2. THEORY

In the following, we summarize our applied non-relativistic quantum mechanical description. The laser field is handled in the classical way via the minimal coupling. The laser beam is taken to be linearly polarized and the dipole approximation is used. If the dimensionless intensity parameter (or the normalized vector potential) ${a_0} = 8.55 \times {10^{ - 10}}\sqrt {I({\rm W}/{\rm c}{{\rm m}^2})} {\rm \lambda} ({\rm \mu m})$ of the laser field is smaller than unity the non-relativistic description in dipole approximation is valid. For 800 nm laser wavelength this means a critical intensity of I = 2.13 × 10¹⁸ W/cm². In case of protons a ₀ is replaced by a _p = [(m _p/m _e)⁻¹]a ₀, where the proton to electron mass ratio is (m _p/m _e) = 1836. Accordingly, for 800 nm wavelength the critical intensity for protons is I _crit = 3.91 × 10²¹ W/cm².

Beyond the optical regime we investigate the scattering process in an X-ray laser field as well. Typical X-ray lasers can have photon energy in the range of 1–10 keV, pulse energy of 3 mJ and 10¹²–10¹³ photons/s photon number and the pulse duration is between 10 and 100 fs (the wavelength of a 10 keV X-ray photon is 18.2 nm). In the X-ray laser community, the photon number is the crucial parameter and not the intensity. However, the maximal achievable intensity can be calculated when the maximal focal spot is known. Focusing of X-ray laser pulses gives up numerous not trivial questions for experimentalist and still under development therefore we consider a maximal available intensity at 10¹⁶ W/cm² for a 10 keV laser pulse in our last model, where the dimensionless intensity parameter a ₀ = 1, 5 × 10⁻⁵. Note, that this is a small value compared to optical frequencies. The critical intensity for the 10 keV photon is a factor of 1836 times higher than for the 800 nm optical frequency.

Additionally, we consider moderate proton kinetic energy, not so much above the Coulomb barrier and neglect the interchange term between the proton projectile an the target carbon protons. This proton exchange effect could be included in the presented model with the help of Woods–Saxon potentials of non-local type (Barna et al., Reference Barna, Apagyi and Scheid2000) but not in the scope of the recent study.

To describe the non-relativistic scattering process of a proton on a nucleus in a spherically symmetric external field the following Schrödinger equation has to be solved,

(1)$$\left[ {{1 \over {2m}}{{\left( {\hat{\bf p} - {e \over c}{\bf A}} \right)}^2} + U({\bf r})} \right]{\rm \Psi} = i\hbar {{{\rm \partial} {\rm \Psi} } \over {{\rm \partial} t}}, $$

where $\hat{\bf p}=- i\hbar {\rm {\rm \partial}} /{\rm \partial} {\bf r}$ is the momentum operator of the proton, and U(r) represents the scattering potential of the nucleon, $${\bf A}(t)={A_0}{\rm {\rm \epsilon} }\cos ({\rm \omega} t)$ is the vector potential of the external laser field with unit polarization vector ε. Figure 1 presents the scattering geometry for a better understanding. The p_i and p_f are the initial and final proton momenta, θ is the scattering angle of the proton, the laser is linearly polarized in the x–z plane, and the propagation of the laser field is parallel to the x-axis.

Fig. 1.

The geometry of the scattering process. The ¹²C nucleus is in the center of the circle, p_i and p_f stand for the initial and final scattered proton momenta, θ is the proton scattering angle, laser pulse propagates parallel to the x-axis and linearly polarized in the x–z plane. The χ angle is needed for the laser-proton momentum transfer.

