Patterson wave

Remembering that Fflkl is simply the measured intensity / <, a scaler quantity, then the expression can be recognized as simply the electron density equation (see Chapter 5) with squared coefficients and all phases 4>hkt set equal to zero. The normalization constant is here l/V2 because of the squared coefficient, where V is the volume of the unit cell. The units it implies for the function, something per volume squared, immediately indicates that P(u, v, w) is not electron density but some other spatial function. Because the equation yields something other than electron density, existing in some unique space, we cannot denote it by p(x, y, z) in jc, y, z (real space) we must designate it by P(u, v, w) in some alternative coordinate space whose variables are u, v, w. Otherwise, P(u, v, w) is the equation for aperiodic function in u, v, w space. The Patterson function, or Patterson wave... 

Duvall, G.E., Shock Waves and Equations of State, in Dynamic Response of Materials to Intense Impulsive Loading (edited by Chou, P.C. and Hopkins, A.K.), US Air Force Materials Laboratory, Wright-Patterson AFB, 1973), pp. 89-121. 

Study of the Initiation of Detonation Waves by Deflagration , USAF Inst of Tech, Wright-Patterson AFB, (Feb, 1954) 16a) B.L. 

CA 49, 11281 (1955) (Calculation of the velocity of shock wave produced by an explosion) 25) C.W. Jones, PrRoySoc 221A, 257-67(1954) (On gas flow in one dimension following a normal shock of variable strength) 26) S.L. Lipsitz, "Thickness of Shock Waves , AF Inst of Tech, Wright-Patterson AFB Rept GAE 54 (1954) 27) A.H. Shapiro S.J. Kline,... 

Keulegan and Patterson (K16), 1940 Determination of criterion for wave formation in turbulent flow in steep channels N . > f or Ner > 2, depending on use of Manning or Chdzy coefficients for resistance term. 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

Dubova, Waves in Soil During a Surface Blast and their Interaction with Obstacles , FTD-HT-23-744-72 (Transi), Foreign Technol Div, Wright-Patterson AFB, Dayton (1972) 115) L,... 

The phase problem can be solved, that is, phases of the scattered waves determined, either by Patterson function or by direct methods. The Patterson function P is a self-convolution of the electron density p, and its magnitude at a point u, v, w can be obtained by multiplying p (x, y, z) hy p (x + u, y + V, z + w) and summing these products for every point of the unit cell. In practice, it is calculated as... 

Patterson and Greene have decomposed BrCN in shock waves over a temperature range 2500-7000 °K. Kinetic data consisted of emission profiles of excited CN, v = Q, y = 1) and C2( Hg, r = 0 v = 0)... 

Even Davisson and Germer s first work on the reflection of slow electrons by crystal lattices made it clear that the facts could not be accurately represented by equations (3) and (5) on the contrary, definite deviations from Bragg s law of reflection occur. These were first explained by Patterson as being due to a diminution of the distance between the lattice planes at the surface. Bethe has shown, however, that better agreement with experiment is obtained by expressing the action of the crystal on the electrons by means of a mean lattice potential V. Schrodinger s equation for the de Broglie waves with an internal lattice potential is then... 

It may not be obvious how we would locate the x, y, z coordinates of the heavy atom in the unit cell. Indeed it is sometimes not a simple matter to find those coordinates, but as for the heavy atom method described above, it can be achieved using Patterson methods (described in Chapter 9). As we will see later, Patterson maps were used for many years to deduce the positions of heavy atoms in small molecule crystals, and with only some modest modification they can be used to locate heavy atoms substituted into macromolecular crystals as well. Another point. It is not necessary to have only a single heavy atom in the unit cell. In fact, because of symmetry, there will almost always be several. This, however, is not a major concern. Because of the structure factor equation, even if there are many heavy atoms, we can still calculate Juki, the amplitude and phase of the ensemble. This provides just as good a reference wave as a single atom. The only complication may lie in finding the positions of multiple heavy atoms, as this becomes increasingly difficult as their number increases. 

