2. The source–target design for table-top fusion

Our exploration of the maximization of table-top fusion yields^{[Reference Ron, Last and Jortner26]} established that an increase of the table-top fusion efficiencies by 3–5 orders of magnitude for NFDCE of nanodroplets, relative to those attained inside or outside a plasma filament^{[Reference Heidenreich, Last and Jortner2,Reference Madison, Patel, Price, Edens, Allen, Cowan, Zweiback and Ditmire6,Reference Madison, Patel, Allen, Price, Fitzpatrick and Ditmire7,Reference Last and Jortner11,Reference Last and Jortner19,Reference Heidenreich, Last, Jortner, Kühn and Wöste23]} , can be attained by transcending the macroscopic plasma filament as a reaction medium for table-top fusion and by considering a source–target design, which was advanced in our previous work^{[Reference Last, Ron and Jortner25,Reference Ron, Last and Jortner26]} , with the following operational conditions.

1. (1) The source–target design is based on the selection of an appropriate source (where high-energy deuterons or protons are produced by CE) and a target (where the fusion reaction occurs). For fusion between two distinct nuclei, high-energy deuterons (or protons) are produced with the source by CE of homonuclear deuterium or hydrogen nanodroplets^{[Reference Last, Ron and Jortner25,Reference Ron, Last and Jortner26]} . The ions react with a solid target of the second reagent.

2. (2) Regarding the properties of the source within the source–target design, a key element for efficient fusion rests on the production of high-energy (up to 15 MeV) deuterons or protons^{[Reference Last and Jortner24]}.

3. (3) The beneficial properties of the cylindrical hollow solid target within the source–target design originate from the efficient collection of high-energy deuterons and protons from the source, together with the moderately low stopping power and large penetration depth of deuterons within the solid^{[Reference Last, Ron and Jortner25,Reference Ron, Last and Jortner26]} .

3. Table-top fusion yields

High table-top fusion yields were calculated for reactions of deuterons with several light nuclei, i.e., ${}^{7} \mathrm{Li} $, ${}^{6} \mathrm{Li} $, and D, within the source–target reaction design^{[Reference Last, Ron and Jortner25,Reference Ron, Last and Jortner26]} . The source consists of deuterons produced by CE of deuterium nano-droplets, ${R}_{0} = 70{\unicode{x2013}} 300~\mathrm{nm} $), impinging on a hollow solid cylinder target containing the ${}^{7} \mathrm{Li} $, ${}^{6} \mathrm{Li} $, and D atoms. The fusion reactions with the highest cross sections in the relevant energy domains^{[Reference Rose and Clark31–Reference Eliezer, Henis and Martinez-Val33]} were considered. The cylindrical solid target involves ${}^{7} \mathrm{Li} $ or ${}^{6} \mathrm{Li} $ (pure metal or LiF ionic solid) for reactions of D with Li isotopes, and low-temperature ($T\lt 20~\mathrm{K} $) deuterium film or deuterated $({\mathrm{CD} }_{2} )$ polymer polyethylene^{[Reference Heidenreich, Last and Jortner2,Reference Davis, Petrov and Velikovich13,Reference Ron, Last and Jortner26]} at room temperature for the $\mathrm{D} {+ }\mathrm{D} $ reaction.

The fusion reaction yield $Y$ per laser pulse is

(1)$$\begin{eqnarray}\displaystyle Y= N\langle y\rangle , &&\displaystyle\end{eqnarray}$$

where $N$ is the number of deuterons produced from the source and $\langle y\rangle $ is the average reaction probability:

(2)$$\begin{eqnarray}\langle y\rangle = \int \nolimits \nolimits_{0}^{{E}_{\max } } P(E)y(E)dE.\end{eqnarray}$$

Here $P(E)$ is the energy distribution function of the ions with a maximal energy ${E}_{\max } $, obtained from scaled electron and ion dynamics (SEID) simulations described in our previous work^{[Reference Last and Jortner24,Reference Last and Jortner34,Reference Last and Jortner35]} , and $y(E)$ is the reaction probability per ion with an initial energy $E$ penetrating into the solid target, given by

