An ON-OFF Magneto-Optical Probe of Anisotropic Biofluid Crystals: A β-Hematin Case Study

Abstract

We have designed, developed and evaluated an innovative portable magneto-optical detector (MOD) in which a light beam with variable polarization passes through a fluid sample immersed in a variable magnetic field. The light intensity is measured downstream along the forward scattering direction. The field is turned on and off through the in-and-out motion of nearby permanent magnets. As a result, for sufficiently magnetically and optically anisotropic samples, the optical absorption is sensitive to changes in the light polarization. Both detection and characterization applications are therefore available. For instance, both the degree of malaria infection can be measured and hemozoin crystalline properties can be studied. We present experimental results for synthetic hemozoin, and describe them in terms of the basic physics and chemistry underlying the correlations of the directions of the external magnetic field and the light beam polarization. We connect this work to a commercialized product for malaria detection and compare it to other magneto-optical instruments and methods. We conduct tests of absorption parameters, the electric polarizability tensor, and we discuss the connection to magnetic and electric dipole moments.

I. Introduction

THE disruption of light beams as they pass through fluids can reveal the presence of anomalies, such as disease in biofluids or contaminants in industrial liquids. A particular embodiment is a magneto-optical application in which a light beam penetrates a fluid in the presence of an external magnetic field and its attenuation is measured with the field turned on and off. (The attenuation in intensity, also referred to variously as extinction, is due to some combination of scattering and absorption.) The interplay between the light beam polarization and the direction of the magnetic field can yield information about any anisotropy in the anomaly, as is the case with paramagnetic crystals. The anomaly must have a magnetic susceptibility and an electric polarizability sufficiently different from the background fluid composition, in order for the magnetic field and light-beam interplay to be effective.

Different versions of magneto-optical detection (MOD) have been proposed and carried out with success in malaria diagnostics; see, for example, [1]–[9]. The parasite responsible for a malarial infection is also the reason for a significant change in the magnetic susceptibility. It digests hemoglobin but the iron-containing heme compound would be toxic to the parasite, which therefore sequesters it into hemozoin crystals over a 50–1000 nm scale in size (and longer than wider rod-like structures). These crystals are paramagnetic, in contrast to the original weakly diamagnetic hemoglobin [10]. Hence, infected blood is rather more magnetic than healthy blood, and the presence of the crystalline anomaly can be found with MOD. Such diagnostics leads to simple early warning detectors such as one recently developed by Grimberg et al. [3], [4]. This instrument has been commercialized and manufactured by Hemex Health for use in the developing world [3], [4].

The thresholds, sensitivity, and specificity in the successful commercial application of the Hemex Health product in the detection of malaria are found in recent Lancet and other references [11]–[13]. In this paper, we present the complementary discussion of how the instrument works, which has not heretofore been published. We describe a) technical details and operation, b) polarization experiments carried out, and c) analysis for the extraction of crystal properties, all for the MOD system developed by Grimberg et al. The system used is a prototype of the commercialized product, the operation of which is based on the variation of detected intensity for forward-angle light penetration when we look at the different combinations of magnetic field ON or OFF, and polarizations parallel or perpendicular to the magnetic field direction.

This work should be compared to pioneering systems developed by Newman et al. [5], (see also Mens et al. [14]) and Butykai et al. [2]. In the former, study is made of the change in intensity caused by the optical polarization modulated electro-optically for a fixed magnetic field, as well as the change in intensity caused by the field modulated with an electromagnet for fixed polarization. In the latter, a uniform field is produced by a Halbach ring of magnets, which itself is uniformly rotated at adjustable low frequencies. The homogeneous field and optimized frequencies play critical roles in rotating the crystals. These rotating-crystal magneto-optical detection (RMOD) developments are reported to rival laboratory “research-grade” instrumentation. A recent report is on the use of RMOD in the field by Arndt et al. [1].

