. Heavy water's extra neutron decreases its ability to absorb excess neutrons, resulting in a better neutron economy. This allows CANDU to run on unenriched natural uranium, or uranium mixed with a wide variety of other materials such as plutonium and thorium. This was a major goal of the CANDU design; by operating on natural uranium the cost of enrichment is removed. This also presents an advantage in nuclear proliferation terms, as there is no need for enrichment facilities, which might also be used for weapons.

Calandria and fuel design

In conventional light-water reactor (LWR) designs, the entire fissile core is placed in a large pressure vessel. The amount of heat that can be removed by a unit of a coolant is a function of the temperature; by pressurizing the core, the water can be heated to much greater temperatures before boiling, thereby removing more heat and allowing the core to be smaller and more efficient.

Building a pressure vessel of the required size is a significant challenge, and at the time of the CANDU's design, Canada's heavy industry lacked the requisite experience and capability to cast and machine reactor pressure vessels of the required size. This problem is amplified by natural uranium fuel's lower fissile density, which requires a larger reactor core. This issue was so major that even the relatively small pressure vessel originally intended for use in the NPD prior to its mid-construction redesign could not be fabricated domestically and had to be manufactured in Scotland instead. Domestic development of the technology required to produce pressure vessels of the size required for commercial-scale heavy water moderated power reactors was thought to be very unlikely.

In CANDU the fuel bundles of about 10 cm diameter are composed of many smaller metal tubes. The bundles are contained in pressure tubes within a larger vessel containing additional heavy water acting purely as a moderator. This larger vessel, known as a calandria, is not pressurized and remains at much lower temperatures, making it much easier to fabricate. In order to prevent the heat from the pressure tubes from leaking into the surrounding moderator, each pressure tube is enclosed in a calandria tube. Carbon dioxide gas in the gap between the two tubes acts as an insulator. The moderator tank also acts as a large heat sink that provides an additional safety feature.

In a conventional pressurized water reactor, refuelling the system requires to shut down the core and to open the pressure vessel. In CANDU, only the single tube being refuelled needs to be depressurized. This allows the CANDU system to be continually refuelled without shutting down, another major design goal. In modern systems, two robotic machines attach to the reactor faces and open the end caps of a pressure tube. One machine pushes in the new fuel, whereby the depleted fuel is pushed out and collected at the other end. A significant operational advantage of online refuelling is that a failed or leaking fuel bundle can be removed from the core once it has been located, thus reducing the radiation levels in the primary cooling loop.

Each fuel bundle is a cylinder assembled from thin tubes filled with ceramic pellets of uranium oxide fuel (fuel elements). In older designs, the bundle had 28 or 37 half-meter-long fuel elements with 12–13 such assemblies lying end-to-end in a pressure tube. The newer CANFLEX bundle has 43 fuel elements, with two element sizes (so the power rating can be increased without melting the hottest fuel elements). It is about 10 cm in diameter, 0.5 m long, weighs about 20 kg, and is intended to eventually replace the 37-element bundle. To allow the neutrons to flow freely between the bundles, the tubes and bundles are made of neutron-transparent zircaloy (zirconium + 2.5% wt niobium).

Purpose of using heavy water

Natural uranium is a mix of isotopes: approximately 99.28% uranium-238 and 0.72% uranium-235 by atom fraction. Nuclear power reactors are usually operated at constant power for long periods of time, which requires a constant rate of fission over time. In order to keep the fission rate constant, the neutrons released by fission must produce an equal number of fissions in other fuel atoms. This balance is referred to as "criticality." Neutrons released by nuclear fission are fairly energetic and don't readily get absorbed (or "captured") by the surrounding fissile material. In order to improve the capture rate, the neutron energy must be reduced, or "moderated", to be as low as possible. In practice, the lower energy limit is the energy where the neutrons are in thermal equilibrium with the moderator. When neutrons approach this lower energy limit, they are referred to as "thermal neutrons."

During moderation it helps to separate the neutrons and uranium, since 238U has a large affinity for intermediate-energy neutrons ("resonance" absorption), but is only easily fissioned by the few energetic neutrons above ≈1.5–2 MeV. Since most of the fuel material is usually 238U, most reactor designs are based on thin fuel rods separated by moderator, allowing the neutrons to travel in the moderator before entering the fuel again. More neutrons are released than the minimum needed to maintain the chain reaction; when uranium-238 absorbs neutrons, plutonium is created, which helps to make up for the depletion of uranium-235. Eventually the build-up of fission products that are more neutron-absorbing than 238U slows the reaction and calls for refuelling.

Light water makes an excellent moderator: the light hydrogen atoms are very close in mass to a neutron and can absorb a lot of energy in a single collision (like a collision of two billiard balls). However, light hydrogen can absorb neutrons, reducing the number available to react with the small amount of 235U in natural uranium, preventing criticality. In order to allow criticality, the fuel must be enriched, increasing the amount of 235U to a usable level. In light-water reactors, the fuel is typically enriched to between 2% and 5% 235U (the leftover fraction with less 235U is called depleted uranium). Enrichment facilities are expensive to build and operate. They may also pose a proliferation concern, as they can be used to enrich the 235U much further, up to weapons-grade material (90% or more 235U). This can be remedied if the fuel is supplied and reprocessed by an internationally approved supplier.

The main advantage of heavy water moderator over light water is the reduced absorption of the neutrons that sustain the chain reaction, allowing a lower concentration of fissile atoms (to the point of using unenriched natural uranium fuel). Deuterium ("heavy hydrogen") already has the extra neutron that light hydrogen would absorb, reducing the tendency to capture neutrons. Deuterium has twice the mass of a single neutron (vs light hydrogen, which has about the same mass); the mismatch means that more collisions are needed to moderate the neutrons, requiring a larger thickness of moderator between the fuel rods. This increases the size of the reactor core and the leakage of neutrons. It is also the practical reason for the calandria design, otherwise, a very large pressure vessel would be needed. The low 235U density in natural uranium also implies that less of the fuel will be consumed before the fission rate drops too low to sustain criticality, because the ratio of 235U to fission products + 238U is lower. In CANDU most of the moderator is at lower temperatures than in other designs, reducing the spread of speeds and the overall speed of the moderator particles. This means that most of the neutrons will end up at a lower energy and be more likely to cause fission, so CANDU not only "burns" natural uranium, but it does so more effectively as well. Overall, CANDU reactors use 30–40% less mined uranium than light-water reactors per unit of electricity produced. This is a major advantage of the heavy-water design; it not only requires less fuel, but as the fuel does not have to be enriched, it is much less expensive as well.

A further unique feature of heavy-water moderation is the greater stability of the chain reaction. This is due to the relatively low binding energy of the deuterium nucleus (2.2 MeV), leading to some energetic neutrons and especially gamma rays breaking the deuterium nuclei apart to produce extra neutrons. Both gammas produced directly by fission and by the decay of fission fragments have enough energy, and the half-lives of the fission fragments range from seconds to hours or even years. The slow response of these gamma-generated neutrons delays the response of the reactor and gives the operators extra time in case of an emergency. Since gamma rays travel for meters through water, an increased rate of chain reaction in one part of the reactor will produce a response from the rest of the reactor, allowing various negative feedbacks to stabilize the reaction.

On the other hand, the fission neutrons are thoroughly slowed down before they reach another fuel rod, meaning that it takes neutrons a longer time to get from one part of the reactor to the other. Thus if the chain reaction accelerates in one section of the reactor, the change will propagate itself only slowly to the rest of the core, giving time to respond in an emergency. The independence of the neutrons' energies from the nuclear fuel used is what allows such fuel flexibility in a CANDU reactor, since every fuel bundle will experience the same environment and affect its neighbors in the same way, whether the fissile material is uranium-235, uranium-233 or plutonium.

Canada developed the heavy-water-moderated design in the post–World War II era to explore nuclear energy while lacking access to enrichment facilities. War-era enrichment systems were extremely expensive to build and operate, whereas the heavy water solution allowed the use of natural uranium in the experimental ZEEP reactor. A much less expensive enrichment system was developed, but the United States classified work on the cheaper gas centrifuge process. The CANDU was therefore designed to use natural uranium.

Safety features
The CANDU includes a number of active and passive safety features in its design. Some of these are a side effect of the physical layout of the system.

CANDU designs have a positive void coefficient, as well as a small power coefficient, normally considered bad in reactor design. This implies that steam generated in the coolant will increase the reaction rate, which in turn would generate more steam. This is one of the many reasons for the cooler mass of moderator in the calandria, as even a serious steam incident in the core would not have a major impact on the overall moderation cycle. Only if the moderator itself starts to boil would there be any significant effect, and the large thermal mass ensures that this will occur slowly. The deliberately "sluggish" response of the fission process in CANDU allows controllers more time to diagnose and deal with problems.

The fuel channels can only maintain criticality if they are mechanically sound. If the temperature of the fuel bundles increases to the point where they are mechanically unstable, their horizontal layout means that they will bend under gravity, shifting the layout of the bundles and reducing the efficiency of the reactions. Because the original fuel arrangement is optimal for a chain reaction, and the natural uranium fuel has little excess reactivity, any significant deformation will stop the inter-fuel pellet fission reaction. This will not stop heat production from fission product decay, which would continue to supply a considerable heat output. If this process further weakens the fuel bundles, the pressure tube they are in will eventually bend far enough to touch the calandria tube, allowing heat to be efficiently transferred into the moderator tank. The moderator vessel has a considerable thermal capability on its own and is normally kept relatively cool.

