Friedmann equations

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Friedmann_equations

Alexander Friedmann

The Friedmann equations, also known as the Friedmann–Lemaître (FL) equations, are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p. The equations for negative spatial curvature were given by Friedmann in 1924.

Assumptions

Main article: Friedmann–Lemaître–Robertson–Walker metric

The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, that is, the cosmological principle; empirically, this is justified on scales larger than the order of 100 Mpc. The cosmological principle implies that the metric of the universe must be of the form $${\displaystyle -\mathrm{d}{s}^2=a(t{)}^2{\mathrm{d}{s}_3}^2-c^2\mathrm{d}{t}^2}$$ where ds₃² is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. This metric is called the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The parameter k discussed below takes the value 0, 1, −1, or the Gaussian curvature, in these three cases respectively. It is this fact that allows us to sensibly speak of a "scale factor" a(t).

Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute Christoffel symbols, then the Ricci tensor. With the stress–energy tensor for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below.

Equations

There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is: $${\displaystyle \frac{{\dot{a}}^2+k{c}^2}{a^2}=\frac{8\pi G\rho +\mathrm{\Lambda }{c}^2}{3},}$$ which is derived from the 00 component of the Einstein field equations. The second is: $${\displaystyle \frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho +\frac{3p}{c^2}\right)+\frac{\mathrm{\Lambda }{c}^2}{3}}$$ which is derived from the first together with the trace of Einstein's field equations (the dimension of the two equations is time⁻²).

a is the scale factor, G, Λ, and c are universal constants (G is the Newtonian constant of gravitation, Λ is the cosmological constant with dimension length⁻², and c is the speed of light in vacuum). ρ and p are the volumetric mass density (and not the volumetric energy density) and the pressure, respectively. k is constant throughout a particular solution, but may vary from one solution to another.

In previous equations, a, ρ, and p are functions of time. ⁠k/a²⁠ is the spatial curvature in any time-slice of the universe; it is equal to one-sixth of the spatial Ricci curvature scalar R since $${\displaystyle R=\frac{6}{c^2{a}^2}(\ddot{a}a+{\dot{a}}^2+k{c}^2)}$$ in the Friedmann model. H ≡ ⁠ȧ/a⁠ is the Hubble parameter.

We see that in the Friedmann equations, a(t) does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for a and k which describe the same physics:

• k = +1, 0 or −1 depending on whether the shape of the universe is a closed 3-sphere, flat (Euclidean space) or an open 3-hyperboloid, respectively. If k = +1, then a is the radius of curvature of the universe. If k = 0, then a may be fixed to any arbitrary positive number at one particular time. If k = −1, then (loosely speaking) one can say that i · a is the radius of curvature of the universe.
• a is the scale factor which is taken to be 1 at the present time. k is the current spatial curvature (when a = 1). If the shape of the universe is hyperspherical and R_t is the radius of curvature (R₀ at the present), then a = ⁠R_t/R₀⁠. If k is positive, then the universe is hyperspherical. If k = 0, then the universe is flat. If k is negative, then the universe is hyperbolic.

Using the first equation, the second equation can be re-expressed as $${\displaystyle \dot{\rho }=-3H\left(\rho +\frac{p}{c^2}\right),}$$ which eliminates Λ and expresses the conservation of mass–energy: $${\displaystyle T^{\alpha \beta }{}_{;\beta }=0.}$$

These equations are sometimes simplified by replacing $${\displaystyle \begin{array}{rlrl}\rho & \to \rho -\frac{\mathrm{\Lambda }{c}^2}{8\pi G}& p& \to p+\frac{\mathrm{\Lambda }{c}^4}{8\pi G}\end{array}}$$ to give: $${\displaystyle \begin{array}{rl}{H}^2={\left(\frac{\dot{a}}{a}\right)}^2& =\frac{8\pi G}{3}\rho -\frac{k{c}^2}{a^2}\\ \dot{H}+H^2=\frac{\ddot{a}}{a}& =-\frac{4\pi G}{3}\left(\rho +\frac{3p}{c^2}\right).\end{array}}$$

The simplified form of the second equation is invariant under this transformation.

