Muonic-atom scattering from hydrogenic liquids in an incoherent approach

The differential cross sections for low-energy muonic hydrogen atom scattering in liquid hydrogenic targets have been calculated in the incoherent approximation using the Van Hove response function. A simple model of liquids and the available experimental parameters have been employed for a description of the diffusive and vibrational modes in these targets. At collision energies below about 10 meV, the obtained cross sections are very different from the analogous cross sections for scattering in hydrogenic gases.


I. INTRODUCTION
Many experiments in low-energy muon physics were performed in the liquid mixtures of hydrogen isotopes. In particular, such hydrogenic targets were employed for studying the muon-catalyzed pd and pt fusion in the muonic molecules pdµ and ptµ (see, e.g., [1][2][3][4][5]). The yields of different products from these fusion reactions strongly depend on the populations of hyperfine states of the muonic molecules. These yields are functions of the concentrations of deuterium and tritium admixtures in the H 2 targets, which is known as the Wolfenstein-Gershtein effect [6,7]. This is due to a complicated chain of processes from the Coulomb capture of the negative muons to formation of the muonic molecules. The time evolution of the spin and kinetic energy of the dµ and tµ atoms is of particular importance. In order to compare the theoretical fusion rates with the experimental data, it is necessary to accurately describe the kinetics of muon-catalyzed fusion. Detailed kinetics calculations [3][4][5], were performed using then available cross sections for scattering of the muonic atoms from the hydrogen-isotope nuclei [8][9][10]. However, in the considered experiments, these processes take place in the molecular liquid targets. One may thus expect that the realistic cross sections are very different from that for the scattering from free nuclei or molecules, which has already been shown in the case of solid hydrogenic targets [11]. An estimation of the differential cross sections for muonic atom scattering from the liquid hydrogenic targets in a simple incoherent approximation is the aim of this work. The obtained results can be applied for an accurate simulation of the new experimental study of muon-catalyzed pt fusion, which is underway at JINR in Dubna. This experiment has been planned in order to clarify the large discrepancies between theory and experiment in the ptµ case, and to observe more fusion channels (e.g., ptµ → 4 He + e + + e − ) for the first time [12,13].
A method of estimating the differential cross sections for muonic hydrogen atom scattering in liquid hydrogenic targets in the incoherent approximation is presented in Sec. II. Some examples of the calculated cross sections for the liquid hydrogen, deuterium, and hydrogen with a small admixture of deuterium or tritium, at the temperature T = 22 K and saturatedvapor pressure, are shown in Sec. III. A summary of the obtained results and conclusions are given in Sec. IV.

II. METHOD OF CALCULATION
Since a muonic hydrogen atom aµ (a = p, d, or t) is a small neutral object, the cross section for aµ scattering in a condensed target can be calculated using the methods that were developed for a description of the neutron scattering. In the incoherent approximation, the partial differential cross section per a single target molecule can be expressed in terms of the incoherent fraction S i (κ, ω) of the Van Hove response function S(κ, ω) [14] The squared amplitude for aµ scattering from a free hydrogenic molecule is denoted here by σ mol , the energy transfer ω and momentum transfer κ to the condensed target are defined as where ε and ε ′ are the initial and final kinetic energies of the atom; vectors k and k ′ denote the corresponding momenta. The energy transfer due to internal transitions in the atom and the target molecule is denoted by ∆E. Equation (1) is exact for the incoherent processes of internal transitions in the atom (spin flip) or molecule (rotational-vibrational excitations). In the case of strictly elastic scattering, the approximation S ≈ S i is valid for higher momentum transfers, when the coherent processes are relatively small. It has been shown in Ref. [11] that the total cross sections for aµ scattering in solid hydrogens are practically unaffected by the coherent processes at collision energy ε 1-2 meV. In Eq. (1), the rotational-vibrational transitions in the target molecules and the spin-flip transitions in the scattered atom aµ are taken into account in the free-molecule squared amplitudes σ mol , which were calculated in Ref. [15] for all combinations of the hydrogen isotopes. These calculations employed the amplitudes for aµ scattering from the bare hydrogen-isotope nuclei [9,10] as the input. The response function S i in Eq. (1) describes the translative motion of molecules in a liquid target. The studies of liquid H 2 and D 2 using slow neutron scattering showed that both the diffusive and collective modes are present in the dynamics of these quantum liquids (see, e.g., Refs. [16][17][18][19] and references therein). These studies revealed the phonon spectra similar to those characteristic for polycrystalline powders. Thus, despite the lack of a periodic structure in the liquid hydrogens, it is reasonable to describe their properties in terms of the Debye temperature for a certain range of the momentum transfers. At lowest energies, the scattering in liquids is dominated by the diffusive motion of target particles, which results in the presence of a broad quasielastic peak centered at the incident energy. At the liquid hydrogen density, the interactions between the neighboring molecules are important. This leads to a recoil-less scattering at lowest κ and ω. For example, such effect was observed in the lowest rotational excitation of H 2 in collision with neutrons [20].
