## Chain reaction

Let $$\ce{^{A}n}$$ represent a neutron isotope containing $$\ce{A}$$ neutrons. In our view a neutron isotope reactor is activated by a chain reaction that includes an exothermic isotope growth reaction such as $\ce{^{A}n + {}^{2}H -> {}^{A+1}n + {}^{1}H}$ that transfers a neutron from $${}^2\text{H}$$ fuel to a neutron isotope. It also includes exothermic isotope fission reactions such as the sequential beta/beta/alpha combination ${}^{\text{A}}\text{n} \ \longrightarrow\ {}^{\text{B}}\text{n} + {}^{\text{A}-\text{B} - 4}\text{n} + {}^4\text{He} .$ Together these reactions can support a chain reaction that builds and maintains the environment of ambient neutron isotopes.

## Transmutation

In the general transmutation reaction a neutron isotope interacts with an ordinary nucleus $${}^\text{B}\text{Z}$$ by transferring one or more neutrons to or from it in a reaction of the form $$\label{first}\tag{\dagger} {}^{\text{A}}\text{n} + {}^{\text{B}}\text{Z} \ \longrightarrow\ {}^{\text{A}-\text{C}}\text{n} + {}^{\text{B}+\text{C}}\text{Z}.$$ In this reaction the number of transferred neutrons $$\text{C}$$ is usually positive and $${}^{\text{B}+\text{C}}\text{Z}$$ is a neutron-rich isotope of $${}^{\text{B}}\text{Z}$$.

Physically we expect that a neutron isotope diffuses at random throughout the active reaction volume, occasionally encountering an ordinary nucleus that serves as a target nucleus for transmutation. It then recoils elastically or it reacts by neutron transfer.

Following transfer of neutrons, a target nucleus becomes a heavier isotope of the target element. That isotope may be stable, or it may undergo decay to an isotope of a new element, and so on through a chain of decays until it reaches a stable isotope.

## Liquid drop model

Quantitative analysis requires knowledge of the mass excesses of neutron isotopes. We write $$\Delta({}^\text{A}\text{n})$$ for the mass excess of isotope $${}^\text{A}\text{n}$$. We expect that a neutron isotope can be described by a liquid drop model in which $\Delta({}^\text{A}\text{n}) = \text{A}\, \Delta(\text{n}) - a_v\, \text{A} + a_s\, \text{A}^{2/3}$ over the range of $$\text{A}$$ for neutron isotopes participating in a steady state reaction. In this formula $$\Delta(\text{n})$$ is the mass excess of a neutron, $$a_v > 0$$ is the binding energy per neutron, and $$a_s > 0$$ is a surface energy parameter. The change in mass excess on adding one neutron is $\theta = \Delta({}^{\text{A}+1}\text{n}) - \Delta({}^\text{A}\text{n}) = \Delta(\text{n}) - a_v + (2/3)\, a_s\, \text{A}^{-1/3}.$ Because the parameter $$\theta$$ is a slowly-varying function of A that grows smaller as A grows larger, we consider a range of values of $$\theta$$ that corresponds to the range of values of A for the neutron isotopes that participate in the chain reaction.

## Energy release and branching ratios

The transmutation reaction \eqref{first} releases energy $$E$$ that depends on the value of $$\text{C}$$ $\begin{split} E &= \Delta({}^\text{A}\text{n}) - \Delta({}^{\text{A}-\text{C}}\text{n}) + \Delta({}^{\text{B}}\text{Z}) - \Delta({}^{\text{B}+\text{C}}\text{Z}) \\ &= \text{C}\,\theta + \Delta({}^{\text{B}}\text{Z}) - \Delta({}^{\text{B}+\text{C}}\text{Z}) . \end{split}$ For any target isotope $${}^{\text{B}}\text{Z}$$ having charge $$\text{Z}$$, and for any value of $$\theta$$ in an allowed range there is a value of $$\text{C}$$ that produces maximum released energy. If the maximum is positive we assume that reaction \eqref{first} occurs and target isotope $${}^{\text{B}}\text{Z}$$ is transmuted to product isotope $${}^{\text{B}+\text{C}}\text{Z}$$.

