Nuclear fusion without laser fields

We began with nuclear fusion in the absence of laser fields. The nuclear fusion process is usually divided into three regions according to the relative distance between the two nuclei, as illustrated in Fig. 1(a). From the rest frame of one nucleus (noted nucleus 1 for convenience), the other nucleus (nucleus 2) is initially in region III with an asymptotic energy E, which is usually between 1 and 10 keV depending on the temperature of the fusion environment. As it approaches, nucleus 2 will reach a classical turning point, where the Coulomb repulsive energy between the two nuclei equals the energy E. Via the quantum tunneling effect, nucleus 2 enters and passes through region II with a small probability. In region I, the two nuclei are very close to each other, and fusion reactions occur. The spatial range of region I is on the order of 1 fm (10^-15 m), and the spatial range of region II is on the order of 100 fm for the typical energies of controlled fusion research.

Fig. 1Full-screen View

(a) Schematic illustration of the three-region division of a nuclear fusion process. (b) The deuteron-triton fusion cross section as a function of the relative collision energy E. (c) The corresponding S function

Indeed, a fusion cross section is usually written in the following form corresponding to the three-region division [50](1)The first factor on the right hand side, S(E), is the result of many-body nuclear physics in region I. The exponential factor is the result of the tunneling process occurring in region II [51]. The 1/E factor, called the geometrical factor, is related to region III. The constant B_G is known as the Gamow constant. For the deuteron-triton (DT) fusion reaction (D + T → ⁴He + n + 17.6 MeV), . An ab initio calculation of the function S(E) is extremely demanding, if possible. For most purposes, including that of the current article, it suffices to adopt an empirical form [52](2)where the parameters Ai and Bi are determined by fitting the experimental data. For DT fusion, the values of these parameters are given as follows [52]:Note that these values are accompanied by the value of E in keV when Eq. (2) is employed. The DT fusion cross section σ(E) and the function S(E) are shown in Fig. 1(b) and (c), respectively. S(E) is a slow-varying function, whereas σ(E) is an exponential function owing to the tunneling process. σ(E) depends on E very sensitively, particularly for relatively low E values.

Two nuclei in a laser field, the center-of-mass reference frame

Now, let us consider the effects of an external laser field on the fusion process. Consider two nuclei with charge and mass and placed in a plane wave laser field. Two commonly used gauges are employed in light-matter interactions. One is the so-called velocity gauge, in which the laser field is characterized by a vector potential . The total Hamiltonian is written as(3)where p₁ and p₂ are the momenta of the two nuclei, and is the distance between the two nuclei. Atomic units have been used, with 4πϵ₀=1, hence the form of the Coulomb potential. We assumed the validity of the long-wavelength approximation by omitting the spatial dependency of the vector potential. This is justified by the fact that the spatial range relevant to nuclear fusion is much smaller than the wavelengths of the available intense lasers. Further discussion on this point is provided in Sect. 4.

For a fusion process, the relative motion between the two nuclei is the most relevant. Therefore, it is more convenient to work in the center-of-mass (CM) reference frame. Define(4)(5)(6)(7)Then, after some straightforward algebra, the Hamiltonian in Eq. (3) can be rewritten as(8)where(9)(10)(11)(12)One sees from Eq. (8), the Hamiltonian can be separated into the motion of the CM with charge and mass and the motion of a virtual particle with charge and mass . The mutual Coulomb potential is unaffected. The motion of the CM is not of concern here because it is not relevant to the fusion process.

Alternatively one may use the so-called length gauge and the Hamiltonian is given as(13)where denotes the laser electric field. Again, the long-wavelength approximation was assumed by neglecting the spatial dependency of the laser electric field. The above Hamiltonian can also be written in the CM reference frame as(14)The motion of the two nuclei is equivalent to the motion of the CM with plus the motion of a virtual relative-motion particle with .

Assumption on region I

The size of region I is on the order of 1 fm. In this region, the two nuclei fuse, and new particles are generated. For the DT fusion reaction, an α particle and a neutron are generated. This is a complex many-body nuclear process, and an ab initio treatment of this process is usually impossible. The addition of influences from an external laser field complicates the process.