(3)$$\begin{eqnarray}y(E)= \int \nolimits \nolimits_{0}^{E} \frac{\sigma ({E}^{\prime } )}{S({E}^{\prime } )} d{E}^{\prime } ,\end{eqnarray}$$

where $\sigma (E)$ is the reaction cross section^{[Reference Rose and Clark31–Reference Eliezer, Henis and Martinez-Val33]} and $S(E)$ is the stopping power normalized to the atomic density of the target^{[Reference Andersen and Ziegler36]}. The energy dependence of $y(E)$ (inset to Figure 1 ) over the relevant energy domain up to 15 MeV (which corresponds to the CE energies) is determined by the cumulative contributions of $\sigma (E)$ and $S(E)$. $y(E)$ exhibits a nearly power-law dependence on $E$ (inset to Figure 1), in the form

(4)$$\begin{eqnarray}\displaystyle y(E)= b{E}^{\xi } , &&\displaystyle\end{eqnarray}$$

where $b$ is a constant and $\xi $ is a scaling parameter. The data of the inset to Figure 1 result in $\xi = 1. 8\pm 0. 3$ for $\mathrm{D} {+ }\mathrm{D} $, $\xi = 2. 2\pm 0. 7$ for $\mathrm{D} {+ }{\text{} }^{7} \mathrm{Li} $ and $\xi = 2. 9\pm 0. 7$ for $\mathrm{D} {+ }{\text{} }^{6} \mathrm{Li} $. For the conditions of complete vertical outer ionization (CVI) of the nanodroplet^{[Reference Heidenreich, Last and Jortner2,Reference Last and Jortner22–Reference Last and Jortner24]} , the CE energetics is determined by electrostatic models^{[Reference Heidenreich, Last and Jortner2,Reference Last and Jortner22,Reference Heidenreich, Last, Jortner, Kühn and Wöste23]} . Under CVI conditions the kinetic energy distribution of the deuterons is^{[Reference Heidenreich, Last and Jortner2,Reference Last and Jortner22,Reference Heidenreich, Last, Jortner, Kühn and Wöste23]}  $P(E)= (3/ 2{E}_{\max } )\mathop{(E/ {E}_{\max } )}\nolimits ^{1/ 2} $ for $0\lt E\leq {E}_{\max } $, with the maximal kinetic energy being^{[Reference Heidenreich, Last and Jortner2,29,Reference Yu-dong, Tian-Xuan, Huang, Xia-Yu, Xiao-Shi, Tang-Qi, Zi-Feng, Jia-Bin, Tian-Ming, Ming, Rui-Zhen, Xiao-An, Chao-Guang, Lu, Jia-Hua, Long-Fei, Bo-Lun, Ming, Wei, Bo, Ji, Ping, Hai-Le, Shao-En and Yong-Kun30]}  ${E}_{\max } = a{ R}_{0}^{2} $, where $a= (4\pi / 3)\bar {B} {\rho }_{\mathrm{mol} } {q}^{2} $, with $\bar {B} = 1. 44\times 1{0}^{- 3} ~\mathrm{keV} ~\mathrm{nm} $, ${\rho }_{\mathrm{mol} } $ is the initial density of the nanostructure, and $q= 1$ is the ion charge. The validity of the CVI relations for the energetics of CE is borne out of SEID simulations^{[Reference Last and Jortner24,Reference Ron, Last and Jortner26,Reference Last and Jortner34,Reference Last and Jortner35]} , which include intra-nanodroplet intensity attenuation^{[Reference Last and Jortner24]} and relativistic effects^{[Reference Last and Jortner37,Reference Last and Jortner38]} . Equations (1), (2) and (4), together with the CVI relations for ${E}_{\max } $ and $P(E)$, result in

(5)$$\begin{eqnarray}Y= [N/ (\zeta + 3/ 2)] ba{ R}_{0}^{\zeta }\end{eqnarray}$$

with

(6)$$\begin{eqnarray}\zeta = 2\xi .\end{eqnarray}$$

Equation (5) predicts a power law for the nanodroplet size dependence of the fusion yields, with the scaling parameter $\zeta $ being given by Equation (6).

Figure 1. Nanodroplet size dependence of the table-top fusion yields $Y$, Equation (1), within the source–target design for the fusion of deuterons with a solid hollow cylinder of ${}^{7} \mathrm{Li} $, ${}^{6} \mathrm{Li} $, solid deuterium, and deuterated polyethylene $({\mathrm{CD} }_{2} )$, as marked on the curves. The laser parameters are ${I}_{M} = 5\times 1{0}^{19} ~\mathrm{W} \cdot {\mathrm{cm} }^{- 2} $, $\tau = 30~\mathrm{fs} $, and $\boldsymbol{W}= 0. 6~\mathrm{J} $. The inset shows the energy dependence of the fusion reaction probability $y(E)$.

