Fragmentation of the ⁴He₂ dimer by relativistic highly charged projectiles in collisions with small kinetic energy release

Let the He₂ dimer collide with a projectile, which has a charge $Z_{p}$ and moves with a velocity ${\bm{v}}$ whose magnitude approaches the speed of light $c\approx 137$ a.u. At $v\sim c$ the parameter $\eta=Z_{p}/v$, characterizing the effective strength of the projectile field in the collision, is well below $1$, which indicates that the field of the projectile in the collision is on overall weak rather than strong. Consequently, the breakup of the He₂ will occur with a non-negligible probability only provided the number of ’steps’ in the interaction of the projectile with the constituents of the dimer is reduced to a necessary minimum. Also, in relativistic collisions with very light atoms the processes of radiative and nonradiative electron capture from the atom by the projectile are already extremely weak having cross sections which are by several orders of magnitude smaller that atomic ionization and excitation cross sections el-cpt ; eic ; we-1998 ; He-ioniz .

The above features inherent to high-energy collisions with light targets restrict the number of the main fragmentation mechanisms, which govern the breakup of the He₂ dimer into He⁺ ions

by high-energy charged projectiles, to four He2-rel-HCI-lett .

We shall consider collisions between the He₂ dimer and the projectile using the semi-classical approximation in which the relative motion of the heavy particles (nuclei) is treated classically whereas the electrons are considered quantum mechanically.

We choose a reference frame in which the dimer is at rest and take the nucleus of one of its atoms as the origin. We shall refer to this atom as atom $A$ whereas the other will be denoted by $B$. In this frame, the coordinates of the nucleus of atom $B$ are given by the inter-nuclear vector ${\bm{R}}$ of the dimer and the projectile moves along a classical straight-line trajectory $\mathbf{R}_{p}(t)=\bm{b}+\bm{v}t$, where $\bm{b}=(b_{x},b_{y},0)$ is the impact parameter with respect to the nucleus of atom $A$ and $\bm{v}=(0,0,v)$ the collision velocity.

Since the size of the He₂ dimer is very large, the ionization of atoms $A$ and $B$ occurs independently of each other and we can use the independent electron approximation. According to it the probability $P_{A^{+}B^{+}}$ for single ionization of atoms $A$ and $B$ in the collision where the projectile moves with an impact parameter $\bm{b}$ is given by

Here, $P_{A^{+}}(\bm{b})$ and $P_{B^{+}}(\bm{b}^{\prime})$ are the probabilities for single ionization of atoms $A$ and $B$, respectively, and ${\bm{b}^{\prime}}={\bm{b}}-{\bm{R}}_{\perp}$ is the collision impact parameter with respect to atom $B$, where ${\bm{R}}_{\perp}$ is the part of the inter-nuclear vector ${\bm{R}}$ of the dimer which is perpendicular to the projectile velocity ${\bm{v}}$.

Within the independent electron approximation the probabilities for single ionization of helium atoms $A$ and $B$ read

where $w(\bm{b})$ ($w(\bm{b}^{\prime})$) is the probability to remove one electron from helium atom in the collision with a projectile having an impact parameter $\bm{b}$ ($\bm{b}^{\prime}$) new-approach .

The quantity

represents the cross section for the production of two singly charged helium atom by the projectile in the collision with the He₂ dimer at a given inter-nuclear vector ${\bm{R}}$ of the latter. Since the probabilities $P_{A^{+}}(\bm{b})$ and $P_{B^{+}}(\bm{b}^{\prime})$ depend just on the absolute value of the respective impact parameters, $P_{A^{+}}(\bm{b})\equiv P_{A^{+}}(b)$ and $P_{B^{+}}(\bm{b}^{\prime})\equiv P_{B^{+}}(b^{\prime})$, the cross section (5) depends only on the absolute value $R_{\perp}$ of the two-dimensional vector ${\bm{R}}_{\perp}$: $\sigma^{\text{DF}}_{A^{+}B^{+}}=\sigma^{\text{DF}}_{A^{+}B^{+}}(R_{\perp})$.

