Nuclear Forces and Interacting Current Loops
Quotes in this article (in Italics) are taken from The Feynman Lectures on Physics, V2, Sect. 8.4, ‘Electrostatic energy in nuclei’.
"The force is not a simple function of the distance between the two protons."
When nuclear/high-energy physicists began investigating the forces between nucleons (protons and neutrons) in the atomic nucleus, it became evident that there is no simple force law for these interactions. As Richard Feynman put it, "… the force is as complicated as it can be." On the other hand a striking simplification was also discovered. The non-electric part of the force between two protons is the same as the force between (a) a proton and a neutron, and (b) a neutron and a neutron.
One attribute that protons and neutrons have in common is their practically identical magnetic dipole moments (a fact that suggests an internal charged structure in the case of the neutron, since its net charge is zero). And it seems likely in Feynman’s discussion that the several possible alignments of these dipole moments are a factor in the complexity of the nuclear force. It is noteworthy that Feynman discusses only cases where the dipole moments of two nucleons are parallel or antiparallel (pointing in the same or opposite directions). He does not consider cases where the dipole moments might be perpendicular, or more generally where the angle between them might be other than zero or p.
In this article a nucleon’s magnetic dipole moment is attributed to an uncharged current loop. The objective is to compute the interactive magnetic forces between two such current loops in a few cases taken from the Feynman text. Only magnetostatic models are considered. Feynman does note that "…when a proton is moving near another proton, the force is different when the orbital motion has the same direction of rotation as the spin, … than when it has the opposite direction of rotation… This is called the ‘spin orbit’ part of the force." In such cases of course the fields are not static. And as noted elsewhere on this site, when an uncharged current loop translates in its plane, the circulating charge is electrically polarized; the translating loop has a nonzero electric dipole moment (in addition to its magnetic dipole moment). In brief, translating, uncharged current loops may have nonzero electric fields … yet another complication (and one that Feynman may not have been aware of).
For computation purposes the current in each loop is I=1 amp, and the current loop radii are mutually R=1 meter. The actual parameters in the case of real nucleons are of course much smaller.
The general approach is to consider Loop A to be permanently at rest, and to compute |F(d)|, the magnetic force experienced by Loop B when it too is at rest and when the centers of Loops A and B are separated by the distance d. Biot-Savart is used to compute the magnetic field of Loop A at points around Loop B. The magnetic force on any circulating charge increment of Loop B is dF = (dqB)(vB x BA). The net force on Loop B, at separation d, is then the sum of these incremental forces.
2. Loops A and B Lie in the xz-plane.
"…the force depends on the orientation of the protons’ spin … the force is different when the spins are parallel from what it is when they are antiparallel …"
In these cases two possibilities are considered: (1) the magnetic dipole moments of Loop A and B both point in the positive y-direction, and (2) the magnetic dipole moment of A points in the positive y-direction, and that of B points in the negative y-direction. The software assumes that the center of Loop A is at rest at the origin, and the center of Loop B is at rest on the x-axis, at x=d.
2_1. Magnetic Dipole Moments Both Point in Positive y-direction.
The magnetic field of Loop A has only a negative y-component at all points in the xz-plane at distances greater than R from the origin. It turns out that the net force on Loop B is in the positive x-direction (repulsive) for all d>2R. Fig. 2_1a plots Fx(d) over the range 2.01R<d<2.1R.
Fx(d), 2.01R<d<2.1R, Moments Both Point in +y-direction
It is clear that the force increments, on individual circulating charge increments of Loop B, are repulsive at points on B that lie close to A, and that they are attractive at points further removed from A. The repulsive force increments evidently dominate, and the net force on B is repulsive.
It is also evident in Fig. 2_1a that the repulsive force approaches infinity as the distance between the closest increments of the two loops approaches zero (i.e. as d approaches 2R). This result is consistent with the fact that the magnetic field magnitude of Loop A approaches infinity as one approaches x=d from more positive values of x.
2_2. Magnetic Dipole Moment of A Points in Positive y-direction; That of B Points in Negative y-direction.
Fig. 2_2a plots Fx(d) over the range 2.01R<d<2.1R for antiparallel dipole moments. In these cases the magnetic force on Loop B is attractive and the force magnitude approaches negative infinity as d approaches 2R.
Fx(d), 2.01R<d<2.1R, Moments Point in Opposite DIrections
2_3. When Loops A and B Overlap.
It might be wondered what the force on Loop B will be when |d|<2R (i.e. when A and B overlap). Loop A’s magnetic field points in the positive y-direction at all points inside the loop.
A problem with the circular line current model is the infinite values of |B| at distances arbitrarily close to Loop A’s perimeter. This is not a problem if Loop A’s current is actually a circular, solid cylinder of charge. But adopting this model introduces other computational difficulties. We shall deal with the very large values of |B|, encountered by a circulating charge increment of Loop B when it is close to the perimeter of Loop A, simply by assuming that the increment of force in such cases has a magnitude of zero.