The system that is presented here addresses the question of whether there is an alternative low-cost, rapid, portable, and accurate instrument for both diagnosing magnetic crystals in solution and researching some of their properties. The answer involves a non-rotating, non-uniform permanent magnet providing the on-off field noted above. We want to emphasize its usefulness in characterizing the magneto-optical properties of such crystals. The magnetic crystal samples studied are synthetic β-hematin crystals suspended in phosphate buffered saline (PBS). The β-hematin have the same structure as hemozoin crystals and are seen to be roughly rod-like in shape with lengths on the order of 1000nm and an approximately square-like cross section on the order of 100nm on a side. The β-hematin are larger while the hemozoin are more spear-like [2], [5], [15]–[18]. A sample from [19] of TEM images of the synthetic crystals are observed in Fig. 1 Our research is concentrated on β-hematin and henceforth it is β-hematin we are discussing in the details below.

Fig. 1.

Artificial hemozoin crystallites in an external magnetic field via TEM pictures. Although the field direction is unspecified, the conclusion is that there is no overall alignment of the crystalline rods despite the internal hard axis alignment perpendicular to the applied magnetic field. See the text discussion. Permission from Butykai et al. [2] through http://creativecommons.org/licenses/by-nc-sa/3.0/

We present the results for different concentrations and light polarizations at a given OFF or ON magnetic field strength and light beam wavelength. These results are expressed in terms of the theoretical modeling available to us for the interaction of the crystals with both the magnetic field and the light beam. The discussions can be made in terms of the induced magnetic and electric dipole moments, and the optical attenuation coefficients.

In the following modeling section, we first consider the magnetic response of the crystals to field strengths approaching Tesla levels at room temperatures. We write its Hamiltonian and connect it to the notion of induced magnetic dipole moments. This is followed by a description of the attenuation of an electromagnetic wave by the crystals, and the connection here is to induced electric dipole moments. We find relationships between the magnetic-principal-axis extinction cross sections and the laboratory extinction cross sections for light polarizations parallel and perpendicular to the magnetic field. The experimental setup and results are presented in the next two sections. The signal shows a larger (smaller) extinction when the magnetic field and electric polarization are parallel (perpendicular); the associated relative cross section sizes are the subject of the discussion section. A fundamental connection to measuring the electric polarizability tensor is made (Appendix). The last section comprises conclusions and issues for future work.

II. Modeling

A. Crystals in a magnetic field: induced magnetization

In the literature, an established model can be found for interpreting our experimental results. We consider the interaction of electromagnetic fields with a basic crystallographic cell in the β-hematin molecular structure. The β-hematin consist of stacks of dimers, with their monomers made up of crystalline (porphyrin) planes ([17], [18], [20]–[22]) coupled by iron oxide (iron carboxylate) links and stabilized by hydrogen bonds. Figs. 2 and 3 illustrate coupled porphyrin planes and their packing in the β-hematin crystallite. These planes have Fe³⁺ cores in the center and slightly above four nitrogen atoms surrounded by a ring of carbon and hydrogen. That is, they consist of four smaller rings (pyrroles) made from four carbons, one nitrogen, and an iron center. These pyrrole molecules combine to create the larger ring through single and double bonds to form the porphyrin planes. An approximate symmetry, the point group C_4v, is associated with four-fold rotations around and perpendicular to this plane at its center and with its axis, as shown in Fig. 4.

Fig. 2.

The porphyrin planes of β-hematin. Reprinted from https://en.wikipedia.org/wiki/Hemozoin. This work has been released into the public domain by its author, Iwona Tesarowicz, and based on CSD data of Pagola et al. [18].

Fig. 3.

Packing diagram of a β-hematin crystal showing the angle between the hard axis, which is perpendicular to the porphyrin plane, and the c-axis, which is along the longitudinal axis of the crystalline rod. The orientation of the packing diagram was chosen such that both the hard and c axes lie in the plane of the paper. It was created with the use of Mercury, a crystal structure visualization program, and the cif data file of Pagola et al. [18], both of which were downloaded from the Cambridge Structural Database, https://www.ccdc.cam.ac.uk/.

Fig. 4.

The iron-nitrogen (approximately planar) core of the porphyrin plane of Fig. 2. Permission from Butykai et al. as in Fig. 1.