However, it is reasonable to expect that the effect of the laser field in this region is negligibly. From the uncertainty principle, the spatial confinement of fm =1.9 × 10^-5 a.u. indicates an uncertainty in momentum of a.u., or an uncertainty in energy of a.u. ≈17 MeV. Here, μ = 2,204 a.u. is the reduced mass of the deuteron and triton. In comparison, the effect of an intense laser field on the fusion process is on the order of 1 keV in energy, which is approximately four orders of magnitude smaller. It seems safe to neglect effects of external laser fields on the nuclear processes happening in region I. This is the basic assumption on which our analyses are based.

Effects of laser fields on region II

The size of region II is on the order of 100 fm. In this section, we show that an intense laser field has a finite but small effect on the tunneling process occurring in this region. The probability of tunneling, or the “penetrability”, through a laser-modified Coulomb potential barrier can be calculated using the Wentzel-Kramers-Brillouin (WKB) method as(15)where is the Coulomb potential and is the laser-induced interaction potential from Eq. (14). Here, θ is the angle between the laser polarization direction and the DT collision (i.e., relative motion) direction. The integration is performed between two classical turning points, r₁ and r₂, which are also the two ends of region II.

What is implicit in writing Eq. (15) is a quasi-static approximation of the pressure. That is, the laser potential can be viewed as static at each time. This approximation is valid when the period of the laser field is much longer than the timescale of the tunneling process. More discussion on this point is provided in Sect. 4.

It can be estimated that the magnitude of V_I is much smaller than that of V_C or . The Coulomb potential V_C in region II can be estimated to be on the order of 100 keV, and the relative collision energy E is on the order of 1 keV in typical nuclear fusion experiments. With an intensity of 10²⁰ W/cm², the magnitude of V_I can be estimated to be on the order of 10 eV. With an intensity of 10²² W/cm², the magnitude of V_I is approximately 100 eV. Therefore we may expand Eq. (15) as(16)where is the penetrability without external laser fields, and γ⁽¹⁾ denotes the first-order correction induced by the laser field. Substituting the expression of V_I we get an explicit formula(17)The magnitude of γ⁽¹⁾ is maximum when reaches peaks and when cos θ = 1. The upper limit of integration r₂ is determined by equating the Coulomb potential to the collision energy E. For E = 5 keV, we get r₂ = 288.3 fm. The lower limit of integration r₁ is determined by a DT touching condition: fm = 3.05 fm. We find that for an intensity of 10²⁰ W/cm², γ⁽¹⁾ takes a maximum value of 0.18%. For an intensity of 10²² W/cm², the maximum value was 1.8%. For E = 10 keV, r₂ = 144.2 fm. The corresponding maximum value of γ⁽¹⁾ is 0.03% for 10²⁰ W/cm², and 0.3% for 10²² W/cm².

Therefore one can see, for currently available state-of-the-art laser intensities, the effects of laser fields on region II are finite but small.

Effects of laser fields on region III

The major effects of a laser field on the fusion process originate from region III. The laser field can substantially influence the collision energy E.

Without external laser fields, the incoming relative-motion virtual particle is usually described asymptotically as a plane wave(18)where the momentum has a magnitude . This plane wave state has a well-defined energy E.

In the presence of a laser field, the asymptotic plane-wave state becomes a Volkov state [53](19)where H_I is the interaction Hamiltonian with an external laser field. It is convenient to use the velocity gauge here, and from Eq. (8)(20)Let us assume that the laser field is linearly polarized along the z axis, and the vector potential . The cases of elliptical or circular polarization are discussed in Sect. 4. Note that because the laser field can be very intense, the A² term cannot be simply ignored as in low-intensity cases.

The Volkov state can be expanded in terms of photon numbers(21)where the coefficient is given by the following integral(22)For convenience we have defined (the ponderomotive energy), , and . Here, θ is the angle between p and the +z axis, and θ enters into the formalism through u. In a thermal environment, the direction between p and the laser polarization axis is random.