The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.

Some cosmologists call the second of these two equations the Friedmann acceleration equation and reserve the term Friedmann equation for only the first equation.

Density parameter

The density parameter Ω is defined as the ratio of the actual (or observed) density ρ to the critical density ρ_c of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclidean). In earlier models, which did not include a cosmological constant term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.

To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2–0.25 atoms per cubic metre.

Estimated relative distribution for components of the energy density of the universe. Dark energy dominates the total energy (74%) while dark matter (22%) constitutes most of the mass. Of the remaining baryonic matter (4%), only one tenth is compact. In February 2015, the European-led research team behind the Planck cosmology probe released new data refining these values to 4.9% ordinary matter, 25.9% dark matter and 69.1% dark energy.

A much greater density comes from the unidentified dark matter, although both ordinary and dark matter contribute in favour of contraction of the universe. However, the largest part comes from so-called dark energy, which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), dark energy does not lead to contraction of the universe but rather may accelerate its expansion.

An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find: $${\displaystyle {\rho }_{\mathrm{c}}=\frac{3H^2}{8\pi G}=1.8788\times {10}^{-26}{h}^2\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}=2.7754\times {10}^{11}{h}^2{M}_{\odot }{\mathrm{M}\mathrm{p}\mathrm{c}}^{-3},}$$

(where h = H₀/(100 km/s/Mpc). For H_o = 67.4 km/s/Mpc, i.e. h = 0.674, ρ_c = 8.5×10⁻²⁷ kg/m³).

The density parameter (useful for comparing different cosmological models) is then defined as: $${\displaystyle \mathrm{\Omega }\equiv \frac{\rho }{{\rho }_c}=\frac{8\pi G\rho }{3H^2}.}$$

This term originally was used as a means to determine the spatial geometry of the universe, where ρ_c is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if Ω is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If Ω is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for Ω in which case this density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the ΛCDM model, there are important components of Ω due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter k is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see.

The first Friedmann equation is often seen in terms of the present values of the density parameters, that is $${\displaystyle \frac{H^2}{H_0^2}={\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{-4}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{-3}+{\mathrm{\Omega }}_{0,k}{a}^{-2}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}.}$$ Here Ω_0,R is the radiation density today (when a = 1), Ω_0,M is the matter (dark plus baryonic) density today, Ω_0,k = 1 − Ω₀ is the "spatial curvature density" today, and Ω_0,Λ is the cosmological constant or vacuum density today.

Useful solutions

The Friedmann equations can be solved exactly in presence of a perfect fluid with equation of state $${\displaystyle p=w\rho c^2,}$$ where p is the pressure, ρ is the mass density of the fluid in the comoving frame and w is some constant.

In spatially flat case (k = 0), the solution for the scale factor is $${\displaystyle a(t)=a_0{t}^{\frac{2}{3(w+1)}}}$$ where a₀ is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by w is extremely important for cosmology. For example, w = 0 describes a matter-dominated universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as $${\displaystyle a(t)\propto t^{2/3}}$$ matter-dominated Another important example is the case of a radiation-dominated universe, namely when w = ⁠1/3⁠. This leads to $${\displaystyle a(t)\propto t^{1/2}}$$ radiation-dominate.

Note that this solution is not valid for domination of the cosmological constant, which corresponds to an w = −1. In this case the energy density is constant and the scale factor grows exponentially.

Solutions for other values of k can be found at Tersic, Balsa. "Lecture Notes on Astrophysics". Retrieved 24 February 2022.