For the interaction with an external magnetic field, we need to understand the magnetic properties of the β-hematin crystal in terms of its magnetization. During the dimerization process, the iron (II) atoms in heme - which are diamagnetic - are changed into iron (III) atoms making the β-hematin a paramagnetic crystal. While the Fe²⁺ ions have “low-spin” with dominant electron pairing of the remaining even number of 3d and of 4s electrons, the Fe³⁺ ions have “high-spin” with five unpaired electron spins maximally aligned to give spin 5/2 by Hund’s rule.

The crystal magnetization (average magnetic dipole moment density) is related to the magnetic H-field through $M_i={\chi }_{ij}{H}_j$ with diagonal components (${\chi }_{xx},{\chi }_{yy},{\chi }_{zz}$) referenced along the magnetization (or magnetic susceptibility or magnetic) principal axes (x, y, z). We shall often refer to the magnetic principal axes as simply the principal axes depicted in Fig. 5. From experiment [20], [21], [23] we find the values ${\chi }_{xx}={\chi }_{yy}=4.61\times {10}^{-4}>{\chi }_{zz}=3.95\times {10}^{-4}$. Correspondingly, the x − y plane is an easy plane and z is a hard axis. Connecting back to the molecular picture, the easy plane is therefore the porphyrin plane.

Fig. 5.

General orientation of the magnetic principal axes (x, y, z) with respect to the laboratory axes (x′, y′, z′) in terms of Euler angle rotations (ϕ, θ, ψ). The magnetic field B is in the z′ direction. N denotes the line of nodes, which is the intersection of the x − y (porphyrin) plane and x′ − y′ plane. The porphyrin plane – see Figs. 2 and 4 - includes four nitrogen atoms (green spheres) and an iron atom (larger red sphere) lying slightly above the plane (elevation not shown). The transformation from the laboratory axes (primed) to the principal axes (unprimed) is accomplished by the following sequence of rotations: about the z′ axis by ϕ, then about N by θ, and finally about the z axis by ψ. In a strong magnetic field, the hard axis (z axis) is perpendicular to the field direction (θ = π/2) which lies in the easy plane (x − y plane). The Euler convention is that of Weisstein, Eric W. “Euler Angles.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/EulerAngles.html.

We turn now to magnetic energetics underlying the correctness of the easy and hard designation. A standard Hamiltonian for spin interactions can be used to demonstrate the tendency for the S = 5/2 spins of Fe³⁺ ions in β-hematin to lie in the easy plane [24]. Consider the individual spin-spin quadratic interaction written in terms of the principal-axis spin coordinates, singling out S_z, the hard-axis component due to the axial anisotropy [2], [20], [23], [24]. Adding in the Zeeman splitting due to an external magnetic field with an isotropic g-factor [2], [23], [24], we have the result (in the notation of the references)

$$H=D\left(S_z^2-\frac{S(S+1)}{3}\right)+E\left(S_x^2-S_y^2\right)-{\mu }_{B}g\overrightarrow{S}\cdot \overrightarrow{B}.$$ (1)

Assuming there is little C_4v symmetry breaking, the E term is negligible. Hence we indeed see the D term is minimized for S_z = 0 and the system spin settles into the easy plane. For the isotropic Zeeman term, μ_B is the Bohr magneton and an approximate value g = 2 is implied by experiments [23]–[27]. Its minimum depends on a thermal average for a given external magnetic B-field (where $\overrightarrow{B}={\mu }_0\overrightarrow{H}$), which is discussed next.

In the present experiments, the external magnetic field is predominantly uniform in what we will define as the z′ direction in a laboratory reference frame with primed axes $\left(x^{\prime },y^{\prime },z^{\prime }\right)$. See Fig. 5 for a comparison of the primed and unprimed coordinate frames. The Zeeman term leads to the energy density $-\overrightarrow{M}\cdot \overrightarrow{B}$, in terms of the magnetization $\overrightarrow{M}$, for a constant external magnetic field $\overrightarrow{B}$. We assume the susceptibility tensor $\overleftrightarrow{\chi }$ is constant (field independent) in which case the magnetization is linear in the magnetic field. In the principal frame (x, y, z), $M_x={\chi }_{xx}{B}_x/{\mu }_0$, $M_y={\chi }_{yy}{B}_y/{\mu }_0$, $M_z={\chi }_{zz}{B}_z/{\mu }_0$ and we recall the symmetry χ_yy = χ_xx. Using laboratory spherical coordinates θ, ϕ for the magnetization vector, θ is the angle between the field direction z′ and the hard axis z, and ϕ is the azimuth defined relative to the y′ axis as in Fig. 5 (the Euler angle ψ is unnecessary to determine a point on the z axis).