One sees from Eq. (21) that in the presence of a laser field, the collision energy is no longer a well-defined single value. Instead, the energy becomes a distribution, which is centered at E+U_p (ponderomotive shift) and separated by the photon energy. The energy of the particle can be higher or lower than E+U_p, corresponding to the absorption or emission of photons. The probability of finding the system with energy is(23)The total probability summing over the photon number n is equal to unity(24)

Energy distribution with different laser parameters

The energy distribution Pn depends sensitively on the laser parameters, particularly the frequency (photon energy). At the same laser intensity, the energy distribution can vary significantly for lasers with different photon energies. This is illustrated in Fig. 2, which shows the energy distributions for six different photon energies under the same intensity. The bare collision energy without the laser fields was assumed to be 5 keV (corresponding to a temperature of 5.8×10⁷ K).

Fig. 2Full-screen View

Energy distributions of the Volkov state with different laser photon energies (frequencies) as labeled on the upper-right corner of each figure. The lasers were assumed to have the same intensity of 10²⁰ W/cm². The bare collision energy E without the laser field is set to be 5 keV for all the cases

One can see that for a high-frequency laser with a photon energy of 1 keV (1,000 eV), almost the entire population still has the original collision energy of 5 keV. The probability of absorbing (emitting) a photon and changing the energy to 6 keV (4 keV) is very small, with a value of 2.6×10^-6. This probability is not visually distinguishable on a linear scale, as shown in Fig. 2(a). The probability of absorbing (emitting) two photons and changing the energy to 7 keV (3 keV) is on the order of 10^-12. It is very difficult to absorb (emit) energy from (to) a high-frequency laser field, even if the intensity is high. The probability of absorbing or emitting more photons exhibits a perturbative feature. That is, the probability decreases substantially as the number of photons increases.

As the photon energy decreases to 100 eV (Fig. 2(b)), the probability of absorbing (emitting) a photon increases to about 0.025. This indicates that 2.5% of the population has an energy of 5.1 keV, and another 2.5% has an energy of 4.9 keV. The probability of absorbing (emitting) two photons is on the order of 10^-4. As the photon energy decreased to 50 eV (Fig. 2(c)), the probability of absorbing (emitting) one photon is about 27%, and that of absorbing (emitting) two photons is about 3.3%. The probability of remaining with the original collision energy decreased to 39%.

As the photon energy decreased to 30 eV (Fig. 2(d)), the energy distribution shows clear nonperturbative features. For example, the probability of absorbing (emitting) two photons is higher than that of absorbing (emitting) a single photon. The number of photons absorbed or emitted was approximately 5, and the energy range populated was approximately 4.85–5.15 keV.

As the photon energy decreases to 10 eV (Fig. 2(e)), the number of photons absorbed or emitted is about 40. The populated energy range was between 4.6 and 5.4 keV. In addition, the distribution shows an overall structure with peaks near the two ends and a valley in the middle. This indicates that the collision has substantial probabilities with energies away from the original bare energy. As the photon energy decreased to 1.55 eV (Fig. 2(f)), which corresponds to a wavelength of 800 nm from Ti:sapphire intense lasers, the number of photons absorbed or emitted is over 1,300, and energy range between 3.1 and 7.3 keV is substantially populated. In this case, the ponderomotive shift U_p = 109 eV, and the distribution is not exactly symmetric.

The main message from the above results is that it is easier for low-frequency lasers to deliver energy to a fusion system. Although the energy of a single photon is small, the number of participating photons can be very large, so the populated energy range is wide.

Note that the energy distribution Pn depends on the parameter u, which depends on the angle θ between the laser polarization direction and collision (relative motion) direction. We set θ=0° for all cases shown in Fig. 2. As θ increases from 0° to 90°, the energy distribution becomes narrower, as shown in Fig. 3 for the case of 1.55 eV (800 nm). Pn for θ>90° is the same as that for (180°-θ). It is easier for the laser to deliver energy to the fusion system if the axis of collision and the axis of laser polarization are aligned.