In our calculations of the probabilities $w(b)$ and $w(b^{\prime})$ we regarded the helium atom in its initial and final states as an effectively single-electron system, where the ’active’ electron moves in the effective field, created by the ’frozen’ atomic core consisting of the atomic nucleus and the ’passive’ electron. This field is approximated by the potential

where $r$ is the distance between the active electron and the atomic nucleus, and $\alpha=3.36$ and $\beta=1.665$ param . We note that with this choice of $\alpha$ and $\beta$, $V({\bm{r}})$ almost coincides with the exact Hartree-Fock potential.

The cross section (5) can be only computed numerically and the computation is quite time consuming. However, a simple estimate can be obtained for this cross section at $R_{\perp}\gg 1$ a.u. He2-rel-HCI-lett

Here, $K_{0}$ and $K_{1}$ are the modified Bessel function A-S , $R_{a}=\frac{\gamma v}{\overline{\omega}}$ is the adiabatic collision radius, where $\gamma=1/\sqrt{1-v^{2}/c^{2}}$ is the collision Lorentz factor, $\overline{\omega}\approx 1.2$ a.u. is the mean transition frequency for single ionization of a helium atom, and $C$ is a free (fit) parameter which is a very slowly varying function of $R_{\perp}$ param-C .

Taking into account that $K_{0}(x)\sim\ln(1.12/x)$ and $K_{1}(x)\sim 1/x$ if $x<1$ whereas $K_{0}(x)\sim K_{1}(x)\sim\sqrt{\frac{\pi}{2x}}\exp(-x)$ at $x>1$, it follows from Eq. (6) that at $1\ll R_{\perp}\lesssim R_{a}$ the cross section decreases with $R_{\perp}$ relatively slowly, $\sigma^{\text{DF}}\sim R^{-2}_{\perp}$, whereas at $R_{\perp}>R_{a}$ the decrease is already exponential. This means that the projectile is able to efficiently irradiate both atoms of the dimer only provided that the dimer transverse size $R_{\perp}$ does not exceed the adiabatic collision radius $R_{a}$. Since $R_{a}\sim\gamma v$, an ultrafast projectile possesses a very large effective interaction range and can probe systems with large dimensions. Simple estimates show that, beginning with impact energies of a few GeV/u, the adiabatic collision radius exceeds the size of the He₂ dimer He2-rel-HCI-lett .

The cross section (5) refers to the situation where two helium atoms in the dimer are separated by the vector ${\bm{R}}$ with the absolute value of its transverse projection $R_{\perp}$. It yields an important information about the collisions giving an insight into the basic physics of the fragmentation process He2-rel-HCI-lett but cannot be (directly) measured in experiment. Therefore, let us ’convert’ the cross section (5) into quantities which can be measured. We start with the expression

where $\Psi_{i}({\bm{R}})$ is the wave function describing the relative motion of the atoms in the ground state of the He₂ dimer.

As was already mentioned, we assume that before the collision with the projectile the He₂ dimer was at rest. Let ${\bm{P}}^{\text{rec}}_{A}$ and ${\bm{P}}^{\text{rec}}_{B}$ be the recoil momenta of the helium ions, which they acquire during the collision because of the interaction with the projectile and electron emission, and let ${\bm{P}}_{A}$ and ${\bm{P}}_{B}$ be the final momenta of these ions after the Coulomb explosion. Then the energy conservation for the relative motion of the ionic fragments reads

Here, $E_{K}={\bm{K}}^{2}/(2\mu)$ is the final kinetic energy of the relative motion of the ionic fragments, ${\bm{K}}=\left({\bm{P}}_{A}-{\bm{P}}_{B}\right)/2$ and $\mu$ are the momentum of their relative motion and the reduced mass, respectively, $Q_{A}$ and $Q_{B}$ ($Q_{A}=Q_{B}=1$) are their charges and $R$ is the distance between the ions when the Coulomb explosion began and which in the case under consideration coincides with the size of the initial He₂ dimer at the collision instant. In the energy balance (8) we have neglected the kinetic energy of the He⁺ ions, which they had due to the nuclear motion before the collision and which is very small because the depth of the potential well in the He₂ dimer is just $\approx 1$ meV.