Mixtures

If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then $${\displaystyle {\dot{\rho }}_f=-3H\left({\rho }_f+\frac{p_f}{c^2}\right)}$$ holds separately for each such fluid f. In each case, $${\displaystyle {\dot{\rho }}_f=-3H\left({\rho }_f+w_f{\rho }_f\right)}$$ from which we get $${\displaystyle {\rho }_f\propto a^{-3\left(1+w_f\right)}.}$$

For example, one can form a linear combination of such terms $${\displaystyle \rho =A{a}^{-3}+B{a}^{-4}+C{a}^0}$$ where A is the density of "dust" (ordinary matter, w = 0) when a = 1; B is the density of radiation (w = ⁠1/3⁠) when a = 1; and C is the density of "dark energy" (w = −1). One then substitutes this into $${\displaystyle {\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho -\frac{k{c}^2}{a^2}}$$ and solves for a as a function of time.

Detailed derivation

To make the solutions more explicit, we can derive the full relationships from the first Friedmann equation: $${\displaystyle \frac{H^2}{H_0^2}={\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{-4}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{-3}+{\mathrm{\Omega }}_{0,k}{a}^{-2}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}}$$ with $${\displaystyle \begin{array}{rl}H& =\frac{\dot{a}}{a}\\ H^2& =H_0^2\left({\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{-4}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{-3}+{\mathrm{\Omega }}_{0,k}{a}^{-2}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}\right)\\ H& =H_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{-4}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{-3}+{\mathrm{\Omega }}_{0,k}{a}^{-2}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}}\\ \frac{\dot{a}}{a}& =H_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{-4}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{-3}+{\mathrm{\Omega }}_{0,k}{a}^{-2}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}}\\ \frac{\mathrm{d}a}{\mathrm{d}t}& =H_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{-2}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{-1}+{\mathrm{\Omega }}_{0,k}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}{a}^2}\\ \mathrm{d}a& =\mathrm{d}t{H}_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{-2}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{-1}+{\mathrm{\Omega }}_{0,k}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}{a}^2}\end{array}}$$

Rearranging and changing to use variables a′ and t′ for the integration $${\displaystyle t{H}_0={\int }_0^a\frac{\mathrm{d}{a}^{\prime }}{\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{\prime -2}+{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{\prime -1}+{\mathrm{\Omega }}_{0,k}+{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}{a}^{\prime 2}}}}$$

Solutions for the dependence of the scale factor with respect to time for universes dominated by each component can be found. In each we also have assumed that Ω_0,k ≈ 0, which is the same as assuming that the dominating source of energy density is approximately 1.

For matter-dominated universes, where Ω_0,M ≫ Ω_0,R and Ω_0,Λ, as well as Ω_0,M ≈ 1: $${\displaystyle \begin{array}{rl}t{H}_0& ={\int }_0^a\frac{\mathrm{d}{a}^{\prime }}{\sqrt{{\mathrm{\Omega }}_{0,\mathrm{M}}{a}^{\prime -1}}}\\ t{H}_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{M}}}& ={\left(\frac{2}{3}{a^{\prime }}^{3/2}\right)|}_0^a\\ {\left(\frac{3}{2}t{H}_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{M}}}\right)}^{2/3}& =a(t)\end{array}}$$ which recovers the aforementioned a ∝ t^2/3

For radiation-dominated universes, where Ω_0,R ≫ Ω_0,M and Ω_0,Λ, as well as Ω_0,R ≈ 1: $${\displaystyle \begin{array}{rl}t{H}_0& ={\int }_0^a\frac{\mathrm{d}{a}^{\prime }}{\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}{a}^{\prime -2}}}\\ t{H}_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}}& ={\frac{a^{\prime 2}}{2}|}_0^a\\ {\left(2t{H}_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{R}}}\right)}^{1/2}& =a(t)\end{array}}$$

For Λ-dominated universes, where Ω_0,Λ ≫ Ω_0,R and Ω_0,M, as well as Ω_0,Λ ≈ 1, and where we now will change our bounds of integration from t_i to t and likewise a_i to a: $${\displaystyle \begin{array}{rl}\left(t-t_i\right)H_0& ={\int }_{a_i}^a\frac{\mathrm{d}{a}^{\prime }}{\sqrt{({\mathrm{\Omega }}_{0,\mathrm{\Lambda }}{a}^{\prime 2})}}\\ \left(t-t_i\right)H_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}}& =\ln |a^{\prime }|{|}_{a_i}^a\\ a_i\exp \left((t-t_i)H_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}}\right)& =a(t)\end{array}}$$

