Agreed, the optimal point is maximum T for E << V .
In a message dated 30/09/2012 19:27:50 GMT Daylight Time, writes:
Numerical integration should not be an issue.
DougOn Sun, Sep 30, 2012 at 10:44 AM, <EMyrone> wrote:
This is not analyutical, but it is well behaved, so numerical integration to machine precision is recommended.
See following posts.
See following entry.
This material is in the public domain and so is the attached article by Mr Hetherington, published more than a year ago. Under The Limitations Act of 1980, no action for defamation may be brought against Mr Hetherington and no action was brought against him. Actions for defamation must be brought within one year. Mr Justice Collins is of course protected by court privilege. No action for defamation may be brought against a Judge in court.
This is not analyutical, but it is well behaved, so numerical integration to machine precision is recommended.
These results look most interesting, all goes well.
MaximumTransmissionCoef-part1-1.pdf
MaximumTransmissionCoef-part2-1.pdf
There is no analytical solution for the complete integral:
sqrt (1 – 1 / (1 + b exp (x / a)) + c / x squared – E)
but it could be approximated, or fitted with a three or four variable least mean squares routine, or integrated numerically. It looks approximately like a bell shaped curve or gaussian. We can take plenty of time with this problem, because it is so important. We have hundreds of pairs of eyes looking at us as we work (in humour).
Yes it exists as in the attached, calculated by Wolfram integrator. Google keywords “Wolfram integrator” , first site. This is quite a simple result, so things look good. It is
Integral = – 2a inverse tanh (( b exp (x / a) + 1) power half)
In a message dated 30/09/2012 14:17:06 GMT Daylight Time, writes:
ok, but we have to integrate
sqrt(V_WS + E) dx
with V_WS = 1 / (1 +exp(ax + b))
does this integral exist?
Horst
Am 30.09.2012 15:12, schrieb EMyrone
The Wolfram integrator using Mathematica gives:
integral dx / (1 + b exp(x)) = x – a loge sub e ( b exp (x / a) + 1)
so it is analytical. Here
b = – R sub 0 / a
The Wolfram integrator using Mathematica gives:
integral dx / (1 + b exp(x)) = x – a loge sub e ( b exp (x / a) + 1)
so it is analytical. Here
b = – R sub 0 / a
Many thansk to Doug Lindstrom and Horst Eckardt. The Coulombic part of eq. (1) is integrable by hand and the strong force is of the type 1 – 1 / ( 1 + exp (x / a) exp (- R0 / a)). There may be an answer in the Wolfram automatic integrator or tables of standard integrals. I will have a look around.
In a message dated 30/09/2012 13:48:31 GMT Daylight Time, writes:
Horst:
I spent yesterday doing similar tasks working directly with the expression for T in equations 1 through 4 of note 229-5. Mathematica was not able to do an analytic solution either. I also tried a power series expansion, but convergence was not obvious. I will work on a few transformations today to see if something comes out of it. After that I will try direct numerical computation of some specific heavy nucleii impact.
DougOn Sun, Sep 30, 2012 at 12:32 AM, <EMyrone> wrote:
There is the important result that the transmission coefficient T in the WKB approxiamtion always has a maximum for any potential. So LENR design should tune to this maximum. The Lennard Jones does not have an analytical integral, and the Woods Saxon integral is complicated. I suggest that Horst go on to note 229(5), which uses the fusion barrier potential used in routine fusion theory, in order to try to find numerically whether quantum tunnelling can result in a high T for E << V. The fusion barrier potential is sketched in Fig. (1) of note 229(5). At present all these notes are being studied intensely off the blog around the world as they are produced. It might be possible to fit the fusion barrier potential with a curve fitting program, then integrate it numerically. The fusion barrier potential is given in eq. (1) of note 229(5), and is a combination of a Coulombic proton proton repulsion and a nuclear strong force attraction modelled by a modified Woods Saxon potential so that V = 0 at r = 0, goes through a maximum at the fusion barrier, then decreases again to zero. The key question is whether quantum tunnelling through the fusion barrier can occur for E << V, i.e. for low energies E. That would be low energy nuclear fusion by quantum tunnelling alone. In the next note I plan to give the ECE theory of quantum tunnelling accompanied by absorption of a wave of spacetime. In the Brillioun Company design this is a phonon wave.
In a message dated 29/09/2012 18:01:47 GMT Daylight Time, writes:
I did some calculations with the Lennard-Jones and Woods-Saxon potenital.
The first general result is: the maximum of the transmission coefficient
in WKB approximation is valid vor all forms of potentials. This comes
out from the general formT = 4 / (2*theta + 1/(2 theta))^2
(eq.(3)) which gives a maximum at theta=1/2. In the preceding email I
used the abbreviationy = theta^2
where consequently the maximum is at y=1/4.
The Lennard-Jones potential has to be inserted in eq.(4) for theta. The
integral is quite problematic, there is no analytical solution. I tried
a Taylor series expansion of the integrand (Maxima handled this), but
this is highly complicated already in second order and not practical.
The situation is better for the Woods-Saxon potential. The integral is
solvable analytically in the approximation E=0. I integrated from 0 to
infinity where the integral exists. As a result one obtains a
complicated expression with log functions of the parameters of the
potential. When evaluating the y value for T=1 in the same way as for
the Coulumb potential, it comes out that m is inversely proportional to
V0 (eq. %o25, E=0 was assumed as reported, y1 and y2 depend on the
Woods-Saxon parameters only). This is not so easy to fulfill because
with increasing m, the potential constant V0 increases (i.e. the
potential becomes more negative for r=0). This is just the inverse
behaviour compared to the calculational result.Horst
