Collision lengths

The amplitude is a complex number, whose absolute value has the dimension of a length. In the limit k0 -> 0, this number becomes real. It is called the collision length. 

With a nucleus of spin zero, we can associate a potential V0. If this nucleus is bound, the collision length is denoted by b and we have... 

If the nucleus is free, the modulus of the collision length is smaller than the... 

The first situation corresponds to a state with total spin (S + 1/2). It is not especially interesting because the corresponding collision length b+ is not very different from the collision length of spinless nuclei like carbon also present in the target. 

The second situation corresponds to a mixture of states of spin (S + 1/2) and of spin (S — 1/2). In this case, the collision length b- which is strongly negative plays an important and useful role, by creating contrast (see Chapter 7). It is the latter situation which will now be studied. 

Starting from (6.4.38) and using relations (6.4.27), we may, in this case, write the coherent collision length in the form... 

For protons (S = 1/2), this length is —1.82 x 10 5 nm (see Section 3.7.7.1) and we see that its modulus is about five times larger than the coherent collision length of protons in a non-polarized target. For the coherent cross-section, a factor 25 is won and this is why this type of experiment is valuable. The nuclear... 

Let us also recall (see Section 4.4.1.2) that the coherent collision length associated with a proton belonging to a target made up of polarized protons is... 

The correlations between sites rj and rt come into play in (6.6.2). The collision lengths byoh are determined independently for instance, by measuring a refractive index n (photon or neutron refraction) ... 

The reciprocal space interval (qmin, that can be explored is indicated in Fig. 6.4. It depends on the type of radiation used. In the case of neutrons and of a sample containing nuclei with non-zero spin (for instance, protons), the coherent collision length associated with these nuclei depends on the state of polarization of the neutron-nuclei system. The incoherent random noise ( , nc) can then be large. 

For light scattering, contrast results from a difference in polarizability of the molecules. For neutron scattering, it results from a difference in nuclear interaction (different collision lengths). In a general way, the contrast can be defined from the refraction indices corresponding to the radiation and to the materials constituting the mixture. 

In general, the monomers belonging to the same type a have the same collision length... 

This grouping has the advantage of generating new factors which are independent of the collision lengths. Let us set... 

The set is made here of N identical molecules. The molecules contain various types a,. .. of monomers. The collision length of a monomer of type a is ba. The concentration of monomers of type a is Ca. The cross-section reads... 

The molecules of species si = 1, 2 are distinguishable only through their collision length brf. In this case... 

Table 7.1. Constants concerning a few usual monomers M = molecular mass, b = collision length, w = volume per mass, b = collision length per unit volume, b j. is calculated with the help of formula (7.2.3) by using as coherent collision lengths for the nuclei, the values of b 11, bQ0h, b 11 given by Table 6.12 in Chapter 6. (non-polarized sample). The partial volumes (per mass) wrf are extracted from ref. 9, and we have v M j(A) (see Chapter 5, Section 2.1).

In this case, the collision lengths by are directly related to the polarizabilities a [see (6.3.68)] and to the wave number k... 

Fig. 7.2. Collision length by per unit volume for a few molecular species (non-polarized sample). The contrast lengths (per unit volume) of a solute with respect to a solvent are obtained by subtraction of the quantities that appear respectively to right and to the left of the central line. (From Ionescu.10)...

Figure 7.2 gives values of the ratio (collision length)/(partial volume) for various solvents and various solutes. The specific contrast of a solute with respect to a solvent can be read directly from the figure. In the same way, the variation domains of a specific contrast obtained by isotopic modification of the solvent or of the solute, can be obtained from this figure. 

If there exists a composition x0 for which (x0) vanishes, it can be calculated theoretically if the collision lengths b of the monomer, b0 of the deuterated solvent, and bt of the non-deuterated solvent are known and if the partial volumes of a monomer and vs of a solvent molecule are also known. Nevertheless, it is necessary to verify that the signal really vanishes for this composition x0. In fact, it happens quite often, that a poor knowledge of the partial volumes leads to an uncertainty as to the value of x0. 

Usually, the values of the collision lengths are such that the experimentalist cannot explore the interval 0 Y 1 by changing x (see 7.3.52). However, this difficulty can be overcome by polarizing the sample (see Chapter 6 C. Fermon and H. Glattli, to be published). 

We then consider a mixture of rather short chains denoted by index zero and of longer chains denoted by index 1. The monomers constituting these chains are chemically identical but isotopically different. Hence, different collision lengths, b0 and are associated with the two types of monomers. We note that in the limit Ct - 0 (see Chapter 7, Section 3.1.1)... 

One-dimensional velocity distribution Specific conductivity, hard-sphere diameter for a collision, length parameter in Lennard-Jones potential, symmetry number of a molecule Intrinsic lifetime of a photoexcited state Azimuthal angular velocity in spherical polar coordinates, azimuthal angle in spherical polar coordinates, angle of deflection Quantum yield at wavelength A Fluorescence (phosphorescence) quantum efficiency... 

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