Heat generated by fission products would initially be at about 7% of full reactor power, which requires significant cooling. The CANDU designs have several emergency cooling systems, as well as having limited self-pumping capability through thermal means (the steam generator is well above the reactor). Even in the event of a catastrophic accident and core meltdown, the fuel is not critical in light water. This means that cooling the core with water from nearby sources will not add to the reactivity of the fuel mass.

Normally the rate of fission is controlled by light-water compartments called liquid zone controllers, which absorb excess neutrons, and by adjuster rods, which can be raised or lowered in the core to control the neutron flux. These are used for normal operation, allowing the controllers to adjust reactivity across the fuel mass, as different portions would normally burn at different rates depending on their position. The adjuster rods can also be used to slow or stop criticality. Because these rods are inserted into the low-pressure calandria, not the high-pressure fuel tubes, they would not be "ejected" by steam, a design issue for many pressurized-water reactors.

There are two independent, fast-acting safety shutdown systems as well. Shutoff rods are held above the reactor by electromagnets and drop under gravity into the core to quickly end criticality. This system works even in the event of a complete power failure, as the electromagnets only hold the rods out of the reactor when power is available. A secondary system injects a high-pressure gadolinium nitrate neutron absorber solution into the calandria.

Fuel cycle

A heavy-water design can sustain a chain reaction with a lower concentration of fissile atoms than light-water reactors, allowing it to use some alternative fuels; for example, "recovered uranium" (RU) from used LWR fuel. CANDU was designed for natural uranium with only 0.7% 235U, so reprocessed uranium with 0.9% 235U is a comparatively rich fuel. This extracts a further 30–40% energy from the uranium. The Qinshan CANDU reactor in China has used recovered uranium. The DUPIC (Direct Use of spent PWR fuel in CANDU) process under development can recycle it even without reprocessing. The fuel is sintered in air (oxidized), then in hydrogen (reduced) to break it into a powder, which is then formed into CANDU fuel pellets.

CANDU reactors can also breed fuel from the more abundant thorium. This is being investigated by India to take advantage of its natural thorium reserves.

Even better than LWRs, CANDU can utilize a mix of uranium and plutonium oxides (MOX fuel), the plutonium either from dismantled nuclear weapons or reprocessed reactor fuel. The mix of isotopes in reprocessed plutonium is not attractive for weapons, but can be used as fuel (instead of being simply nuclear waste), while consuming weapons-grade plutonium eliminates a proliferation hazard. If the aim is explicitly to utilize plutonium or other actinides from spent fuel, then special inert-matrix fuels are proposed to do this more efficiently than MOX. Since they contain no uranium, these fuels do not breed any extra plutonium.

Economics
The neutron economy of heavy-water moderation and precise control of on-line refueling allow CANDU to use a wide range of fuels other than enriched uranium, e.g., natural uranium, reprocessed uranium, thorium, plutonium, and used LWR fuel. Given the expense of enrichment, this can make fuel much cheaper. There is an initial investment into the tonnes of 99.75% pure heavy water to fill the core and heat-transfer system. In the case of the Darlington plant, costs released as part of a freedom of information act request put the overnight cost of the plant (four reactors totalling 3,512 MWe net capacity) at $5.117 billion CAD (about US$4.2 billion at early-1990s exchange rates). Total capital costs including interest were $14.319 billion CAD (about US$11.9 billion) with the heavy water accounting for $1.528 billion, or 11%, of this.

Since heavy water is less efficient than light water at slowing neutrons, CANDU needs a larger moderator-to-fuel ratio and a larger core for the same power output. Although a calandria-based core is cheaper to build, its size increases the cost for standard features like the containment building. Generally nuclear plant construction and operations are ≈65% of overall lifetime cost; for CANDU, costs are dominated by construction even more. Fueling CANDU is cheaper than other reactors, costing only ≈10% of the total, so the overall price per kWh electricity is comparable. The next-generation Advanced CANDU reactor (ACR) mitigates these disadvantages by having light-water coolant and using a more compact core with less moderator.

When first introduced, CANDUs offered much better capacity factor (ratio of power generated to what would be generated by running at full power, 100% of the time) than LWRs of a similar generation. The light-water designs spent, on average, about half the time being refueled or maintained. Since the 1980s, dramatic improvements in LWR outage management have narrowed the gap, with several units achieving capacity factors ~90% and higher, with an overall US fleet performance of 92% in 2010. The latest-generation CANDU 6 reactors have an 88–90% CF, but overall performance is dominated by the older Canadian units with CFs on the order of 80%. Refurbished units had historically demonstrated poor performance, on the order of 65%. This has since improved with the return of Bruce units A1 and A2 to operation, which have post-refurbishment (2013+) capacity factors of 90.78% and 90.38%, respectively.

Some CANDU plants suffered from cost overruns during construction, often from external factors such as government action. For instance, a number of imposed construction delays led to roughly a doubling of the cost of the Darlington Nuclear Generating Station near Toronto, Ontario. Technical problems and redesigns added about another billion to the resulting $14.4 billion price. In contrast, in 2002 two CANDU 6 reactors at Qinshan in China were completed on-schedule and on-budget, an achievement attributed to tight control over scope and schedule.

Nuclear nonproliferation
In terms of safeguards against nuclear weapons proliferation, CANDUs meet a similar level of international certification as other reactors. The plutonium for India's first nuclear detonation, Operation Smiling Buddha in 1974, was produced in a CIRUS reactor supplied by Canada and partially paid for by the Canadian government using heavy water supplied by the United States. In addition to its two PHWR reactors, India has some safeguarded pressurised heavy-water reactors (PHWRs) based on the CANDU design, and two safeguarded light-water reactors supplied by the US. Plutonium has been extracted from the spent fuel from all of these reactors; India mainly relies on an Indian designed and built military reactor called Dhruva. The design is believed to be derived from the CIRUS reactor, with the Dhruva being scaled-up for more efficient plutonium production. It is this reactor which is thought to have produced the plutonium for India's more recent (1998) Operation Shakti nuclear tests.

Although heavy water is relatively immune to neutron capture, a small amount of the deuterium turns into tritium in this way. This tritium is extracted from some CANDU plants in Canada, mainly to improve safety in case of heavy-water leakage. The gas is stockpiled and used in a variety of commercial products, notably "powerless" lighting systems and medical devices. In 1985 what was then Ontario Hydro sparked controversy in Ontario due to its plans to sell tritium to the United States. The plan, by law, involved sales to non-military applications only, but some speculated that the exports could have freed American tritium for the United States nuclear weapons program. Future demands appear to outstrip production, in particular the demands of future generations of experimental fusion reactors like ITER, with up to 10kg of tritium being required in order to start up a fusion reactor and so dozens of kilograms being required for a fleet. Between 1.5 to 2.1 kg of tritium were recovered annually at the Darlington separation facility by 2003, of which a minor fraction was sold. Consequently, the Canadian Nuclear Laboratories in 2024 announced a decades-long program to refurbish existing CANDU plants and equip them with tritium breeding facilities.

The 1998 Operation Shakti test series in India included one bomb of about 45 ktonTNT yield that India has publicly claimed was a hydrogen bomb. An offhand comment in the BARC publication Heavy Water – Properties, Production and Analysis appears to suggest that the tritium was extracted from the heavy water in the CANDU and PHWR reactors in commercial operation. Janes Intelligence Review quotes the Chairman of the Indian Atomic Energy Commission as admitting to the tritium extraction plant, but refusing to comment on its use. India is also capable of creating tritium more efficiently by irradiation of lithium-6 in reactors.

Tritium production
Tritium, 3H, is a radioactive isotope of hydrogen, with a half-life of 12.3 years. It is produced in small amounts in nature (about 4 kg per year globally) by cosmic ray interactions in the upper atmosphere. Tritium is considered a weak radionuclide because of its low-energy radioactive emissions (beta particle energy up to 18.6 keV). The beta particles travel 6 mm in air and only penetrate skin up to 6 micrometers. The biological half-life of inhaled, ingested, or absorbed tritium is 10–12 days.

Tritium is generated in the fuel of all reactors; CANDU reactors generate tritium also in their coolant and moderator, due to neutron capture in heavy hydrogen. Some of this tritium escapes into containment and is generally recovered; a small percentage (about 1%) escapes containment and is considered a routine radioactive emission (also higher than from an LWR of comparable size). Responsible operation of a CANDU plant therefore includes monitoring tritium in the surrounding environment (and publishing the results).

In some CANDU reactors the tritium is periodically extracted. Typical emissions from CANDU plants in Canada are less than 1% of the national regulatory limit, which is based on International Commission on Radiological Protection (ICRP) guidelines (for example, the maximal permitted drinking-water concentration for tritium in Canada, 7,000 Bq/L, corresponds to 1/10 of the ICRP's dose limit for members of the public). Tritium emissions from other CANDU plants are similarly low.

