Cartesian laboratory frame

END theory treats the collisional system in a Cartesian laboratory frame and each level of approximation [3] is defined by a choice of system wave function characterized by a set of time-dependent parameters and choice of basis set. Minimal END employs a system state vector of the form [3]... 

Fig. 3.14 Euler angles of the molecular frame r, C with respect to the Cartesian laboratory frame x, y, z...

Generally, any vector of a surface force acting on a body can be decomposed into tangential (fx,fy, shear) and normal (Z, pressure) components (index j = x, y, z) as shown in Fig. 8.2a. In its turn, the element of surface A is a vector characterized by its area A and outward-directed unit vector s that has also three projections in the Cartesian laboratory frame (index /). Therefore, the second rank stress tensor [1] is defined as... 

Laboratory frame The Cartesian coordinates (x, y, and z) are stationary with respect to the observer, in contrast to the rotating frame, in which they rotate at the spectrometer frequency. 

The Cartesian indices refer to an arbitrarily chosen laboratory frame. For certain NLO processes intrinsic permutation symmetry can be used to reduce further the number of independent components. In the case of the Kerr susceptibility, (-w w,0,0), intrinsic permutation symmetry in the last two indices holds, xltJ zx X xx- The most general Kerr susceptibility of an isotropic medium therefore has only two independent components, x9 zz and x9 xx Likewise, the EFISHG susceptibility (-2w w, w,0), important for the evaluation of second-order molecular polarizabilities in solution (see pp. 158 and 162), has only two independent components, x zz and x9]txz, because of intrinsic permutation symmetry in the second and third indices. 

The Cartesian indices refer to an arbitrarily chosen laboratory frame. For certain NLO processes intrinsic permutation symmetry can be used to reduce further the number of independent components. In the case of the Kerr susceptibility, w,0,0), intrinsic permutation symmetry in the last two... 

Fig. 5.4.1 Relationship between the Cartesian laboratory coordinate frame with axes x and y and the rotated Cartesian frame with axes r and s. The field gradient is applied parallel to r at an angle

Here the space variables r and s in the Cartesian coordinate frame of the projection have been replaced by the Cartesian coordinates x and y in the laboratory frame (cf. Fig. 5.4.1), and

rotation angle between both frames. This equation is another formulation of the projection cross-section theorem (cf. eqn (5.4.12)), which states that the Fourier transform p(k, cp) of a projection P(r, (p) is defined on a line p kcos(p, k sin p) at an angle

[Pg.203]

Finally, END uses a laboratory frame of Cartesian coordinates, thus avoiding transformations to internal coordinates. This provides great simplifications and generality and is possible since the formalism guarantees strict adherence to fundamental conservation laws (28). For instance, the conservation of total momentum makes it possible to eliminate the overall translational motion at any time point in the evolution. It should also be noted that there is no restriction in principle on the complexity of either the target or the projectile, and that all accessible product channels, including breakup of cluster or molecular projectiles and targets, can be reached. 

The Cartesian reference frames p, a, b, and r are defined as follows p denotes a laboratory-fixed frame in which the laser polarization vectors are fixed a signifies a frame of reference that has been chosen to have its origin at center A similarly b has its origin at center B r represents a frame in whieh the R vectors are rotationally invariant. 

Consider a set of Cartesian co-ordinate axes, as in Fig. 1, which are fixed in space these are often called the laboratory frame (Bloch, 1946). The z direction is taken to be that of a large polarizing field H0 about which the angular momentum vector and associated magnetic... 

The integration of this set of coupled first-order differential equation can be done in a number of ways. Care must be taken since there are basically rather two different time scales involved, i.e. that of the nuclear dynamics and that of the normally considerably faster electron dynamics. It should be observed that this END takes place in a Cartesian laboratory reference frame, which means that the overall translation as well as overall rotation of the molecular system is included. This offers no complications since the equations of motion satisfy basic conservation laws and, thus, total momentum and angular momentum are conserved. At any time in the evolution of the molecular system can the overall translation be isolated and eliminated if so should be deemed necessary. This level of theory [16,19] is implemented in the program system ENDyne [20], and has been applied to atomic and molecular reactive collisions. Calculations of cross sections, differential as well as integral, yield results in excellent agreement with the best experiments. 

Where XYZ stands for a specified Cartesian coordinate frame. Thus once a Cartesian coordinate frame is chosen, the nine spherical components can be determined using Eq. (7.4.1) from the nine Cartesian components. Clearly the spherical components and the Cartesian components change if the coordinate axes are rotated. Suppose we know the values of the Cartesian components of the polarizability tensor in a coordinate frame rigidly fixed within the molecule6 (the body-fixed frame OXY Z). Then the problem confronting us is to determine the Cartesian components of the polarizability tensor in a coordinate system rigidly fixed in the laboratory (the laboratory frame OX Y Z ). The relative orientations of the molecular and laboratory-fixed... 

Let us designate the laboratory Cartesian axes as x, y, z and the rotating frame axes as x, y, z We might visualize the laboratory frame as the three axes, x, y, z, chalked in the corner of the laboratory with the z direction up and the x and y directions on the floor. If we have a phonograph turntable in the laboratory, consider the spindle to be the z axis and two mutually perpendicular lines on the turntable be the x and y axes. As the turntable rotates, the z and z axes continue to point in the same direction, but the x and y axes rotate with respect to the x and y axes. If there is a magnetization rotating at the Larmor frequency about the z axis as seen in the laboratory frame, and we now climb on the turntable rotating at the Larmor frequency, the mag-... 

Figure 2.4 illustrates the concept of the rotating frame. The rotating frame is distinct from the laboratory (static) frame. The rotating frame is a frame of reference we can use to view the net magnetization vector without having to worry about how it precesses at its Larmor (NMR) frequency (in this example 125 MHz). It is a second Cartesian (xyz) coordinate system in which the z-axis is stationary and parallel to the z-axis of the laboratory frame of reference, but in which the x- and y-axes remain perpendicular to each other (and to the z-axis) and rotate at the Larmor frequency in the laboratory frame s xy plane. The axes of the rotating frame are denoted x , /, and z. ... 

A, of an n-dimensional Euclidean space X is generated as follows. Each point x of a set A, is represented by suitable hyperpolar coordinates of one radial coordinate and n — I angle coordinates, where n is the dimension of the underlying space X and the origin is attached to a specified point c of set A In three dimensions, the usual polar coordinates r, 0, and 0 can be used, with respect to the center c of each set A, and with reference to Cartesian coordinate axes defined parallel to axes of a coordinate system of the laboratory frame. 

In concluding this section we explore the relationship between the anisotropic properties of a nematic and the orientational order parameters. It is convenient to do this using irreducible spherical tensors since they have particularly straightforward transformation properties under rotation, as we shall discover. We begin with the tensor components set in the molecular frame which are necessarily independent of the molecular orientation in the laboratory frame. For a tensor of rank L there are (2L + 1) components which are distinguished by the label m, taking values L, L - I,. .. -L + 1, -L these are denoted by T [2]. They are simply linear combinations of the components of the symmetric Cartesian tensor of the same rank. The components of the tensor in the laboratory frame are T, and are related to T by... 

---