Without the external scattering potential U(r) the particular solution of (1) can be immediately written down as non-relativistic Volkov states φ_p(r, t) which exactly incorporate the interaction with the laser field,

(2)$$\matrix{ {{{\rm \varphi} _{\rm p}}({\bf r},t) = } \hfill & {\displaystyle{1 \over {{{(2{\rm \pi} \hbar )}^{3/2}}}}{\rm exp}\left[ {\displaystyle{i \over \hbar }{\bf p} \cdot {\bf r} - \int_{{t_0}}^t dt^{\prime}\displaystyle{1 \over {2m}}{{\left( {{\bf p} - \displaystyle{e \over c}{\bf A}(t^{\prime})} \right)}^2}} \right].} \hfill \cr } $$

Volkov states, which are modulated de Broglie waves, parameterized by momenta p and form an orthonormal and complete set,

(3)$$\eqalign{& \int {d^3}r{\rm \varphi} _{\rm p}^{\rm \ast } ({\bf r},t){{\rm \varphi} _{{\rm p^{\prime}}}}({\bf r},t) = {{\rm \delta} _3}({\bf p} - {\bf p^{\prime}}), \cr & \int {d^3}p{{\rm \varphi} _{\rm p}}({\bf r},t){\rm \varphi} _{\rm p}^{\rm \ast } ({\bf r^{\prime}},t) = {{\rm \delta} _3}({\bf r} - {\bf r^{\prime}}).} $$

To solve the original problem of Eq. (1) we write the exact wave function as a superposition of an incoming Volkov state and a correction term, which vanishes at the beginning of the interaction (in the remote past t ₀→−∞). The correction term can also be expressed in terms of the Volkov states, since these form a complete set [see the equation of (3)],

(4)$${\rm \Psi} ({\bf r},t) = {{\rm \varphi} _{{{\rm p}_{\rm i}}}}({\bf r},t) + \int {d^3}p{a_{\rm p}}(t){{\rm \varphi} _{\rm p}}({\bf r},t),\quad {a_{\rm p}}({t_0}) = 0. $$

It is clear that the unknown expansion coefficients a _p(t) describe the non-trivial transition symbolized as p_i→p, from a Volkov state of momentum p_i to another Volkov state with momentum p. If we take the projection of Ψ into some Volkov state φ_p(t), we get

(5)$$\int {d^3}r{\rm \varphi} _{\rm p}^{\rm \ast } ({\bf r},t){\rm \Psi} ({\bf r},t) = {{\rm \delta} _3}({\bf p} - {{\bf p}_{\rm i}}) + {a_{\rm p}}(t). $$

Bye inserting Ψ of Eq. (4) into the complete Schrödinger equation (1), we receive the following integro-differential equation for the coefficients a _p(t):

(6)$$\eqalign{i\hbar {{\dot a}_{{\rm p^{\prime}}}}(t) & =\int {d^3}r{\rm \varphi} _{{\rm p^{\prime}}}^{\rm \ast } ({\bf r},t^{\prime})U({\bf r}){{\rm \varphi} _{{{\rm p}_{\rm i}}}}({\bf r},t^{\prime}) \cr & \quad +\int {d^3}p{a_{\rm p}}(t)\int {d^3}r{\rm \varphi} _{{\rm p^{\prime}}}^{\ast} ({\bf r},t^{\prime})U({\bf r}){{\rm \varphi} _{\rm p}}({\bf r},t^{\prime}),} $$

where the scalar product was taken with ${{\rm \varphi} _{{\rm p^{\prime}}}}(t)$ on both sides of the resulting equation and the orthogonality property of the Volkov sates was taken after all [see the first equation of (3)]. Owing to the initial condition a _p(t ₀) = 0, displayed already in Eq. (4) the formal solution of (6) can be written as

(7)$$\eqalign{{a_{{\rm p^{\prime}}}}(t) & = - \displaystyle{i \over \hbar }\int_{{t_0}}^t dt^{\prime}\int {d^3}r{\rm \varphi} _{{\rm p^{\prime}}}^{\ast} ({\bf r},t^{\prime})U({\bf r}){{\rm \varphi} _{{{\rm p}_{\rm i}}}}({\bf r},t^{\prime}) \cr & \quad - \displaystyle{i \over \hbar }\int_{{t_0}}^tdt^{\prime}\int {d^3}p{a_{\rm p}}(t^{\prime})\int {d^3}r{\rm \varphi} _{{\rm p^{\prime}}}^{\ast} ({\bf r},t^{\prime})U({\bf r}){{\rm \varphi} _{\rm p}}({\bf r},t^{\prime}).} $$

In the spirit of the iteration procedure used in scattering theory the (k+1)th iterate of a _p(t) is expresses by the kth iterate on the right-hand side in (7) like