In general, there is significant public controversy about radioactive emissions from nuclear power plants, and for CANDU plants one of the main concerns is tritium. In 2007 Greenpeace published a critique of tritium emissions from Canadian nuclear power plants by Ian Fairlie. This report was criticized by Richard Osborne.

History
The CANDU development effort has gone through four major stages over time. The first systems were experimental and prototype machines of limited power. These were replaced by a second generation of machines of 500 to 600 MWe (the CANDU 6), a series of larger machines of 900 MWe, and finally developing into the CANDU 9 and ACR-1000 effort.

Early efforts
The first heavy-water-moderated design in Canada was the ZEEP, which started operation just after the end of World War II. ZEEP was joined by several other experimental machines, including the NRX in 1947 and NRU in 1957. These efforts led to the first CANDU-type reactor, the Nuclear Power Demonstration (NPD), in Rolphton, Ontario. It was intended as a proof-of-concept and rated for only 22 MWe, a very low power for a commercial power reactor. NPD produced the first nuclear-generated electricity in Canada and ran successfully from 1962 to 1987.

The second CANDU was the Douglas Point reactor, a more powerful version rated at roughly 200 MWe and located near Kincardine, Ontario. It went into service in 1968 and ran until 1984. Uniquely among CANDU stations, Douglas Point had an oil-filled window with a view of the east reactor face, even when the reactor was operating. Douglas Point was originally planned to be a two-unit station, but the second unit was cancelled because of the success of the larger 515 MWe units at Pickering.

Gentilly-1, in Bécancour, Quebec, near Trois-Rivières, Quebec, was also an experimental version of CANDU, using a boiling light-water coolant and vertical pressure tubes, but was not considered successful and closed after seven years of fitful operation. Gentilly-2, a CANDU-6 reactor, began operating in 1983. Following statements from the in-coming Parti Québécois government in September 2012 that Gentilly would close, the operator, Hydro-Québec, decided to cancel a previously announced refurbishment of the plant and announced its shutdown at the end of 2012, citing economic reasons for the decision. The company has started a 50-year decommissioning process estimated to cost $1.8 billion.

In parallel with the classic CANDU design, experimental variants were being developed. WR-1, located at the AECL's Whiteshell Laboratories in Pinawa, Manitoba, used vertical pressure tubes and organic oil as the primary coolant. The oil used has a higher boiling point than water, allowing the reactor to operate at higher temperatures and lower pressures than a conventional reactor. WR-1's outlet temperature was about 490 °C compared to the CANDU 6's nominal 310 °C; the higher temperature and thus thermodynamic efficiency offsets to some degree the fact that oils have about half the heat capacity of water. The higher temperatures also result in more efficient conversion to steam, and ultimately, electricity. WR-1 operated successfully for many years and promised a significantly higher efficiency than water-cooled versions.

600 MWe designs
The successes at NPD and Douglas Point led to the decision to construct the first multi-unit station in Pickering, Ontario. Pickering A, consisting of Units 1 to 4, went into service in 1971. Pickering B with units 5 to 8 came online in 1983, giving a full-station capacity of 4,120 MWe. The station is very close to the city of Toronto, in order to reduce transmission costs.

A series of improvements to the basic Pickering design led to the CANDU 6 design, which first went into operation in the early 1980s. CANDU 6 was essentially a version of the Pickering power plant that was redesigned to be able to be built in single-reactor units. CANDU 6 was used in several installations outside Ontario, including the Gentilly-2 in Quebec, and Point Lepreau Nuclear Generating Station in New Brunswick. CANDU 6 forms the majority of foreign CANDU systems, including the designs exported to Argentina, Romania, China and South Korea. Only India operates a CANDU system that is not based on the CANDU 6 design.

900 MWe designs
The economics of nuclear power plants generally scale well with size. This improvement at larger sizes is offset by the sudden appearance of large quantities of power on the grid, which leads to a lowering of electricity prices through supply and demand effects. Predictions in the late 1960s suggested that growth in electricity demand would overwhelm these downward pricing pressures, leading most designers to introduce plants in the 1000 MWe range.

Pickering A was quickly followed by such an upscaling effort for the Bruce Nuclear Generating Station, constructed in stages between 1970 and 1987. It is the largest nuclear facility in North America and second largest in the world (after Kashiwazaki-Kariwa in Japan), with eight reactors at around 800 MWe each, in total 6,232 MW (net) and 7,276 MW (gross). Another, smaller, upscaling led to the Darlington Nuclear Generating Station design, similar to the Bruce plant, but delivering about 880 MWe per reactor in a four-reactor station.

As was the case for the development of the Pickering design into the CANDU 6, the Bruce design was also developed into the similar CANDU 9. Like the CANDU 6, the CANDU 9 is essentially a repackaging of the Bruce design, so that it can be built as a single-reactor unit. No CANDU 9 reactors have been built.

Generation III+ designs
Through the 1980s and 1990s the nuclear power market suffered a major crash, with few new plants being constructed in North America or Europe. Design work continued throughout, and new design concepts were introduced that dramatically improved safety, capital costs, economics and overall performance. These generation III+ and generation IV machines became a topic of considerable interest in the early 2000s, as it appeared that a nuclear renaissance was underway and large numbers of new reactors would be built over the next decade.

AECL had been working on a design known as the ACR-700, using elements of the latest versions of the CANDU 6 and CANDU 9, with a design power of 700 MWe. During the nuclear renaissance, the upscaling seen in the earlier years re-expressed itself, and the ACR-700 was developed into the 1200 MWe ACR-1000. ACR-1000 is the next-generation (officially, "generation III+") CANDU technology, which makes some significant modifications to the existing CANDU design.

The main change, and the most radical among the CANDU generations, is the use of pressurized light water as the coolant. This significantly reduces the cost of implementing the primary cooling loop, which no longer has to be filled with expensive heavy water. The ACR-1000 uses about 1/3rd the heavy water needed in earlier-generation designs. It also eliminates tritium production in the coolant loop, the major source of tritium leaks in operational CANDU designs. The redesign also allows a slightly negative void reactivity, a major design goal of all Gen III+ machines.

The design also requires the use of slightly enriched uranium, enriched by about 1 or 2%. The main reason for this is to increase the burn-up ratio, allowing bundles to remain in the reactor longer, so that only a third as much spent fuel is produced. This also has effects on operational costs and timetables, as the refuelling frequency is reduced. As is the case with earlier CANDU designs, the ACR-1000 also offers online refuelling.

Outside of the reactor, the ACR-1000 has a number of design changes that are expected to dramatically lower capital and operational costs. Primary among these changes is the design lifetime of 60 years, which dramatically lowers the price of the electricity generated over the lifetime of the plant. The design also has an expected capacity factor of 90%. Higher-pressure steam generators and turbines improve efficiency downstream of the reactor.

Many of the operational design changes were also applied to the existing CANDU 6 to produce the Enhanced CANDU 6. Also known as CANDU 6e or EC 6, this was an evolutionary upgrade of the CANDU 6 design with a gross output of 740 MWe per unit. The reactors are designed with a lifetime of over 50 years, with a mid-life program to replace some of the key components e.g. the fuel channels. The projected average annual capacity factor is more than 90%. Improvements to construction techniques (including modular, open-top assembly) decrease construction costs. The CANDU 6e is designed to operate at power settings as low as 50%, allowing them to adjust to load demand much better than the previous designs.

Sales efforts in Canada
By most measures, the CANDU is "the Ontario reactor". The system was developed almost entirely in Ontario, and only two experimental designs were built in other provinces. Of the 29 commercial CANDU reactors built, 22 are in Ontario. Of these 22, a number of reactors have been removed from service. Two new CANDU reactors have been proposed for Darlington with Canadian government help with financing, but these plans ended in 2009 due to high costs.

AECL has heavily marketed CANDU within Canada, but has found a limited reception. To date, only two non-experimental reactors have been built in other provinces, one each in Quebec and New Brunswick, other provinces have concentrated on hydro and coal-fired plants. Several Canadian provinces have developed large amounts of hydro power. Alberta and Saskatchewan do not have extensive hydro resources, and use mainly fossil fuels to generate electric power.

Interest has been expressed in Western Canada, where CANDU reactors are being considered as heat and electricity sources for the energy-intensive oil sands extraction process, which currently uses natural gas. Energy Alberta Corporation announced 27 August 2007 that they had applied for a licence to build a new nuclear plant at Lac Cardinal (30 km west of the town of Peace River, Alberta), with two ACR-1000 reactors going online in 2017 producing 2.2 gigawatts (electric). A 2007 parliamentary review suggested placing the development efforts on hold. The company was later purchased by Bruce Power, who proposed expanding the plant to four units of a total 4.4 gigawatts. These plans were upset and Bruce later withdrew its application for the Lac Cardinal, proposing instead a new site about 60 km away. The plans are currently moribund after a wide consultation with the public demonstrated that while about $1/undefined$ of the population were open to reactors, $1/undefined$ were opposed.

