Small component magnitude

Table I also contains an analysis of the orbital character of these five energy levels. These were determined from the four-component spinors by neglecting the two lower, "small," components, and by assuming that the radial functions depend only upon , i.e. that the radial functions for pi/2 and p3/2> or for da/2 and ds/2> are the same. The orbitals may then be written in "Pauli" form as products of (complex) spherical harmonics and spin functions. Populations are equal to the squares of the absolute magnitudes of the coefficients listed in Table I. [For all but 17e3g, an additional orbital (not shown) is occupied which has the same energy but the opposite spin pattern (i.e. a and 3 are interchanged).]...

The model diagnostic tools suggest the inherent dimensionality of this data set is 2. th two principal components, the percent variance described is 99.4% and the residuals appear reasonably random and are small in magnitude. The RMSECV PCA is not revealing, but also does not contradict this conclusion. 

A small component of the short-term variability falls under the category of anomalies or transients. This includes a short-lived but abrapt negative excursion 55 Myr ago (Figure 2(b)). The magnitude of the change implies a 4-5 °C transient... 

The use of radioactively labelled Ca to monitor changes in the magnitude and duration of the slow Ca + current is also difficult, partly because of the small amounts of Ca2+ that are involved. When considered on a beat to beat basis the Ca + that enters heart muscle cells by way of the slow Ca + current represents less than 2 percent of the total tissue Ca + and if, as seems probable, this small component is rapidly recycled to the exterior its accurate detection during the time course of the action potential presents substantial technical difficulties. 

After the fpFW transformation the magnitude of the small component as compared to the large component has decreased in leading order in 1/c,... 

We know that for a positive-energy bound state (or an unbound state with W close to me ) the lower component is (except possibly very close to a nucleus, for atoms depending on the sign of k) smaller than the upper component by a factor of 0(c ) (hence the alternative names large and small components for p> and respectively - which are only valid for this kind of state). In order to deal with quantities of the same order of magnitude we introduce [33]... 

These functions satisfy the normalization condition (20). It should be noted that the ratio of the scale factors in (48) and (49) is - (1 — e /c )/ I + Cmc/c ) 0iZl2n. Thus, Qnit r) is several orders of magnitude smaller than Pn ( ) for Z = 1. For this reason, P K and QnK are referred to as the large and small components of the radial Dirac wave function, respectively. 

Hertz is too small in magnitude for the NMR instrument to be able to distinguish the separate peak components.) Nonequivalence of Jab and 7bc> of course, happens most often when the chemical shifts of the protons on carbons A, B, and C are all quite different. 

Figure 6.1 also shows the increasing size of the small component QisCf) with nuclear charge Ze. For superheavy atoms — say, for Z > 100 — the small component is no longer small. For Z = c 137, Pis(r) and Qis(p) are of similar magnitude in absolute value but different sign at any distance r. 

The influence of the small component is to some extent shifted into the Hamiltonian /i in such a way that the magnitude of the transformed small component is significantly decreased. This is most easily seen by a restriction to the leading order in 1/c. Then the relation between the large and small component of the untransformed Dirac spinor is given by... 

We see that the magnetic dipole operator connects large and small components, but the prefactor c ensures that the results are of the same magnitude as expectation values between two large components. We can demonstrate this point by substituting for the small component using the approximate expression from the nonrelativistic limit... 

The Pauli expansion results from taking 2mc out of the denominator of the equation for the elimination of the small component (ESC). The problem with this is that both E and V can potentially be larger in magnitude than 2mc and so the expansion is not valid in some region of space. In particular, there is always a region close to the nucleus where V - E /2mc > 1. An alternative operator to extract from the denominator... 

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