The Λ-dominated universe solution is of particular interest because the second derivative with respect to time is positive, non-zero; in other words implying an accelerating expansion of the universe, making ρ_Λ a candidate for dark energy: $${\displaystyle \begin{array}{rl}a(t)& =a_i\exp \left((t-t_i)H_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}}\right)\\ \frac{{\mathrm{d}}^{2}a(t)}{\mathrm{d}{t}^2}& =a_i{H_0}^2{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}\exp \left((t-t_i)H_0\sqrt{{\mathrm{\Omega }}_{0,\mathrm{\Lambda }}}\right)\end{array}}$$

Where by construction a_i > 0, our assumptions were Ω_0,Λ ≈ 1, and H₀ has been measured to be positive, forcing the acceleration to be greater than zero.

Rescaled Friedmann equation

Set $${\displaystyle \tilde{a}=\frac{a}{a_0},{\rho }_c=\frac{3H_0^2}{8\pi G},\mathrm{\Omega }=\frac{\rho }{{\rho }_{\mathrm{c}}},t=\frac{\tilde{t}}{H_0},{\mathrm{\Omega }}_{\mathrm{k}}=-\frac{k{c}^2}{H_0^2{a}_0^2},}$$ where a₀ and H₀ are separately the scale factor and the Hubble parameter today. Then we can have $${\displaystyle \frac{1}{2}{\left(\frac{d\tilde{a}}{d\tilde{t}}\right)}^2+U_{\text{eff}}(\tilde{a})=\frac{1}{2}{\mathrm{\Omega }}_{\mathrm{k}}}$$ where $${\displaystyle U_{\text{eff}}(\tilde{a})=\frac{-\mathrm{\Omega }{\tilde{a}}^2}{2}.}$$

For any form of the effective potential U_eff(ã), there is an equation of state p = p(ρ) that will produce it.

In popular culture

Several students at Tsinghua University (CCP leader Xi Jinping's alma mater) participating in the 2022 COVID-19 protests in China carried placards with Friedmann equations scrawled on them, interpreted by some as a play on the words "Free man". Others have interpreted the use of the equations as a call to “open up” China and stop its Zero Covid policy, as the Friedmann equations relate to the expansion, or “opening” of the universe.

Spent nuclear fuel

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Spent_nuclear_fuel

Spent fuel pool at a nuclear power plant

Spent nuclear fuel, occasionally called used nuclear fuel, is nuclear fuel that has been irradiated in a nuclear reactor (usually at a nuclear power plant). It is no longer useful in sustaining a nuclear reaction in an ordinary thermal reactor and, depending on its point along the nuclear fuel cycle, it will have different isotopic constituents than when it started.

Nuclear fuel rods become progressively more radioactive (and less thermally useful) due to neutron activation as they are fissioned, or "burnt", in the reactor. A fresh rod of low enriched uranium pellets (which can be safely handled with gloved hands) will become a highly lethal gamma emitter after 1–2 years of core irradiation, unsafe to approach unless under many feet of water shielding. This makes their invariable accumulation and safe temporary storage in spent fuel pools a prime source of high level radioactive waste and a major ongoing issue for future permanent disposal.

Nature of spent fuel

Nanomaterial properties

In the oxide fuel, intense temperature gradients exist that cause fission products to migrate. The zirconium tends to move to the centre of the fuel pellet where the temperature is highest, while the lower-boiling fission products move to the edge of the pellet. The pellet is likely to contain many small bubble-like pores that form during use; the fission product xenon migrates to these voids. Some of this xenon will then decay to form caesium, hence many of these bubbles contain a large concentration of ¹³⁵
Cs.

