Resonance function

B3.4.7.2 NUMERICALLY EXTRACTING BOUND STATES AND RESONANCE FUNCTIONS... 

An embodiment having a cylindrical structure is shown above. The detector element is built on a substrate 24 of p-doped CdTe, CdMnTe, InSb or ZnTe. To facilitate the removal of holes, p+-type regions 30, 30 are embedded in the layer 7. A resonant cavity structure is also provided. The width of the layers 7, 16 and 24 combine to equal one-quarter of the wavelength of the frequency of energy the detector is designed to receive. These layers form a resonating function not unlike the concept of resonating cavities used in optics. 

The matched element resonators function using the sympathetic oscillation in an optical element when it is in contact with a piezoelectric driver, the frequency of which is set at the natural frequency of the optical element. An example of a matched two element resonator would consist of a quartz block, x-cut and gold plated, that is cemented to a block of optical material such as silica or calcium fluoride. The dimensions of the optical block are cut to match the natural frequency of the gold plated quartz block. The system is... 

Figure 4. Multipexed detection of interleukins, (a) Spectra for resonators labeled 1-5 that correspond to control, streptavidin-functionalized control, anti-interleukin 6, anti-interleukin 4, and anti-interleukin 8, respectively. The trace in blue shows the initial baseline spectrum. The red trace corresponds to the test spectrum after introducing 10 pg/ml of interleukin 6 along with 1 pg/ml of interleukin 8, followed by the sequential association of secondary antibodies corresponding to each of these interleukins. We clearly see shifts corresponding to the resonators functionalized with anti-interleukin 6 and 8 (Resonance 3 and 5, respectively) while the other resonances do not shift appreciably thus indicating the lack of non-specific binding. Fabry-Perot resonances were filtered out in both spectra by performing a fast Fourier transform, (b) Reaction stages at each of five resonators.

Figure B3.4.12. A schematic ID vibrational pre-dissociation potential curve (wide full line) with a superimposed plot of the two bound functions and the resonance function. Note that the resonance wavefunction is associated with a complex wavevector and is slowly increasing at very large values o R. In practice this increase is avoided by using absorbing potentials, complex scaling, or stabilization.

Specifically, in Ref. [92b], in response to earlier criticism by Bransden as to the validity of our proposed variational method for resonances [92a, 93], we commented on certain properties of resonance states, emphasizing that what is important is to have a consistent definition of matrix elements. We focused on the argument that the essence of the difference between the resonance function and Fo lies in the change of the boundary conditions asymptotically [11,37]. Thus, given a wavefunction calculation of any type in a finite region of configuration space of radius R, it was argued that matrix... 

Since the absolute modulus square of the resonance function varies with time, it is of interest to see the result of applying the continuity equation, or more precisely of the continuity equation integrated along the internal region of the interaction. 

One sees from Eq. (26) that the residue is proportional to the resonant functions Un r) and M (r )- The factors 2mlh and 2/< , that appear respectively, in the numerator and denominator, are there because the derivation was made in the k plane. As shown by Eq. (A.16) these factors are absent if the derivation performed in the complex energy plane. 

When loaded with a thin film, a TSM resonator functions as a gravimetric probe of the populations of surface-attached species. In this context, thin implies a film whose combination of thickness and shear stiffness is such that the acoustic wave does not deform it consequently, the film moves synchronously with the resonator. Such a film is termed acoustically thin. Monolayer adsorbates can generally be expected to conform to this requirement. Since such systems have been described in a companion article in this series, we describe this simpler case (as it relates to multilayer films) first and in less detail. 

In contrast, when loaded with a sufficiently thick film, a TSM resonator functions as a rheological probe of the properties of surface-attached species. An acoustically thick film is one whose combination of thickness and shear stiffness is such that the acoustic wave significantly deforms it. Consequently, displacement of the outer regions of the film exhibits a phase delay with respect to displacement of the underlying resonator. Multilayer adsorbates may behave in this manner if they are either intrinsically soft or are plasticized by the permeation of a solvent or other small molecules. This situation is more complicated and it is only recently that the theoretical aspects have been treated in quantitative fashion for fluid-immersed films. Accordingly, this case is described subsequently to the simpler gravimetric case and in greater detail. 

As is commonly the case in energy-transfer results [3.10] (3.30) reveals a "resonance-function" behavior. That is, a (c d) is of the form... 

Under general circumstances, where both radiative and collisional damping processes must be considered, the molecular resonances are then represented by the adjoint resonant functions v> and [Pg.291]

We can thus summarize the necessary conditions for an efficient energy transfer (i.e., a resonance function near unity). At a given energy gap, the resonance function 7 ( ) will be exponentially small at low velocities, and will increase with increasing velocities. This version is the form most useful for colhsion theory. An indirect illustration of this exponential gap principle is the strong positive temperature dependence of the V—T relaxation rate. For most diatomic molecules the temperature dependence is best fitted 1 a so-called Landau-Teller equation... 

Aspect ratio, gold 800 nm plasmon resonance functionalized with aEGFR 800 nm 4 min lOWcm 2.4 kj cm Half laser energy required to kill labeled malignant cells compared to normal cells 2006(5]... 

C, D, and E were found to. be resonant functions of the static magnetic field B with resonances occurring at B=Bg=ti(i)g/gjiJg. 

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