Rigid-body rotation

Returning now to the rigid-body rotational Hamiltonian shown above, there are two special cases for which exact eigenfunctions and energy levels can be found using the general properties of angular momentum operators. 

A proper orthogonal tensor represents a rigid-body rotation, and R is called the material rotation tensor. It has the properties... 

In a vortex (rigid-body rotation), Ljungqvist has shown that the mean value of the concentration over the entire region inside the streamline where the point of emission is situated is considerably higher than that of the outside. This allows us to use the concept of contamination accumulation in the context of vortices. 

We describe as rigid-body rotation any molecular motion that leaves the centre of mass at rest, leaves the internal coordinates unaltered, but otherwise changes the positions of the atomic nuclei with respect to a reference frame. Whereas in a simple molecule, such as carbon monoxide, it is easy to visualize the two atoms vibrating about a mean position, i.e. with the bond length changing periodically, we may sometimes find it easier to see the vibration in our mind s eye if we think of one atom being stationary while the other atom moves relative to it. 

We note that there is a gradient term in Eq. (30), which is independent of the vector potentials so the rotation of the condensate produces a lattice of neutral vortex lines [12, 19], which simulates the rigid body rotation. We emphasize also that the equations for the current (19) and (20) do not change, because... 

The case of polyatomics, which is considered in more detail below, follows analogously. The point to keep in mind is that the symmetry restricts the number of distinguishable states. The symmetry number is the number of equivalent (indistinguishable) positions into which a molecule can be carried by rigid body rotation. For example s = 12 for CH4 since the molecule can be held by a CH bond and rotated into three equivalent positions, and there are four CH bonds. Similarly for benzene s = 12 since there are six indistinguishable positions for rotation about an axis perpendicular to the plane of the molecule (and through its center), and six more when the molecule is flipped over. 

These operations do not occur separately and in any particular sequence but are simply a convenient way to conceptualize the transformation as a series of operations, each of which can be analyzed separately, but which working together produce a martensitic structure containing an invariant plane. As such, they can be imagined to occur in any sequence. For purposes of analysis, it is convenient to imagine that the lattice-invariant deformation occurs first, followed by the lattice deformation, followed finally by the rigid-body rotation. We now show that a lattice-invariant shear by slip followed by the lattice deformation analyzed above can produce an undistorted plane. 

Invariant Plane by Addition of Rigid-Body Rotation... 

The plane containing a and c in Fig. 24.9 is the plane in the f.c.c. phase that initially contained the vectors a" and c". If the b.c.t. phase is now given a rigid-body rotation so that a" —> a and c" —> c, the undistorted plane in the b.c.t. phase will be returned to its original inclination in the f.c.c.-axis system and will therefore be an invariant plane of the overall deformation. In the present case, this can be achieved by a rotation around the axis indicated by u in Fig. 24.9 (see Exercise 24.3). The solution of the problem is now complete. The invariant plane is known, and the orientation relationship between the two phases and total shape change can be determined from the combined effects of the known lattice-invariant deformation, lattice deformation, and rigid-body rotation. 

If S is the lattice-invariant deformation tensor and R the rigid-body rotation tensor, the total shape deformation tensor, E, producing the invariant plane can be expressed as... 

Section 24.2.3 claims that the rotation axis in the final rigid-body rotation, R, which rotates a" — a and c" —> c in Fig. 24.9 is located at the position u. By using the stereographic method, show (within the recognized rather low accuracy of the method) that this is indeed the case. 

The axis of rotation required to bring a" —> a by a rigid-body rotation must lie somewhere on a plane normal to the vector (S" — a). 

Figure 24.19 Stereogram showing the method for locating the rotation axis, u, for the rigid-body rotation, R. in Section 24.2.3. From Li b rman [19].

One physical restriction, translated into a mathematical requirement, must be satisfied that is that the simple fluid relation must be objective, which means that its predictions should not depend on whether the fluid rotates as a rigid body or deforms. This can be achieved by casting the constitutive equation (expressing its terms) in special frames. One is the co-rotational frame, which follows (translates with) each particle and rotates with it. The other is the co-deformational frame, which translates, rotates, and deforms with the flowing particles. In either frame, the observer is oblivious to rigid-body rotation. Thus, a constitutive equation cast in either frame is objective or, as it is commonly expressed, obeys the principle of material objectivity . Both can be transformed into fixed (laboratory) frame in which the balance equations appear and where experimental results are obtained. The transformations are similar to, but more complex than, those from the substantial frame to the fixed (see Chapter 2). Finally, a co-rotational constitutive equation can be transformed to a co-deformational one. 

In order to determine the shear strain, the contributions to these angles from rigid body rotation must be subtracted. That angle simply corresponds to that swept in moving A to A, that is gg/r. Hence the shear strain is ... 

For example, if one-third of the A (or B) crystal lattice sites are coincidence points belonging to both the A and B lattices, then E = 1 / = 3. The value of also gives the ratio between the areas enclosed by the CSL unit cell and crystal unit cell. The value of E is a function of the lattice types and grain misorientation. The two grains need not have the same crystal structure or unit cell parameters. Hence, they need not be related by a rigid body rotation. The boundary plane intersects the CSL and will have the same periodicity as that portion of the CSL along which the intersection occurs (Lalena and Cleary, 2005). 

The rigid body rotational Hamiltonian can be written in the form... 

We will now proceed to combine our knowledge of the rigid body rotation with the Zeeman interactions discussed in part ( ). 

It is customary to take the spin orbit and rigid body rotation terms together, for reasons which will become immediately apparent. On expansion of the rigid body term (8.350) we obtain... 

Complete matrices for the parity-conserved 2 If fine-structure states (exclusive of nuclear spin terms) may now be constructed by combining the A-doubling matrices given above with the spin-orbit and rigid body rotation matrices given in our discussion of the LiO spectrum. The matrix representation is block diagonal for each value of J and each parity. The results for the positive and negative parity states are as follows. 

We now recall that the spin-orbit coupling and rigid body rotation terms in the effective Hamiltonian are... 

---