Foreign sales
During the 1970s, the international nuclear sales market was extremely competitive, with many national nuclear companies being supported by their governments' foreign embassies. In addition, the pace of construction in the United States had meant that cost overruns and delayed completion was generally over, and subsequent reactors would be cheaper. Canada, a relatively new player on the international market, had numerous disadvantages in these efforts. The CANDU was deliberately designed to reduce the need for very large machined parts, making it suitable for construction by countries without a major industrial base. Sales efforts have had their most success in countries that could not locally build designs from other firms.

In the late 1970s, AECL noted that each reactor sale would employ 3,600 Canadians and result in $300 million in balance-of-payments income. These sales efforts were aimed primarily at countries being run by dictatorships or similar, a fact that led to serious concerns in parliament. These efforts also led to a scandal when it was discovered millions of dollars had been given to foreign sales agents, with little or no record of who they were, or what they did to earn the money. This led to a Royal Canadian Mounted Police investigation after questions were raised about sales efforts in Argentina, and new regulations on full disclosure of fees for future sales.

CANDU's first success was the sale of early CANDU designs to India. In 1963, an agreement was signed for export of a 200 MWe power reactor based on the Douglas Point reactor. The success of the deal led to the 1966 sale of a second reactor of the same design. The first reactor, then known as RAPP-1 for "Rajasthan Atomic Power Project", began operation in 1972. A serious problem with cracking of the reactor's end shield led to the reactor being shut down for long periods, and the reactor was finally downrated to 100 MW. Construction of the RAPP-2 reactor was still underway when India detonated its first atomic bomb in 1974, leading to Canada ending nuclear dealings with the country. Part of the sales agreement was a technology transfer process. When Canada withdrew from development, India continued construction of CANDU-like plants across the country. By 2010, CANDU-based reactors were operational at the following sites: Kaiga (3), Kakrapar (2), Madras (2), Narora (2), Rajasthan (6), and Tarapur (2).

In Pakistan, the Karachi Nuclear Power Plant with a gross capacity of 137 MWe was built between 1966 and 1971.

In 1972, AECL submitted a design based on the Pickering plant to Argentina's Comision Nacional de Energia Atomica process, in partnership with the Italian company Italimpianti. High inflation during construction led to massive losses, and efforts to re-negotiate the deal were interrupted by the March 1976 coup led by General Videla. The Embalse Nuclear Power Station began commercial operation in January 1984. There have been ongoing negotiations to open more CANDU 6 reactors in the country, including a 2007 deal between Canada, China and Argentina, but to date no firm plans have been announced.

A licensing agreement with Romania was signed in 1977, selling the CANDU 6 design for $5 million per reactor for the first four reactors, and then $2 million each for the next twelve. In addition, Canadian companies would supply a varying amount of equipment for the reactors, about $100 million of the first reactor's $800 million price tag, and then falling over time. In 1980, Nicolae Ceaușescu asked for a modification to provide goods instead of cash, in exchange the amount of Canadian content was increased and a second reactor would be built with Canadian help. Economic troubles in the country worsened throughout the construction phase. The first reactor of the Cernavodă Nuclear Power Plant only came online in April 1996, a decade after its December 1985 predicted startup. Further loans were arranged for completion of the second reactor, which went online in November 2007.

In January 1975, a deal was announced for a single CANDU 6 reactor to be built in South Korea, now known as the Wolsong-1 Power Reactor. Construction started in 1977 and commercial operation began in April 1983. In December 1990 a further deal was announced for three additional units at the same site, which began operation in the period 1997–1999. South Korea also negotiated development and technology transfer deals with Westinghouse for their advanced System-80 reactor design, and all future development is based on locally built versions of this reactor.

In June 1998, construction started on a CANDU 6 reactor in Qinshan China Qinshan Nuclear Power Plant, as Phase III (units 4 and 5) of the planned 11 unit facility. Commercial operation began in December 2002 and July 2003, respectively. These are the first heavy water reactors in China. Qinshan is the first CANDU-6 project to use open-top reactor building construction, and the first project where commercial operation began earlier than the projected date.

CANDU Energy is continuing marketing efforts in China. In addition, China and Argentina have agreed a contract to build a 700 MWe CANDU-6 derived reactor. Construction is planned to start in 2018 at Atucha.

Economic performance
The cost of electricity from any power plant can be calculated by roughly the same selection of factors: capital costs for construction or the payments on loans made to secure that capital, the cost of fuel on a per-watt-hour basis, and fixed and variable maintenance fees. In the case of nuclear power, one normally includes two additional costs, the cost of permanent waste disposal, and the cost of decommissioning the plant when its useful lifetime is over. Generally, the capital costs dominate the price of nuclear power, as the amount of power produced is so large that it overwhelms the cost of fuel and maintenance. The World Nuclear Association calculates that the cost of fuel, including all processing, accounts for less than one cent (US$0.01) per kWh.

Information on economic performance on CANDU is somewhat lopsided; the majority of reactors are in Ontario, which is also the "most public" among the major CANDU operators. Several anti-nuclear organizations like the Ontario Clean Air Alliance (OCAA) and Pembina have claimed that every CANDU design in Ontario went over budget by at least 25%, and average over 150% higher than estimated. However, this is predicated on using "dollar of the day" figures that are not adjusted for inflation. With inflation accounted for, all plants were on or under budget with the exception of Darlington. Even accounting for inflation, Darlington went far over budget, at almost double the original estimate, but this project was stopped in-progress thereby incurring additional interest charges during a period of high interest rates, which is a special situation that was not expected to repeat itself.

In the 1980s, the pressure tubes in the Pickering A reactors were replaced ahead of design life due to unexpected deterioration caused by hydrogen embrittlement. Extensive inspection and maintenance has avoided this problem in later reactors.

All the Pickering A and Bruce A reactors were shut down in 1999 in order to focus on restoring operational performance in the later generations at Pickering, Bruce, and Darlington. Before restarting the Pickering A reactors, OPG undertook a limited refurbishment program. The original cost and time estimates based on inadequate project scope development were greatly below the actual time and cost and it was determined that Pickering units 2 and 3 would not be restarted for commercial reasons.

These overruns were repeated at Bruce, with Units 3 and 4 running 90% over budget. Similar overruns were experienced at Point Lepreau, and Gentilly-2 plant was shut down on 28 December 2012.

Based on the projected capital costs, and the low cost of fuel and in-service maintenance, in 1994 power from CANDU was predicted to be well under 5 cents/kWh.

In 1999, Ontario Hydro was broken up and its generation facilities re-formed into Ontario Power Generation (OPG). In order to make the successor companies more attractive for private investors, $19.4 billion in "stranded debt" was placed in the control of the Ontario Electricity Financial Corporation. This debt is slowly paid down through a variety of sources, including a 0.7-cent/kWh tariff on all power, all income taxes paid by all operating companies, and all dividends paid by the OPG and Hydro One.

As of October 2022, Darlington is into the final half of the 10-year major refurbishment project of all four units, having reached their design mid-life. The budget is set at $12.5 billion, and planned to produce power at 6 to 8 cents/kWh. The project is currently on-time and on-budget.

Darlington Units 1, 3 and 4 have operated with an average lifetime annual capacity factor of 85% and Unit 2 with a capacity factor of 78%, As of 2010, refurbished units at Pickering and Bruce had lifetime capacity factors between 59 and 69%. This includes periods of several years while the units were shut down for the retubing and refurbishing. Post-refurbishment capacity factors are much higher with Bruce A1 at 90.78%, Bruce A2 at 90.38% (2013+), Pickering A1 at 71.18% and Pickering A4 at 70.38%. In 2009, Bruce A units 3 and 4 had capacity factors of 80.5% and 76.7% respectively, in a year when they had a major Vacuum Building outage.

Active CANDU reactors
Today there are 31 CANDU reactors in use around the world, and 13 "CANDU-derivatives" in India, developed from the CANDU design. After India detonated a nuclear bomb in 1974, Canada stopped nuclear dealings with India. The breakdown is:
 * Canada: 19 and 5 decommissioned.
 * South Korea: 3, and 1 shutdown.
 * China: 2
 * India: 2, 17 active CANDU-derivatives, and 3 CANDU-derivatives under construction.
 * Argentina: 1
 * Romania: 2, and 3 dormant part-constructed.
 * Pakistan: 1 shutdown.

title: CANDU reactor


text:
Cuitláhuac

Cuitláhuac (c. 1476 – 1520)  or Cuitláhuac (in Spanish orthography; Cuitlāhuac,, honorific form: Cuitlahuatzin) was the 10th Huey Tlatoani (emperor) of the Aztec city of Tenochtitlan for 80 days during the year Two Flint (1520). He is credited with leading the resistance to the Spanish and Tlaxcalteca conquest of the Mexica Empire, following the death of his kinsman Moctezuma II.

Biography
Cuitláhuac was the eleventh son of the ruler Axayacatl and a younger brother of Moctezuma II, the late Emperor of Tenochtitlan, who died during the Spanish occupation of the city. His mother's father, also called Cuitlahuac, had been ruler of Iztapalapa, and the younger Cuitláhuac also ruled there initially. Cuitláhuac was an experienced warrior and an adviser to Moctezuma, warning him not to allow the Spaniards to enter Tenochtitlan. Hernán Cortés imprisoned both Moctezuma and Cuitláhuac. Following the massacre of Aztec elites when Cortés was away from Tenochtitlan, the Mexica besieged the Spanish and their indigenous allies. Cuitláhuac was released on the pretense to reopen the market to get food to the invaders. Moctezuma was killed under disputed circumstances, and Cuitláhuac was elected tlatoani following the flight of the Spaniards and their allies from Tenochtitlan on June 30, 1520. Some sources claim he was serving in that role even before Moctezuma's death.