(8)$$\eqalign{a_{\rm p}^{(k+1)} (t)& = - \displaystyle{i \over \hbar }\int_{{t_0}}^t dt^{\prime}\int {d^3}r{\rm \varphi} _{{\rm p^{\prime}}}^{\ast} {\bf (r},t^{\prime})U({\bf r}){{\rm \varphi} _{{{\rm p}_{\rm i}}}}({\bf r},t^{\prime}) \cr & \quad - \displaystyle{i \over \hbar }\int_{{t_0}}^t dt^{\prime}\int {d^3}pa_{\rm p}^{(k)} (t^{\prime})\int {d^3}r{\rm \varphi} _{{\rm p^{\prime}}}^{\rm \ast } ({\bf r},t^{\prime})U({\bf r}){{\rm \varphi} _{\rm p}}({\bf r},t^{\prime}).} $$

In the first Born approximation [where the transition amplitude is linear in the scattering potential U(r)], we receive the transition amplitude in the next form

(9)$$\eqalign{T_{\rm fi} & = \mathop {\lim }\limits_{t \to \infty } \mathop {\lim }\limits_{{t_0} \to - \infty } a_{{{\rm p}_{\rm f}}}^{(1)} (t) \cr & =- \displaystyle{i \over \hbar }\int_{ - \infty }^\infty dt^{\prime}\int {d^3}r{\rm \varphi} _{{{\rm p}_{\rm f}}}^{\rm \ast } ({\bf r},t^{\prime})U({\bf r}){{\rm \varphi} _{{{\rm p}_{\rm i}}}}({\bf r},t^{\prime}).} $$

By taking the explicit form of the Volkov states (2) with the vector potential A(t) = εA ₀cos(ωt) we observe that the A ² term drops out from the transition matrix element (9), and T _fi becomes

(10)$$\eqalign{&{T_{{\rm fi}}} = \sum\limits_{n =- \infty }^\infty T_{{\rm fi}}^{(n)}, \cr & T_{{\rm fi}}^{(n)} = - 2{\rm \pi} i{\rm \delta} \left( {\displaystyle{{\,p_{\rm f}^ 2 - p_{\rm i}^ 2 } \over {2m}} + n\hbar {\rm \omega} } \right){J_n}(z)\displaystyle{{U({\bf q})} \over {{{(2{\rm \pi} \hbar )}^3}}},} $$

before the time integration was done, the exponential expression was expanded into a Fourier series with the help of the Jacobi–Anger formula (Abramowitz & Stegun, Reference Abramowitz and Stegun1972) which gave us the Bessel function

(11)$${e^{iz\,{\rm sin}({\rm \omega} t)}} = \sum\limits_{n = - \infty }^\infty {J_n}(z)\,{e^{in{\rm \omega} t}}. $$

The U(q) is the Fourier transformed of the scattering potential with the momentum transfer of ${\bf q} \equiv {{\bf p}_{\rm i}} - {{\bf p}_{\rm f}}$ where p_i is the initial and p_f is the final proton momenta. The absolute value is $q=\sqrt {\,p_{\rm i}^2 + p_{\rm f}^2 - 2{\,p_{\rm i}}{\,p_{\rm f}}{\rm cos}({{\rm \theta} _{{\,p_{\rm i}},{\,p_{\rm f}}}})} $. In our case, for 49 MeV energy protons absorbing optical photons the following approximation is valid q ≈ 2p _i|sin(θ/2)|.

The Dirac delta describes photon absorptions (n<0) and emissions (n>0) with energy conservation.

J _n(z) is the Bessel function with the argument of

(12)$$z \equiv {{{m_{\rm e}}} \over {{m_{\rm p}}}}{a_0}(\hat{\bf q}{\rm \epsilon }){{2{\,p_i}} \over {\hbar {k_0}}}\vert {\rm sin}({\rm \theta} /2)\vert , $$

where m _e and m _p are the electron and proton masses, a ₀ is the dimensionless intensity parameter (given above), $\hat{\bf q}$ and ε are the unit vectors of the momentum transfer and the laser polarization direction. It can be shown with geometrical means that for low-energy photons where (${E_{{\rm ph}}}\lt {E_{{{\rm p}^+}}} $) the angle in the scalar product of $\hat{\bf q}{\rm \epsilon } \equiv {\rm cos}\,\chi $ is χ = π/2−θ/2 where θ is the scattering angle of the proton varying from 0 to π. See Figure 1.