There may be a single value of $$\text{C}$$ for $$\theta$$ in the allowed range, leading to a single transmutation product, or there may be different values of $$\text{C}$$ within contiguous subranges of $$\theta$$, leading to a mix of transmutation products. These neutron-rich isotopes generally are unstable and each undergo a series of beta decays to stable or long-lived transmutation products $${}^{\text{B}+\text{C}}(\text{Z}+\Delta\text{Z})$$.

When this process is repeated many times, each time with a new target nucleus and a new random value of $$\theta$$, the supply of target isotope $${}^{\text{B}}\text{Z}$$ is gradually depleted and the supplies of transmutation products $${}^{\text{B}+\text{C}}(\text{Z}+\Delta\text{Z})$$ are correspondingly increased.

## Which isotopes are produced by neutron transfer

We can express the transmutation reaction \eqref{first} as $$\tag{\star} {}^{\text{A}}\text{n} + {}^{\text{B}}\text{Z} \ \longrightarrow\ {}^{\text{A}+\text{B}-\text{D}}\text{n} + {}^{\text{D}}\text{Z},$$ where $$\text{D} = \text{B} + \text{C}$$, so that $$\text{D}$$ is the mass number of the product isotope. Similarly, we can express the energy released by the transfer of neutrons in terms of $$\text{D}$$: $E = \text{D}\,\theta - \Delta({}^{\text{D}}\text{Z}) + \big\{\Delta({}^{\text{B}}\text{Z}) - \text{B}\,\theta\big\}.$ We see that the value of $$\text{D}$$ that maximizes the enery released is the same for all $$\text{B}$$ (although the amount of enery released is not the same). As implied by the discussion in the previous section, different values of $$\theta$$ may produce different values of $$\text{D}$$.

Uncertainty regarding the mass excesses of isotopes translates into possible uncertainty about $$\text{D}$$ for a given value of $$\theta$$. Let $$\Delta$$ denote a vector of mass excesses for a given element, and let $$\text{D}(\theta,\Delta)$$ denote the value of $$\text{D}$$ that maximizes the energy released given both $$\theta$$ and $$\Delta$$. Let $$p(\Delta)$$ denote the probability distribution that arrises from uncertainty about the mass execesses. Such uncertainty can be taken into account by computing $p(\text{D}|\theta) = \int \text{D}(\theta,\Delta)\, p(\Delta)\, d\Delta,$ where $$p(\text{D}|\theta)$$ is a discrete probability distribution for each value of $$\theta$$. (If there is no uncertainty about $$\Delta$$ then $$p(\text{D}|\theta)$$ puts all its weight on a single value of $$\text{D}$$.) The branching ratios are obtained via $p(\text{D}) = \int p(\text{D}|\theta)\, p(\theta)\, d\theta.$ where $$p(\theta)$$ is the probability distribution for $$\theta$$ that arrises from the properties of the chain reaction.

## Calibration

Iwamura and associates have produced a large amount of experimental data. (See below for citations.)

We assume that the concentration of ambient neutron isotopes is stable, and that neutron transfer reactions occur at a steady rate that can be quantified by a fixed mean lifetime for survival of target nuclei, which we denote by $$t_m$$.

To match the lifetimes of these processes to the timescale of natural decay values we note that the target survival mean lifetime $$t_m$$ can be determined approximately from the Iwamura data. Overall $$t_m$$ increases as the target mass number A increases, roughly according to the relationship $t_m = 0.05\,\text{A}$ for time measured in days. The expression for $$t_m$$ indicates that initiation of neutron transfer is less probable for larger targets, suggesting that the alterations in nuclear structure required for neutron addition become more complex and therefore less likely.

Our analysis of data suggests that $$\theta$$ lies in the allowed range $2.95 < \theta < 3.82.$ In the absence of guiding evidence we assume that the values of $$\theta$$ are uniformly distributed throughout this range.

## Calculation: System of ODEs

Using the equations above and the specified parameters values, a system of ordinary differential equations is obtained that can be used to calculate the theoretical evolution of neutron transfers and isotope decays through successive generations of transmutation for any target element or isotope.