Many advanced theoretical models and computer programs were developed for a description of neutron scattering from liquid hydrogen and deuterium (see, e.g., Refs. [21][22][23]). These models lead to different forms of the response function, which is usually convoluted with a response from the integral degrees of freedom of the molecules H 2 and D 2 . Such convolution involves the spin correlations that are characteristic for the neutron scattering from these molecules and different from the analogous case of muonic hydrogen atom scattering. In our approach, the spin correlations and rotational-vibrational structure of the molecules are already taken into account in the square amplitudes σ mol . For these reasons, below we evaluate the response function for the specific case of muonic atom scattering, using a simple general model from Ref. [24].
In liquid hydrogens, the diffusive motion at small κ is well described [16] by the Langevin equation where M is the molecular mass, R is a position of the molecule, ζ denotes the strength of a frictional force due to the movement in liquid, and F s is a stochastic force connected with the scattering from other molecules. The coefficient ζ is related to the self-diffusion coefficient D s by the Einstein relation in which k B denotes Boltzmann's constant. Equations (3) and (4) lead to the following diffusion contribution [24]: to the response function S i . Function exp(−2W ) denotes the Debye-Waller factor and the Bose factor n B (ω) is defined as where β T = 1/(k B T ). Equation (5) can be applied for small κ and ω, apart from the limit κ, ω → 0. The width ∆ diff at the half maximum of the quasi-elastic Lorentzian factor in Eq. (5) equals 2D s κ 2 . Since the neutron experiments revealed that ∆ diff is almost constant at higher momentum transfers, we fix its value at κ > κ max . For example, κ max ≈ 1.7Å −1 in the case of 20-K liquid deuterium [16]. The collective-motion contribution S pho to S i can be described using the following incoherent phonon expansion for a harmonic crystal [14,24]: without the strictly elastic term δ(ω), which is characteristic for scattering in solids. Functions g n , which describe the subsequent n-phonon processes, and the exponent 2W (κ 2 ) of the Debye-Waller factor are given in Ref. [11] for the isotropic Debye model of a solid. We estimate the effective Debye temperature Θ D using the relation: where h is Planck's constant, c s denotes the sound velocity, N A is Avogadro's constant, and V represents the molar volume. The corresponding Debye energy is defined as In theory of liquids, a generalized frequency spectrum Z(ω) was introduced [25], which is analogous to the density of vibrational states in solids. The Langevin equation (3) leads to the following contribution from the diffusive motion: to Z(ω). A contribution from the collective vibrations is given by the standard Debye density Z vib (ω) of vibrational states. These two contributions are shown in Fig. 1 for the 22-K liquid hydrogen at saturated-vapor pressure. The values of parameters D s , c s , and V are taken from Ref. [26]. The presented plot exhibits characteristic features, which are observed in liquids. The spectrum at lowest ω is determined by the diffusive modes. In particular, Z(ω) tends to a finite value when ω approaches zero. For solids, this limit equals zero. The Debye frequency Θ D corresponds to the large peaks in liquids, which are due to the finite-frequency collective oscillations. The tail at large ω, which is apparent in the experiments, is consistent the Enskog theory for a fluid of hard spheres [27].
For the first estimation of the cross sections for muonic atom scattering in liquid hydrogens, we use a sum of the functions (5) and (7) as a fair approximation of the total response function. Figure 2 presents the function (10) for the 22-K liquid H 2 at saturated-vapor pressure. In this plot, one can clearly distinguish the diffusive contribution at the lowest ω, and then the subsequent contributions from the one-phonon and two-phonon processes.
For large κ and ω, any system (apart from liquid helium at lowest temperatures) is accurately described by the asymptotic Gaussian form [14,24] in which ω R = κ 2 /(2M) denotes the recoil energy and E T is the mean kinetic energy of a hydrogenic molecule in the considered liquid. In solid and liquid hydrogens, this energy is significantly greater than the Maxwellian energy 3 2 k B T owing to large zero-point vibrations of the light hydrogenic molecules in these quantum systems [28]. The mean kinetics energy can be estimated using the following formula The asymptotic response function (11) leads to a correct free-molecule limit at large energy transfers.