Since in the process of single ionization of helium atoms by relativistic projectiles the recoil momenta of the helium ions do not noticeably exceed $1$ a.u. rhci -r-sea , the first term on the right hand side of Eq. (8) is in the meV range. Such values are comparable to the Coulomb potential energy $Q_{A}Q_{B}/R$ at the inter-nuclear distances $R$ of the order of $10^{4}$ a.u. which is much larger than the size of the He₂ (and any known) dimer. Therefore, the neglect of the recoil energy, given by the first term on the right-hand side of Eq. (8), does not have any substantial impact on the energy balance. Thus, the relation

between the kinetic energy release and the size of the dimer at the collision instant, which neglects the recoil energies, is expected to be very accurate down to energies $E_{K}\simeq 10$ meV.

Let us now make a more restrictive assumption that not only the recoil energies are much smaller than the Coulomb energy $Q_{A}Q_{B}/R$ but also that the absolute values of the recoil momenta, $|{\bm{P}}^{\text{rec}}_{A}|$ and $|{\bm{P}}^{\text{rec}}_{B}|$, of the fragments are significantly less than $K=|{\bm{K}}|$. Since these values do not exceed $1$ a.u. the assumption will be fulfilled beginning with $K\gtrsim 4$–$5$ a.u. that corresponds to the energies $E_{K}\gtrsim 60$ meV. Under such conditions the momentum $\bm{K}$ will be directed essentially along the inter-nuclear vector $\bm{R}$ of the initial He₂ dimer and, taking also into account (9), we get

Using (7) and (9)-(10) we obtain the fragmentation cross section differential in the relative momentum ${\bm{K}}$ of the ionic fragments

where $\hat{\bm{K}}={\bm{K}}/K$, $R_{\perp}(\bm{K})=(Q_{A}Q_{B}\,\sin\Theta_{\bm{K}})/E_{K}$ and $\Theta_{\bm{K}}$ is the polar angle of the momentum $\bm{K}$ (the $z$-axis is along the projectile velocity $\bm{v}$). Since the bound state of the He₂ dimer is spherically symmetric, the wave function $\Psi_{i}$ does not depend on $\hat{\bm{K}}$: $\Psi_{i}\bigg{(}\frac{Q_{A}Q_{B}}{E_{K}}\hat{\bm{K}}\bigg{)}=\Psi_{i}\bigg{(}\frac{Q_{A}Q_{B}}{E_{K}}\bigg{)}$.

Taking into account that $d^{3}{\bm{K}}=K^{2}dK\,d\Omega_{\bm{K}}=\mu KdE_{K}\,\sin\Theta_{\bm{K}}d{\bm{K}}d\varphi_{\bm{K}}$, where $\varphi_{\bm{K}}$ is the azimuthal angle of $\bm{K}$, and that the right-hand side of Eq. (11) does not depend on $\varphi_{\bm{K}}$ we obtain the fragmentation cross section differential in the kinetic energy release $E_{K}$ and the polar angle $\Theta_{\bm{K}}$

In our consideration of the direct fragmentation process, its two steps – ’instantaneous’ production of two He⁺ ions and the consequent Coulomb explosion of the He⁺–He⁺ system – are fully disentangled, even though the energy of the emitted electrons and the kinetic energy release have the same source: the energy transferred by the projectile to the He₂ dimer. However, since the electrons emitted from helium atoms in collisions with relativistic projectiles have energies extending up to a few tens of eV whereas, as will be seen below, the spectrum of the kinetic energy release in the DF process peaks at $E_{K}<1$ eV and essentially vanishes already at $E_{K}\simeq 5$–$6$ eV, such a consideration is expected to be quite accurate.

