3. Desarrollo experimental
3.2. Metodología
3.2.1. Síntesis de nanomateriales
Again we seek a better approximation to A^ at the end point of the range for t , this time i = n, than for neighbouring values.
In (6.23) we already have an excellent approximation to A^ ; this can still be improved a little however in respect of the accuracy of the proportionality constant, now that values of A^ are available up to n - 20.. Repetition of the argument in ^5*1 leads to an
extension of Table 5*4 by eight more entries (from n = 1 3 to n = 20) and, by extrapolation), a more precise estimate of -0.522499625 for the proportionality constant. The expression for A^ may therefore be written A% = K,Ca^)! r," [ I + r! + o H A j , (8.8) where />, = 0.0839742526264095 ; c, = I 1.631325858441925 , (8.9) K, = -0.522499625
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To find the approximate form of for neighbouring values of & , we again resort to trial and error checked by numerical comp utation. The conclusion reached is that the variation of A*^ withn and S. for is best represented by
(
8.
10)
where O', is a constant approximately equal to -0.33333. This is justified by the observation that
^ _______! a^cf ^
A (kh i- + pp 2h4.S
tend respectively to the constants and <Jj as n — > for small values of s.
8 ,3 Approximate Form of the Polynomials for X 0 and
It is convenient at this juncture to denote by
A (x)
the n ‘th polynomial derived from the coefficients A^ , thus;A „ ( x ) - f ^ " (8.11)
For very small values of X, A%(x) is dominated by the contributions n d n
associated with A^, A,, A^.., , which are fairly accurately expressed by (8.7). Hence we have, to a good approximation,
' SI--0 (2.Z)! C A X t 0)
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where
0 •= +O W ^ j[ >Po- A ] . (8.13)
On the other hand, when X becomes very large, A^(X) is dominated by A% , Aj^^j , A^^g^ ... , which are fairly accurately expressed by (8 .lO). We then have
^ A»., X
oo)
I
cr,/x^- U"- j ‘ (8.14)
It is easy to check numerically that (8.12) and (8.I4) are good approximations to A^(x) for X 0 and X-5* respectively, and that they fail for intermediate values of X, Far more important than this,
however, is that (8.12) predicts exactly the right form for in the region 0 •$ X 4 1. In Chapter 5 we described how A y ,(x ) oscillates
with variation of X in this region, the amplitude of the oscillation decreasing as X increases from 0, becoming vanishingly small near X = 1, with the period of the oscillation simultaneously increasing. All these features are inherent in (8.12) provided @ is an increasing
‘Ill
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function of X with values 0 and If/ 2 at X = 0 and 1 respectively. The required value at X = 1 is not satisfied by (8.I3), of course, but we are entitled to hope that the actual form of
(x)
whenfinally elucidated might be hardly more complicated than (8.12),where
0(x) meets the specification described above and is approximated by (8,13) when X is small. Similarly it is a reasonable expectation that a slightly modified version of (8.I4) might be the correct form for A ^
(x)
for the region 1 < X < , the principal modificationconsisting of replacing X/(l- /X^ )by a more general (and, as yet, unknown) function of X, one which reduces to X (l-Gj/X^ ) in the limit of very large X. These hypotheses we now put to the test.
8.4 Approximate Form of the Polynomials for All Values of X
Following the discussion of the previous section" we assume A^ (X) to be given approximately by
/‘'Ceos e) e
6sO«+i
) 0^
0 4x
4I ^
(8.15)
[('(^)] ,
I < X < ^ ,(8.16)
where 0
(x)
and h(X) are to be determined.The simplest way to find 0 (X) is to locate numerically the zeros of Ay^
(x).
These occur at approximately1 ) ^ > 3 E , S'Jl , . (8.17)
(^Approximately, because (8.I5) is inexact - higher order terms will shift the zeros slightly from these positions.^ For given n, (8.I7)
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determines r the. values 9 @(X%%)... corresponding to the observed zeros X^^^ , X%%... of
(x).
When we plot all the values of 0(x)
versus X for all the different values of n, a continuous curve will result only if the original hypothesis is correct.Fig.l displays the. plot of (2/ IT) 0 (x) versus X, incorporating data corresponding to values of n.between 4 and 2 0. The curve is quite smooth, and it is easily confirmed that 6 (,X) ^ tan ( JFFo' X) for X ^ 0»4. As X —> 1, the gradient of the curve increases rapidly and it is not easy to find the value of G (l) by extrapolation; how ever, the curve is perfectly consistent with our expectation © (1)=T/ 2
To find the function
h(x)
numerically we calculate values of A„^x)for fixed X and variable,n and extrapolate to l/n = 0, repeating for other values of X. Consistency between the limits predicted by the various Salzer formulae serves as a guide to the true limit for that value of X, Table 8*3 displays the values of
h(x)/x
for a selection of values of l/x, Due to the difficulty of accurately computing high* order A(x)
near X = 1 (at this point numerical cancellation is prac tically complete) the estimates ofh(x) gradually decrease inprecision as l/X runs from 0 to 1*. The number in brackets after each entry indicates the likely error in the last significant figure quoted. As expected,
h(x)/x
ùi l/( 1- /X^) for l/X 0*5*We now require to find the analytic form of ^(X) and h(X). Happily we possess one important clue to this problem, in the numer ical value of the constant which appears in the approximation to