Size correctness

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

Siepmann J I, McDonald I R and Frenkel D 1992 Finite-size corrections to the chemical potential J. Phys. Oondens. Matter 4 679-91... 

If the line art and text originals are sized correcdy ia relation to one another, these can be assembled together onto a stiff paper or acetate base to form a paste-up or mechanical and photographed as a unit. Liae art or text that is not sized correctly is enlarged or reduced on the camera and assembly is done at a later stage. Continuous-tone originals are treated separately and assembled later. 

Droplet Size Corrections. The majority of correlations found in the Hterature deal with mean droplet diameters. A useflil equation for Sauter... 

L. E. Levine, K. Lakshmi Narayan, K. F. Kelton. Finite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equation. J Mater Res 72 124, 1997. 

Tube Size Correction Factors for Turbuient Fiow... 

Heated Length Correction Factors, Streamline Flow Tube Size Correction Factors for Streamline Flow ... 

Select a fan from the tables at (1) required inlet cfm at inlet temperature and pressure, (2) at fan tables required operating rpm approved by manufacturers for fan size, correcting (reducing) for manufacturer s safe speed for temperature and materials of wheel construction. Generally, standard high-strength steel construction has a correction on speed of 1.0 up to 400°F and graduated down to 0.87 for 800°F, and only a 1.0 factor for aluminum at 70°F and 0.97 at 200°F. 

It is essential when designing the pipe layout for gas distribution that unavoidable pressure losses are not incurred. For low-pressure gas, the pressure available at the meter inlet will be only 21 mbar, and the allowable pressure loss to the point of use only 1 mbar, although higher pressures may be available in some circumstances. If such a low-pressure loss is not to be exceeded it is essential that the pipework be sized correctly. It is preferable to oversize pipework rather than undersize, particularly as this allows... 

Calculating the effect size of a therapeutic intervention is central (step 3 in Box 3.3). Different ways to calculate effects sizes can be applied as described in Table 3.2. All statements in this box actually describe the effect sizes correctly. Is the efficacy higher for drug A than for drug B Probably not since the relative risk reduction is not identical. Instead the result probably reflects other differences such as higher morbidity (blood pressure, other risk factors, or diseases) in case A. 

As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

TABLE 3. Nuclear finite-size correction to the energy (in cm ) for the low transitions of Li-like ions, and values of the effective nuclear radius (in 10 cm). 

In the Breit Hamiltonian in (3.2) we have omitted all terms which depend on spin variables of the heavy particle. As a result the corrections to the energy levels in (3.4) do not depend on the relative orientation of the spins of the heavy and light particles (in other words they do not describe hyperfine splitting). Moreover, almost all contributions in (3.4) are independent not only of the mutual orientation of spins of the heavy and light particles but also of the magnitude of the spin of the heavy particle. The only exception is the small contribution proportional to the term Sio, called the Darwin-Foldy contribution. This term arises in the matrix element of the Breit Hamiltonian only for the spin one-half nucleus and should be omitted for spinless or spin one nuclei. This contribution combines naturally with the nuclear size correction, and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear size contribution. 

Fig. 6.2. Diagrams for elastic nuclear size corrections of order Za) m with one form factor insertion. Empty dot corresponds to factor Gb(—fc ) — 1...

Parametrically the result in (6.13) is of order m Za) m/A), where A is the form factor scale. Hence, this correction is suppressed in comparison with the leading proton size contribution not only by an extra factor Za but also by the extra small factor m/A. This explains the smallness of this contribution, even in comparison with the proton size correction of order (Za) (see below Subsect. 6.3.2), since one factor m/A in (6.13) is traded for a much larger factor Za in that logarithmically enhanced contribution. 

The description of nuclear structure corrections of order Za) m in terms of nuclear size and nuclear polarizability contributions is somewhat artificial. As we have seen above the nuclear size correction of this order depends not on the charge radius of the nucleus but on the third Zemach moment in (6.15). One might expect the inelastic intermediate nuclear states in Fig. 6.4 would... 

Nuclear size corrections of order (Za) may be obtained in a quite straightforward way in the framework of the quantum mechanical third order perturbation theory. In this approach one considers the difference between the electric field generated by the nonlocal charge density described by the nuclear form factor and the field of the pointlike charge as a perturbation operator [16, 17]. 

In the Schrodinger-Coulomb approximation the expression in (6.33) reduces to the leading nuclear size correction in (6.3). New results arise if we take into account Dirac corrections to the Schrodinger-Coulomb wave functions of relative order (Za). For the nS states the product of the wave functions in (6.33) has the form (see, e.g, [17])... 

Due to the large magnitude of the leading nuclear size correction in (6.3) at the current level of experimental accuracy one also has to take into account... 

The nuclear size correction of order Za) m in muonic hydrogen in the external field approximation is given by (6.13). Unlike ordinary hydrogen, in muonic hydrogen it makes a difference if we use mj or mmf in this expression (compare footnote after (6.13)). We will use the factor mj as obtained in [53]... 

Nuclear size corrections of order (Za) m to the S levels were calculated in [59, 53] and were discussed above in Subsect. 6.3.2 for electronic hydrogen. Respective formulae may be directly used in the case of muonic hydrogen. Due to the smallness of this correction it is sufficient to consider only the leading logarithmically enhanced contribution to the energy shift from (6.35) [21]... 

The nuclear size correction of order Za) m to P levels from (6.39) gives an additional contribution 4 x 10 meV to the 2Pi — 2Si energy splitting and may safely be neglected. 

Using the dipole form factor one can connect the third Zemach moment with the proton rms radius, and include the nuclear size correction of order Za) m in (7.62) on par with other contributions in (7.58) and (7.74) depending on the proton radius. Then the total dependence of the Lamb shift on Vp acquires the form [3, 25]... 

This expression also may be used for determination of the proton rms radius from the experimental data. Numerically it makes almost no difference because contributions in (7.63) and (7.64) with high accuracy coincide. However, the coefficient before r in (7.76) is modef dependent, so it is conceptuaffy advantageous to use the experimentaf vafue of the third Zemach moment obtained in [56] for cafculation of the nucfear size correction of order Za) m, and use the expression in (7.75) for determination of the proton radius from experimentaf data. 

Fig. 11.2. Elastic nuclear size correction of order Za)Ep with two form factor insertions. Empty dot corresponds either to Gsi—k ) — 1 or Gm(—A )/(l + /t) — 1...

The Zemach correction is essentially a nontrivial weighted integral of the product of electric and magnetic densities, normalized to unity. It cannot be measured directly, like the rms proton charge radius which determines the main proton size correction to the Lamb shift (compare the case of the proton size correction to the Lamb shift of order Za) in (6.13) which depends on the third Zemach moment). This means that the correction in (11.4) may only conditionally be called the proton size contribution. 

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