Approximation Fraunhofer

Fraunhofer rules do not include the influence of refraction, reflection, polarization and other optical effects. Early Iziser particle analyzers used Fraunhofer approximations because the computers of that time could not handle the storage cuid memory requirements of the Mie method. For example, it has been found that the Fraunhofer-based instrumentation cannot be used to measure the particle size of a suspension of lactose (R.I. = 1.533) in iso-octane (R.I. = 1.391) because the relative refractive index is 1.10, i.e.- 1.533/1.391. This is due to the fact that diffraction of light passing through the particles is nearly the same as that passing around the particles, creating a combined interference pattern which is not indicative of the true... 

According to the Fraunhofer approximation of kinematic scattering theory the real space and the reciprocal space are related to each other by an integral transform known by the name Fourier transform, which shall be indicated by the operator The n-dimensional (nD) Fourier transform of h (r) is defined by... 

In this case m = 1 (t is simply a dummy variable of integration). If A is the wavelength, aT the radius of the transducer, L the distance from the transducer, and r the cylindrical radial coordinate at that distance, then u = 2nLaT/Xr. Since jinc(0) = 5, the amplitude distribution is A + 2A0 )m.c 2nLai/ r), where Ao is the amplitude on the axis at distance L. Whether the Fraunhofer approximation is valid is determined by the dimensionless distance spi which is defined as... 

The Fraunhofer approximation is useful when sF 5 1, i.e. well beyond the Fresnel distance F = a jX. Closer to the transducer the field must be calculated numerically (Zemanek 1971). If aj/X > 1, the amplitude and phase fluctuate considerably in the region between the face of the transducer and the plane sF = 1 in particular there are nulls along the axis in the region 0 < s <0.5. The final maximum in the amplitude on the axis occurs at sp = 1, i.e. F = a /X, which is known as the Fresnel length. In the plane perpendicular to the axis at sp = 1 the field is reasonably well behaved in both amplitude and phase. 

To monitor nanoparticle swelling in salt environment, we employed laser diffraction Mastersizer Micro Particle Analyzer MAF5000 (Malvern) with a dynamic range of 0.3 to 300 pm. This instrument utilizes Mie scattering algorithm with the Fraunhofer approximation. 

The newer la.ser diffraction instrument allows measurement for particle sizes ranging from 0.1 pm to 8 mm (7). Most of the laser diffraction instruments in the pharmaceutical industry use the optical model based on several theories, either Fraunhofer, (near-) forward light scattering, low-angle laser light scattering, Mie, Fraunhofer approximation, or anomalous diffraction. These laser diffraction instruments assume that the particles measured are spherical. Hence, the instrument will convert the scattering pattern into an equivalent volume diameter. A typical laser diffraction instrument consists of a laser, a sample presentation system, and a series of detectors. 

In principle, the diffraction patterns can be quantitatively understood within the Fraunhofer approximation of Kirchhoff s diffraction theory as described in any optics textbook (e.g., [Hecht 1994]). However, Fraunhofer s optical diffraction theory misses an important point of our experiments with matter waves and material gratings the attractive interaction between the molecule and the wall results in an additional phase of the molecular wavefunction [Grisenti 1999], Although the details of the calculations are somewhat involved2, the qualitative effect of this attractive force on far-field diffraction can be understood as a narrowing of the real slit width to an effective slit width [Briihl 2002], For our fullerene molecules the reduction can be as big as 20 nm for the unselected molecular beam and almost 30 nm for the slower, velocity selected beam. The stronger effect on slower molecules is due to the longer and therefore more influential interaction between the molecules and the wall. 

Figure 7-11 Calculated response curves for the intensity of diffracted light based on Fraunhofer approximation for different minimum collection angles (maximum collection angle 4.76°, laser wave length 514.5 ran)...

Fraunhofer rules do not include the influence of refraction, reflection, polarization and other optical effects. Early laser particle analyzers used Fraunhofer approximations because the computers of that time could not handle the storage and memory requirements of the Mle method. 

For a given C(x ), Equation 3 can be evaluated to produce a diffraction pattern appearing at the screen. The integral may be solved for any set of experimental conditions, but the problem is simplified significantly if the screen is placed at some large distance away from the electrode. This is a standard approximation in solving diffraction problems known as the Fraunhofer approximation,13 and can easily be realized experimentally. When this approximation is made, the expression for f-7 is simplified, and Equation 3 becomes... 

The approximation of Fraunhofer has been employed in some implementations. Flowever, it does not include amplitude and polarization dependence and also ignores light transmitted through the particle. Thus, the Fraunhofer approximation is prone to predicting incorrect quantities of small particles having a specific size. A matrix of the predicted light scattering data is precalculated and stored in memory. 

The nature of the errors observed when using the Fraunhofer approximation are not always predictable. 

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