From $({\,p_{\rm i}}/\hbar {k_0})=\sqrt {({m_{\rm p}}/{m_{\rm e}})} \sqrt {(2{m_{\rm e}}{c^2}{E_{\rm i}}/{\hbar ^2}{\rm \omega} _0^2 )} $ collecting the constants together the final formula for z reads

(13)$$z = {{1.4166 \times {{10}^{ - 3}}} \over {\hbar {{\rm \omega} _0}}}\sqrt {{{{E_{\rm p}}} \over {1836}}} \sqrt I \times {\rm cos}(\rm \chi ) \times \vert {\rm sin}({\rm \theta} /2)\vert , $$

where the laser energy ħω₀ is measured in eV, the proton energy E _p in MeV, and the laser intensity I in W/cm² (note that this formula is valid for any kind of external laser field. For a 49 MeV proton projectile even the 10 keV X-ray laser has a negligible energy). The final differential cross-section formula for the laser associated collision with simultaneous nth-order photon absorption and emission processes is

(14)$${{d{{\rm \sigma} ^{(n)}}} \over {d{\rm \Omega} }} = {{{\,p_{\rm f}}} \over {{\,p_{\rm i}}}}J_n^2 (z){{d{{\rm \sigma} _{\rm B}}} \over {d{\rm \Omega} }}. $$

The ${d \sigma_B}/{d \Omega} = \lpar {m}/{2\pi \hbar^{2}} \rpar^2 \vert U({\bf{q}}) \vert^2 $ is the usual Born cross-section for the scattering on the potential U(r) alone (without the laser field). The expression Eq. (14) was calculated with different authors using different methods (Bunkin & Fedorov, Reference Bunkin and Fedorov1965; Bunkin et al., Reference Bunkin, Kazakov and Fedorov1973; Faisal, Reference Faisal1973; Reference Faisal1987; Kroll & Watson, Reference Kroll and Watson1973; Gontier & Rahman, Reference Gontier and Rahman1974; Bergou, Reference Bergou1980; Bergou & Varró, Reference Bergou and Varró1980).

In our case the scattering interaction U(r) is a central field U(r) which is the sum of the Coulomb potential of a uniform charged sphere (Rudchik et al., Reference Rudchik, Shyrma, Kemper, Rusek, Koshchy, Kliczewski, Novatsky, Ponkratenko, Piasecki, Romanyshyna, Stepanenko, Strojek, Sakuta, Budzanowski, Głowacka, Skwirczyńska, Siudak, Choiński and Szczurek2010) and a short range optical (Woods & Saxon, Reference Woods and Saxon1954) potential

(15)$$U(r) = {V_{\rm c}}(r) + {V_{{\rm ws}}}(r) + i[W(r) + {W_{\rm s}}(r)] + {V_{{\rm ls}}}(r){\bf l} \cdot {\rm \sigma}, $$

where the Coulomb term is

(16)$$\eqalign{{V_{\rm c}}& =\displaystyle{{{Z_{\rm p}}{Z_{\rm t}}{e^2}} \over {2{R_0}}}\left( {3 - \displaystyle{{{r^2}} \over {R_{\rm c}^2 }}} \right),\quad r\lt {R_{\rm c}}, \cr {V_{\rm c}} & =\displaystyle{{{Z_{\rm p}}{Z_{\rm t}}{e^2}} \over r},\quad r \geq {R_{\rm c}},} $$

where ${R_{\rm c}}={r_0}A_{\rm t}^{1/3} $ is the target radius calculated from the mass number of the target with r ₀ = 1.25 fm. Z _p and Z _t are the charge of the projectile and the target and e is the elementary charge. This kind of regularized Coulomb potential helps us to avoid singular cross-sections and routinely used in nuclear physics.