Cuitláhuac was ritually married to Moctezuma's eldest daughter, a ten- or eleven-year-old girl, who later was called Isabel Moctezuma.

Cuitláhuac ruled just 80 days, perhaps dying from smallpox that had been introduced to the New World by an African suffering from the disease who was part of Pánfilo de Narváez's expedition to capture Cortés. The early sources do not explicitly say from what he succumbed. Immediately after Cuitláhuac's death, Cuauhtémoc was made the next tlatoani.

Legacy
The modern Mexican municipality of Cuitláhuac, Veracruz and the Mexico City Metro station Metro Cuitláhuac are named in honor of Cuitláhuac. The asteroid 2275 Cuitláhuac is also named after this ruler.

There is an Avenue in Mexico City Called Cuitláhuac (Eje 3 Norte) that runs from Avenue Insurgentes to Avenue Mexico-Tacuba and that is part of an inner ring; also many streets in other towns and villages in Mexico are so called.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Cuitláhuac


text:
Cuauhtémoc

Cuauhtémoc, also known as Cuauhtemotzín, Guatimozín, or Guatémoc, was the Aztec ruler (tlatoani) of Tenochtitlan from 1520 to 1521, making him the last Aztec Emperor. The name Cuauhtemōc means "one who has descended like an eagle", and is commonly rendered in English as "Descending Eagle", as in the moment when an eagle folds its wings and plummets down to strike its prey. This is a name that implies aggressiveness and determination.

Cuauhtémoc took power in 1520 as successor of Cuitláhuac and was a cousin of the late emperor Moctezuma II. His young wife, who was later known as Isabel Moctezuma, was one of Moctezuma's daughters. He ascended to the throne when he was around 25 years old, while Tenochtitlan was being besieged by the Spanish and devastated by an epidemic of smallpox brought to the Americas by Spanish conquerors. After the killings in the Great Temple, there were probably few Aztec captains available to take the position.

Early life
Cuauhtemoc's date of birth is unknown, as he does not enter the historical record until he became emperor. He was the eldest legitimate son of Emperor Ahuitzotl and may well have attended the last New Fire ceremony, marking the beginning of a new 52-year cycle in the Aztec calendar. According to several sources his mother, Tiyacapantzin, was a Tlatelolcan princess. Like the rest of Cuauhtemoc's early biography, that is inferred from knowledge of his age, and the likely events and life path of someone of his rank. Following education in the calmecac, the school for elite boys, and then his military service, he was named ruler of Tlatelolco, with the title cuauhtlatoani ("eagle ruler") in 1515. To have reached this position of rulership, Cuauhtemoc had to be a male of high birth and a warrior who had captured enemies for sacrifice. Cuauhtemoc married the Aztec princess who later became known as Isabel Moctezuma.

Rule
When Cuauhtemoc was elected tlatoani in 1520, Tenochtitlan had already been rocked by the invasion of the Spanish and their indigenous allies, the death of Moctezuma II, and the death of Moctezuma's brother Cuitlahuac, who succeeded him as ruler, but died of smallpox shortly afterwards. In keeping with traditional practice, the most able candidate among the high noblemen was chosen by vote of the highest noblemen, and Cuauhtemoc assumed the rulership. Although under Cuitlahuac Tenochtitlan began mounting a defense against the invaders, it was increasingly isolated militarily and largely faced the crisis alone, as the numbers of Spanish allies increased with the desertion of many polities previously under its control.

Cuauhtémoc called for reinforcements from the countryside to aid the defense of Tenochtitlán, after eighty days of warfare against the Spanish. Of all the Nahuas, only Tlatelolcas remained loyal, and the surviving Tenochcas looked for refuge in Tlatelolco, where even women took part in the battle. Cuauhtémoc was captured on August 13, 1521, while fleeing Tenochtitlán by crossing Lake Texcoco with his wife, family, and friends.

He surrendered to Hernán Cortés along with the surviving pipiltin (nobles) and, according to Spanish sources, he asked Cortés to take his knife and "strike me dead immediately". According to the same Spanish accounts, Cortés refused the offer and treated his foe magnanimously. "You have defended your capital like a brave warrior," he declared. "A Spaniard knows how to respect valor, even in an enemy."

At Cuauhtémoc's request, Cortés also allowed the defeated Mexica to depart the city unmolested. Subsequently, however, when the booty found did not measure up to the Spaniards' expectations, Cuauhtémoc was subjected to "torture by fire", whereby the soles of his bare feet were slowly broiled over red-hot coals, in an unsuccessful attempt to discover its whereabouts. On the statue to Cuauhtemoc, on the Paseo de la Reforma in Mexico City, there is a bas relief showing the Spaniards' torture of the emperor. Eventually, some gold was recovered but far less than Cortés and his men expected.

Cuauhtémoc, now baptized as Fernando Cuauhtémotzín, continued to hold his position under the Spanish, keeping the title of tlatoani, but he was no longer the sovereign ruler. From his surrender until his death, Cuauhtémoc was mostly kept in guarded custody by the Spaniards.

Execution
In 1525, Cortés took Cuauhtémoc and several other indigenous nobles on his expedition to Honduras, as he feared that Cuauhtémoc could have led an insurrection in his absence. While the expedition was stopped in the Chontal Maya capital of Itzamkanac, known as Acalan in Nahuatl, Cortés had Cuauhtémoc executed for allegedly conspiring to kill him and the other Spaniards.

There are a number of discrepancies in the various accounts of the event. According to Cortés himself, on 27 February 1525, he learned from a citizen of Tenochtitlan, Mexicalcingo, that Cuauhtémoc, Coanacoch (the ruler of Texcoco), and Tetlepanquetzal, the ruler of Tlacopan, were plotting his death. Cortés interrogated them until each confessed and then had Cuauhtémoc, Tetlepanquetzal, and another lord, Tlacatlec, hanged. Cortés wrote that the other lords would be too frightened to plot against him again, as they believed he had uncovered the plan through magic powers. Cortés's account was accepted by contemporary historian Francisco López de Gómara.

According to Bernal Díaz del Castillo, a conquistador serving under Cortés who recorded his experiences in his book The True History of the Conquest of New Spain, the supposed plot was revealed by two men, named Tapia and Juan Velásquez. Díaz portrays the executions as unjust and based on no evidence, and he admits to having liked Cuauhtémoc personally. He also records Cuauhtémoc giving the following speech to Cortés through his interpreter Malinche: "Oh Malinzin [i.e., Cortés]! Now I understand your false promises and the kind of death you have had in store for me. For you are killing me unjustly. May God demand justice from you, as it was taken from me when I entrusted myself to you in my city of Mexico!" Díaz wrote that afterwards, Cortés suffered from insomnia because of guilt and badly injured himself while he was wandering at night.

Fernando de Alva Cortés Ixtlilxóchitl, a castizo historian and descendant of Coanacoch, wrote an account of the executions in the 17th century partly based on Texcocan oral tradition. According to Ixtlilxóchitl, the three lords were joking cheerfully with one another because of a rumor that Cortés had decided to return the expedition to Mexico, when Cortés asked a spy to tell him what they were talking about. The spy reported honestly, but Cortés invented the plot himself. Cuauhtémoc, Coanacoch, and Tetlepanquetzal were hanged as well as eight others. However, Cortés cut down Coanacoch, the last to be hanged, after his brother began rallying his warriors. Coanacoch did not have long to enjoy his reprieve, as Ixtlilxóchitl wrote that he died a few days later.

Tlacotzin, Cuauhtémoc's cihuacoatl, was appointed his successor as tlatoani. He died the next year before he could return to Tenochtitlan.

Bones
The modern-day town of Ixcateopan in the state of Guerrero is home to an ossuary purportedly containing Cuauhtémoc's remains. Archeologist Eulalia Guzmán, a "passionate indigenista", excavated the bones in 1949, which were discovered shortly after bones of Cortés, found in Mexico City, had been authenticated by the Instituto Nacional de Antropología e Historia (INAH). Initially, Mexican scholars congratulated Guzmán, but after a similar examination by scholars at INAH, their authenticity as Cuauhtemoc's was rejected, as the bones in the ossuary belonged to several different persons, several of them seemingly women. The finding caused a public uproar. A panel assembled by Guzmán gave support to the initial contention. The Secretariat of Public Education (SEP) had another panel examine the bones, which gave support to INAH's original finding, but did not report on the finding publicly. A scholarly study of the controversy was published in 2011 and argued that the available data suggests that the grave is an elaborate hoax prepared by a local of Ichcateopan as a way of generating publicity, and that subsequently supported by Mexican nationalists such as Guzman who wished to use the find for political purposes.