III. RESULTS OF CALCULATIONS
Our calculations of the cross sections have been performed for liquid hydrogen and deuterium at 22 K and saturated-vapor pressures, which corresponds to the conditions of the experiments performed at PSI [3,4] and JINR [12]. At 22 K, practically all symmetric molecules H 2 and D 2 are in the ground rotational state K = 0. Thus, we are dealing with the liquid parahydrogen (para-H 2 ) and orthodeuterium (ortho-D 2 ). Only freshly prepared targets are statistical mixtures of the rotational states K = 0 and K = 1, due to a slow rotational deexcitation of the symmetric molecules [26]. Such targets are often called normal hydrogens and labeled as nH 2 and nD 2 , respectively.
An estimation of the Debye temperature using Eq. (8) and the parameters c s and V from Ref. [26] results in Θ D ≈ 66 K and ω D ≈ 5.7 meV, for both hydrogen and deuterium. For our conditions, the value of self-diffusion coefficient D s equals 1.12×10 −4 cm 2 /s for hydrogen and 0.49 × 10 −4 cm 2 /s for deuterium [26]. These parameters have been applied for plotting Figs. 1 and 2, and for calculating the cross sections. In order to illustrate contributions from the diffusive and vibration modes to the total cross section, the following function is shown in Fig. 3. In Eq. (13), M aµ denotes the mass of aµ and M is a reduced mass of the system which consists of the impinging muonic atom and a single hydrogenic molecule. Function C inc is a ratio of the cross section for aµ scattering from this molecule in the liquid target and the corresponding cross section for the same free molecule. In definition (13), it is assumed that the latter cross section is constant. Function C inc falls from the maximal value of (M aµ /M) 2 , at ε → 0, to the free-molecule limit of 1 at ε 100 meV. The quasielastic diffusive mode is most important at lowest energies. The vibration mode includes annihilation and creation of one phonon and many phonons. The plotted one-phonon contribution to C inc has a maximum at the Debye energy. The total cross section for scattering of the pµ atom from liquid parahydrogen is shown in Fig. 4. The atom is in the ground spin state F = 0 during the scattering process. The contributions to the cross section from the molecular motion in para-H 2 and from the rotational-vibrational excitations of the target molecule H 2 are shown separately. The latter contribution begin to appear at ε ≈ 40 meV, which corresponds to the first rotational excitation K = 0 → 2 of H 2 . The transition K = 0 → 1 is forbidden for the state F = 0 of pµ. The doubled total cross section for pµ(F = 0) scattering from a bare proton [9] is shown for a comparison. In this Figure, three regimes of scattering are visible. For ε 100 meV, effects of the target molecule interactions with the neighboring H 2 molecules are important. Above this energy, the scattering passes to the free-molecule regime. Finally, above about 1 eV (depending on the choice of hydrogen isotopes [15]), the free-nuclei regime is achieved.  The total cross sections for pµ(F = 0) scattering in the liquid and gaseous parahydrogen are shown in Fig. 5. One can see that the cross sections begin to significantly diverge below 10 meV. The differential cross sections for aµ scattering from hydrogenic targets are strongly anisotropic. This can be conveniently shown using the transport cross sections where ϑ denotes the scattering angle. These cross sections are employed for a simple description of the slowing down process of different particles. The transport cross sections for pµ(F = 0) scattering in the liquid and gaseous para-H 2 are plotted in Fig. 6. The cross section for scattering of the pµ atom in the excited spin state F = 1 significantly differs from that for the ground spin state F = 0. The amplitudes of pµ(F = 0) and  pµ(F = 1) scattering from a proton have the opposite signs and quite different magnitudes. Moreover, in the case of pµ(F = 1) scattering from the H 2 molecule, the ortho-para rotational transitions are allowed due to the proton exchange between the atom and the H 2 molecule. A difference of the both cross sections is apparent while comparing Fig. 5 with Fig. 7. Other differences between the cross sections are due to the scattering in hydrogenic targets with various populations of the molecular rotational states. The cross sections for the pµ(F = 1) scattering in liquid and gaseous normal hydrogen are presented in Fig. 8 as an example. In nH 2 , a population of the excited rotational state K = 1 is equal to 75%. As a result, the rotational deexcitation K = 1 → 0 is possible, which significantly changes the cross sections at lowest energies (see Figs. 7 and 8).