The fusion yields, Equation (1), were calculated from

1. (i) the $y(E)$ data of Equation (3) (presented in the inset to Figure 1), and the $P(E)$ functions obtained from SEID simulations, which result in $\langle y\rangle $, Equation (2);

2. (ii) the number $N$ of the deuterons produced from the Coulomb-exploding source.

$N$ is governed by laser energy deposition inside the plasma filament within the laser focal volume^{[Reference Last, Ron and Jortner25,Reference Ron, Last and Jortner26]} . The fraction $\beta $ of the laser energy acquisition by the assembly of nanodroplets is^{[Reference Last, Ron and Jortner25,Reference Ron, Last and Jortner26]} $\beta = N{E}_{\mathrm{abs} } / \boldsymbol{W}$ with $0\leq \beta \leq 1$, where $\boldsymbol{W}$ is the laser pulse energy and ${E}_{\mathrm{abs} } $ is the laser energy absorbed per atom within a nanodroplet, which was obtained from SEID simulations for exploding deuterium nanodroplets, while the laser parameters are the peak intensity ${I}_{M} = 5\times 1{0}^{19} ~\mathrm{W} \cdot {\mathrm{cm} }^{- 2} $, pulse duration $\tau = 3\times 1{0}^{- 14} ~\mathrm{s} $, pulse energy $\boldsymbol{W}= 0. 6~\mathrm{J} $^{[Reference Ter-Avetisyan, Schnürer, Hilscher, Jahnke, Busch, Nicles and Sandner17]}, and laser wavelength $\lambda = 8\times 1{0}^{- 5} ~\mathrm{cm} $. Following our previous work^{[Reference Ron, Last and Jortner26]}, the number of deuterons within the macroscopic plasma filament is

(7a)$$\begin{eqnarray}N= 3. 54(\rho / \lambda )\mathop{(\boldsymbol{W}/ {I}_{M} \tau )}\nolimits ^{2} ; \quad \beta \lt 1\end{eqnarray}$$

for the weak assembly intensity attenuation, and

(7b)$$\begin{eqnarray}N= \boldsymbol{W}/ {E}_{\mathrm{abs} } ; \quad \beta = 1\end{eqnarray}$$

for the strong assembly intensity attenuation. Here, the macroscopic plasma filament is characterized by the deuteron density, where $\rho = 3\times 1{0}^{18} ~{\mathrm{cm} }^{- 3} $^{[Reference Ter-Avetisyan, Schnürer, Hilscher, Jahnke, Busch, Nicles and Sandner17]}.

The nanodroplet size dependence of the fusion yields was calculated for the laser and nanoplasma parameters given above. For these input data, the weak assembly attenuation limit $\beta \lt 1$ is strictly applicable over the entire size domain^{[Reference Ron, Last and Jortner26]}. Furthermore, for the highest laser intensity ${I}_{M} = 5\times 1{0}^{19} $, the CVI relation is nearly applicable up to ${R}_{0} = 300~\mathrm{nm} $, whereupon Equation (5) is applicable for the analysis of the yield data. The nanodroplet size dependence of $Y$ portrayed in Figure 1 exhibits a nearly linear dependence of log $Y$ versus $\log {R}_{0} $, resulting in a power-law size dependence of $Y$ on ${R}_{0} $, of the form $Y\propto { R}_{0}^{\varsigma } $. The scaling parameters $\zeta $ obtained for Figure 1 are $\zeta = 3. 3\pm 0. 5$ for $\mathrm{D} {+ }\mathrm{D} $, $\zeta = 4. 5\pm 0. 7$ for $\mathrm{D} {+ }{\text{} }^{7} \mathrm{Li} $, and $\zeta = 5. 4\pm 0. 6$ for $\mathrm{D} {+ }{\text{} }^{6} \mathrm{Li} $. These scaling parameters $\zeta $, obtained for the nanodroplet size dependence of $Y$, obey the relation $\zeta = 2\xi $, where $\xi $, Equation (4), are the scaling parameters for the energy dependence reaction probability, Equation (3), which are presented above. This result is in accord with the relation predicted by Equation (6).