The short-range nuclear part is given via

(17)$$\eqalign{{V_{{\rm ws}}}(r)& = - {V_{\rm r}}{\,f_{{\rm ws}}}(r,{R_0},{a_0}), \cr W(r) & = - {V_{\rm v}}{\,f_{{\rm ws}}}(r,{R_{\rm s}},{a_{\rm s}}), \cr {W_{\rm s}}(r)& = - {W_{\rm s}}( - 4{a_{\rm s}}){{\,f^{\prime}}_{{\rm ws}}}(r,{R_{\rm s}},{a_{\rm s}}), \cr {V_{{\rm ls}}}(r)& = - ({V_{{\rm so}}} + i{W_{{\rm so}}})( - 2){g_{{\rm ws}}}(r,{R_{{\rm so}}},{a_{{\rm so}}}), \cr {\,f_{{\rm ws}}}(r,R,a)& = {1 \over {1 + {\rm exp}\left( {\displaystyle{{r - R} \over a}} \right)}}, \cr {{\,f^{\prime}}_{{\rm ws}}}(r,R,a)& = {d \over {dr}}{\,f_{{\rm ws}}}(r,R,a), \cr {g_{{\rm ws}}}(r,R,a)& = {{\,f^{\prime}}_{{\hskip-4pt \rm ws}}}(r,R,a)/r.} $$

The constants V _r, W _v, V _so, and W_so are the strength parameters, and a _0,s,so, R _0,s,so are the diffuseness and the radius parameters given for large number of nuclei. The f function is called the shape function of the interaction. As we will see at moderate collisions energies the complex terms become zero. In the last part of the present paper we will use the numerical parameters of Abdul-Jalil and Jackson (Reference Abdul-Jalil and Jackson1979) for proton–carbon collision. According to the work of Hlophe et al. (Reference Hlophe, Elster, Johnson, Upadhyay, Nunes, Arbanas, Eremenko, Escher and Thompson2013) the complete analytic form of the Fourier transform of the WS potential can be calculated via the following kind of complex integrals $V(q)=\int_0^{\infty} dz\left({(z\, exp(i{\rm \rho}_{\rm k}z))/(1+exp(z - {\rm \alpha}_{\rm k}))}\right)$, where ρ_k = qa _k, α_k = R _k/a _k and z = r/a _k are dimensionless variables. The integrals can be evaluated by contour integration using the residuum theorem. For exhaustive details, see Hlophe et al. (Reference Hlophe, Elster, Johnson, Upadhyay, Nunes, Arbanas, Eremenko, Escher and Thompson2013). The Fourier transformed second term of Eq. (15) reads

(18)$$\eqalign{V_{\rm ws}(q) & = {V_{\rm r} \over {\rm \pi ^2}} \left\{ {{\rm \pi} {a_0}e^{ - {\rm \pi} {a_0}q} \over {q{{(1 - {e^{ - 2{\rm \pi} {a_0}q}})}^2}}} \left[R_0(1 - e^{ - 2\pi {a_0}{q}})\,{\rm cos}(q{R_0}) \right. \right. \cr & \left. \left.-{{\rm \pi} {a_0}(1+{e^{ - 2{\rm \pi} {a_0}q}})\,{\rm sin}(q{R_0})} \right] \right. \cr & \left. - a_0^3 {e^{ - ({R_0}/{a_0})}}\left[ {\displaystyle{1 \over {{{(1+a_0^2 {q^2})}^2}}} - \displaystyle{{2{e^{ - ({R_0}/{a_0})}}} \over {{{(4 + a_0^2 {q^2})}^2}}}} \right] \right\}.} $$

For the W(q) imaginary term, the same expression was derived with W _v, a _s, and R _s instead of V _r, a ₀, and R ₀. The surface term W _s (r) [fourth term in Eq. (15)] gives the following formula in the momentum space:

