Spin Simulator: Neutron 720° Rotation, Einstein-de Haas and Barnett Effects

A free spin simulator: a neutron interferometer where the spin must turn 720 degrees, not 360, to come back (Rauch and Werner, 1975), with live counts and a field scan; the Einstein-de Haas effect, where flipping a hanging iron rod's magnetization makes it turn; and the Barnett effect, where spinning iron magnetizes it.

A slow neutron beam hits a silicon crystal interferometer cut from one block: the first plate splits each neutron into two paths, the middle plate bends them back together and the last plate makes them interfere. One path runs through a magnetic field that turns (precesses) the neutron's spin by an angle α. Turn it by 360° and the forward detector goes dark: the neutron's wave has changed sign. Only at 720° is it back to normal. Drag the picture to turn it.

Magnetic field:

Length of the field region: cm

Neutron wavelength:

Phase shifter in the other path: A thin plate (aluminium in the real experiments) that shifts path I's wave.

Contrast of the fringes: Real interferometers of 1975 managed about 20-40%.

Count rate: neutrons per secondCounts come in at random (Poisson), so the numbers jitter like real data.

Pretend the neutron were a classical spinning top (a vector: back to normal after 360°) Animation speed:

Scan the field, like the 1975 experiments

Scan up to:

Number of field settings:

Counting time per setting:

Background: counts/s

Show the classical vector prediction Plot against the turn angle instead of the field

How the spin simulator works

Everything runs in your browser from the textbook equations; nothing is uploaded. The three experiments all show that spin is real angular momentum, and the first one shows that an electron-like or neutron-like particle is not a little spinning ball: its wave must turn twice before it is back to where it started.

Larmor precession \(\displaystyle \alpha = \gamma_n B\,\frac{L}{v},\qquad \gamma_n = 1.832\times10^{8}\ \text{rad s}^{-1}\text{T}^{-1},\qquad v = \frac{h}{m_n\lambda}\)

the spinor after turning \(\displaystyle \psi \to e^{-i\alpha\sigma_z/2}\,\psi,\qquad \alpha = 2\pi:\ \psi \to -\psi\)

forward (O) beam \(\displaystyle I_O \propto 1 + V\cos\!\left(\frac{\alpha}{2} + \chi\right)\quad\text{(spin }\tfrac12\text{)},\qquad 1 + V\cos(\alpha + \chi)\quad\text{(a classical vector)}\)

magnetism is angular momentum \(\displaystyle \mu = -g'\frac{e}{2m_e}\,L_e,\qquad \Delta L_{\text{rod}} = -\Delta L_e = \frac{\Delta\mu}{\gamma'},\quad \gamma' = g'\frac{e}{2m_e}\)

rod on a torsion fibre \(\displaystyle I\ddot\theta = -\kappa\theta - c\dot\theta - \frac{dL_e}{dt},\qquad \theta_{\max} = \frac{\Delta L}{\sqrt{I\kappa}}\)

Barnett: rotation as a field \(\displaystyle B_{\text{eq}} = \frac{\omega}{\gamma'},\qquad M = -\chi\,\frac{\omega}{\mu_0\gamma'}\)

The neutron interferometer and the 720° turn

A perfect silicon crystal, cut so that three parallel plates stand on one base (the Laue-Laue-Laue or "LLL" interferometer of Bonse and Hart, used for neutrons by Rauch, Treimer and Bonse in 1974), splits a neutron's wave by Bragg diffraction, sends the two halves about 4 cm apart and brings them together again. Where they meet, the waves add: if both halves arrive in step, most neutrons go into the forward beam (O); if one half has changed sign, they go into the other beam (H). A magnetic field across one path makes the neutron's spin precess (turn) at the Larmor frequency, 29.16 MHz per tesla, so the turn angle is the field times the time spent in it. Quantum mechanics says the state of a spin-½ particle turned by α gets the factor e^−iασ/2: after one full turn it is multiplied by −1. The page uses an unpolarized beam (both spin states give the same intensity) with the given fringe contrast V and a flat background.