Legacy
Cuauhtemoc is the embodiment of indigenist nationalism in Mexico, being the only Aztec emperor who survived the conquest by the Spanish Empire (and their native allies). He is honored by a monument on the Paseo de la Reforma, his face has appeared on Mexican coins, banknotes, and he is celebrated in paintings, music, and popular culture.

Many places in Mexico are named in honour of Cuauhtémoc. These include Ciudad Cuauhtémoc in Chihuahua and the Cuauhtémoc borough of Mexico City. Smaller towns include Ciudad Cuauhtémoc, Veracruz and Ciudad Cuauhtémoc, Chiapas.

The Cuauhtémoc is a vessel of the Mexican Navy that serves as a cultural ambassador with frequent visits to world ports. There is a Cuauhtémoc station on Line 1 of the Mexico City metro as well as one for Moctezuma. There is also a metro station in Monterrey named after him.

Cuauhtémoc is also one of the few non-Spanish given names for Mexican boys that is perennially popular. Individuals with this name include the politician Cuauhtémoc Cárdenas and footballer Cuauhtémoc Blanco.

In the Aztec campaign of the PC game Age of Empires II: The Conquerors, the player plays as Cuauhtémoc, despite the name Montezuma for the campaign itself, and Cuauhtémoc narrates the openings and closings to each scenario. In the next installment to the series, Age of Empires 3: The War Chiefs, Cuauhtémoc is the leader of Aztecs.

In the 1996 Rage Against the Machine single People of the Sun, lyricist Zack De La Rocha rhymes "When the fifth sun sets get back reclaimed, The spirit of Cuauhtémoc alive and untamed".

Cuauhtémoc, in the name Guatemoc, is portrayed sympathetically in the adventure novel Montezuma's Daughter, by H. Rider Haggard. First appearing in Chapter XIV, he becomes friends with the protagonist after they save each other's lives. His coronation, torture, and death are described in the novel.

title: Cuauhtémoc


text:
Cross section (physics)

In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted $σ$ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

When two discrete particles interact in classical physics, their mutual cross section is the area transverse to their relative motion within which they must meet in order to scatter from each other. If the particles are hard inelastic spheres that interact only upon contact, their scattering cross section is related to their geometric size. If the particles interact through some action-at-a-distance force, such as electromagnetism or gravity, their scattering cross section is generally larger than their geometric size.

When a cross section is specified as the differential limit of a function of some final-state variable, such as particle angle or energy, it is called a differential cross section (see detailed discussion below). When a cross section is integrated over all scattering angles (and possibly other variables), it is called a total cross section or integrated total cross section. For example, in Rayleigh scattering, the intensity scattered at the forward and backward angles is greater than the intensity scattered sideways, so the forward differential scattering cross section is greater than the perpendicular differential cross section, and by adding all of the infinitesimal cross sections over the whole range of angles with integral calculus, we can find the total cross section.

Scattering cross sections may be defined in nuclear, atomic, and particle physics for collisions of accelerated beams of one type of particle with targets (either stationary or moving) of a second type of particle. The probability for any given reaction to occur is in proportion to its cross section. Thus, specifying the cross section for a given reaction is a proxy for stating the probability that a given scattering process will occur.

The measured reaction rate of a given process depends strongly on experimental variables such as the density of the target material, the intensity of the beam, the detection efficiency of the apparatus, or the angle setting of the detection apparatus. However, these quantities can be factored away, allowing measurement of the underlying two-particle collisional cross section.

Differential and total scattering cross sections are among the most important measurable quantities in nuclear, atomic, and particle physics.

With light scattering off of a particle, the cross section specifies the amount of optical power scattered from light of a given irradiance (power per area). It is important to note that although the cross section has the same units as area, the cross section may not necessarily correspond to the actual physical size of the target given by other forms of measurement. It is not uncommon for the actual cross-sectional area of a scattering object to be much larger or smaller than the cross section relative to some physical process. For example, plasmonic nanoparticles can have light scattering cross sections for particular frequencies that are much larger than their actual cross-sectional areas.

Collision among gas particles

In a gas of finite-sized particles there are collisions among particles that depend on their cross-sectional size. The average distance that a particle travels between collisions depends on the density of gas particles. These quantities are related by
 * $$\sigma = \frac{1}{n \lambda},$$

where
 * $2r$ is the cross section of a two-particle collision (SI unit: m2),
 * $σ$ is the mean free path between collisions (SI unit: m),
 * $n$ is the number density of the target particles (SI unit: m−3).

If the particles in the gas can be treated as hard spheres of radius $λ$ that interact by direct contact, as illustrated in Figure 1, then the effective cross section for the collision of a pair is
 * $$\sigma = \pi \left(2r\right)^2$$

If the particles in the gas interact by a force with a larger range than their physical size, then the cross section is a larger effective area that may depend on a variety of variables such as the energy of the particles.

Cross sections can be computed for atomic collisions but also are used in the subatomic realm. For example, in nuclear physics a "gas" of low-energy neutrons collides with nuclei in a reactor or other nuclear device, with a cross section that is energy-dependent and hence also with well-defined mean free path between collisions.

Attenuation of a beam of particles
If a beam of particles enters a thin layer of material of thickness $σ$, the flux $λ$ of the beam will decrease by $n$ according to
 * $$\frac{\mathrm d \Phi}{\mathrm d z} = -n \sigma \Phi,$$

where $r$ is the total cross section of all events, including scattering, absorption, or transformation to another species. The volumetric number density of scattering centers is designated by $dz$. Solving this equation exhibits the exponential attenuation of the beam intensity:
 * $$\Phi = \Phi_0 e^{-n \sigma z},$$

where $Φ$ is the initial flux, and $dΦ$ is the total thickness of the material. For light, this is called the Beer–Lambert law.

Differential cross section
Consider a classical measurement where a single particle is scattered off a single stationary target particle. Conventionally, a spherical coordinate system is used, with the target placed at the origin and the $σ$ axis of this coordinate system aligned with the incident beam. The angle $n$ is the scattering angle, measured between the incident beam and the scattered beam, and the $Φ_{0}$ is the azimuthal angle.
 * Differential cross section.svg

The impact parameter $z$ is the perpendicular offset of the trajectory of the incoming particle, and the outgoing particle emerges at an angle $z$. For a given interaction (coulombic, magnetic, gravitational, contact, etc.), the impact parameter and the scattering angle have a definite one-to-one functional dependence on each other. Generally the impact parameter can neither be controlled nor measured from event to event and is assumed to take all possible values when averaging over many scattering events. The differential size of the cross section is the area element in the plane of the impact parameter, i.e. $θ$. The differential angular range of the scattered particle at angle $φ$ is the solid angle element $b$. The differential cross section is the quotient of these quantities, $θ$.

It is a function of the scattering angle (and therefore also the impact parameter), plus other observables such as the momentum of the incoming particle. The differential cross section is always taken to be positive, even though larger impact parameters generally produce less deflection. In cylindrically symmetric situations (about the beam axis), the azimuthal angle $dσ = b dφ db$ is not changed by the scattering process, and the differential cross section can be written as
 * $$ \frac{\mathrm{d} \sigma}{\mathrm{d}(\cos \theta)} =\int_0^{2\pi} \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} \,\mathrm{d}\varphi $$.

In situations where the scattering process is not azimuthally symmetric, such as when the beam or target particles possess magnetic moments oriented perpendicular to the beam axis, the differential cross section must also be expressed as a function of the azimuthal angle.

For scattering of particles of incident flux $θ$ off a stationary target consisting of many particles, the differential cross section $dΩ = sin θ dθ dφ$ at an angle $dσ⁄dΩ$ is related to the flux of scattered particle detection $φ$ in particles per unit time by
 * $$\frac{\mathrm d \sigma}{\mathrm d \Omega}(\theta,\varphi) = \frac{1}{n t \Delta\Omega} \frac{F_\text{out}(\theta,\varphi)}{F_\text{inc}}.$$

Here $F_{inc}$ is the finite angular size of the detector (SI unit: sr), $dσ⁄dΩ$ is the number density of the target particles (SI unit: m−3), and $(θ,φ)$ is the thickness of the stationary target (SI unit: m). This formula assumes that the target is thin enough that each beam particle will interact with at most one target particle.

The total cross section $F_{out}(θ,φ)$ may be recovered by integrating the differential cross section $ΔΩ$ over the full solid angle ($n$ steradians):
 * $$\sigma = \oint_{4\pi} \frac{\mathrm d \sigma}{\mathrm d \Omega} \, \mathrm d \Omega = \int_0^{2\pi} \int_0^\pi \frac{\mathrm d \sigma}{\mathrm d \Omega} \sin \theta \, \mathrm d \theta \, \mathrm d \varphi.$$

It is common to omit the "differential" qualifier when the type of cross section can be inferred from context. In this case, $t$ may be referred to as the integral cross section or total cross section. The latter term may be confusing in contexts where multiple events are involved, since "total" can also refer to the sum of cross sections over all events.

The differential cross section is extremely useful quantity in many fields of physics, as measuring it can reveal a great amount of information about the internal structure of the target particles. For example, the differential cross section of Rutherford scattering provided strong evidence for the existence of the atomic nucleus.

Instead of the solid angle, the momentum transfer may be used as the independent variable of differential cross sections.