## Iwamura experimental data

• Yasuhiro Iwamura, Takehiko Itoh and Ichiro Toyoda, Observations of Anomalous Nuclear Effects in D2-Pd System,'' Proc.\ ICCF-4, Maui, Hawaii, December 6--9, 1994, vol.\ 2, p12.
• Takehiko Itoh, Yasuhiro Iwamura, Nobuaki Gotoh and Ichiro Toyoda, Observation of Nuclear Products under Vacuum Condition from Deuterated Palladium with High Loading Ratio,'' Proc.\ ICCF-5, Monte Carlo, Monaco, April 9--13, 1995, p189.
• Yasuhiro Iwamura, Nobuaki Gotoh, Takehiko Itoh and Ichiro Toyoda, Characteristic X-ray and Neutron Emission from Electrochemically Deuterated Palladium,'' Proc.\ ICCF-5, Monte Carlo, Monaco, April 9--13, 1995, p197.
• Yasuhiro Iwamura, Takehiko Itoh, Nobuaki Gotoh and Ichiro Toyoda, Correlation between Behavior of Deuterium in Palladium and Occurrence of Nuclear Reactions Observed by Simultaneous Measurement of Excess Heat and Nuclear Products,'' Proc.\ ICCF-6, Toya, Japan, October 13--18, 1996, p679.
• Yasuhiro Iwamura, Takehiko Itoh, Nobuaki Gotoh, Mitsuru Sakano, Ichiro Toyoda and Hiroshi Sakata, Detection of anomalous elements, X-ray and excess heat induced by continuous diffusion of deuterium through multilayer cathode (Pd/CaO/Pd),'' Proc.\ ICCF-7, Vancouver, Canada, April 19--24, 1998, p167.
• Yasuhiro Iwamura, Takehiko Itoh and Mitsuru Sakano, Nuclear products and their time dependence induced by continuous diffusion of deuterium through multi-layer palladium containing low work function material,'' Proc.\ ICCF-8, Lerici (La Spezia), Italy, May 21--26, 2000, p141.
• Yasuhiro Iwamura, Takehiko Itoh, Mitsuru Sakano and Satoshi Sakai, Observation of Low Energy Nuclear Reactions Induced by D2 Gas Permeation Through Pd Complexes,'' Proc.\ ICCF-9, Beijing, China, May 19--24, 2002, p141.
• Yasuhiro Iwamura, Mitsuru Sakano and Takehiko Itoh, Elemental Analysis of Pd Complexes: Effects of D2 Gas Permeation,'' Jpn.\ J.\ Appl.\ Phys.\ Vol.41 (2002).
• Yasuhiro Iwamura, Takehiko Itoh and Mitsuru Sakano, Nuclide Transmutation Device and Nuclide Transmutation Method,'' United States Patent Application Publication, US 2002/0080903 A1, June 27, 2002.
• Yasuhiro Iwamura, Takehiko Itoh, Mitsuru Sakano, Satoshi Sakai and Shizuma Kuribayashi, Low Energy Nuclear Transmutation in Condensed Matter Induced by D2 Gas Permeation Through Pd Complexes: Correlation Between Deuterium Flux and Nuclear Products,'' Proc.\ ICCF-10, Cambridge, Massachusetts, USA, August 24--29, 2003, p435.
• Yasuhiro Iwamura, Takehiko Itoh, Mitsuru Sakano, Noriko Yamazaki, Shizuma Kuribayashi, Yasuko Terada, Tetsuya Ishikawa and Jirohta Kasagi, Observation of Nuclear Transmutation Reactions Induced by D2 Gas Permeation Through Pd Complexes,'' Proc.\ ICCF-11, Marseilles, France, Oct 31--Nov 5, 2004, p339.
• Yasuhiro Iwamura, Takehiko Itoh, Noriko Yamazaki, N. Watari, K. Muta, S. Tsuruga, H.Yonemura, K. Fukutani and D. Sekiba, Transmutation Experiments induced by Deuterium Permeation through Nano-structured Pd Multilayer Thin Film,'' JCF12 Abstracts, 12th Meeting of Japan CF-Research Society, Kobe University, Japan, Dec 17--18, 2011, p14.
• Yasuhiro Iwamura, Takehiko Itoh, Y. Terada, T. Ishikawa, Transmutation Reactions Induced by Deuterium Permeation through Nano-structured Pd Multilayer Thin Film,'' Transactions of the American Nuclear Society, vol. 107, pp. 422--425 San Diego, California, November 11--15, 2012.