In this section we report our results for the fragmentation cross sections. In our calculations the ground state of the He₂ dimer was described by a wave function $\Psi_{i}$ corresponding to the dimer binding energy $I_{b}$ of $139$ neV (this value is very close to that given in 139.2 ): this wave function was used to get results presented below in figures 1-7. In addition, in order to explore the sensitivity of the cross sections to the value of $I_{b}$, also wave functions $\Psi_{i}$ corresponding to $I_{b}=130$ neV and $148$ neV were applied for obtaining results shown in figure 8.

Four different approximations were employed for computing the probabilities $w$ for single electron removal from a helium atom (see Eq. (4)). They include: i) the nonrelativistic first Born approximation (nr-FBA); ii) the nonrelativistic symmetric eikonal approximation (nr-SEA) nr-sea ; iii) the relativistic first Born approximation (r-FBA) and iv) the relativistic symmetric eikonal approximation (r-SEA) r-sea .

The differences between results obtained by using the first Born and symmetric eikonal approximations offer an idea about the importance of the higher-order effects in the interaction between the projectile and the atoms of the dimer. On the other hand, the differences between results of the relativistic and nonrelativistic approximations yield the information about the role of relativistic effects in the He₂ fragmentation.

In particular, in the nr-FBA and nr-SEA the speed of light $c$ is assumed to be infinite which means that all relativistic effects caused by very high collision velocities vanish. In the r-FBA and r-SEA $c\approx 137$ a.u. and relativistic effects arising due to high impact energies are taken into account. Since even in very high-energy collisions with helium targets the overwhelming majority of emitted electrons have energies well below $100$ eV we-jpb-2005 , the main relativistic effects, which are due to high impact energies and which may influence the He₂ fragmentation, are caused by the deviation of the electric field of the projectile from the (unretarded) Coulomb form.

One should say that, in addition to relativistic effects due to high impact velocities, there are also relativistic effects in the ground state of the free He₂ which – according to 139.2 – reduce the binding energy of the dimer by about $14\%$. This affects the form of its wave function (especially at very large inter-atomic distances) that in turn may have an impact on the shape of the spectrum of the fragments He2-rel-HCI-lett .

In figures 1 and 2 we show the doubly differential cross section $\frac{d\sigma^{\text{DF}}_{\text{fr}}}{dE_{K}d\Theta_{\bm{K}}}$ for the fragmentation of the He₂ dimer into He⁺ ions occurring in collisions with $1$ GeV/u and $7$ GeV/u U⁹²⁺ projectiles, respectively. The corresponding collision velocities and the Lorentz factor are $v=120$ a.u., $\gamma=2.08$ and $v=136$ a.u., $\gamma=8.5$, respectively. The cross section is plotted as a function of the kinetic energy release $E_{K}$ and the fragmentation angle $\Theta_{\bm{K}}$ presenting a general picture of the fragmentation spectrum. Taking into account that the spectrum is symmetric with respect to the transformation $\Theta_{\bm{K}}\leftrightarrow\pi-\Theta_{\bm{K}}$, only the interval of angles $0\leq\Theta_{\bm{K}}\leq\pi/2$ is considered.

It is seen in these figures that the spectrum is predominantly localized in the energy interval $0.1$ eV $\lesssim E_{K}\lesssim 2$ eV and at fragmentation angles $\Theta_{\bm{K}}$ not significantly exceeding $20^{\circ}$. It is also seen that at very small energies $E_{K}$ the spectrum is restricted to smaller fragmentation angles $\Theta_{\bm{K}}$ whereas with increasing the energy the spectrum shifts to larger $\Theta_{\bm{K}}$.

The interval of kinetic energies $0.1$ eV $\lesssim E_{K}\lesssim 2$ eV corresponds to the instantaneous size $R$ of the He₂ dimer during the collision in the range $14$ a.u. $\lesssim R\lesssim 270$ a.u. The probability to find the He₂ dimer in this range is above $90\%$ that is reflected in the dominance of the above energy interval in the spectrum. At energies $\gtrsim 5$–$6$ eV the spectrum practically vanishes since they correspond to the inter-nuclear distances $R\lesssim 4$ a.u., where the probability to find the He₂ dimer is negligibly small.