Energies of the hyperfine splitting in the muonic hydrogen atoms (182 meV for pµ and 48.5 meV for dµ) are much greater than the Debye energy of about 6 meV. Therefore, the energy transfer ω in the spin-flip transitions is relatively large. As a result, the spin-flip cross sections in aµ scattering from the liquid hydrogens only slightly differ from the corresponding cross sections for hydrogenic gases. The reaction pµ(F = 1) + H 2 → pµ(F = 0) + H 2 in the liquid and gaseous parahydrogen is plotted in Fig. 9. An significant difference between the spin-flip cross sections is apparent only below 2 meV. Let us note that the effective spin-flip reaction in the scattering pµ + H 2 is due to the muon exchange between the protons [29]. Figures 10 and 11 present the total and transport cross sections for scattering of the dµ atom in the ground spin state F = 1/2 from the liquid and gaseous orthodeuterium at 22 K. The corresponding cross sections for dµ scattering in the upper spin state F = 3/2 are similar since the mean amplitudes for the scattering dµ(F = 1/2) + d and dµ(F = 3/2) + d are quite close. The mean amplitude denotes here averaging over the total spin of the system dµ + d. Moreover, the rotational transition K = 1 → 0 in the scattering dµ + D 2 is allowed for the both states F = 1/2 and 3/2. As a result, the cross sections for dµ(F = 1/2) and dµ(F = 3/2) scattering in nD 2 , where the population of the rotational ortho-state K = 0 equals 2/3, are similar. The cross sections of the spin-flip process dµ(F = 3/2) + D 2 → dµ(F = 1/2) + D 2 in the 22-K liquid and gaseous ortho-D 2 are shown in Fig. 12. A relative difference between these cross sections is significantly greater (17% at 1 meV) than in the pµ + H 2 case. This is caused by a smaller hyperfine splitting in the dµ atom.
The experimental studies of fusion reactions in the muonic molecules pdµ and ptµ are often performed in liquid H 2 targets with small admixtures ( 1%) of deuterium or tritium [2-   are presented in Figs. 15 and 16. In the kinetics equations for the muon-catalyzed fusion in H/D/T mixtures, the rates of different important reactions are employed. In particular, the spin-flip rates are needed for a correct description of the time spectra of fusion products. In Fig. 17 we present the calculated spin-flip rates for dµ(F = 3/2) scattering from the HD molecule and tµ(F = 1) scattering from the HT molecule in liquid hydrogen. The rates are normalized to the density of 2.12 × 10 22 molecules/cm 3 , which is common in low-energy muon physics. This is the number density of liquid hydrogen at 20.4 K and saturated-vapor pressure.

IV. CONCLUSIONS
The differential cross sections for low-energy scattering of the muonic hydrogen atoms from liquid hydrogens have been estimated in the incoherent approximation. The Van Howe response function and the simple model of liquid hydrogen, which takes the diffusive and  vibrational degrees of freedom into account, have been employed. Our calculations show that effects associated with the presence of liquid are very important for kinetic energies below 10 meV. Above 100 meV, the corresponding cross sections for scattering in liquid and gaseous hydrogens are practically identical. In the case of spin-flip reactions, an appreciable difference between these cross sections is apparent only below a few meV, which is caused by a relatively high (compared to the Debye energy) energy transfers to the target. This difference (12% for the scattering pµ(F = 1) + H 2 and 17% for dµ(F = 3/2) + D 2 at 1 meV) is however significant for the spin-dependent processes, before a steady-state condition is reached. Thermalization of the energetic (∼ 1 eV) pµ atoms in a pure liquid hydrogen is fast ( 10 ns) owing to high magnitudes of the scattering cross sections and a large number density of the target molecules. The same conclusion holds for thermalization of the dµ atoms in a pure liquid deuterium. As a result, the time spectra, such as the spectrum of dd-fusion products, are affected by properties of the liquid only at short times ( 10 ns).  Another situation can be observed in the case of liquid hydrogen with small admixtures of deuterium and tritium. The negative muons from a beam are captured mostly by the H 2 molecules, which leads to formation of the pµ atoms and subsequent formation of the dµ or tµ atoms in the muon exchange reactions pµ + d → dµ + p or pµ + t → tµ + p. The released energy greatly accelerates the new atoms. They cannot be effectively slowed down to thermal energies in scattering from the abundant H 2 molecules because of the deep Ramsauer-Townsend minima in the elastic scattering processes dµ + p and tµ + p, at energies of a few eV [10] (cf., Figs. 13 and 15). Thus, the scattering from the molecules HD and HT establishes the effective mechanism of dµ and tµ deceleration (see Figs. 14 and 16) and spin deexcitation due to the presence of deuteron or triton in the target molecules. Since the concentrations of deuterium and tritium are small, the region of thermal energies is reached at much larger times than in the case of pure liquid H 2 and D 2 . As a result, in such H/D/T mixtures, one can expect at large times significant effects associated with the dynamics of the liquid.