In the case of mixed oxide (MOX) fuel, the xenon tends to diffuse out of the plutonium-rich areas of the fuel, and it is then trapped in the surrounding uranium dioxide. The neodymium tends to not be mobile.

Also metallic particles of an alloy of Mo-Tc-Ru-Pd tend to form in the fuel. Other solids form at the boundary between the uranium dioxide grains, but the majority of the fission products remain in the uranium dioxide as solid solutions. A paper describing a method of making a non-radioactive "uranium active" simulation of spent oxide fuel exists.

Fission products

3% of the mass consists of fission products of ²³⁵U and ²³⁹Pu (also indirect products in the decay chain); these are considered radioactive waste or may be separated further for various industrial and medical uses. The fission products include every element from zinc through to the lanthanides; much of the fission yield is concentrated in two peaks, one in the second transition row (Zr, Mo, Tc, Ru, Rh, Pd, Ag) and the other later in the periodic table (I, Xe, Cs, Ba, La, Ce, Nd). Many of the fission products are either non-radioactive or only short-lived radioisotopes, but a considerable number are medium to long-lived radioisotopes such as ⁹⁰Sr, ¹³⁷Cs, ⁹⁹Tc and ¹²⁹I. Research has been conducted by several different countries into segregating the rare isotopes in fission waste including the "fission platinoids" (Ru, Rh, Pd) and silver (Ag) as a way of offsetting the cost of reprocessing; this is not currently being done commercially.

The fission products can modify the thermal properties of the uranium dioxide; the lanthanide oxides tend to lower the thermal conductivity of the fuel, while the metallic nanoparticles slightly increase the thermal conductivity of the fuel.

Table of chemical data

The chemical forms of fission products in uranium dioxide

Element	Gas	Metal	Oxide	Solid solution
Br Kr	Yes	-	-	-
Rb	Yes	-	Yes	-
Sr	-	-	Yes	Yes
Y	-	-	-	Yes
Zr	-	-	Yes	Yes
Nb	-	-	Yes	-
Mo	-	Yes	Yes	-
Tc Ru Rh Pd Ag Cd In Sb	-	Yes	-	-
Te	Yes	Yes	Yes	Yes
I Xe	Yes	-	-	-
Cs	Yes	-	Yes	-
Ba	-	-	Yes	Yes
La Ce Pr Nd Pm Sm Eu	-	-	-	Yes

Plutonium

Spent nuclear fuel stored underwater and uncapped at the Hanford site in Washington, US

About 1% of the mass is ²³⁹Pu and ²⁴⁰Pu resulting from conversion of ²³⁸U, which may be considered either as a useful byproduct, or as dangerous and inconvenient waste. One of the main concerns regarding nuclear proliferation is to prevent this plutonium from being used by states, other than those already established as nuclear weapons states, to produce nuclear weapons. If the reactor has been used normally, the plutonium is reactor-grade, not weapons-grade: it contains more than 19% ²⁴⁰Pu and less than 80% ²³⁹Pu, which makes it not ideal for making bombs. If the irradiation period has been short then the plutonium is weapons-grade (more than 93%).

Uranium

96% of the mass is the remaining uranium: most of the original ²³⁸U and a little ²³⁵U. Usually ²³⁵U would be less than 0.8% of the mass along with 0.4% ²³⁶U.

Reprocessed uranium will contain ²³⁶U, which is not found in nature; this is one isotope that can be used as a fingerprint for spent reactor fuel.

If using a thorium fuel to produce fissile ²³³U, the SNF (Spent Nuclear Fuel) will have ²³³U, with a half-life of 159,200 years (unless this uranium is removed from the spent fuel by a chemical process). The presence of ²³³U will affect the long-term radioactive decay of the spent fuel. If compared with MOX fuel, the activity around one million years in the cycles with thorium will be higher due to the presence of the not fully decayed ²³³U.

For natural uranium fuel, fissile component starts at 0.7% ²³⁵U concentration in natural uranium. At discharge, total fissile component is still 0.5% (0.2% ²³⁵U, 0.3% fissile ²³⁹Pu, ²⁴¹Pu). Fuel is discharged not because fissile material is fully used-up, but because the neutron-absorbing fission products have built up and the fuel becomes significantly less able to sustain a nuclear reaction.