Rauch et al. 1975 and Werner et al. 1975 did this almost at the same time. Werner's group, using 1.445 Å neutrons at the University of Missouri reactor, found the intensity came back after a precession of 704 ± 38°, against 720° predicted and 360° for anything that behaves like an ordinary vector. Later runs (Rauch et al. 1978) gave 716.8 ± 3.8°.
The scan uses the same steps: set a field, count for a while, move on. The fit is a weighted least-squares fit of a + b cos(2πB/P) + c sin(2πB/P), with the period P found by searching, and its error where χ² rises by one (scaled up when the fit is poor). The period is then turned into degrees of precession with the same α = γBL/v the apparatus uses.
"Classical spinning top" means a vector that turns with the field. It is back to its old self after 360°, so it would show a period of 360°. The neutron does not.

Einstein-de Haas

Ampère thought magnetism came from tiny loops of current inside iron. A loop of moving electrons has angular momentum as well as a magnetic moment, so lining the loops up (magnetizing the rod) should make the rest of the rod turn the other way. Albert Einstein and Wander de Haas measured it in 1915 by reversing the field in time with the swing of the fibre so the tiny kicks added up, and reported g′ ≈ 1, the value for orbiting electrons. Careful later work (Barnett, Beck, Scott and others, up to the 1960s) found g′ ≈ 1.9 for iron: almost all of iron's magnetism comes from electron spin, whose g is 2.0023. The 1915 result is a famous case of an experiment agreeing with the expected answer for the wrong reason.

The rod is a solid cylinder, moment of inertia I = ½mr², fully magnetized (iron 1.71 MA/m) with a 10 ms response. The fibre's stiffness κ comes from the period you choose (κ = I(2π/T)²).
The g′ readout is what an experimenter would get: the measured kick to the rod (with the fibre's own torque added back and any disturbances left in), divided into the change of moment.
The plot keeps a full account: rod + electrons + what the fibre passed to the room = 0 at every moment.

Barnett effect

In a frame turning at ω, a magnetic moment that comes with angular momentum behaves as if a field ω/γ′ were added (Larmor's theorem). The rod's electrons therefore magnetize it as a field of ω/γ′ would: about 4 nanotesla at 6000 rpm for iron, more than ten thousand times weaker than the Earth's field, which is why Barnett had to cancel the Earth's field with great care. The magnetization is χ times the equivalent H; χ here is the rod's effective susceptibility, which for a real rod is limited by its shape (the demagnetizing field). The magnetometer reading is the field of a point dipole at the chosen distance, side-on.

Further

Chantal Roth's talks on spin and spinors: youtu.be/p4s5CRSihNM and youtu.be/q15Zem6zqcI, and her interactive jsfiddles a7kLuhx0 and p7cx23sk. The ideas for this page came from them.
H. Rauch, A. Zeilinger, G. Badurek, A. Wilfing, W. Bauspiess, U. Bonse (1975). Verification of coherent spinor rotation of fermions. Physics Letters A 54, 425-427.
S. A. Werner, R. Colella, A. W. Overhauser, C. F. Eagen (1975). Observation of the phase shift of a neutron due to precession in a magnetic field. Physical Review Letters 35, 1053-1055.
H. Rauch, A. Wilfing, W. Bauspiess, U. Bonse (1978). Precise determination of the 4π-periodicity factor of a spinor wave function. Zeitschrift für Physik B 29, 281-284.
A. Einstein, W. J. de Haas (1915). Experimenteller Nachweis der Ampèreschen Molekularströme. Verhandlungen der Deutschen Physikalischen Gesellschaft 17, 152-170.
S. J. Barnett (1915). Magnetization by rotation. Physical Review 6, 239-270. S. J. Barnett (1935). Gyromagnetic and electron-inertia effects. Reviews of Modern Physics 7, 129-166.
G. G. Scott (1962). Review of gyromagnetic ratio experiments. Reviews of Modern Physics 34, 102-109.

Comments