4. Fusion efficiencies and their dependence on the laser pulse energy

The fusion efficiency^{[Reference Li, Liu, Ni, Li and Xu15,Reference Ron, Last and Jortner26,Reference Davis and Petrov39]} is

(8)$$\begin{eqnarray}\displaystyle \Phi = Y/ \boldsymbol{W}. &&\displaystyle\end{eqnarray}$$

The fusion yields and efficiencies were maximized for the nanodroplet size and the laser parameters. Our results for $Y$ (Figure 1) and $\Phi $ were obtained at a fixed laser pulse energy of ${\boldsymbol{W}}_{0} = 0. 6~\mathrm{J} $^{[Reference Ter-Avetisyan, Schnürer, Hilscher, Jahnke, Busch, Nicles and Sandner17]} and at a high laser intensity of ${I}_{M} = 5\times 1{0}^{19} ~\mathrm{W} \cdot {\mathrm{cm} }^{- 2} $. We shall now advance a scaling method for the dependence of $Y$ and $\Phi $ on the laser pulse energy $\boldsymbol{W}$ for the domains of weak assembly intensity attenuation ($\beta \lt 1$) and strong assembly intensity attenuation ($\beta = 1$). Increasing $\boldsymbol{W}$ beyond ${\boldsymbol{W}}_{0} $ is expected to increase $Y$ and $\Phi $ for $\beta \lt 1$ in the range ${\boldsymbol{W}}_{0} \lt \boldsymbol{W}\leq {\boldsymbol{W}}_{M} $, while for $\beta = 1$ a distinct dependence of the parameters on $\boldsymbol{W}$ is realized in the range $\boldsymbol{W}\gt {\boldsymbol{W}}_{M} $. ${\boldsymbol{W}}_{M} $ marks the laser power for the ‘transition’ from $\beta \lt 1$ to $\beta = 1$, which is given by^{[Reference Ron, Last and Jortner26]}

(9)$$\begin{eqnarray}{\boldsymbol{W}}_{M} = \mathop{({I}_{M} \tau )}\nolimits ^{2} / 3. 54(\rho / \lambda ){E}_{\mathrm{abs} } .\end{eqnarray}$$

A typical value of ${\boldsymbol{W}}_{M} = 8~\mathrm{J} $ for the largest nanodroplet size and highest intensity, i.e., ${R}_{0} = 300~\mathrm{nm} $ and ${I}_{M} = 5\times 1{0}^{19} ~\mathrm{W} \cdot {\mathrm{cm} }^{- 2} $, was estimated from Equation (9). The $\boldsymbol{W}$ scaling of $Y(\boldsymbol{W})$ and of $\Phi (\boldsymbol{W})$ is obtained in the form^{[Reference Ron, Last and Jortner26]}

(10a)$$\begin{eqnarray}\displaystyle Y(\boldsymbol{W})/ Y({\boldsymbol{W}}_{M} )= \mathop{(\boldsymbol{W}/ {\boldsymbol{W}}_{M} )}\nolimits ^{2} ; \quad {\boldsymbol{W}}_{0} \lt \boldsymbol{W}\leq {\boldsymbol{W}}_{M} , &&\displaystyle\end{eqnarray}$$

(11a)$$\begin{eqnarray}\displaystyle \Phi (\boldsymbol{W})/ \Phi ({\boldsymbol{W}}_{M} )= (\boldsymbol{W}/ {\boldsymbol{W}}_{M} ); \quad {\boldsymbol{W}}_{0} \lt \boldsymbol{W}\leq {\boldsymbol{W}}_{M} , &&\displaystyle\end{eqnarray}$$

and

(10b)$$\begin{eqnarray}\displaystyle Y(\boldsymbol{W})/ Y({\boldsymbol{W}}_{M} )= (\boldsymbol{W}/ {\boldsymbol{W}}_{M} ); \quad \boldsymbol{W}\gt {\boldsymbol{W}}_{M} , &&\displaystyle\end{eqnarray}$$

(11b)$$\begin{eqnarray}\displaystyle \Phi (\boldsymbol{W})/ \Phi ({\boldsymbol{W}}_{M} )= 1; \quad \boldsymbol{W}\gt {\boldsymbol{W}}_{M} . &&\displaystyle\end{eqnarray}$$