Some natural uranium fuels use chemically active cladding, such as Magnox, and need to be reprocessed because long-term storage and disposal is difficult.

Minor actinides

Spent reactor fuel contains traces of the minor actinides. These are actinides other than uranium and plutonium and include neptunium, americium and curium. The amount formed depends greatly upon the nature of the fuel used and the conditions under which it was used. For instance, the use of MOX fuel (²³⁹Pu in a ²³⁸U matrix) is likely to lead to the production of more ²⁴¹Am and heavier nuclides than a uranium/thorium based fuel (²³³U in a ²³²Th matrix).

For highly enriched fuels used in marine reactors and research reactors, the isotope inventory will vary based on in-core fuel management and reactor operating conditions.

Spent fuel decay heat

Decay heat as fraction of full power for a reactor SCRAMed from full power at time 0, using two different correlations

Main article: Decay heat

When a nuclear reactor has been shut down and the nuclear fission chain reaction has ceased, a significant amount of heat will still be produced in the fuel due to the beta decay of fission products. For this reason, at the moment of reactor shutdown, decay heat will be about 7% of the previous core power if the reactor has had a long and steady power history. About 1 hour after shutdown, the decay heat will be about 1.5% of the previous core power. After a day, the decay heat falls to 0.4%, and after a week it will be 0.2%. The decay heat production rate will continue to slowly decrease over time.

Spent fuel that has been removed from a reactor is ordinarily stored in a water-filled spent fuel pool for a year or more (in some sites 10 to 20 years) in order to cool it and provide shielding from its radioactivity. Practical spent fuel pool designs generally do not rely on passive cooling but rather require that the water be actively pumped through heat exchangers. If there is a prolonged interruption of active cooling due to emergency situations, the water in the spent fuel pools may therefore boil off, possibly resulting in radioactive elements being released into the atmosphere.

Fuel composition and long term radioactivity

Main article: Long-lived fission product

Activity of U-233 for three fuel types. In the case of MOX, the U-233 increases for the first 650,000 years as it is produced by decay of Np-237 that was created in the reactor by absorption of neutrons by U-235.

Total activity for three fuel types. In region 1 we have radiation from short-lived nuclides, and in region 2 from Sr-90 and Cs-137. On the far right we see the decay of Np-237 and U-233.

The use of different fuels in nuclear reactors results in different SNF composition, with varying activity curves.

Long-lived radioactive waste from the back end of the fuel cycle is especially relevant when designing a complete waste management plan for SNF. When looking at long-term radioactive decay, the actinides in the SNF have a significant influence due to their characteristically long half-lives. Depending on what a nuclear reactor is fueled with, the actinide composition in the SNF will be different.

An example of this effect is the use of nuclear fuels with thorium. Th-232 is a fertile material that can undergo a neutron capture reaction and two beta minus decays, resulting in the production of fissile U-233. Its radioactive decay will strongly influence the long-term activity curve of the SNF around a million years. A comparison of the activity associated to U-233 for three different SNF types can be seen in the figure on the top right. The burnt fuels are Thorium with Reactor-Grade Plutonium (RGPu), Thorium with Weapons-Grade Plutonium (WGPu) and Mixed Oxide fuel (MOX, no thorium). For RGPu and WGPu, the initial amount of U-233 and its decay around a million years can be seen. This has an effect in the total activity curve of the three fuel types. The initial absence of U-233 and its daughter products in the MOX fuel results in a lower activity in region 3 of the figure on the bottom right, whereas for RGPu and WGPu the curve is maintained higher due to the presence of U-233 that has not fully decayed. Nuclear reprocessing can remove the actinides from the spent fuel so they can be used or destroyed (see Long-lived fission product#Actinides).