For the energy, there is no azimuthal angle dependence and the θ (polar) angular dependence is given by the portion

$$E(\theta )=+\frac{1}{2{\mu }_0}({\chi }_{xx}-{\chi }_{zz})B^{2}V{\cos }^2\theta$$ (2)

with sample volume V. In the zero-temperature limit, the energy minimum is at θ = π/2 for ${\chi }_{xx}>{\chi }_{zz}$; then the direction of the hard-axis (z-axis) is perpendicular to the external magnetic field, and the easy plane must contain the external magnetic field vector. At nonzero T, the thermal motion fights the 90° alignment, so the principal frame average orientation depends on the relative strength of the thermal energy (k_BT) and the magnetic energy. The average cos²θ value is found through the Boltzmann factor

$$f(\theta )=e^{-E/\left(k_{B}T\right)}/Z$$ (3)

with the partition function

$$Z=2\pi {\int }_0^{\pi }{e}^{-E/\left(k_{B}T\right)}\sin \theta d\theta .$$ (4)

In the experiments carried out in which the magnetic field is fractions of Tesla, we have $E\gg k_{B}T$, and we are indeed in the low-temperature or high-field limit, where $\theta \to \pi /2$. For more general experiments, the integral (4) is given in specific function form for finite T and B in the Appendix, along with numerical estimates confirming that the low-temperature (high-field) limit is achieved in this work.

To complete the modeling, we ask how the principal-frame orientation affects the orientation of the full β-hematin crystal lattice and body (“rod”). The unit dimer cells building up the lattice are shown in Figs. 2 and 3 and include pairs of porphyrin units. The unit cells have three unequal crystallographic axes (standard a,b,c labeling) at oblique angles with each other; i.e., it is a triclinic system [2], [18], [28]. In the crystal lattice, the β-hematin dimers stack up such that the individual hard axes throughout the rod are aligned with each other and their porphyrin (easy) planes are parallel to each other, see Fig. 3. Importantly, while the c-axes throughout the crystal lattice are aligned with the longitudinal rod direction, this is not the hard-axis direction. From [2], [17], it is learned that the angle between the c-axis and the hard-axis is approximately 60°, as shown in Fig. 3.

For magnetic energies large compared to thermal energies, it is the overall hard-axis direction that is perpendicular to the magnetic field, which thus lies parallel to the common easy plane. However, rotations of the principal frame through ϕ and ψ, as defined in Fig. 5, for fixed θ implies a spread of crystal orientations for the same thermal equilibrium temperature. Therefore, at high external magnetic fields, the rod (the common crystal c-axis of Fig. 3) does not settle into a plane perpendicular to the field. Even with a strong magnetic field, the rods are still quite random, as exemplified in Fig. 1, since they can freely rotate around the internal hard axis, which, as we have noted, is predominantly perpendicular to the field direction. It can be seen that as ϕ and ψ go randomly over 2π, the longitudinal rod direction ranges from 30° to 150° with respect to the z′ axis. Still, and most importantly, the orientation of the magnetization principal frame is controllable via the external magnetic field, and hence it can be probed in experiments such as ours.

B. Light absorption by crystals – induced electric dipole moments

Anisotropic β-hematin crystals lead to changes in the transmission of light for different polarizations. The crystal suspension is birefringent (the real part of the index of refraction is polarization dependent) and dichroic (the imaginary part is polarization dependent). Our study focuses on the attenuation of the light beam in the forward direction, presumably dominated by dichroic absorption in accordance with previous research [2], [29], [30]. In general, we are measuring attenuation coefficients, which are proportional to extinction cross sections, and their dependence on the angle between the polarization and the external magnetic field. Of special interest are the two cases of polarization parallel and perpendicular to the applied magnetic field.

