Gaussian radius

The time dependence of such a signal is presented in Fig. 3.22. For curves 1-4 the width of the entrance slit l is five times larger than the Gaussian radius of the beam ro for curves 5-8 it is equal to it, and lower... 

In certain cases a higher degree of irreversibility is observed. This is the case for the so-called pseudo-brushes where the monomer adsorption on the surface is supposed to be irreversible and where the loop size distribution remains fixed under swelling. A pseudobrush is obtained by placing the adsorbing surface in contact with a molten polymer (or a concentrated semidilute solution). All the chains within one Gaussian radius of gyration R N I a adsorb on the surface and the adsorbance is The adsorption is... 

Wangsoub and Mitchell 2009). The extracted results are shown in Fig. 5.9 with plots of the mean radius which is largely unchanged and the Gaussian radius distribution b. The width reduces sharply and reaches a plateau value between 5000 and 10,000 shear units (shear rate x time of shearing). 

Fig. 5.9 Plots of the mean diameter and the standard deviation of the Gaussian radius distribution as a function of increasing shear strain obtained from SAXS data using a Monte Carlo approach (redrawn from Wangsoub and Mitchell (2009))...

Hence, the A blocks are weak stretched, their average size is larger than the Gaussian radius whereas the shell size scales with the... 

One can introduce a distributed micleation rate J(R, f)dR for nucleating clusters of radius between R and R + dR. Its integral overi is the total micleation rate J(0- Equation (A3,3.1031 can be viewed as a radius-dependent droplet energy which has a maximum at = R. If one assumes j R, t) to be a Gaussian fiinction, then... 

For a continuous distribution, summation may be replaced by integration and by assuming a Gaussian distribution of size, Stoeckli arrives at a somewhat complicated expression (not given here) which enables the total micropore volume IFo, a structural constant Bq and the spread A of size distribution to be obtained from the isotherm. He suggests that Bq may be related to the radius of gyration of the micropores by the expression... 

The TEMqq mode is a specially desirable case. This transverse mode, the Gaussian mode, is symmetric and has no nulls. It is preferred for many apphcations. Its intensity / as a function of radius r from the center of the beam is given by the equation... 

The surface-averaged Gaussian curvature, K y, introduces the length scale, describing an average radius of curvature of the single,... 

The symbol ao is the first Bohr radius, approximately 52.9 pm, and to is the permittivity of free space.) As we will see in later chapters, Gaussian orbitals are... 

Pettinger et al. observed TERS with a sharp Au tip for MGITG dye on Au(l 11) with a side illumination [28]. They studied the bleaching of the dye and fitted the data by taking into account the radial varying intensity distribution of the field as a Gaussian profile instead of a Heaviside profile. For the former profile, the TERS radius is smaller by a factor of 1/2 than for the latter profile. They obtained a TERS enhancement factor of 6.25 x 10. The radius of the enhanced field Rfieid is about 50 nm, which results in a TERS radius of 25 nm, which is smaller than the radius of the tip apex (about 30 nm). 

The conformation of polymer chains in an ultra-thin film has been an attractive subject in the field of polymer physics. The chain conformation has been extensively discussed theoretically and experimentally [6-11] however, the experimental technique to study an ultra-thin film is limited because it is difficult to obtain a signal from a specimen due to the low sample volume. The conformation of polymer chains in an ultra-thin film has been examined by small angle neutron scattering (SANS), and contradictory results have been reported. With decreasing film thickness, the radius of gyration, Rg, parallel to the film plane increases when the thickness is less than the unperturbed chain dimension in the bulk state [12-14]. On the other hand, Jones et al. reported that a polystyrene chain in an ultra-thin film takes a Gaussian conformation with a similar in-plane Rg to that in the bulk state [15, 16]. 

The abbreviation au stands for atomic units , which is a system of units meant to simplify the equations of molecular and atomic quantum mechanics. The units of the au system are combinations of the fundamental units of mass (mass of the electron), charge (charge of the electron), and Planck s constant. By setting these three quantities equal to unity one gets simpler equations. The au system has a simple relation to the SI and Gaussian (cgs) systems of units. For example, 1 au of length = aQ (Bohr radius) = 5.29 x 10"9 cm =... 

Rg)cai2 is the calculated radius of gyration with the parameters for PIP, taken from P.J. Flory (Statistical Mechanics of Chain Molecules, Wiley Intersciences, New York 1969) and the assumption of Gaussian chain statistics. 

Here x=qNb/6=qR where Ru is the radius of an undeformed Gaussian random flight chain. I deriving Eq. 10, the sum in Eq. 7 is replaced by an integral. The effect of the free chain segments, exclusive of the position of the junctions appears in the first two terms in the exponential. The deformations of the chains depend on the constraints on the junctions. Results are immediately derivable from Eq. 10. 

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