Spent fuel corrosion

Noble metal nanoparticles and hydrogen

According to the work of corrosion electrochemist David W. Shoesmith, the nanoparticles of Mo-Tc-Ru-Pd have a strong effect on the corrosion of uranium dioxide fuel. For instance his work suggests that when hydrogen (H₂) concentration is high (due to the anaerobic corrosion of the steel waste can), the oxidation of hydrogen at the nanoparticles will exert a protective effect on the uranium dioxide. This effect can be thought of as an example of protection by a sacrificial anode, where instead of a metal anode reacting and dissolving it is the hydrogen gas that is consumed.

Storage, treatment, and disposal

See also: List of nuclear waste treatment technologies

Spent fuel pool at TEPCO's Fukushima Daiichi Nuclear Power Plant on 27 November 2013

Spent nuclear fuel is stored either in spent fuel pools (SFPs) or in dry casks. In the United States, SFPs and casks containing spent fuel are located either directly on nuclear power plant sites or on Independent Spent Fuel Storage Installations (ISFSIs). ISFSIs can be adjacent to a nuclear power plant site, or may reside away-from-reactor (AFR ISFSI). The vast majority of ISFSIs store spent fuel in dry casks. The Morris Operation is currently the only ISFSI with a spent fuel pool in the United States.

Nuclear reprocessing can separate spent fuel into various combinations of reprocessed uranium, plutonium, minor actinides, fission products, remnants of zirconium or steel cladding, activation products, and the reagents or solidifiers introduced in the reprocessing itself. If these constituent portions of spent fuel were reused, and additional wastes that may come as a byproduct of reprocessing are limited, reprocessing could ultimately reduce the volume of waste that needs to be disposed.

Alternatively, the intact spent nuclear fuel can be directly disposed of as high-level radioactive waste. The United States has planned disposal in deep geological formations, such as the Yucca Mountain nuclear waste repository, where it has to be shielded and packaged to prevent its migration to humans' immediate environment for thousands of years. On March 5, 2009, however, Energy Secretary Steven Chu told a Senate hearing that "the Yucca Mountain site no longer was viewed as an option for storing reactor waste."

Geological disposal has been approved in Finland, using the KBS-3 process.

In Switzerland, the Federal Council approved in 2008, the plan for the deep geological repository for radioactive waste.

Remediation

Algae has shown selectivity for strontium in studies, where most plants used in bioremediation have not shown selectivity between calcium and strontium, often becoming saturated with calcium, which is present in greater quantities in nuclear waste. Strontium-90 is a radioactive byproduct produced by nuclear reactors used in nuclear power. It is a component of nuclear waste and spent nuclear fuel. The half life is long, around 30 years, and is classified as high-level waste.

Researchers have looked at the bioaccumulation of strontium by Scenedesmus spinosus (algae) in simulated wastewater. The study claims a highly selective biosorption capacity for strontium of S. spinosus, suggesting that it may be appropriate for use of nuclear wastewater. A study of the pond alga Closterium moniliferum using non-radioactive strontium found that varying the ratio of barium to strontium in water improved strontium selectivity.

Risks

See also: Spent fuel pool

Spent nuclear fuel stays a radiation hazard for extended periods of time with half-lifes as high as 24,000 years. For example 10 years after removal from a reactor, the surface dose rate for a typical spent fuel assembly still exceeds 10,000 rem/hour—far greater than the fatal whole-body dose for humans of about 500 rem received all at once.

There is debate over whether spent fuel stored in a pool is susceptible to incidents such as earthquakes or terrorist attacks that could potentially result in a release of radiation.

In the rare occurrence of a fuel failure during normal operation, the primary coolant can enter the element. Visual techniques are normally used for the postirradiation inspection of fuel bundles.

Since the September 11 attacks the Nuclear Regulatory Commission has instituted a series of rules mandating that all fuel pools be impervious to natural disaster and terrorist attack. As a result, used fuel pools are encased in a steel liner and thick concrete, and are regularly inspected to ensure resilience to earthquakes, tornadoes, hurricanes, and seiches.