(19)$$\eqalign{{W_{\rm s}}(q) = & - 4{a_{\rm s}}{W_{\rm s} \over {\rm \pi ^2}} \left\{{{{\rm \pi} {a_{\rm s}}{e^{ - \pi {a_{\rm s}}q}}} \over {{(1 - {e^{ - 2{\rm \pi} {a_{\rm s}}q}})}^2}} \left[{({\rm \pi} {a_{\rm s}}(1+{e^{ - 2{\rm \pi} {a_{\rm s}}q}})}\right. \right. \cr & \left. \left. - {1 \over q}(1 - {e^{ - 2{\rm \pi} {a_{\rm s}}q}))}\,{\rm cos}(q{R_{\rm s}}) + {R_{\rm s}}(1 - {e^{ - 2{\rm \pi} {a_{\rm s}}q}})\,{\rm sin}(q{R_{\rm s}})\right] \right. \cr & \left.+ a_{\rm s}^2 {e^{ - ({R_{\rm s}}/{a_{\rm s}})}} \left[ {\displaystyle{1 \over {{{(1 + a_{\rm s}^ 2 {q^2})}^2}}} - \displaystyle{{4{e^{ - ({R_{\rm s}}/{a_{\rm s}})}}} \over {{{(4 + a_{\rm s}^2 {q^2})}^2}}}} \right] \right\}.} $$

The last term in Eq. (15), the transformed spin–orbit coupling term leads to

(20)$$\eqalign{& V_{\rm ls}(q) = - {a_{\rm so} \over {\rm \pi} ^2} (V_{\rm so} + i{W_{\rm so}}) \left\{{{2{\rm \pi} {e^{ - {\rm \pi} {a_{{\rm so}}}q}}} \over {1 - {e^{ - 2{\rm \pi} {a_{{\rm so}}}q}}}} {\rm sin}(q{R_{{\rm so}}})\right. \cr & \quad \left. + {e^{( - {R_{\rm so}}/{a_{\rm so}})}}\left( {{1 \over {1 + a_{\rm so}^2 {q^2}}} - {{2{e^{ - {R_{\rm so}}}}} \over {4+a_{\rm so}^2 {q^2}}}}\right)\right\}.} $$

where the momentum transfer is defined as above q ≡ p_i−p_f. The low-energy transfer approximation formula q ≈ 2p _i|sin(θ/2)| is valid.

The Fourier transform of the charged sphere Coulomb field is also far from being trivial

(21)$$\eqalign{V_{\rm c}(q) & = \displaystyle{{Z_{\rm p}Z_{\rm t}e^2} \over {2^{(5/6)} \sqrt {\rm \pi} q^3}} \bigg(- 2 \!\cdot\! 3^{(1/3)}q{\rm \cos} [2^{(2/3)}3^{(1/3)}q] \cr & \quad +2^{(1/3)}(1+2 \!\cdot \!2^{(1/3)}3^{(2/3)}q^2){\rm \sin} [2^{(2/3)}3^{(1/3)}q] \bigg) \cr & \quad +3Z_{\rm p}Z_{\rm t}e^2 \sqrt {\displaystyle{2 \over {\rm \pi}}} \bigg( \displaystyle{{i{\rm \pi} \vert q\vert } \over {2q}} - {\rm Ci}[2^{(2/3)}3^{(1/3)}q]+\log (q). \cr & \quad - \log \vert q \vert - i{\rm Si}[2^{(2/3)}3^{(1/3)}q]\bigg),} $$

where Ci and Si are the cosine and the sine integral functions, respectively; for details see Abramowitz and Stegun (Reference Abramowitz and Stegun1972).

4. SUMMARY

We presented a formalism which gives an analytic angular differential cross-section model for laser-assisted proton nucleon scattering on a Woods–Saxon optical potential where the nth-order photon absorption is taken into account simultaneously. We coupled the mathematical description of multi-photon processes to the well-established low-energy nuclear physics description. As an example the physically relevant proton–¹²Ca collision system was investigated at moderate 49 MeV proton energies. Two different kinds of laser fields are investigated. The first one is the optical Ti:sapphire system with wavelength of 800 nm with intensities in the range of 10¹¹–10²¹ W/cm². As a second system we took a 10 keV X-ray laser field with 10¹⁶ W/cm² intensity. The calculated cross-sections are much lower than the elastic cross-sections in all cases. We hope that our study will give a strong impetus and couple the nuclear and laser physics community together to perform such experiments in the ELI or X-FEL facilities which will be available in a couple of years.