Differential cross sections in inelastic scattering contain resonance peaks that indicate the creation of metastable states and contain information about their energy and lifetime.

Quantum scattering
In the time-independent formalism of quantum scattering, the initial wave function (before scattering) is taken to be a plane wave with definite momentum $σ$:
 * $$\phi_-(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; e^{i k z},$$

where $dσ⁄dΩ$ and $4π$ are the relative coordinates between the projectile and the target. The arrow indicates that this only describes the asymptotic behavior of the wave function when the projectile and target are too far apart for the interaction to have any effect.

After scattering takes place it is expected that the wave function takes on the following asymptotic form:
 * $$\phi_+(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; f(\theta,\phi) \frac{e^{i k r}}{r},$$

where $σ$ is some function of the angular coordinates known as the scattering amplitude. This general form is valid for any short-ranged, energy-conserving interaction. It is not true for long-ranged interactions, so there are additional complications when dealing with electromagnetic interactions.

The full wave function of the system behaves asymptotically as the sum
 * $$\phi(\mathbf r) \;\stackrel{r \to \infty}{\longrightarrow}\; \phi_-(\mathbf r) + \phi_+(\mathbf r).$$

The differential cross section is related to the scattering amplitude:
 * $$\frac{\mathrm d \sigma}{\mathrm d \Omega}(\theta, \phi) = \bigl|f(\theta, \phi)\bigr|^2.$$

This has the simple interpretation as the probability density for finding the scattered projectile at a given angle.

A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. The cross section of two particles (i.e. observed when the two particles are colliding with each other) is a measure of the interaction event between the two particles. The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles $k$) depends only on the number of incident particles per unit of time (current of incident particles $z$), the characteristics of target (for example the number of particles per unit of surface $r$), and the type of interaction. For $f$ we have
 * $$\begin{align}

I_\text{r} &= I_\text{i}N\sigma, \\ \sigma &= \frac{I_\text{r}}{I_\text{i}} \frac{1}{N} \\ &= \text{probability of interaction} \times \frac{1}{N}. \end{align}$$

Relation to the S-matrix
If the reduced masses and momenta of the colliding system are $I_{r}$, $I_{i}$ and $N$, $Nσ ≪ 1$ before and after the collision respectively, the differential cross section is given by
 * $$\frac{\mathrm d\sigma}{\mathrm d\Omega} = \left(2\pi\right)^4 m_\text{i} m_\text{f} \frac{p_\text{f}}{p_\text{i}} \bigl|T_{\text{f}\text{i}}\bigr|^2,$$

where the on-shell $m_{i}$ matrix is defined by
 * $$S_{\text{f}\text{i}} = \delta_{\text{f}\text{i}} - 2\pi i \delta\left(E_\text{f} - E_\text{i}\right) \delta\left(\mathbf{p}_\text{i} - \mathbf{p}_\text{f}\right) T_{\text{f}\text{i}}$$

in terms of the S-matrix. Here $p_{i}$ is the Dirac delta function. The computation of the S-matrix is the main goal of the scattering theory.

Units
Although the SI unit of total cross sections is m2, a smaller unit is usually used in practice.

In nuclear and particle physics, the conventional unit is the barn b, where 1 b = 10−28 m2 = 100 fm2. Smaller prefixed units such as mb and μb are also widely used. Correspondingly, the differential cross section can be measured in units such as mb/sr.

When the scattered radiation is visible light, it is conventional to measure the path length in centimetres. To avoid the need for conversion factors, the scattering cross section is expressed in cm2, and the number concentration in cm−3. The measurement of the scattering of visible light is known as nephelometry, and is effective for particles of 2–50 μm in diameter: as such, it is widely used in meteorology and in the measurement of atmospheric pollution.

The scattering of X-rays can also be described in terms of scattering cross sections, in which case the square ångström is a convenient unit: 1 Å2 = 10−20 m2 = $10,000 pm^{2}$ = 108 b. The sum of the scattering, photoelectric, and pair-production cross-sections (in barns) is charted as the "atomic attenuation coefficient" (narrow-beam), in barns.

Scattering of light
For light, as in other settings, the scattering cross section for particles is generally different from the geometrical cross section of the particle, and it depends upon the wavelength of light and the permittivity, shape, and size of the particle. The total amount of scattering in a sparse medium is proportional to the product of the scattering cross section and the number of particles present.

In the interaction of light with particles, many processes occur, each with their own cross sections, including absorption, scattering, and photoluminescence. The sum of the absorption and scattering cross sections is sometimes referred to as the attenuation or extinction cross section.
 * $$\sigma = \sigma_\text{abs} + \sigma_\text{sc} + \sigma_\text{lum}.$$

The total extinction cross section is related to the attenuation of the light intensity through the Beer–Lambert law, which says that attenuation is proportional to particle concentration:
 * $$A_\lambda = C l \sigma,$$

where $m_{f}$ is the attenuation at a given wavelength $p_{f}$, $T$ is the particle concentration as a number density, and $δ$ is the path length. The absorbance of the radiation is the logarithm (decadic or, more usually, natural) of the reciprocal of the transmittance $\mathcal{T}$:
 * $$A_\lambda = -\log \mathcal{T}.$$

Combining the scattering and absorption cross sections in this manner is often necessitated by the inability to distinguish them experimentally, and much research effort has been put into developing models that allow them to be distinguished, the Kubelka-Munk theory being one of the most important in this area.

Cross section and Mie theory
Cross sections commonly calculated using Mie theory include efficiency coefficients for extinction $Q_\text{ext}$, scattering $Q_\text{sc}$ , and Absorption $Q_\text{abs}$ cross sections. These are normalized by the geometrical cross sections of the particle $\sigma_\text{geom} = \pi a^2$ as $$ Q_\alpha = \frac{\sigma_\alpha}{\sigma_\text{geom}}, \qquad \alpha = \text{ext}, \text{sc}, \text{abs}. $$ The cross section is defined by

\sigma_\alpha = \frac{W_\alpha}{I_{\text{inc}}} $$ where $$\left[W_\alpha \right] = \left[ \text{W} \right]$$ is the energy flow through the surrounding surface, and $$ \left[I_{\text{inc}}\right] = \left[ \frac{\text{W}}{\text{m}^2} \right]$$ is the intensity of the incident wave. For a plane wave the intensity is going to be $$I_{\text{inc}} = |\mathbf{E}|^2 / (2 \eta)$$, where $$\eta = \sqrt{\mu \mu_0 / (\varepsilon \varepsilon_0)}$$ is the impedance of the host medium.

The main approach is based on the following. Firstly, we construct an imaginary sphere of radius $$r$$ (surface $$A$$) around the particle (the scatterer). The net rate of electromagnetic energy crosses the surface $$A$$ is

W_\text{a} = - \oint_A \mathbf{\Pi} \cdot \hat{\mathbf{r}} dA $$ where $\mathbf{\Pi} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}^* \times \mathbf{H} \right]$ is the time averaged Poynting vector. If $$W_\text{a} > 0$$ energy is absorbed within the sphere, otherwise energy is being created within the sphere. We will not consider this case here. If the host medium is non-absorbing, the energy must be absorbed by the particle. We decompose the total field into incident and scattered parts $$\mathbf{E} = \mathbf{E}_\text{i} + \mathbf{E}_\text{s}$$, and the same for the magnetic field $$\mathbf{H}$$. Thus, we can decompose $$W_a$$ into the three terms $$ W_\text{a} = W_\text{i} - W_\text{s} + W_{\text{ext}} $$, where

W_\text{i} = - \oint_A \mathbf{\Pi}_\text{i} \cdot \hat{\mathbf{r}} dA \equiv 0, \qquad W_\text{s} = \oint_A \mathbf{\Pi}_\text{s} \cdot \hat{\mathbf{r}} dA, \qquad W_{\text{ext}} = \oint_A \mathbf{\Pi}_{\text{ext}} \cdot \hat{\mathbf{r}} dA. $$ where $$\mathbf{\Pi}_\text{i} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_\text{i}^* \times \mathbf{H}_\text{i} \right] $$, $$\mathbf{\Pi}_\text{s} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_\text{s}^* \times \mathbf{H}_\text{s} \right] $$, and $$\mathbf{\Pi}_{\text{ext}} = \frac{1}{2} \operatorname{Re} \left[ \mathbf{E}_s^* \times \mathbf{H}_i + \mathbf{E}_i^* \times \mathbf{H}_s \right] $$.

