Cross section stopping

Referring to Fig. 2.3, one can easily visualize that when the incident projectile gets scattered from an atom inside the target material (instead of the atom at the surface), the stopping cross-section factor is given by 

The He stopping cross-sections for a few elements at a few selected energies are given in Table 2.2. 

When a beam slows down in a target composed of more than one element, the energy loss can be calculated using Bragg s rule, which states that the total energy loss eAB in a compound AmB is given by 

The stopping power of a material is defined as the energy loss per distance travelled in the material, dE/dz, and the Bragg rule may be expressed more generally, if the stopping cross-sections for each element are known  

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The easiest way to obtain the concentration m of the isotope A is to use a reference sample ( standard ), that contains the isotope A with a known atomic fraction/ for comparison. Knowing the NRA yield of the standard Yst and its stopping cross-section gt, the atomic fraction m in the sample can easily be evaluated from the yield Ya of the sample for the same projectile energy, taking the different stopping cross-sections into account ... 

Thus the Bethe sum rule is fulfilled exactly in the RPA at all values of the momentum transferred, provided that a complete basis set is used. Therefore, as in the case of the TRK sum rule when optical transition properties (q = 0) are considered, we expect that the BSR sum rule will be useful in evaluating basis set completeness when generalized oscillator strength distributions are calculated, for example for use in calculating stopping cross sections. It should be noted [12] that the completeness of the computational basis set is dependent on q, and thus care needs be taken to evaluate the BSR at various values of q. 

Any energy difference can thus be converted into a depth z using this equation. If the stopping cross-section [s] is used instead of dE/dz, the equation becomes ... 

Calculation based on the stopping cross section implied by the experiments of Michaud and Sanche (1987) in solid water, giving tA 2x l(h14 s. 

Calculation of many trajectories at different impact parameters for each given incident energy yields the energy-dependent deflection function and energy loss, which can then, through equation (1), be used to calculate the stopping cross section. 

This level of theory outhned above is implemented in the ENDyne code [18]. The explicit time dependence of the electronic and nuclear dynamics permits illustrative animated representations of trajectories and of the evolution of molecular properties. These animations reveal reaction mechanisms and details of dynamics otherwise difficult to discern, making the approach particularly suitable for the study of the subtleties of contributions to the stopping cross section. 

The total electronic stopping cross sections for the three independent directions in which the water molecule has been oriented with respect to the beam axis are presented in Fig. 2, along with the experimental results of Reynolds et al. [28] for comparison. 

Fig. 2. Stopping cross section for protons impinging on oriented H2O as a function of proton velocity. See Fig. 1 for the definition of target orientations.

Fig. 3. Orientationally averaged total stopping cross section St for protons impinging on H2O as a function of proton velocity along with the experimental results of Reynolds et ah (filled circles) [28]. 5 e, and represent the electronic, nuclear, and rovibrational contributions to the stopping, respectively.

More interesting, perhaps, is the observation that St(c) is very close to the total stopping cross section, This would imply that in the Bethe formulation, the mean excitation energy associated with the random orientation and that associated... 

Fig. 2. Stopping cross section for hydrogen in argon. Calculated from binary theory with and without shell correction. Experimental data from numerous laboratories compiled in Ref. [6].

One must expect the presence of mixed terms of the form k B in the expansion. The term of lowest order a —2, d = l), contributing oczf to the stopping cross section, would indicate a difference between the Barkas-Andersen correction evaluated from the Born series and the Bohr model, respectively. While such a comparison has not been performed in general terms, a numerical evaluation for the specific case of Li in C revealed a negligible difference [24]. 

In practice, series expansions of the stopping cross section in powers of Zi are only useful at high beam velocities where neither the Bloch nor... 

In the present context the most important conclusion emerging from all theoretical schemes, whether classical or quantal, is that the dependence of the stopping cross section on the ion charge tends to be weaker than q. This dependence is sensitive primarily to the ratio Z1/Z2 and less pronouncedly so to the projectile speed [43]. 

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