TDTR热力学模型

T ( x) P( x0) g (x x0)dx0 P( x) g ( x)
Apply Fourier transFra Baidu bibliotekorm
F (T ( x)) F ( P( x)) F ( g ( x))
Then the real temperature field can be solved as the inverse Fourier transform of F(T(x)).
A point heating source with unit power modulated with frequency ω, (boundary conduction):
2r 2 rT |r 0 1 T ( r , t ) |r 0
Solve for this equation in spherical case, the point spread function is
In real case we assume pump and probe beams have the same diameter (w0=w1)
Multi-layer Solution
r-1 r r+1
We number the layers by n=1 for the layer that terminates at the surface of the solid. The iteration starts with the layer farthest from the surface. In practical applications of this method to analysis of TDTR data, heat cannot reach the other side of this bottom layer at rates comparable to the modulation frequency, therefore, B+ =0 and B− =1 for the final layer.
Temperature Field in a single layer
The heat point spread function:
g (r ) exp( qr ) 2r
q 2 i / D
Since the laser beams of TDTR experiment have cylindrical symmetry, we use hankel transform. The hankel transform of g(r) is
Where w0 is 1/e2 of the pump beam, where A is the amplitude of the heat absorbed by the sample at frequency w.
The hankel transform of p(r) is
Temperature Field in a single layer
g (r ) exp( qr ) 2r
Fourier transform and Hankel transform
Assuming I have a point heat source with unit power, described by the point-spread-function (PSD) g(x).Then in the real case of Gaussian distributed laser spot, the temperature field will be a convolution of the power distribution of the heating source P(x) with the g(x).
J0 is the zero order Bessel function
Temperature Field in a single layer
The surface is heated by a bump beam with a Gaussian distribution of intensive p(r);
文献总结 TDTR的热力学计算模型
Heat conduction model
Point heat source
Heat flux
Heat conduction equation in an isotropic material without a heat source
tT q 2 2T