For comparison with our measurements of the optical extinction of a fluid with a dilute population of β-hematin crystals in the absence or presence of an external magnetic field, we adapt a model for the forward transmission of light through a thin slab containing a collection of subunits [31]. Specifically, we consider the interaction of light with the iron-nitrogen cores of β-hematin dimers through induced electric dipole moments. These subunits are small compared with the optical wavelengths, yet we ultimately have a collection of them making up our crystals so that the overall crystals are not small compared to optical wavelengths.

The intensity of the detected light wave is the bilinear electromagnetic Poynting vector comprising the incident flux, scattered flux, and interference (extinction term) of the incident and scattered amplitudes. In a certain spectroscopic range, the extinction is connected to the spatial dependence of the transition amplitudes involving porphyrin π and π * orbitals and Fe³⁺ d orbitals. Anti-bonding π * molecular orbitals are normally higher in energy than bonding π molecular orbitals. Such electron wave functions making up the transition amplitudes lead to the polarization effects that are key to detecting malaria infections.

The background and details for the cross section modeling are found in the Appendix. We assume the scattered amplitude is dominated by the crystalline collection of individual electric dipole moments over scales less than the light wavelength. Consistent with our experiments and the cited literature, the scattered flux can then be neglected such that the main extinction mechanism is absorption. The experimentally determined β-hematin attenuation coefficients are used to calculate the extinction cross sections, which are in turn related to the polarizability tensor. Referring back to Fig. 5 and the previous primed and unprimed coordinate discussion, we define y′ as the initial-beam direction (and final forward direction) and z′ as the uniform magnetic field direction. We define the cross sections σ_i in which the subscript refers to the linear polarization direction. Hence i = x′, z′ in the laboratory and i = x, z on the principal axes. For induced electric dipole moments defined along the β-hematin principal axes, the corresponding polarizability (electric dipole density) tensor $\overleftrightarrow{\alpha }$ is diagonal. The cross section σ_x(σ_z), which is proportional to Imα_xx (Imα_zz) for absorption dominated extinction, corresponds to polarization parallel (perpendicular) to the previously defined easy plane. We assume this plane is the porphyrin plane for β-hematin. The general rotation matrix representing an axisymmetric magnetic principal orientation with respect to the laboratory can be written in terms of the same polar angle used in (2) and an azimuthal angle that is random. The energy (2) is independent of this azimuth and an average over it yields

$${\sigma }_{x^{\prime }}=\frac{1}{2}{\sigma }_x\left(1+{\cos }^2\theta \right)+\frac{1}{2}{\sigma }_z{\sin }^2\theta ,$$ (5)

$${\sigma }_{z^{\prime }}={\sigma }_x{\sin }^2\theta +{\sigma }_z{\cos }^2\theta .$$ (6)

It is noticed that the combination is independent of cos θ

$$2{\sigma }_{x^{\prime }}+{\sigma }_{z^{\prime }}=2{\sigma }_x+{\sigma }_z$$ (7)

corresponding to invariance under rotations of the trace of the polarization matrix.

The θ thermal averaging of (5) and (6) through the Boltzmann factor (3) models the light intensity measurements described in the next section. If the magnetic field is turned off in either equation, the 3D random average of cos²θ is 1/3 and we have for the thermal average

$${\langle {\sigma }_{x^{\prime }}\rangle }_0={\langle {\sigma }_{z^{\prime }}\rangle }_0=\frac{2}{3}{\sigma }_x+\frac{1}{3}{\sigma }_z,B=0$$ (8)

checking with the simple picture of completely random motion for ${\sigma }_x={\sigma }_y$. The limit (8) is redundant with (7). Inverting these gives information about the extinction cross sections for the principal axes. The laboratory extinction cross sections enter the attenuation coefficients in the Beer-Lambert exponential decrease in the intensity as light travels a distance ℓ straight through (i.e., the forward direction) a uniform sample

$$I(\ell )=I_0{e}^{-n\sigma \ell }$$ (9)

assuming a uniform number density n and an average β-hematin extinction cross section σ for the crystals. The factor I₀ includes the attenuation in the absence of β-hematin in the suspension, and hence has an implicit dependence on ℓ. But in taking ratios we need only the explicit exponent in (9). The dependence of the extinction cross section depends on the beam polarization, which has been left understood in (9).