Patterson SG, Petrich SG, Ram RJ, Kolodiejski R (1999) Continuous-wave room temperature operation of bipolar cascade laser. Electron Lett 35 397-397... 

The wave-lengths for tungsten were measured in cooperation with Dr. R. A. Patterson. 

Hill and Martin (2002) presented a review of conventional analytical methods for CWAs. They discussed various. sensors, such as surface acoustic wave sensors, electrochemical sen.sors, spectrophotometric sensors, immunochemical sen.sors, and IMS detector. For OP nerve agents. FPD and MS are the detectors of choice when coupled with GC or LC. Miniature ion trap ma.ss spectrometer has been described for the detection of nerve agents in the field (Patterson et at., 2002 Riter et al., 2002). A book published by the Institute of Medicine and the National Research Council explains the use of various types of detectors for nerve agents as well as CWAs (lOM, 1999). 

The Rayleigh-Taylor (RT) component has been added to the breakup model by Patterson et al. [11] to improve predictions of the secondary breakup of the droplets. The RT model predicts instabilities on the surface of the drop that grow until a certain characteristic breakup time is reached, when the drop finally breaks up. The RT waves are only allowed to form on droplets with diameters larger than the wavelength of the fastest growing disturbance. When the disturbances exceed the elapsed breakup time, the droplet is split into smaller droplets, with diameters proportional to the wavelength of the disturbances. 

Single compartment models are desirable in this work, because any interaction between the mathematical models and the corresponding experimental systems should be based on the same known quantities. It would not be reasonable to include variables in the models that either cannot be determined from separate equations or cannot be provided as an input from the available experimental data. To check and validate models, we are using microscopy and imaging techniques to capture dynamic calcium events taking place within a cell as a whole, as opposed to calcium diffusion across a cell or intracellular calcium waves, for example (Patterson et al., 2007). 

The distribution of intensity of scattered radiation in a diffraction pattern is related by a Fourier transformation to the autocorrelation function of scattering density, p r)p(r )), where ( ) indicates an average over the sample. In crystallography the autocorrelation function is known as the Patterson function. It is very useful to factor out contributions to the total intensity from interfering waves scattered by single particles and from interparticle interferences ... 

The rotatory dispersion of amino acids in the visible range of the spectrum has been studied by several workers (Karrer and Kaase, 1919 Waser, 1923 Pertzoff, 1927). Recently, Patterson and Erode (1943) have published detailed results with thirteen a-amino acids. With most compounds studied, the dispersion is normal, i.e., the specific rotation [a] at a particular wave length can be expressed by a one-term Drude equation which has the form a = a/(X — Xo ), where a and Xo are constants. It is likely that, if measurements were extended to the far ultraviolet, amino acids, like many other optically active compounds, would exhibit anomalous dispersion, especially in those parts of the spectrum in which absorption of amino and carboxyl groups becomes significant. It appears, especially from the work of Patterson and Erode (1943) that the l forms of amino acids fall into three classes Group I consists of those amino acids which have a normal and positive dispersion for this class, to which most purely aliphatic amino acids belong, Xo has... 

In Fig. 3, line A, which tenninates in squares, represents the potential-cycling experiments of Patterson (Patterson 2002). These were square-wave cycles with a period of 60s between 0.87 and 1.2V for lOOh at 50°C resulting in a loss of 50% of the initial surface area. Line B, which terminates in filled triangles, represents the potential-cycling experiments of Mathias et al. (2005). They found a surface area loss of approximately 50% for Pt/C after lOOh of square-wave cycling the potential of the electrode between 0.7 and 0.9 V with a period of 60s at 80°C. Comparison of lines A and B snggests that the rate of surface area loss is a strong function of temperature. 

Fuller, J.A., Taylor, T.S., Elfe, T.B. and Hill, G.N., Dielectric properties of ceramics for millimeter-wave tubes. Technical Report AFWAL-TR-84-1005, Air Force Avionics Laboratory, Wright-Patterson AFB, (1984). 

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