Fig. 3Full-screen View

(Color online) Energy distributions for three different θ angles, as labeled on figure, between the laser polarization direction and the DT collision direction. The laser has a photon energy of 1.55 eV (wavelength of 800 nm) and an intensity of 10²⁰ W/cm². The 0° case is the same as Fig. 2(f)

Enhancement of fusion

In the presence of a laser field, the collision energy E changes from a single value to a distribution, the character of which depends on the laser parameters. The fusion system can either absorb energy from the laser field, leading to collision energies higher than E, or lose energy to the laser field, leading to energies lower than E (the center of the energy distribution is shifted to E+U_p). Energies higher than E lead to higher fusion yields, and energies lower than E lead to lower fusion yields.

However, the net effect is an enhancement of the fusion yield. This is because the cross section function in Eq. (1) depends exponentially on the collision energy (concave upward), as shown in Fig. 4 in linear scale. The fusion yields gained at higher energies are greater than those lost at lower energies. This is the mechanism of fusion-yield enhancement in the presence of laser fields.

Fig. 4Full-screen View

(Color online) Illustration of fusion enhancement in a laser field. Without laser fields, the bare collision energy is assumed to be 5 keV, corresponding to a fusion cross section of 0.538×10^-3 barn (dashed horizontal line). In the presence of a laser field, the collision energy becomes a distribution (an example is shown as the red curve on the left axis), with energies higher and lower than 5 keV. Higher energies lead to higher fusion yields, whereas lower energies lead to lower fusion yields. However, the fusion yields gained with higher energies are greater than the fusion yields lost with lower energies because the cross section function is an exponential function, concaving upward (black solid curve, right axis)

Effective fusion cross section

It is sensible to define an effective fusion cross section in the presence of a laser field. We denote this laser-assisted cross section as σ_L (E), which can be calculated by averaging over all θ angles between the collision direction and the laser polarization direction(25)where σ_L (E, θ) is defined as(26)The effective cross section σ_L (E), in comparison to the laser-free cross section σ(E), provides a quantitative measure of the effect of a laser field on the nuclear fusion process. It should be emphasized that the laser field mainly affects the fusion process by modifying the collision energy in region III before the tunneling process. It has very small effects on the processes occurring in regions I and II, as explained above.

Figure 5 shows the laser-assisted effective DT fusion cross section for three bare energies (E = 1, 5, and 10 keV) and different laser intensities and photon energies. These three energies represent the typical temperatures in thermonuclear fusion experiments. One sees from Fig. 5 that to have noticeable effects on nuclear fusion, the laser intensity needs to be higher than 10¹⁸ or 10¹⁹ W/cm².

Fig. 5Full-screen View

(Color online) Laser-assisted effective DT fusion cross sections for bare energies (a) 1, (b) 5, and (c) 10 keV. For each case, different laser intensities and photon energies were used, as labeled in the figure. The cross section values without laser fields are marked in each figure as a horizontal dashed line

One can also see that σ_L is larger for lower laser frequencies at the same intensity. For all three cases and with the intensity range shown, the fusion enhancement for the higher frequency cases (100 eV and 1,000 eV) is very small. This is the direct consequence of the point explained above, that low-frequency lasers are more efficient in delivering energy to the fusion system. It is difficult to absorb energy from high-frequency laser fields.

Substantial enhancements can be observed with low-frequency 1.55 eV lasers. For E = 1 keV, the enhancement is three orders of magnitude at an intensity of 10²⁰ W/cm² and nine orders of magnitude at an intensity 5×10²¹ W/cm². The enhancement ratio decreases as E increases. This is because the cross section function σ(E) in Eq. (1) is more sensitive to E for smaller E values. The effect of the laser field was more pronounced for smaller E values.

The possibility of using lasers to reduce the temperature requirements of fusion reactions is evident. For example, without laser fields, the DT fusion cross section at E = 1 keV is 1.37×10^-11 barn. With an 800-nm (1.55-eV) laser field of intensity 10²⁰ W/cm², the effective fusion cross section becomes 1.02×10^-8 barn, which is equal to the cross section value at E = 1.6 keV without laser fields. If the laser intensity is 5×10²¹ W/cm², the effective cross section becomes 0.027 barn, which is equal to the cross section value at E = 10 keV without laser fields. In other words, the enormous gap in the DT fusion cross section between 1 keV (1.16×10⁷ K) and 10 keV (1.16×10⁸ K) is filled or compensated completely by the intense laser field.