III. EXPERIMENTAL MATERIAL AND METHODS

The sample material is commercially synthesized β-hematin prepared with an acidic method [25], [32]. The dark-brown β-hematin crystals were suspended in sterile phosphate buffered saline (PBS). For each of the range of concentrations studied, the suspensions were diluted to make a given concentration and vortexed before every scan for a more homogeneous dispersion of β-hematin.

With a schematic and photograph shown in Fig. 6, we next describe the components of the MOD: light beam probe, polarizer, photodiode, cuvettes, and magnet/motor. The light beam probe has a fixed wavelength distribution. Specifically, the MOD uses a red-light LED with a dominant wavelength of about 635nm. The light then passes through a polarizer (Thorlabs, Newton, NJ) before reaching the sample. Then, a photodiode with a computer readout is placed in the forward direction to detect the transmitted beam.

Fig. 6.

a) Schematic of the MOD device. Moving the magnets to the left (right) yields the ON (OFF) position. b) Prototype of the MOD device

The sample containers of the suspensions are clear plastic FireflySci cuvettes made from PMMA (polymethyl methacrylate) or acrylic plastic. This material has a minimal effect on the polarization of the transmitted light. The optical activity of the PMMA cuvettes was tested and was determined to cause a negligible change (< 1°) in the polarization of the incident light within the sample. The cuvettes are placed in the MOD perpendicular to the light beam, which traverses the pathlength ℓ inside the suspension (the inner diameter of the cuvette). It is measured to be ℓ = 10.00 ± 0.02mm in (9) from a SD average over 20 cuvettes using vernier calipers.

Permanent magnets provide the magnetic field. The magnetic field is turned on and off by the motion in and out of the motorized magnets. Thus, the label B-ON corresponds to the in-position where the magnet sides are centered on the cuvette containing the sample. The field strength at the center position between the magnets is found to be 0.58 ± 0.01T by a Gaussmeter measurement. In the “away” position (B-OFF), the measured field drops to 41 ± 2G.

In Fig. 5, the magnetic field direction at the cuvette defines the z′-axis of the primed coordinate system and the light beam travels along the positive y′-axis, so x′ is orthogonal to both the field and the beam. A linear polarizer for either x′ (vertical) or z′ (horizontal) polarization is placed in the beam path. The photodiode voltage signal is analyzed using a LABVIEW program. The computer also controls the motor of the magnet effecting the B-ON and B-OFF positions. For each of the transmission experiments (the two linearly polarized cases described above), the magnet was moved between B-ON and B-OFF positions and a series of different concentrations were measured over the range of 4.0 – 64.0μg/mL. All data measurements were collected and displayed using a MATLAB code.

IV. EXPERIMENTAL RESULTS

All data are from the photodiode voltage measurements made with the magnetic field turned alternately on and off. Fig. 7 shows such an alternating-field time series for the voltage with horizontally and vertically polarized light for various concentrations of the β-hematin crystals suspended in PBS. As the external magnetic field is alternately applied and removed, the intensity of the horizontally polarized light is seen to decrease in Fig. 7. The opposite occurs for the vertically polarized light, for which the intensity increases, at constant concentration.

Fig. 7.

The voltage (signal intensity) changes for the two polarizations for various concentrations as the magnetic field is turned ON and OFF.

Since the measured voltage at the photodiode is proportional to the intensity of the beam, the voltage will also satisfy for a given magnetic field the exponential behavior of (9) with the exponent rewritten as

$$V(C)=V_0{e}^{-aC}$$ (10)

in which C is the mass concentration and the attenuation coefficient is

$$a=\sigma \ell /m$$ (11)

with m the average mass of a β-hematin crystal.

Numerical fits to the data in Fig. 8 show for a given magnetic field (on or off) that the voltage is indeed decreased exponentially as the β-hematin concentration is increased. Attenuation coefficients were determined by taking the logarithm of (10) and doing least-squares linear fits. Table 1 gives the values for the constants in (10), as well as the R² values. As seen from the plots and from the R² values, the fits are excellent. It should be noted that the values of a for B off are slightly different for horizontal and vertical polarization, although theoretically they should be the same. This error is small, being less than 3%. It is believed to be the result of systematic errors such as voltage drift or slight degradations in the sample between the times the data were collected for horizontal and vertical polarizations. In consequence, for B off the average of the two values are taken.

