Line of nodes

Fig. 24. Magnetic field dependence of the electronic thermal conductivity at T - 0, normalized to its value at Hc2- Circles are for LuNi2B2C, squares for UPt3 and diamonds for Nb. Note the qualitative difference between the activated thermal conductivity of the s-wave superconductor Nb and the roughly linear growth seen in UPt3, a superconductor with a line of nodes (Boaknin et al. 2001).

The line of intersection of the XY and ab planes is called the line of nodes , in Fig. 5.2, this is line ON. (Unfortunately, there is no uniformity in the definition of the Eulerian angles.)... 

Figure 5.3. The Euler angles , LX OX" = 0 and LY Oy = x-...

The inclination of Mercury s orbit to that of the ecliptic plane (the plane of Earth s orbit about the Sun) is 7.0°. This slight orbital tut dictates that when Mercury is at inferior conjunction it is only rarely silhouetted against the Sun s disk as seen from Earth. On those rare occasions when Earth, Mercury, and the Sun are in perfect alignment, however, a solar transit of Mercury can take place, and a terrestrial observer will see Mercury move in front of, and across the Sun s disk. A transit of Mercury can only occur when the planet is at inferior conjunction during the months of May and November. During these months Earth is near the line along which the orbit of Mercury intersects the ecliptic plane— this is the line of nodes for Mercury s orbit. Approximately a dozen solar transits of Mercury occur each century, and the final transit of the twentieth century occurred on 15 November 1999. 

Second, the resulting vector is rotated by Rq = e l,n/1 about the line of nodes... 

The motion is therefore doubly degenerate, since for a given value of J the energy is independent of Ja (the total angular momentum) as well as of J. Not only the longitude of the node, but also the angular distance of the perihelion from the line of nodes, remains unaltered. We have only one quantum condition,... 

We now examine the secular motions caused by the electric field. The perihelion of the orbital ellipse alters its position relatively to the line of nodes, and the latter itself moves uniformly about the axis of the field. It follows from (5) that two periods of the perihelion motion occur during one revolution of the line of nodes. 

The secular motions of the orbit under the influence of the electric field are thus as follows while the line of nodes revolves once, the perihelion of the orbital ellipse performs two oscillations about the meridian plane perpendicular to the line of nodes. For a transit through this meridian plane in one direction, the total momentum J2°/27t is a maximum and consequently the eccentricity is a minimum for a transit in the other direction the eccentricity is a maximum. Since the component Js°/2it of the angular momentum in the direction of the field remains constant, the inclination of the orbital plane oscillates with the same frequency as the eccentricity. It has its maximum or minimum value when the perihelion passes through the equilibrium position, and it assumes both its maximum and minimum value twice during one revolution of the line of nodes. The major axis remains constant during this oscillation of orbital plane and perihelion (since Jj0 remains constant) the eccentricity varies in such a way that the electrical centre of gravity always remains in the plane... 

In this plane it describes a curve about the direction of the field since the inclination and rotation of the line of nodes have the frequency ratio 2 1, the curve is closed and, in the course of one revolution, the electrical centre of gravity attains its maximum distance from the axis twice and also its minimum distance twice. We shall show later ( 38) that the electric centre of gravity executes an harmonic oscillation about the axis of the fi d. 

We consider first the case in which only an electric field E acts, r x and r 2 both rotate with the same velocity about the direction of the field, but in opposite directions. In the course of a complete rotation of each of the vectors they come twice into a configuration in which E is coplanar with them and they both lie on the same side of E. In this position their difference, and therefore the resultant angular momentum P, is a minimum, the eccentricity attains its maximum and the plane of the orbit deviates least from the equatorial plane of the field. Between these positions there are two others where rx, r 2, and E likewise lie in a plane, but with rx and r 2 on opposite sides of E. P is then a maximum and the eccentricity a minimum, while the plane of the orbit has its greatest inclination with the equatorial plane. While the magnitude of P goes through two librations during such a revolution, the direction P completes only one rotation, i.e. the line of nodes of the orbital plane completes one revolution. 

When both fields are acting, the rotations of rx and r 2 occur about different axes. Thus the simple phase relation, which we had in the case of an electric field only, between the rotation of the line of nodes on the one hand and the orbital eccentricity and inclination on the other, will be destroyed and a much more complex motion sets in. Special difficulties arise when the two cones described by the vectors rx and r2 intersect. If the rotation frequencies are incommensurable, the vectors rx and r 2 will then approach indefinitely close to one another, and, therefore, the angular momentum becomes indefinitely small. If now the frequency of rotation in the ellipse is incommensurable with the other two frequencies, the electron approaches indefinitely close to the nucleus. On the basis of the fundamental principles we have previously used, we should have to exclude such motions. We shall see later, however, when fixing the quantum conditions, that such orbits may be transformed adia-batically into those of the pure Stark or Zeeman effect which we must allow. 

In what follows we shall again omit the bars over w1 and J x 2-nw is then the angular distance of the electron in its orbit from the line of nodes and -q are zero in the unperturbed motion. 

If the p jlar axis of the co-ordinate system be taken in the direction of the resultant angular momentum P=, /2ir, the angular separation of the line of nodes from a fixed line in the invariable plane is a cyclic variable conjugate to P. For the other co-ordinates let us take the radius vector r of the outer electron and the conjugate momentum p together with the angular separation ifi of the outer electron from the line of nodes and the conjugate momentum... 

ON = positive direction of line of nodes, the intersection of the X Y and xy planes. Also positive sense of a rotation of OZ to Oz. 

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