Coordinate systems laboratory frame

Both the mathematical treatments and pictorial representations of many physical phenomena can be simplified by transforming to a coordinate system that moves in some way, rather than one which is fixed in the laboratory. We shall find it convenient to define a coordinate system, or frame of reference, that rotates about B0 at a rate revolutions/second, or corf radians/second, where the frequencies... 

Fig. 6. (a) Different coordinate systems (laboratory L, director D, and magnetic m) nsed to define motion parameters for a nitroxide spin label, (b) Diffusion rotation angles used to define the magnetic axes relative to the diffusion axes. Note that the reference system for these angles is the diffusion frame, whereas the reference system is the magnetic (g) frame for the magnetic tilt angles (cf. Fig. 3). 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

Theorists calculate cross sections in the CM frame while experimentalists usually measure cross sections in the laboratory frame of reference. The laboratory (Lab) system is the coordinate frame in which the target particle B is at rest before the collision i.e. Vg = 0. The centre of mass (CM) system (or barycentric system) is the coordinate frame in which the CM is at rest, i.e. v = 0. Since each scattering of projectile A into (v[i, (ji) is accompanied by a recoil of target B into (it - i[/, ([) + n) in the CM frame, the cross sections for scattering of A and B are related by... 

In a crossed-beam experiment the angular and velocity distributions are measured in the laboratory coordinate system, while scattering events are most conveniently described in a reference frame moving with the velocity of the centre-of-mass of the system. It is thus necessary to transfonn the measured velocity flux contour maps into the center-of-mass coordmate (CM) system [13]. Figure B2.3.2 illustrates the reagent and product velocities in the laboratory and CM coordinate systems. The CM coordinate system is travelling at the velocity c of the centre of mass... 

These simple relations motivate a more formal approximation in which we first re-expand the interaction potential in a space-fixed ("laboratory-frame") coordinate system as... 

Laboratory frame model A means of visualising the processes taking place in an NMR experiment by observing these processes at a distance, i.e., with a static coordinate system. See Rotating frame model. 

Physics and chemistry are carried out in laboratory frames using coordinate systems to set up experimental devices. Before discussing quantum mechanical processes let us recall the form of the total Hamiltonian for a set of particles having charges qa and masses ma interacting with an electromagnetic field A. This Hamiltonian is given by ... 

Let ijs now apply this concept of the RRF to the case where an rf field Hi is present. We choose a Cartesian coordinate system with tlje z axis along the dc field Hq and the y axis along the rf field Hi. The total field is given in the laboratory reference frame by... 

In (3.1) not all tensors are necessarily coaxial or diagonal. If the principal axes system of the g tensor is chosen as the molecular coordinate system eM, g has diagonal form. The laboratory frame eL is then related to eM by the rotation matrix R according to... 

Fig. 4. Schematic diagram of the layered model for a pore (47). The two nuclear spins diffuse in an infinite layer of finite thickness d between two flat surfaces. The M axes are fixed in the layer system. The L axes are fixed in the laboratory frame, with Bq oriented at the angle P from the normal axis n. The cylindrical polar relative coordinates p, (p, and z are based on the M axis. The smallest value of p corresponding to the distance of minimal approach between the two spin bearing molecules is 5.

Figure 1. Schematic of the mesomorphous structure in a lamellar phase. The different coordinate systems used in the text are outlined laboratory frame (L), director frame (D), and molecular frame (M). 0LD and 0DM are angles between the z axis in laboratory-director systems and director-molecular systems, respectively (13).

By a nuclear configuration (NC) we understand the set of informations NC Xk, Zk, Mk consisting of the coordinates Xk, the masses Mk and charge numbers Zk of the nuclei 1, 2,..., K of a molecular system. The coordinate vectors will be referred to a coordinate system, which will be defined when required. Important coordinate systems will be the laboratory system (LS, basis e 1) and the frame system (FS, basis e"f). The latter is attached to the nuclear configuration by a prescription to be defined in each case. The relation between e1 and may be expressed by... 

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... 

The differential cross-section d/i/dfl is not invariant when we change the description from one coordinate system to another. Clearly, due to the relation in Eq. (4.55) a change in y will not lead to the same change in 0 and the space angle d 2c.m. = si n ydycif/) is not identical to the space angle dfl = sin GdGd in the laboratory frame. Thus,... 

Figure 8.11. Cartesian coordinate system for describing the position vectors of the particles (electrons and nuclei) in a molecule. 0(X, Y, Z) is the laboratory-fixed frame of arbitrary origin, and c.m. is the centre-of-mass in the molecule-fixed frame. For the purposes of illustration four particles are indicated, but for most molecular systems there will be many more than four.

In Figure 2.8 are shown four representations of M in the old way, the so-called laboratory frame of reference, the normal x,y,z coordinate system as viewed by a stationary... 

Instead, suppose the x and y axes were themselves precess-ing clockwise (when viewed from above) around the z axis at the same frequency the nuclear spins are precessing. Further suppose we, the observers, were precessing around the z axis at the same frequency. To differentiate this rotating coordinate system from the fixed (i.e., laboratory frame) system, we will use labels and z to represent the three rotating axes (the z axis is coincident on and equivalent to the z axis). To us rotating observers, the rotating axes and B, appear stationary, and M will rotate in the plane perpendicular to B. These relationships are shown in Figure 2.9,... 

In Section 11.2 we note that p is usually expressed in a rotating frame. Unless there is a specific application in which it is important to distinguish between the laboratory and rotating frames, we continue to use p as the symbol, regardless of coordinate system. 

In the laboratory frame the motion of the three particles depends on nine variables, three of which define the position of the center-of-mass. Other three coordinates are needed to describe the rotation of the system in the space and therefore the internal motion is described by the three remaining coordinates. For example, in molecular dynamics the potential energy surface in general is calculated and presented using geometrical coordinates, such the interparticle distances, or two bond distances and an angle. But it is convenient and necessary to use different coordinate systems to describe and understand the dynamics of the particles, because of the rotational terms which appear in the full Hamiltonian. In this context, we will present the transformation equations from the interparticle distances to coordinate sets of the hyperspherical and related types, successful in the treatment of the dynamics. 

It appears that the variable k will be convenient shortly calculating the collision term because it allows us to write down the number of scattered particles in a form independent of the particular system of coordinates used in the previous calculations (i.e., the laboratory frame). Trigonometrical considerations imply that ... 

The calculation is simplified considerably by transforming the Bloch equations into a coordinate system which rotates with the rf magnetic field vector (2.2.12) around the 2-axis of the laboratory frame. In this rotating frame the magnetic field including the rf field component appears static, but the magnitude of the Bq field in z-direction is changed, and (2.2.10) turns into... 

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