The cross section $\sigma^{\text{DF}}_{A^{+}B^{+}}$ for the production of two helium ions by the projectile decreases with the transverse size $R_{\perp}$ of the dimer at the collision instant ($\sigma^{\text{DF}}_{A^{+}B^{+}}\sim 1/R_{\perp}^{2}$ at $R_{\perp}<R_{a}$). Since at a fixed $\Theta_{\bf K}$ the value of $R_{\perp}$ is proportional to $R$, the fragmentation with very small $E_{K}$ may occur at very small $\Theta_{\bf K}$ only, whereas at significantly larger values of $E_{K}$ it becomes possible at not very small $\Theta_{\bf K}$ as well. Taking also into account that the cross section $\frac{d\sigma^{\text{DF}}_{\text{fr}}}{dE_{K}d\Theta_{\bm{K}}}$ is proportional to the geometrical factor $\sin\Theta_{\bm{K}}$ we can explain the ’shift’ of the spectrum to larger $\Theta_{\bm{K}}$ with increase in $E_{K}$, which is observed in figures 1 and 2.

It also follows from the results presented in figures 1 and 2 that for collisions with so very highly charged projectiles, as U⁹² ions, the symmetric eikonal approximation predicts a noticeably smaller number of the fragmentation events. This shows that the higher-order effects in the interaction between the projectile and the target remain visible even at very high impact energies. The smaller cross section values, predicted by the symmetric eikonal approximation, reflects the general tendency that in high-energy collisions the higher-order effects reduce cross section values.

Figures 3 and 4 give a more detailed information about the fragmentation process by displaying the cross section $\frac{d\sigma^{\text{DF}}_{\text{fr}}}{dE_{K}d\Theta_{\bm{K}}}$ as a function of the fragmentation angle $\Theta_{\bm{K}}$ at a given value of the kinetic energy release $E_{K}$: $60$ meV, $100$ meV, $1$ eV and $2$ eV. Some conclusions can be drawn from the results presented in these figures.

First, by comparing results of the different calculations we see that the shape of the He₂ fragmentation pattern is in general influenced by both relativistic effects, which arise due to collision velocities approaching the speed of light, and higher-order effects, which originate in a very high charge of the projectile.

Second, the influence of the relativistic effects increases when the kinetic energy release decreases: these effects especially impact the fragmentation events with very small kinetic energy release where they enhance the cross section up to one or two orders of magnitude. On the other hand, the fragmentation with larger values of $E_{K}$ is only very modestly influenced by these effects.

Third, the situation with the higher-order effects is just opposite: their role rises with increasing $E_{K}$ and they mostly affect the fragments having larger energies. However, even for such fragments these effects remains modest, reducing the cross section values not significantly more than $10$-$20\%$. The higher-order effects also somewhat influence the low-energy part of the spectrum: in this case, however, their effect is limited to the very small angles $\Theta_{\bm{K}}$ only.

The above qualitative features of the fragmentation process can be understood by noting that the cross section $\sigma^{\text{DF}}_{A^{+}B^{+}}(R_{\perp})$ is most profoundly influenced by the relativistic effects when the magnitude of $R_{\perp}$ is large ($R_{a}/\gamma<R_{\perp}\lesssim R_{a}$) whereas the higher-order effects are most significant in collisions where the projectile passes close to both atoms of the dimer (the impact parameters $b$ and $b^{\prime}$ do not exceed a few atomic units) that is possible only at relatively small values of $R_{\perp}$.

Very large values of $R_{\perp}$ imply that the instantaneous dimer size $R$ in the collision has to be very large and, in addition, the dimer orientation angle $\Theta_{\bf R}$ should not be small. Since $E_{K}=1/R$ and $\Theta_{\bf R}\approx\Theta_{\bf K}$, it follows that the relativistic effects are most significant for fragmentation events, which are characterized by a very small kinetic energy release and not too small angles $\Theta_{\bm{K}}$.