Fig. 8.

Exponential fits to the voltage data for the magnetic field turned ON or OFF.

TABLE I.

	Horizontal Polarization (along z′ direction)			Vertical Polarization (along x′ direction)
	V0(V)	a(mL/|μg)	R2	V0(V)	a(mL/μg)	R2
B off	8.0074	0.008776	0.99999	7.9753	0.009017	0.99994
B on	8.0011	0.010584	0.99999	7.9957	0.008010	0.99998

The coefficients V₀ (in volts) and a (in mL/μg) for the exponential fit of (10) as shown in Fig. 8, as well as the R² value of the least squares linear fit. The number of significant digits is less than those given here in view of the few percent errors in our measurements.

We obtain the laboratory extinction cross sections directly from the laboratory attenuation coefficients in (11) for a given mass concentration. Then with the measurements of the cross sections for the polarization along the laboratory x′ and z′ directions (σ_x′ and σ_z′) and our modeling and thermal averaging, we can find the cross sections in the principal frame, for polarization along the x-direction (easy plane, σ_x = σ_y) and z-direction (hard axis, σ_z). As thermal averages, (5) and (6) are given by

$$\langle {\sigma }_{x^{\prime }}\rangle =\frac{{\sigma }_x+{\sigma }_z}{2}+\frac{{\sigma }_x-{\sigma }_z}{2}\langle {\cos }^2\theta \rangle ,$$ (12)

$$\langle {\sigma }_{z^{\prime }}\rangle ={\sigma }_x-\left({\sigma }_x-{\sigma }_z\right)\langle {\cos }^2\theta \rangle .$$ (13)

The combination (7) is unchanged by thermal averaging and we recall

$${\langle {\sigma }_{x^{\prime }}\rangle }_0={\langle {\sigma }_{z^{\prime }}\rangle }_0=\frac{2}{3}{\sigma }_x+\frac{1}{3}{\sigma }_z,B=0.$$ (8)

We also recall for $E\gg k_{B}T$ the hard axis z is perpendicular to the field; i.e., the magnetic field lies in the easy plane. The average angle goes to the limit $\langle \theta \rangle =\pi /2$ or $\langle {\cos }^2\theta \rangle =0$ so (12)-(13) reduce to

$$\langle {\sigma }_{x^{\prime }}\rangle =\frac{1}{2}{\sigma }_x+\frac{1}{2}{\sigma }_z,\text{largeB,}$$ (14)

$$\langle {\sigma }_{z^{\prime }}\rangle ={\sigma }_x,\text{largeB}.$$ (15)

These derived results agree with a simple picture where the hard axis z is restricted to but otherwise random in a two-dimensional plane perpendicular to the laboratory z′ axis. The polarization along z′ then lies in the x − y plane and the polarization along x′ lies randomly in a plane perpendicular to z′. Using symmetry in x and y, (14) and (15) follow. The left-hand sides are found experimentally in the laboratory and any two of the three independent equations (8), (14), (15) can be solved for σ_x and σ_z. If the solution of any two give the same result within sufficiently small errors, then the underlying assumption that the crystal hard axes lie randomly in the x′ −y′ plane (and perpendicular to the applied magnetic field), when the strong field is applied, is supported. Because equations (14) and (15) were derived using the assumption that scattering is negligible, so that the extinction cross section is approximately equal to the absorption cross section, the dominance of absorption over scattering in the overall beam extinction would also be supported by sufficiently small errors in σ_x and σ_z when calculated using different combinations of (8), (14), and (15).

Table 2 shows that the solutions of the three equations are experimentally self-consistent. The coefficients a_x and a_z satisfy equations like (8), (14), and (15), since they are proportional to their respective cross sections, and do not depend on the mass and penetration length. Since the solutions from any pair of the equations are equal within small errors, the underlying assumptions used in the derivation of (8), (14), and (15) are assumed to be correct, i.e., that the crystal hard axes lie in the x^′ − y^′ plane when the field is applied and absorption dominates over scattering in the beam attenuation.

