Stopping power density effect

Fermi (1940) pointed out that as /)—-1 the stopping power would power would approach °° were it not for the fact that polarization screening of one medium electron by another reduced the interaction slightly. This effect is important for the condensed phase and is therefore called the density correction it is denoted by adding -Z<5/2 to the stopping number. Fano s (1963) expression for 8 reduces at high velocities to... 

Figure 3 Density-normalized stopping power (MeV cm /g) of gaseous and liquid water as a function of electron energy according to track simulation by Pimblott et al. [35]. There is a noticeable phase effect, while a peak is seen at —100 eV in both phases.

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

If the charged particle is a nucleus with no electron shells around it, the molecular stopping power depends on the velocity of the particle in the following way. At relativistic velocities v = c, the dependence of Se on v is determined by the logarithmic term in the brackets, because at v- c the factor preceding the brackets tends to some finite limit. So the molecular stopping power experiences the so-called relativistic rise. As for Se, its relativistic rise in real dense media is slowed down by the density effect (see Section V.B.2). 

Further developments for slow ions included the application of the density functional theory (DFT) by Echenique et al. [31-33], which yields a more sophisticated (and also non-linear) treatment of many-body effects in dense media. This theory explains also the oscillatory behavior of the stopping powers in the range of low velocities (v < Vq, Vq being the Bohr velocity). But the question of extending the DFT calculations to intermediate or large velocities is still a complicated numerical problem. 

In Fig. 4 we show the experimental stopping power at zo — 1.2 a.u. compared to the theoretical value of the stopping power. At this distance the corresponding density parameter is = 2.2 a.u. The figure shows an excellent quantitative agreement between theory and experiment. Note that at Zo= 1.2 a.u. the electronic density is very close to the bulk value. In this case, effects due to the nonuniformity of the electronic density seem to be negligible, and equation (16), valid for uniform systems, is applicable. 

The stopping power of the detector determines the mean distance the photon travels until complete deposition of its energy and depends on the density and effective atomic number (ZeU) of the detector material. The scintillation... 

As stated earlier, Eqs. 4.2 to 4.4 disregard the effect of forces between atoms and atomic electrons of the attenuating medium. A correction for this density ejfect has been made, but it is small and it will be neglected here. The density effect reduces the stopping power slightly. 

Stopping power vs. relative momentum, py = p/Mc, for muons in copper. The solid curve indicates the total stopping power, the dash-dotted and dashed lines the Bethe-Bloch equation with and without density effect correction. The vertical bands separate the validity regions of various approximations indicated in the figure. The dotted line denoted with p. indicates the Barkas effect. In the Bethe-Bloch region the stopping power scales with the particle mass and Z/A of the medium... 

The power law fluid yield stress is zero, and the fluid is deformed as long as the effect of a small force on the fluid. Particle density is greater than that of the fluid. In addition there is a vertical downward force formed by particle gravity and buoyancy force of the particle fluid. Therefore, particles settle. When the particle diameter is small to a certain extent, it will not overcome the yield stress and get a suspension in the fluid. Then sedimentation does not occur, which is known as natural suspended state. When the fluid stops circulating, it can make the solid phase suspension in the annulus to prevent the deposition of the solid phase at the bottom of the borehole. In this case, accidents can be avoided. Conditions for particles sedimentation is shown as follows ... 

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