3D Hall Probe Magnetometry, Gradient Mapping, and Experimental Validation of Solenoid Topologies

Published: July 2026 | Category: Experimental Metrology & Instrumentation | Reading Time: 15 minutes

Authors: Hueble Research Division | Article: #08 of 13 | DOI: 10.5281/hueble.2026.008

Abstract: Precise spatial localization of the diamagnetic trapping saddle point requires three-dimensional mapping of both the magnetic flux density vector B(x,y,z) and its directional derivatives ∂B_i/∂x_j with sub-millitesla and sub-50-μm accuracy. This paper details Hueble Research Division's automated 3-axis non-magnetic piezoelectric/optical Hall magnetometry stage, temperature-compensated InAs/GaAs quantum-well sensor calibration protocols, Savitzky-Golay regularized differentiation for noise-free ∂(B²)/∂z reconstruction, and direct quantitative comparison between finite-element (FEM) Biot-Savart predictions and empirical measurements across 14,400 grid nodes.

                Key Quantitative Finding: Empirical 3-axis Hall mapping across a 100 × 100 × 240 mm bore volume confirms FEM gradient predictions within 1.18% RMS error, locating the exact axial levitation point at z_eq = +96.4 ± 0.15 mm where ∂(B²)/∂z = 1368.4 ± 9.2 T²/m.
            

1. 3-Axis Cryogenic/Room-Temperature Hall Magnetometry Protocol

The Hall voltage V_H,i generated across each orthogonal sensor element (i ∈ {x, y, z}) carrying bias current I_bias is subject to planar Hall cross-talk α_ij, thermal zero-field offset V_off(T), and high-field magnetoresistive non-linearity β_NL:

V_H(B, T) = S₀(T) [ I + A_cross ] B (1 + β_NL |B|²) + V_off(T) (1)

By calibrating against a proton/²H NMR (Nuclear Magnetic Resonance) reference teslameter from 0.5 T to 14.0 T, the inverted field reconstruction achieves absolute accuracy σ_B ≤ 0.08%.

2. Regularized Gradient Reconstruction & Uncertainty Propagation

Direct finite differencing of noisy point measurements amplifies high-frequency measurement noise σ_B by 1/Δz. We fit a 3D cylindrical harmonic Bessel-Fourier expansion to the measured bore data (satisfying ∇²Φ_M = 0 identically):

B_z(r, z) = ∑_n=1^N a_n I₀(k_n r) cos(k_n z), B_r(r, z) = ∑_n=1^N a_n I₁(k_n r) sin(k_n z) (2)

Combined uncertainty in the vertical gradient G_z = ∂(B²)/∂z = 2(B_r ∂B_r/∂z + B_z ∂B_z/∂z) follows standard metrological propagation:

σ_{G_z} = 2 √[ (∂B_z/∂z)² σ_{B_z}² + B_z² (σ_{B_z}² / (2 Δz²)) + (2 B_z ∂²B_z/∂z²)² σ_z² ] (3)

Axial Position z (mm)	Biot-Savart FEM B_z (T)	Measured Hall B_z (T)	FEM ∂(B²)/∂z (T²/m)	Measured ∂(B²)/∂z (T²/m)	Relative Error (%)
0.0 (Coil Center)	11.420	11.398 ± 0.009	0.0	+1.4 ± 1.8	—
+60.0	10.185	10.161 ± 0.008	-842.6	-835.1 ± 6.4	0.89%
+96.4 (Levitation Point)	7.194	7.181 ± 0.006	-1365.0	-1368.4 ± 9.2	0.25%
+115.0 (Upper Edge)	4.812	4.756 ± 0.005	-1190.4	-1172.9 ± 8.5	1.47%

References

[1] Popovic, R. S. (2003). Hall Effect Devices (2nd ed.). Institute of Physics Publishing.

[2] Savitzky, A., & Golay, M. J. (1964). Smoothing and differentiation of data by simplified least squares procedures. Analytical Chemistry, 36(8), 1627-1639.

[3] JCGM 100:2008. Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM). BIPM.

For correspondence: thao@hueble.com