On the other hand, collisions occurring at small values of $R_{\perp}$, where the higher-order effects can be substantial, are characterized by either small values of the instantaneous dimer size $R$ or very small angles $\Theta_{\bm{R}}\approx\Theta_{\bm{K}}$ (or a combination of both). The kinetic energy release $E_{K}$ of $0.06$ eV, $0.1$ eV, $1$ eV and $2$ eV corresponds to $R\approx 453$ a.u., $272$ a.u., $27.2$ a.u. and $13.6$ a.u., respectively. In order to have substantial higher-order effects at the first two values of $R$ the dimer has to be almost parallel to the collision velocity. As a result, in collisions with very small $E_{K}$ such effects can become visible only at the fragmentation angles $\Theta_{\bm{K}}$ close to zero whose contribution, due to the geometric factor $\sin\Theta_{\bm{K}}$, is very small. For collisions with larger $E_{K}$ the restrictions on the angle $\Theta_{\bm{K}}$ are not so strict and the higher-order effects may be noticeable for the whole range $0^{\circ}\leq\Theta_{\bm{K}}\leq 90^{\circ}$.

The energy $E_{K}$ of $0.06$ eV, $0.1$ eV, $1$ eV and $2$ eV correspond to the relative momentum $K$ of $\approx 4$ a.u., $5.2$ a.u., $16.4$ a.u. and $23.2$ a.u., respectively. The first two momentum values are not very large. Therefore, in the case of $E_{K}=0.06$ eV and $0.1$ eV the very narrow peak at very small angles (see panels ($a$) and ($b$) of figures 3–4) can be noticeably smeared out by the recoil effects.

In figure 5 we display the angular distribution of the He⁺ fragments represented by the cross section $\frac{d\sigma^{\text{DF}}_{\text{fr}}}{d\Theta_{\bm{K}}}$. The cross section was obtained by integrating either over the energy interval $0$ eV $\leq E_{K}\leq 2$ eV, in which the direct fragmentation mechanism is by far the dominant one, or over the broader range $0$ eV $\leq E_{K}\leq 7$ eV, which covers essentially all energies which are possible in the fragmentation events caused by this mechanism (see figures 1-2).

Figure 5 shows that the angular distribution of the He⁺ fragments becomes significantly weaker dependent on the angle $\Theta_{\bm{K}}$ when their energy $E_{K}$ increases. This feature (which is already seen in figures 3–4) can be understood by taking into account that, due to the relation $E_{K}=1/R$, the He⁺ fragments with larger energies emerge when the projectiles hit the He₂ dimers with smaller size $R$. In such collisions the DF mechanism is more efficient causing the He₂ breakup with significant probabilities even when the dimer is oriented at large angles with respect to the impact velocity.

The energy spectrum, $d\sigma^{\text{DF}}_{\text{fr}}/dE_{K}$, of He⁺ fragments produced by bombarding the He₂ dimer by $1$ and $7$ GeV/u U⁹²⁺ projectiles is displayed in figure 6. The spectrum shown in the left panels of this figure was obtained by integrating over all possible fragmentation angles, $0^{\circ}\leq\Theta_{\bm{K}}\leq 180^{\circ}$, whereas that given in the right panels includes only the fragments moving at large angles, $60^{\circ}\leq\Theta_{\bm{K}}\leq 120^{\circ}$, with respect to the projectile velocity.

The results shown in figure 6 confirm the correspondences between the relativistic and higher-order effects and the energy $E_{K}$ and angle $\Theta_{\bm{K}}$ of the fragments which were discussed above. In particular, we again observe that at a given collision velocity the relativistic effects rise when the energy $E_{K}$ decreases, becoming especially strong in collisions where the He⁺ fragments possess quite small energies $E_{K}$ and move at large angles with respect to the projectile velocity.

