A. Putze et al.: An MCMC technique to sample transport and source parameters of Galactic cosmic rays. II. The quantity l$_{rad}$ = √$_{Kγτ}$$_{0}$ is the typical distance on which a radioactive nucleus diffuses before decaying. Using K ≈ 10 28 cm 2 s − 1 and τ ≈ 1 Myr, the diffusion length is l$_{rad}$ 200 pc. Hence, in principal, any underdensity on a scale r$_{h}$ ∼ 100 pc about the Sun leads to an exponential attenuation of the flux of radioactive nuclei. This attenu- ation is both energy-dependent and species-dependent. It is energy-dependent because it decreases with the energy as the time-of-flight of a radioactive nucleus is boosted by both its Lorentz factor and the increase in the diffusion coefficient. It is species-dependent because nuclei half-lives for the standard Z < 30 cosmic-ray clocks range from 0.307 Myr for $^{36}$Cl to 1.51 Myr for $^{10}$Be. In this paper, we model the local bubble to be this sim- ple hole in the gaseous disc, as shown in Fig. 1. The expo- nential decrease in the flux of this modified DM, as given by Eq. (10), is directly plugged into the solutions for the standard DM ( r$_{h}$ = 0). In principle, i) the hole has also an impact on stable species as it decreases the amount of matter available for spallations, and ii) in the 2D geometry, a hole at R ⊙ = 8 kpc breaks down the cylindrical geome- try. However, in practice, Donato et al. (2002) found that the first effect is minor, and that the hole can always be taken to be the origin of the Galaxy (the impact of the R boundary being negligible for radioactive species). Other subleties exist, which were not considered in Donato et al. (2002). Indeed, the damping in the solar neighbourhood—combined with the production of the ra- dioactive species matching the data at low energy—means that at intermediate GeV/n energies, the flux of this ra- dioactive species is higher in the modified model (with r$_{h}$ = 0) than in the standard one (with r$_{h}$ = 0). It also means that everywhere else in the Galactic disc, at all en- ergies, the radioactive fluxes are higher in the modified model (with damping). There are two consequences: i) all spallative products from these radioactive nuclei originate in an effective diffusion region in the disc (Taillet & Maurin 2003), the size of which may be much larger than the size of the underdense bubble. In this case, these products ought to be calculated from the undamped fluxes; ii) the decay products of these radioactive nuclei (e.g., $^{10}$B, which orig- inates from the β -decay of $^{10}$Be) are stable species that originate in an effective diffusion sphere (decay can occur not only in the disc, but in the halo). Both these effects must be considered because their contributions potentially affect the calculation, e.g., of the B/C and Be/B ratios (by means of the B flux), which are used to fit the models. We confirm that taking spallative products from the damped or undamped radioactive fluxes left these ratios unchanged. On the other hand, for the decay products, the effect is of the order of 1 − 10%, which is in general enough to change the values of the best-fit parameters. However, the average flux (over the effective diffusion zone) from which the decay products originate lies between the damped and undamped values: the lower the effective diffusive sphere, the closer the flux is to the damped one. In particular, at low energy, when convection is allowed, the diffusion zone can be small (Taillet & Maurin 2003). To keep the approach simple, we use here the damped flux of radioactive species for all spallative and decay prod- ucts (as was implicitly assumed in Donato et al. 2002). This approach is expected to provide the maximal possible size for r$_{h}$ (if a non-null value is preferred by the fit). We assume that a universal source spectrum for all nu- clei exists, and that it has a simple power-law description. As in Paper I, we assume that Q ( E ) ∝ β $^{η}$R − $^{α}$. The pa- rameter α is the spectral index of the sources and η en- codes the behaviour of the spectrum at low energy. The normalisations of the spectra are given by the source abun- dances q$_{j}$ , which are renormalised during the propagation step to match the data at a specified kinetic energy per nu- cleon (usually ∼ 10 GeV/n). The correlations between the source and the transport parameters and their impact on the transport parameter determination were discussed in Paper I. In this study, we set η = − 1 and γ = α + δ = 2 . 65 (Ave et al. 2008). Constraints on the source spectra from the study of the measured primary fluxes are left to a sub- sequent paper (Donato et al., in preparation). In Paper I, the sets of fragmentation cross-sections were taken from the semi-empirical formulation of Webber et al. (1990) updated in Webber et al. (1998) (see also Maurin et al. 2001 and references therein). In this pa- per, they are replaced by the 2003 version, as given in Webber et al. (2003). Spallations on He are calculated with the parameterisation of Ferrando et al. (1988). The gas density scale height strongly varies with r de- pending on the form considered—neutral, molecular, or ionised (see, e.g., Ferri` ere 2001). We use the surface den- sity measured in the solar neighbourhood as a good es- timate of the average gas in the Galactic disc. We set n$_{ISM}$ = 1 cm − $^{3}$, which corresponds to a surface density Σ$_{ISM}$ = 2 hn$_{ISM}$ ∼ 6 × 10 20 cm − 2 (Ferri` ere 2001). The number fraction of H and He is taken to be 90% and 10%, respectively. The ionised-hydrogen space-averaged density may be identified with the free-electron space- averaged density, which is the sum of the contributions of H$_{II}$ regions and the diffuse component (G´ omez et al. 2001; Ferri` ere 2001). The intensity of the latter is well measured 0 . 018 ± 0 . 002 cm − 3 (Berkhuijsen et al. 2006; Berkhuijsen & M¨ uller 2008), whereas the former depends strongly on the Galactocentric radius r (Anderson & Bania 2009). For the total electron density, we choose to set 〈 n$_{e}$ − 〉 = 0 . 033 and T$_{e}$ ∼ 10 $^{4}$K (Nordgren et al. 1992). The disc half-height is set to be h = 100 pc. It is not a physical parameter per se in the δ ( z ) approximation, al- though it is related to the phenomena occurring in the thin disc. Physical parameters are related to the surface den- sity, which is easily rescaled from that calculated setting h = 100 pc (should we use a different h value). In the 2D geometry, the boundary is set to be R = 20 kpc and the sun is located at R ⊙ = 8 . 0 kpc. 2.4. Input ingredients and free parameters of the study 2.4.1. Gas density 2.4.2. Fragmentation cross-sections 2.4.3. Source spectrum 2.4.4. Free parameters We have two geometrical free parameters – L , the halo size of the Galaxy (kpc); 4