Particle in space

The essence of this idea is that there is a limit to which particles of like-size can occupy a given space, even when arranged in closely packed arrays (e.g., cubic or tetrahedral arrays). The voids that are left are usually smaller than the parent particles and may be filled by particles of smaller size to increase the concentrations of particles in space. Thus, polydispersity can give a lower viscosity at the same volume fraction or permit higher volume loading at the equivalent monodisperse viscosity. 

Gravilation. Two particles in space are attracted toward each other by a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically this may be stated as... 

To provide a mathematical description of a particle in space it is essential to specify not only its mass, but also its position (perhaps with respect to an arbitrary origin), as well as its velocity (and hence its momentum). Its mass is constant and thus independent of its position and velocity, at least in the absence of relativistic effects. It is also independent of the system of coordinates used to locate it in space. Its position and velocity, on the other hand, which have direction as well as magnitude, are vector quantities. Their descriptions depend on the choice of coordinate system. In this chapter Heaviside s notation will be followed, viz. a scalar quantity is represented by a symbol in plain italics, while a vector is printed in bold-face italic type. 

Concentrated (isotropic) solution p3. Once the concentration exceeds p2 the polymers have a tendency to align with each other. They still show a degree of disorder but over a limited concentration range both the dynamics and distribution of particles in space are substantially modified. This isotropic state is maintained up to a concentration pT which is of the same order as (bL2) l ... 

Gilra, D. P., 1972a. Collective excitations and dust particles in space, in The Scientific Results from the Orbiting Astronomical Observatory OAO-2, A. D. Code (Ed.), NASA SP-310, pp. 295-319. 

This concludes the discussion of the diffusive motion of two particles in space. 

This relationship provides the bridge between corpuscular physics and wave physics, since the momentum p = mv (in this case p = me) is then related to the wavelength A. This equation holds much deeper implications, since each particle of mass m and velocity v is associated with a wavelength A which in effect defines the distribution of the particle in space when the wavelength is short, the particle is more localized. 

The quantity qX2 = D plays the role of the diffusion coefficient in the particular case when b — const, i.e., when all the points of the axis are equivalent, as is the case for the motion of particles in space without exterior forces, we get Fick s law,... 

For a set of random variables, such as set of x, y, and z coordinates of a distribution of random particles in space, common metrics such as average position and higher moments can be used to describe the distribution of these variables. For three-dimensional space, the most often used metric is the average position, (jc0, y0, z0), which is found by summing the individual types of coordinates and dividing by the number of positions. For example, x0 would equal the sum of all of... 

The electrostatic, induction, and dispersion terms can be expanded in a convergent series closely related to the multipole expansion, but fully accounting for the charge-overlap effects, the so-called bipolar expansion introduced by Buehler and Hirschfelder199,200. In the local coordinate systems with the origins located at the centers of masses of the monomers A and B, separated by the distance R, and with their x and y axes parallel and aligned along the z axes, the distance between two particles in space can be expressed as follows,... 

Because the random fluctuations in the positions of particles in space are often translational, the kinetics of these processes can be considered comparable to the decay of a concentration gradient by translational Brownian motion. Likewise, as the orientation of any molecule undergoes similar random fluctuations in space,... 

From a quantum-mechanical viewpoint, the problem is not essentially different, but in fact more information is required. Since the nucleus and electrons cannot be treated as point particles, an additional set of three parameters must be specified for each such particle to describe the orientation of the axis of spin of the particle in space. 

The potential energy V of a three-dimensional harmonic oscillator depends on the position of a particle in space, as described by the three coordinates x, y and z. It can thus be considered to be a function of these variables written as V(x,y, z) and defined by the equation... 

Let us now look at a gas so dilute that only two-body collisions must be considered and the distribution of free particles in space is absolutely random, not withstanding their different electrical charges (22). Then evidently the Vik are positive, proportional to the overall particle concentration, and independent of the molar fractions, and each only depends on its corresponding type of collision ik. 

In this paper we consider some coordinate sets used for the treatment in classical and quantum mechanics of the motion of three particles in space. The alternative sets of coordinate systems for the three-body problem have been studied extensively [1-5] and a good choice of the coordinate systems is of crucial importance. Key references for the basic theory are 11 1,6], where also history is sketched and credits are given. 

For the simulation of the described NMR experiments on dispersed nanoparticles, one needs to account for a number of key variables the time, the orientation of a given particle in space and its location with respect to a local reference system. In addition, in case of sample spinning experiments, the time-dependent orientation of the rotor has to be considered. For numerical calculation, aU these parameters are approximated in a finite element scheme ... 

Figme 1 Jacobi vectors for three particles in space. 

Figure 5 The tree represents the hyperspherical parametrization for the components of the Jacobi vectors describing three particles in space. li,mi eind l2,m2 can be put into correspondence with j,my and l,mi of section 4, and the upper part of the graph to the ordinary spherical harmonics. ...

Figure 1.1 Everything in the universe, including particles in space and things around you, is composed of matter. 

Polarimetric measurements provide essential information in space physics that cannot be found through other methods of observations. Obtained and analysed polarimetric results have shown that such measurements provide important information about a) diagnostics of non-thermal radiation mechanisms b) determination of spatial stracture of matter, a field of radiation and magnetic fields of small angular-size objects c) determination of optical, geometrical, physical and chemical properties of dust particles in space -circumstellar, interstellar, and intergalactic. 

In other words, a point-shaped particle does not know about the gradient of the energy. We could introduce the concept that the point-shaped particle does not stand in its position, and the motion is fluctuating around a certain position. In this way, the particle could find out that besides its instantaneous position is a place where it would possess less energy. On the other hand, an extended particle in space could experience a gradient of the energy in space. 

The microstmctural representation requires an initial conhguration of the particles, which can be, in principle, arbitrary (e.g., amorphous, granular, single crystalline, or molecular). Usually their mass is concentrated in their center of gravity, and its coordinates determine the position of the particle in space and their distance relative to each other (cf. Fig. 1). 

The mathematical description of a crystal size distribution and of its change in space and time makes use of the conservative character of the number of particles in space and state (i.e., particle size L). In the respective number balance, the particle size distribution is represented by the number density, see Hulburt and Katz (1964) and Randolph and Larson (1988),... 

What our Earth-bound chemists would like to know is. How do the chemical reactions that create organic molecules and fragments in space occur There are smoke-like dust particles in space, and reactions can occur between fragments briefly trapped on the particle surfaces. The surfaces can act like catalysts, making the reactions possible. Water ice may play a role. Every relevant tool available to our Earth-bound chemists is brought to bear on creating the conditions and experiments to study the reactions. As one chemist has described it, It s a messy problem. ... 

---