All the field can be decomposed into the series of vector spherical harmonics (VSH). After that, all the integrals can be taken. In the case of a uniform sphere of radius $$a$$, permittivity $$\varepsilon$$, and permeability $$\mu$$, the problem has a precise solution. The scattering and extinction coefficients are $$ Q_\text{sc} = \frac{2}{k^2a^2}\sum_{n=1}^\infty (2n+1)(|a_{n}|^2+|b_{n}|^2) $$ $$ Q_\text{ext} = \frac{2}{k^2a^2}\sum_{n=1}^\infty (2n+1)\Re(a_{n}+b_{n}) $$ Where $k = n_\text{host} k_0$. These are connected as $$ \sigma_\text{ext} = \sigma_\text{sc} + \sigma_\text{abs} \qquad \text{or} \qquad Q_\text{ext} = Q_\text{sc} + Q_\text{abs} $$

Dipole approximation for the scattering cross section
Let us assume that a particle supports only electric and magnetic dipole modes with polarizabilities $\mathbf{p} = \alpha^e \mathbf{E}$ and $\mathbf{m} = (\mu \mu_0)^{-1}\alpha^m \mathbf{H}$  (here we use the notation of magnetic polarizability in the manner of Bekshaev et al.  rather than the notation of Nieto-Vesperinas et al. ) expressed through the Mie coefficients as $$ \alpha^e = 4 \pi \varepsilon_0 \cdot i \frac{3 \varepsilon}{2 k^3} a_1, \qquad \alpha^m = 4 \pi \mu_0 \cdot i \frac{3 \mu}{2 k^3} b_1. $$ Then the cross sections are given by $$ \sigma_{\text{ext}} = \sigma_{\text{ext}}^{\text{(e)}} + \sigma_{\text{ext}}^{\text{(m)}} = \frac{1}{4\pi \varepsilon \varepsilon_0} \cdot 4\pi k \Im(\alpha^e) + \frac{1}{4\pi \mu \mu_0} \cdot 4\pi k \Im(\alpha^m) $$ $$ \sigma_{\text{sc}} = \sigma_{\text{sc}}^{\text{(e)}} + \sigma_{\text{sc}}^{\text{(m)}} = \frac{1}{(4\pi \varepsilon \varepsilon_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^e|^2 + \frac{1}{(4\pi \mu \mu_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^m|^2 $$ and, finally, the electric and magnetic absorption cross sections $\sigma_{\text{abs}} = \sigma_{\text{abs}}^{\text{(e)}} + \sigma_{\text{abs}}^{\text{(m)}}$ are $$ \sigma_{\text{abs}}^{\text{(e)}} = \frac{1}{4 \pi \varepsilon \varepsilon_0} \cdot 4\pi k \left[ \Im(\alpha^e) - \frac{k^3}{6 \pi \varepsilon \varepsilon_0} |\alpha^e|^2\right] $$ and $$ \sigma_{\text{abs}}^{\text{(m)}} = \frac{1}{4 \pi \mu \mu_0} \cdot 4\pi k \left[ \Im(\alpha^m) - \frac{k^3}{6 \pi \mu \mu_0} |\alpha^m|^2\right] $$

For the case of a no-inside-gain particle, i.e. no energy is emitted by the particle internally ($\sigma_{\text{abs}} > 0$ ), we have a particular case of the Optical theorem $$ \frac{1}{4\pi \varepsilon \varepsilon_0} \Im(\alpha^e) + \frac{1}{4\pi \mu \mu_0} \Im(\alpha^m) \geq \frac{2 k^3}{3} \left[ \frac{|\alpha^e|^2}{(4\pi \varepsilon \varepsilon_0)^2} + \frac{|\alpha^m|^2}{(4\pi \mu \mu_0)^2} \right] $$ Equality occurs for non-absorbing particles, i.e. for $\Im(\varepsilon) = \Im(\mu) = 0$.

Scattering of light on extended bodies
In the context of scattering light on extended bodies, the scattering cross section, $A_{λ}$, describes the likelihood of light being scattered by a macroscopic particle. In general, the scattering cross section is different from the geometrical cross section of a particle, as it depends upon the wavelength of light and the permittivity in addition to the shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross section and the number of particles present. In terms of area, the total cross section ($λ$) is the sum of the cross sections due to absorption, scattering, and luminescence:
 * $$\sigma = \sigma_\text{abs} + \sigma_\text{sc} + \sigma_\text{lum}.$$

The total cross section is related to the absorbance of the light intensity through the Beer–Lambert law, which says that absorbance is proportional to concentration: $C$, where $l$ is the absorbance at a given wavelength $σ_{sc}$, $σ$ is the concentration as a number density, and $A_{λ} = Clσ$ is the path length. The extinction or absorbance of the radiation is the logarithm (decadic or, more usually, natural) of the reciprocal of the transmittance $\mathcal{T}$:
 * $$A_\lambda = - \log \mathcal{T}.$$

Relation to physical size
There is no simple relationship between the scattering cross section and the physical size of the particles, as the scattering cross section depends on the wavelength of radiation used. This can be seen when looking at a halo surrounding the Moon on a decently foggy evening: Red light photons experience a larger cross sectional area of water droplets than photons of higher energy. The halo around the Moon thus has a perimeter of red light due to lower energy photons being scattering further from the center of the Moon. Photons from the rest of the visible spectrum are left within the center of the halo and perceived as white light.

Meteorological range
The scattering cross section is related to the meteorological range $A_{λ}$:
 * $$L_\text{V} = \frac{3.9}{C \sigma_\text{scat}}.$$

The quantity $λ$ is sometimes denoted $C$, the scattering coefficient per unit length.

Elastic collision of two hard spheres
The following equations apply to two hard spheres that undergo a perfectly elastic collision. Let $l$ and $L_{V}$ denote the radii of the scattering center and scattered sphere, respectively. The differential cross section is

\frac{d\sigma}{d\Omega} = \frac{R^2}{4}, $$ and the total cross section is

\sigma_\text{tot} = \pi \left(r + R\right)^2. $$ In other words, the total scattering cross section is equal to the area of the circle (with radius $Cσ_{scat}$) within which the center of mass of the incoming sphere has to arrive for it to be deflected.

Rutherford scattering
In Rutherford scattering, an incident particle with charge $b_{scat}$ and energy $σ_{scat}$ scatters off a fixed particle with charge $R$. The differential cross section is

\frac{d \sigma}{d \Omega} = \left(\frac{q \, Q}{16\pi\varepsilon_0 E \sin^2(\theta/2)} \right)^2 $$ where $$\varepsilon_0$$ is the vacuum permittivity. The total cross section is infinite unless a cutoff for small scattering angles $$\theta$$ is applied. This is due to the long range of the $$1/r$$ Coulomb potential.

Scattering from a 2D circular mirror
The following example deals with a beam of light scattering off a circle with radius $r$ and a perfectly reflecting boundary. The beam consists of a uniform density of parallel rays, and the beam-circle interaction is modeled within the framework of geometric optics. Because the problem is genuinely two-dimensional, the cross section has unit of length (e.g., metre). Let $r + R$ be the angle between the light ray and the radius joining the reflection point of the ray with the center point of the mirror. Then the increase of the length element perpendicular to the beam is
 * $$\mathrm dx = r \cos \alpha \,\mathrm d \alpha.$$

The reflection angle of this ray with respect to the incoming ray is $q$, and the scattering angle is
 * $$\theta = \pi - 2 \alpha.$$

The differential relationship between incident and reflected intensity $E$ is
 * $$I \,\mathrm d \sigma = I \,\mathrm dx(x) = I r \cos \alpha \,\mathrm d \alpha = I \frac{r}{2} \sin \left(\frac{\theta}{2}\right) \,\mathrm d \theta = I \frac{\mathrm d \sigma}{\mathrm d \theta} \,\mathrm d \theta.$$

The differential cross section is therefore ($Q$)
 * $$\frac{\mathrm d \sigma}{\mathrm d \theta} = \frac{r}{2} \sin \left(\frac{\theta}{2}\right).$$

Its maximum at $r$ corresponds to backward scattering, and its minimum at $α$ corresponds to scattering from the edge of the circle directly forward. This expression confirms the intuitive expectations that the mirror circle acts like a diverging lens. The total cross section is equal to the diameter of the circle:
 * $$\sigma = \int_0^{2 \pi} \frac{\mathrm d \sigma}{\mathrm d \theta} \,\mathrm d \theta = \int_0^{2 \pi} \frac{r}{2} \sin \left(\frac{\theta}{2}\right) \,\mathrm d \theta = 2 r.$$

Scattering from a 3D spherical mirror
The result from the previous example can be used to solve the analogous problem in three dimensions, i.e., scattering from a perfectly reflecting sphere of radius $2α$.

The plane perpendicular to the incoming light beam can be parameterized by cylindrical coordinates $I$ and $dΩ = dθ$. In any plane of the incoming and the reflected ray we can write (from the previous example):
 * $$\begin{align}

r &= a \sin \alpha,\\ \mathrm dr &= a \cos \alpha \,\mathrm d \alpha, \end{align}$$ while the impact area element is
 * $$ \mathrm d \sigma = \mathrm d r(r) \times r \,\mathrm d \varphi = \frac{a^2}{2} \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right) \,\mathrm d \theta \,\mathrm d \varphi.$$

In spherical coordinates,
 * $$\mathrm d\Omega = \sin \theta \,\mathrm d \theta \,\mathrm d \varphi.$$

Together with the trigonometric identity
 * $$\sin \theta = 2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right),$$

we obtain
 * $$\frac{\mathrm d \sigma}{\mathrm d \Omega} = \frac{a^2}{4}.$$

The total cross section is
 * $$\sigma = \oint_{4 \pi} \frac{\mathrm d \sigma}{\mathrm d \Omega} \,\mathrm d \Omega = \pi a^2.$$

title: Cross section (physics)