On Dec 29, 2009, at 6:42 PM, fznidar...@aol.com wrote:
In a message dated 12/29/2009 7:33:04 PM Eastern Standard Time,
hheff...@mtaonline.net writes:
Yes, from the very little information provided above I suspect
there might be. I suspect you are using c as the speed of light
instead of the speed of sound in the medium.
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
No. I tried to get an estimate of phonon frequency from the
resonant freq formula.
It is the sq root of K/M times 1/rwo pie
By the above I take it you mean frequency f is given by:
f = (1/(2*Pi)) (k/M)^0.5
which is the resonant frequency of a mechanical system.
For M I used 2* 1836 *9.1 *10 exp -31 KG
The mass of the deuteron is 3.3444941 x 10^-27 kg.
For K I used 1.45 exp 11 Newtons / meter
What is wrong?
Frank Z
The above k is the spring constant, given by Hooke's law:
F = -k/x
where F is the displacement force and x is the displacement. It
assumes a linear relationship, which does not exist with respect to
Coulomb's law. But, let's ignore that for now. Palladium has a room
temperature lattice constant of about 0.389 nm, or 3.89x10^-10 m.
Let the deuteron move 1/100 of that distance from the zero force
location, or let x = 3.89x10^-12 m. Assume the deuteron is vibrating
at a displacement +- x from the neutral force point.
From Coulomb's law:
F1 = -C_k * q^2/(r1)^2 = -C_k * q^2/(3.89x10^-10 m - 3.89x10^-12
m )^2
F2 = +C_k * q^2/(r1)^2 = -C_k * q^2/(3.89x10^-10 m + 3.89x10^-12
m )^2
where C_k * q^2 = 2.30708 J/m, and:
F1 = -1.55558x10^-9 N
F2 = 1.49459x10^-9 N
F = F1 + F2 = -6.099724x10^-11 N
k = -F/x = (6.099724x10^-11 N)/(3.89x10^-12 m) = 1.568 N/m
and we obtain the frequency by:
f = (1/(2*Pi)) (k/M)^0.5 = (1/(2*Pi)) ((1.568 N/m)/(3.3444941 x
10^-27 kg))^0.5
f = 3.459 x 10^12 Hz
Now that is resolved, there is another problem. The deuterons
actually are screened from each other by lattice electrons. They
exist in lattice potential wells. They essentially move at lattice
speeds due to lattice vibrations, except when tunneling between
lattice sites. These lattice vibrations might be considered to
achieve resonance when lambda = N * 2 * (lattice constant), which for
Pd is 7.78x10^-10 m. The resonant frequency is given by:
f = v / lambda
where v is the speed of sound in the medium, or 3070 m/s, so:
f = ( 3070 m/s) / (7.78x10^-10 m) = 3.946 x 10^12 Hz
which is not far off from the other frequency I gave, which was
computed from the deuteron 1 dimensional mechanical resonance.
In all cases the phonon energy E is quantized to:
E = (N + 1/2)*h*f
where (1/2)*h*f is the zero point energy, and N is an integer.
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