Applicability to other fusion reactions

Although the results presented above are for the DT fusion reaction, our analyses also apply to other fusion reactions, such as the neutronless proton-boron (p-¹¹B) fusion reaction [54-58], which is limited in its practical application because of its significantly lower fusion cross section compared to that of DT fusion under low bare energy conditions.

The p-¹¹B interaction exhibits a larger effective charge, q = 0.5, in contrast to the value of q = 0.2 for the deuteron-triton (DT) system. Furthermore, the reduced mass for the p-¹¹B pair is notably smaller, with a.u., compared to a.u. for the DT pair. Consequently, under identical laser parameters, the ponderomotive energy, given by , was higher for the p-¹¹B system than for the DT system. The parameters and indicate that a greater number of photons are absorbed or emitted by the p-¹¹B pair compared to the DT pair in laser fields.

Figure 6(a) illustrates that the energy distribution of the Volkov state for the p-¹¹B system is broader than that of the DT system, as depicted in Fig. 3. The laser was characterized by a photon energy of 1.55 eV and a peak intensity of 10²⁰ W/cm². Figure 6(b) presents the laser-assisted effective cross section for p–¹¹B fusion as a function of bare collision energy, assuming a laser photon energy of 1.55 eV. Various laser intensities were considered, as indicated in the figure. The cross section in the absence of laser fields is given in Ref. [59], is indicated by the black dashed line. Notably, under the influence of a strong laser field with an intensity of 5 × 10²¹ W/cm², the effective cross section of p–¹¹B fusion can reach the order of 10^-4 barn in the bare energy range of 1–10 keV, representing a significant enhancement in fusion reactivity. The resonant peak of the p–¹¹B fusion cross section, originally located at approximately 150 keV, shifts toward a lower and broader bare energy range under the influence of the laser field. This further indicates an effective enhancement of the fusion reactivity. These results highlight the potential of laser-assisted p–¹¹B fusion as a promising pathway for future research and development in clean, radiation-free energy generation.

Fig. 6Full-screen View

(Color online) (a) Energy distribution of the Volkov state for p-¹¹B fusion reaction. θ denotes the angle between the laser polarization direction and the p-¹¹B collision direction. The laser had a photon energy of 1.55 eV and a peak intensity of 10²⁰ W/cm². (b) Laser-assisted effective p-¹¹B fusion cross sections as a function of bare collision energy, assuming a laser frequency of 1.55 eV. The results for different laser intensities are shown. The black dashed line denotes the cross section in the absence of the laser field

However, laser fields have no effect, at least within the approximations adopted in this study, on fusion reactions with two nuclei having the same charge-to-mass ratio. An example of this type is the deuteron-deuteron fusion reaction. If so, one can find from Eq. (12) that the charge of the relative-motion virtual particle is zero and the laser ceases to have an effect on the relative-motion degree of freedom. This is easily understood because the motion of a charged particle in a laser field is determined by its charge-to-mass ratio. If two nuclei have the same charge-to-mass ratio, then their motion in the laser field will be the same, and the laser field has no tendency to separate them apart or press them closer, that is, the laser field has no effect on the relative motion between the two nuclei.

The long-wavelength approximation

We assumed the validity of the long-wavelength approximation by neglecting the spatial dependency of the laser vector potential or the electric field. This is justified if the spatial range relevant to fusion is significantly smaller than the laser wavelength. The former range can be estimated by the quiver motion amplitude , which is the amplitude of spatial oscillations of a free charged particle in a laser field. The required validity condition of the long-wavelength approximation is(27)Putting in the values of μ and q for the DT fusion we get a.u., or the amplitude of the laser electric field in a.u.. We provide two examples. For photon energy 1,000 eV, ω = 36.8 a.u., so a.u. or equivalently the intensity W/cm². For photon energy 1.55 eV, ω = 0.0569 a.u., so a.u. or equivalently the intensity W/cm². The validity of the long-wavelength approximation can be guaranteed for both the examples.

