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Autocorrelation Due to Isotropic 3D Translational Diffusion

The important points of this section are:

• The three-dimensional detection, or confocal, volume in an FCS instrument in given by

• The autocorrelation functions for three-dimensional isotropic diffusion are given by

Detailed Derivations

Fluctuations of Concentration in a Small Volumeback to top

We start with relatively simple question: What is the autocorrelation function for simple, isotropic, three-dimensional diffusion? We begin by noting that the laser beam near the waist can be approximated by Gaussians in the x, y, and z directions, where the z direction is defined as that of the optical axis. That is

Note that the term κ² takes into account the fact that the radius is larger in the vertical z direction than in the horizontal, x and y, directions.

Now suppose that the concentration of a fluorescent species is

Where <C> is the average concentration and represents , the diffusion-driven stochastic fluctuations within the confocal volume (see What is the Confocal Volume?). Fluctuations relax according to the diffusion equation

If you think of this on a macroscopic scale, you get a fluctuation in C, and diffusion causes it to relax. It would take an infinite amount of time for that fluctuation to smooth out so that it reaches an infinite distance away. As a result we have the boundary condition, that

Fluctuations and Correlation of Detected Photonsback to top

Now suppose that we detect n(t) photons in a time interval Δt, then

Where q is an excitation and emission efficiency coefficient and where we have simplified our notation by abbreviating dxdydz as .

If n(t) fluctuates around some average <n> where

It then follows that

Noting that the intensity, I(t)=n(t)/Δt, the correlation function of the measured fluorescence intensity fluctuations, dI(t), is given by

Where M is the number of measured intervals Δt

Calculating the Autocorrelation Functionback to top

The standard method of a solving partial differential equation of the form of Equation 2 is to apply the Fourier transform (see Fourier Transforms in FCS) as well as the boundary condition of Equation 3. This converts it to a simple differential equation. Equation 3 becomes

which has the solution

We are then in a position to calculate the autocorrelation function of the concentration fluctuations, . Note that this function relates the fluctuation at two arbitrary points in space separated by a time interval, t.

It is significant to note that because we are dealing with dilute, ideal solutions, at time 0, when the system is presumed to be at equilibrium, there is no correlation between the concentration at one point and the concentration at another point. Thus

where we have made use of the fact that ensemble averaging is interchangeable with Fourier transformation (in this case inverse Fourier transformation).

We can then put the Fourier transform of the concentration fluctuation, Equation 12, into Equation 14

We then reverse the Fourier transform process

Upon putting Equation 13 into Equation 16, this becomes

Equation 17 can next be used in Equation 10 to calculate the measured intensity correlation function

We then perform the integrals over d³r and d³r’ and obtain

where is, of course, the Fourier transform of the beam profile, Equation 1. This may be calculated using the laser intensity profile to obtain

Putting Equation 6 and Equation 20 into Equation 19, we obtain

If we define the characteristic correlation time as

then

If we again apply the beam profile we obtain

Finally we note that has units of volume. If we define it to be the confocal volume, V_c, then the number of diffusing particles is given by Np=<C>Vc and