Accordingly, the calculation of the maximal value of $\Phi $ will be achieved by SEID simulations for the optimization of $Y$ and $\Phi $ at a fixed laser pulse energy (${\boldsymbol{W}}_{0} \lt {\boldsymbol{W}}_{M} $), followed by the $\boldsymbol{W}$ scaling of these attributes from ${\boldsymbol{W}}_{0} $ to ${\boldsymbol{W}}_{M} $. The maximal value of $\Phi $ is $\Phi ({\boldsymbol{W}}_{M} )$. Our results for $Y({\boldsymbol{W}}_{M} )$ and $\Phi ({\boldsymbol{W}}_{M} )$ were obtained from the scaling of the SEID simulation results at ${\boldsymbol{W}}_{0} = 0. 6~\mathrm{J} $ (Figure 2). The optimal fusion yields per laser pulse (for the nanodroplet size ${R}_{0} = 300~\mathrm{nm} $, and laser parameters ${I}_{M} = 5\times 1{0}^{19} ~\mathrm{W} \cdot {\mathrm{cm} }^{- 1} $, $\tau = 30~\mathrm{fs} $, and ${\boldsymbol{W}}_{M} = 8~\mathrm{J} $) are $Y({\boldsymbol{W}}_{M} )= 3. 4\times 1{0}^{10} $, $1. 7\times 1{0}^{10} $, and $5. 3\times 1{0}^{9} $ for the fusion of D with ${}^{7} \mathrm{Li} $, ${}^{6} \mathrm{Li} $, and D, respectively. Our analysis then results in the attainment of the maximal high table-top fusion efficiencies, i.e., $\Phi ({\boldsymbol{W}}_{M} )= 4\times 1{0}^{9} ~{\mathrm{J} }^{- 1} , 2\times 1{0}^{9} ~{\mathrm{J} }^{- 1} $, and $7\times 1{0}^{8} ~{\mathrm{J} }^{- 1} $ for the fusion of D with ${}^{7} \mathrm{Li} $, ${}^{6} \mathrm{Li} $, and D, respectively.

Figure 2. A record of the currently available data for the dependence of the efficiency $\Psi $ of conversion of laser energy to nuclear energy on the laser pulse energy $\boldsymbol{W}$ for table-top fusion driven by CE of nanodroplets and in a source–target design. A comparison is presented between experimental data for DD fusion driven by CE of (${\mathrm{D} }_{2} {\mathop{)}\nolimits}_{n} $ and $\mathop{({\mathrm{CD} }_{4} )}\nolimits_{n} $ clusters inside and outside a macroscopic plasma filament^{[Reference Grillon, Balcou, Chambaret, Hulin, Martino, Moustaizis, Notebaert, Pittman, Pussieux, Rousse, Rousseau, Sebban, Sublemontier and Schmidt5–Reference Lu, Liu, Wang, Wang, Zhou, Deng, Xia, Xu, Lu, Jiang, Leng, Liang, Ni, Li and Xu8]}, theoretical–computational data for fusion of deuterium with light atoms ${}^{7} \mathrm{Li} $, ${}^{6} \mathrm{Li} $, T, and D within the source–target design (present work and reference 26), and of experimental data for DT and DD inertial fusion in ‘big science’ inertial fusion setups ^{[Reference Cok, Craxton and McKenty28–Reference Yu-dong, Tian-Xuan, Huang, Xia-Yu, Xiao-Shi, Tang-Qi, Zi-Feng, Jia-Bin, Tian-Ming, Ming, Rui-Zhen, Xiao-An, Chao-Guang, Lu, Jia-Hua, Long-Fei, Bo-Lun, Ming, Wei, Bo, Ji, Ping, Hai-Le, Shao-En and Yong-Kun30]}.

The maximal value of the laser energy to nuclear energy conversion efficiency for table-top fusion is

(12)$$\begin{eqnarray}\Psi ({\boldsymbol{W}}_{M} )= \Phi ({\boldsymbol{W}}_{M} )Q,\end{eqnarray}$$

where $Q$ is the energy release in the nuclear reaction and $\Phi ({\boldsymbol{W}}_{M} )= Y({\boldsymbol{W}}_{M} )/ {\boldsymbol{W}}_{M} $. The estimates for $\Psi ({\boldsymbol{W}}_{M} )$ for the fusion of D with ${}^{7} \mathrm{Li} $, ${}^{6} \mathrm{Li} $, and D are $1. 0\times 1{0}^{- 2} , 1. 1\times 1{0}^{- 3} $, and $3. 9\times 1{0}^{- 4} $, respectively.