Fig. 2_3a plots Fx(d) over the range –1.75R<d<1.75R. (Negative values of d signify that the center of Loop B is to the left or negative x-side of Loop A’s center.) The dipole moments both point toward positive y. Note that, whereas Loop B is repelled from Loop A when d>2R (see Fig. 2_1a), it becomes bound to A when the loops overlap sufficiently! The equilibrium point is evidently when the centers coincide (i.e. when d=0).
Fx(d), -1.75 R<d<1.75 R, Moments Both Point in +y Direction
Fig. 2_3b plots Fx(d) over the range –1.75R<d<1.75R when Loop B’s dipole moment points in the negative y-direction. Note that whereas the loops are attracted to one another when d>2R (Fig. 2_2a) and their magnetic dipole moments point in opposite directions, they are repelled when |d|<2R.
Fx(d), -1.75 R<d<1.75 R, Moments Point in Opposite Directions
Fig. 2_3c plots Fx(d) over the range .25R<d<3.75R, when the dipole moments point in opposite directions. Note in this case that, at d<2R, Loop B is repelled and at d>2R it is attracted. There is binding centered on d=2R.
Fx(d), .25R<d<3.75R, Moments Point in Opposite Directions
3. Center of Loops A and B Lie on the y-axis; Loop Planes are Coincident /Parallel to the xz-plane.
"… the force is considerably different when the separation of the two protons is in the direction parallel to their spins … than it is when the separation is in a direction perpendicular to the spins…"
In these cases two possibilities are again considered: (1) the magnetic dipole moments of Loop A and B mutually point in the positive y-direction, and (2) the magnetic dipole moment of A points in the positive y-direction, whereas that of B points in the negative y-direction. The software assumes that the center of Loop A is at rest at the origin, and the center of Loop B is at rest on the y-axis, at y=d.
3_1. Magnetic Dipole Moments Both Point in Positive y-direction.
At point (R,d,0) the magnetic field of Loop A has positive x- and y-components. By symmetry we need only compute the y-increment of force on the single, circulating charge increment of Loop B at that point. (The y-increment of force on every other Loop B circulating charge increment is the same at other points around the loop.) Fig. 3_1 plots the net Fy(d) acting on Loop B over the range .01R<d<.1R. Like two bar magnets with their opposite poles closest together, the current loops attract. The magnitude of the attractive force evidently approaches infinity as d approaches zero.
Fy(d), .01R<d<.1R, Moments Both Point in +y-direction
3_2. Magnetic Dipole Moments Point in Opposite Directions.
Fig. 3_2 plots the net Fy(d) acting on Loop B, again over the range .01R<d<.1R. Like two bar magnets with their north poles closest together, the current loops repel.
Fy(d), .01R<d<.1R, Moments Point in Opposite Directions
"… the force depends, as it does in magnetism, on the velocity of the protons, only much more strongly than in magnetism …"
Consideration of only a few of the simplest cases indicates that the attractive/repulsive forces between two resting, uncharged current loops, plotted as a function of the distance between the loop centers, is sensitive (a) to the orientations of the loop’s magnetic dipole moments, and (b) to the planes in which the loops lie. In general there is no simple formula as there is for the electrostatic force between two resting point charges. When the loop magnetic dipole moments do not point in the same or opposite directions, and/or when the line between their centers is not parallel/normal to the loop planes, then still other forces (and perhaps torques) would presumably be computed. Like the nuclear forces between nucleons, the force is (as Feynman lamented) complicated.
An interesting result of the modeling exercises is the phenomenon of "magnetic binding." Loops that lie in a common plane and attract one another evidently experience repulsive forces when they overlap. In the case of circular line currents there are admittedly ambiguities associated with the infinite magnetic field strength approached as the distance from a loop’s contour goes to zero. But these ambiguities might be ameliorated by modeling the loops as circular, solid cylindrical distributions of charge. Of course in either case the implicit assumption is that the charges of the two loops can occupy the same space at overlap points without disturbing one another.
The question as to whether the nuclear force is fundamentally magnetic in nature requires further scrutiny. If it is, then there may be interesting implications for nuclear fusion. It may be easier, for example, to get two protons (a) with opposite spins and (b) with hypothetical current loops lying in a common plane, to approach than it is when their magnetic dipole moments point in the same direction. In the case of neutrons the prospect of neutronic bonds is equally interesting.
The software used to obtain the results in this article can easily be modified to consider other situations. In more realistic cases where the modeled current loops translate relative to one’s inertial frame of reference, the added complication of translation-induced electric polarization would need to be factored in.
The computation of current loop interactive forces (and perhaps torques) seems rooted in necessity … a fact that perhaps explains in part why the quest for a general nuclear force law has eluded more analytically-inclined theorists. In general the number of physical problems that can only be solved using a computer are for all practical purposes infinite. And computing solutions in such cases needs hardly to be encouraged, notwithstanding the greater exactness of mathematical solutions in those relatively few cases that lend themselves to mathematical analysis.