In figure 7 we compare the energy spectra of the He⁺ ions produced in collisions with $1$ and $7$ GeV/u U⁹²⁺ and $11.37$ MeV/u S¹⁴⁺ ($v=21.2$ a.u., $\gamma=1.01$). It is seen that the shape of the spectrum profoundly varies with an increase in the impact energy. In particular, the maximum of the energy distribution is shifted from $E_{K}\approx 1.7$ eV in collisions with $11.37$ MeV/u S¹⁴⁺ to $E_{K}\approx 0.7$ eV in collisions with $7$ GeV/u U⁹²⁺ where the low-energy part of the spectrum is strongly enhanced (see figure 7a) and this enhancement becomes even much more pronounced if the fragmentation events with large angles $\Theta_{\bm{K}}$ are selected (figure 7b).

In collisions with $11.37$ MeV/u S¹⁴⁺ and $7$ GeV/u U⁹²⁺ the effective strength of the projectile field, given by the ratio $Z_{p}/v$, is essentially the same ($0.68$ and $0.66$, respectively). Therefore, the strong enhancement of the lower-energy part of the fragmentation spectrum, which reflects the corresponding strong increase in the breakup of dimers with very large instantaneous size $R$, is caused by a much higher impact energy. One should note, however, that since He₂ is a very light target, a large increase in the collision velocity has a more profound overall effect on the spectrum shape and the total number of events than an increase in the Lorentz factor $\gamma$.

The values for the binding energy $I_{b}$ of the He₂ dimer, reported in the literature, vary between $I_{b}=44.8$ neV 44.8 and $I_{b}=161.7$ neV 161.7 . The distribution of the probability density $\rho(R)=|\Psi_{i}(R)|^{2}$ in the ground state of the dimer depends on the value of $I_{b}$ and a variation $\Delta I_{b}$ in the binding energy would be reflected by the corresponding variation $\Delta\rho(R)$ of the probability density $\rho(R)$. However, if $\Delta I_{b}$ is relatively small, $\Delta I_{b}\ll I_{b}$, a large range of $R$ has to be spanned in order for the changes in the shape of $\rho(R)$ to become noticeable.

In this respect, as it was already emphasized in He2-rel-HCI-lett , the fragmentation by ultra-fast projectiles can be of especial interest. Indeed, since the adiabatic collision radius $R_{a}$ increases with the impact energy as $\sim v\gamma$, they possess a very large effective interaction range that is a great advantage in probing the structure of such enormous objects like the He₂ dimer.

In figure 8 we display the energy spectrum $\frac{d\sigma^{\text{DF}}_{\text{fr}}}{dE_{K}}$ for the He₂ fragmentation by $7$ GeV/u U⁹²⁺. In the range of $E_{K}$, shown in the figure, the direct fragmentation is by far the dominant breakup channel and the cross section $\frac{d\sigma^{\text{DF}}_{\text{fr}}}{dE_{K}}$ represents the total energy spectrum.

The spectrum was calculated for three values of the dimer binding energy $I_{b}$: $130$ neV, $139$ neV and $148$ neV. For a better visibility of the variation of the spectrum shape with $I_{b}$, the spectra for $I_{b}=130$ neV and $I_{b}=148$ neV were normalized to the spectrum for $I_{b}=139$ neV at an energy $E_{K}=60$ meV.

In the energy interval, $60$ meV $\leq E_{K}\leq 1$ eV, the ratios $\frac{d\sigma^{\text{DF}}_{\text{fr}}(I_{b}=148)}{dE_{K}}/\frac{d\sigma^{\text{DF}}_{\text{fr}}(I_{b}=139)}{dE_{K}}$, $\frac{d\sigma^{\text{DF}}_{\text{fr}}(I_{b}=139)}{dE_{K}}/\frac{d\sigma^{\text{DF}}_{\text{fr}}(I_{b}=130)}{dE_{K}}$ and $\frac{d\sigma^{\text{DF}}_{\text{fr}}(I_{b}=148)}{dE_{K}}/\frac{d\sigma^{\text{DF}}_{\text{fr}}(I_{b}=130)}{dE_{K}}$ vary by about $18\%$, $20\%$ and $42\%$, respectively. The corresponding relative variations, $\Delta I_{b}/I_{b}$, of the binding energy are approximately $6.5\%$, $7\%$ and $14\%$, respectively. This means that in the energy interval under consideration the shape of the energy spectrum ’magnifies’ the variation of the binding energy by about three times.