The quasi-static approximation

As mentioned in Sect. 2.4, when writing Eq. (15), we have implicitly used the quasi-static approximation. This approximation is valid when the laser period is much longer than the time scale of the tunneling process, which can be estimated using a classical model. For example, for E = 5 keV, the velocity of relative motion is a.u. The tunneling length (from the tunneling entrance point to the tunneling exit point) is 5.38×10^-3 a.u. (285 fm), and the estimated time for the tunneling process is 0.013 a.u. or 0.32 as (1 as = 10^-18 s). The quasi-static approximation is valid as long as the laser period is much longer than 0.32 as or the photon energy is much lower than 13 keV.

We note that the quasi-static approximation is also an important concept in strong-field atomic ionization [60, 61]. The ratio between the classically estimated electron tunneling time and the laser period is called the Keldysh parameter [60]. The quasi-static approximation is valid when the Keldysh parameter , that is, the tunneling time scale is much shorter than the laser period.

The Coulomb Volkov state

For simplicity, we used the (plane-wave) Volkov state to describe the relative motion of the virtual particle. The Volkov state is the solution to the time-dependent Schrödinger equation for a free charged particle in the presence of a laser field. The Coulomb potential between the two nuclei was neglected.

The quantum state of the full Coulomb-plus-laser system does not have a general analytical solution. An approximate solution is the Coulomb-Volkov state [62, 63], which has the same form as the plane-wave Volkov state, except that replace the in Eqs. (19) or (21) with a Coulomb wave function . The temporal part, and hence the energy distribution, remained the same. Therefore, the results and discussions presented above are not affected if Coulomb Volkov states are used.

Elliptical or circular polarization

The extension of the above formalism to elliptically or circularly polarized laser fields is straightforward; therefore, we only outline a few steps here. Assuming that the vector potential is in the z-y plane: , with ellipticity ε and amplitude . The Volkov wavefunction can then be written in the same form as Eq. (21), except that has an additional argument w owing to the additional polarization direction. Here , , and . are the direction angles of momentum p. The coefficient Fn is obtained via the integral(28)Because elliptical polarization breaks the cylindrical symmetry of linear polarization, the effective fusion cross section σ_L must be averaged over both θ and ϕ. Figure 7 shows how the angle-averaged σ_L changes with ε for intensity 1×10²⁰ W/cm². Elliptical or circular polarization did not lead to further enhancements in the fusion yield. The efficiency is (slightly) lower than that of linear polarization, mainly because of the reduction in the field amplitude from A₀ to .

Fig. 7Full-screen View

(Color online) Dependency of angle-averaged effective fusion cross section σ_L on laser ellipticity ε, for three bare energies 1, 5, and 10 keV under intensity 1×10²⁰ W/cm². For each case, σ_L has been normalized to the corresponding value of linear polarization

Effects of plasma screening

In laser-induced plasmas, the range of charge screening effects can be characterized by a Debye sphere, with a radius given by the Debye length , where λ_D is in cm, T_e is in eV, and n_e is in units of cm^-3. Plasma screening becomes significant when the inter-nuclear separation exceeds the Debye length, because the Coulomb interaction is substantially shielded by the surrounding plasma electrons. Conversely, when the inter-nuclear distance is shorter than the Debye length, the screening effect is relatively weak, and the bare nuclear interaction dominates.

Assuming a plasma electron temperature of 2 keV and electron density of 10¹⁷ cm^-3, the corresponding Debye length is approximately 1 μm. This characteristic length scale is significantly larger than the amplitude of the quiver motion between two nuclei in a laser field with an intensity of 5×10²¹ W/cm² and a photon energy of 1.55 eV, which is approximately 0.56 nm. Consequently, the inter-nuclear separation remains well below the Debye length, and the influence of plasma screening is negligible. Therefore, the two-nuclei interaction model employed in this study is well justified in disregarding the effects of the plasma screening.

The plasma environment may also involve additional effects, such as collisional interactions and radiation processes, which can be further investigated through advanced laser-plasma simulations in future studies.