A more detailed analysis of the cross sections shows that the range of quite small $E_{K}$ ($E_{K}\lesssim 0.2$ eV) gives the main contribution to the variation of the cross section ratios. This is not very surprising since the small energy range,   $60$ meV $\leq E_{K}\lesssim 0.2$ eV,   corresponds to a very large interval, $136$ a.u. $\lesssim R\lesssim 454$ a.u., of the interatomic distances in the He₂ dimer, where the shape of its ground state becomes sensitive to even a very modest variation of the binding energy. The projectile is able to span in the collision so large interval of distances since at $7$ GeV/u the adiabatic collision radius $R_{a}\simeq 10^{3}$ a.u. already significantly exceeds the size of the dimer.

According to our r-SEA calculations, the contribution $\sigma^{\text{DF}}_{\text{fr}}$ of the DF mechanism to the total cross section for the breakup of the He₂ dimer in collisions with $1$ and $7$ GeV/u U⁹²⁺ projectiles amounts to $3.65$ Mb and $2.39$ Mb, respectively.

The relativistic effects in the direct fragmentation are most significant when the ’instantaneous’ transverse size $R_{\perp}$ of the He₂ dimer in the collision is very large. Consequently, they most profoundly influence the fragmentation events with small $E_{K}$ and large $\Theta_{\bm{K}}$. However, such events have relatively small probabilities and, as a result, the cross section $\sigma^{\text{DF}}_{\text{fr}}$ is very weakly influenced by the relativistic effects. In particular, for collisions with $1$ and $7$ GeV/u U⁹²⁺ the difference between our results for $\sigma^{\text{DF}}_{\text{fr}}$, obtained using relativistic and nonrelativistic approaches, is about merely $\simeq 1\%$.

The field of the projectile acting on the dimer is strongest when the impact parameters with respect to both atomic sites of the dimer are small, i.e. in collisions where the ’instantaneous’ transverse size $R_{\perp}$ of the dimer is not large. Since such collisions give a more significant contribution to $\sigma^{\text{DF}}_{\text{fr}}$, than those with very large $R_{\perp}$, the influence of the higher-order effects in the interaction between the projectile and the dimer on the magnitude of $\sigma^{\text{DF}}_{\text{fr}}$ is substantially larger reaching about $14\%$ for the fragmentation by $1$ and $7$ GeV/u U⁹²⁺.

In the present paper the He₂ fragmentation via the IE-ICD and DI-RET mechanisms was not calculated. Nevertheless, we can still obtain a rough estimate for the contribution to the total fragmentation cross section due to these two mechanisms. According to the experimental results of thesis_He2_S14+ , in the breakup of the He₂ dimer by $11.37$ MeV S¹⁴⁺ the ratio of the summed contributions of the IE-ICD and DI-RET channels to the contribution of the DF channel is approximately equal to $1.6$. Taking into account that our calculation for the DF contribution in collisions with $11.37$ MeV/u S¹⁴⁺ yields $1.8$ Mb, we obtain that the summed contributions of the IE-ICD and DI-RET to the fragmentation by $11.37$ MeV S¹⁴⁺ is $\approx 1.8\times 1.6=2.9$ Mb.

The cross sections for the fragmentation via the IE-ICD and DI-RET are proportional to the cross sections for simultaneous ionization-excitation and double ionization, respectively, of the helium atom. Both these cross section scale similarly with $Z_{p}$ and $v$. Therefore, by calculating them for collisions with $11.37$ MeV/u S¹⁴⁺ and $1$ and $7$ GeV/u U⁹²⁺ projectiles and using our results for the DF cross sections, we have estimated that the summed contribution of the IE-ICD and DI-RET mechanisms to the fragmentation in collisions with $1$ and $7$ GeV/u U⁹²⁺ is roughly equal to $4.3$ Mb and $3$ Mb, respectively.