TABLE II.

	ax	az
From (8) and (14)	0.01058	0.00544
From (14) and (15)	0.01058	0.00552
From (8) and (15)	0.01067	0.00535
Average	0.01061	0.00544
Relative standard deviation	0.38%	1.27%

Noting that a is proportional to σ, we calculate the x and y (principal axes) components for a from pairs of three equations using the experimental data given in Table 1. As in Table 1, the number of significant digits is less than those given here in view of the few percent errors in our measurements. Units are mL/μg.

From table 2, we also find that a_x/a_z = 2 to within a few percent. Since the cross sections can be defined by a basic proportionality to the attenuation coefficients, see (11), it may be concluded that a_x/a_z = σ_x/σ_z = 2 within a few percent error.

To get an adaptable formula for an estimate of the size of the cross-sections based on the measured attenuation coefficient values, we have generically

$$\sigma =0.1(a\rho V/\ell ){\text{nm}}^2$$ (16)

in terms of the following units: a in mL/μg, ρ in g/cm³, V in nm³, and ℓ in cm. For a β-hematin density of 1.44 g/cm³, and two different representative volumes [17], [20]: 200nm × 200nm × 700nm and 50nm × 50nm × 300nm, respectively, we find the extinction cross sections are σ_x = 4.3 × 10⁴ nm² and σ_z = 2.2 × 10⁴ nm² for the former, while they rescale to σ_x = 11.5 × 10² nm² and σ_z = 5.9 × 10² nm² for the latter. The factor of two is of course preserved for this linear relationship. Notice the extinction cross sections are on the same order of magnitude as the geometric ones. However, it is the electric dipole interactions that determine the sizes of the cross sections; the sizes are not directly related to the geometrical dimensions of the rod.

V. DISCUSSION

All experiments have been carried out at fixed values of room temperature, cuvette penetration depth, and B-ON magnetic field strength. The expected exponential reductions of the light beam in the forward direction as a function of β-hematin concentrations in PBS fluid have been confirmed for different polarizations and with the field turned on and off. The attenuation coefficients are on the order of 10⁻²mL/μg implying that extinction cross sections are on the order of (100nm)². The changes in these numbers for different polarization in the presence and absence of a magnetic field (at the 0.6T level) were measured and are discussed below.

As pointed out in the Results section, the ratio of the x and z components of both the attenuation coefficients and extinction cross sections are given by $a_x/a_z={\sigma }_x/{\sigma }_z=2$ to within a few percent. Since the imaginary part of the polarizability tensor $\overleftrightarrow{\alpha }$ is approximately proportional to the extinction cross section (assuming negligible scattering), i.e. σ_x = kIm (α_xx) and σ_z = kIm (α_zz) (see the Appendix), we also conclude that the ratio of the components of the polarizability tensor satisfy Im(α_xx)/Im(α_zz) = 2 to within a few percent.

We believe the different sources of error in the results presented above include LED power drift, polarizer nonuniformities, sample degradations, variations in concentration preparation, and commercial sample irregularities. The first, for example, could be reduced with a beam splitter. These are systematic sources of error adding up to a few percent. The statistical error is small, as seen in the excellent R² values in Table I. Therefore, the measurement errors appear to be dominated by (small) systematics. We note that the calculated ratios a_x/a_z = σ_x/σ_z = Im{α_xx}Im{α_zz} = 2 are less susceptible to possible systematic errors than the individual measurements of a_i and σ_i. Therefore, the factor of two increase in attenuation for polarization along the easy plane compared to the hard axis is expected to be an accurate standard for β-hematin crystals.

A major assumption made in the theoretical analysis presented in this work is the dominance of absorption over scattering in the total extinction of visible light by the suspension of β-hematin crystals. This assumption led us to three linearly independent equations relating the attenuation coefficients in the magnetic principal basis to those measured in our experiments with vertical polarization, horizontal polarization, and the magnetic field off. Confirming that our experimental results are in agreement with all three of the formulas was a consistency check, which demonstrated that scattering is in fact negligible in comparison to absorption for the experiments performed in this work, so that the extinction cross section, which is proportional to the attenuation coefficient, is approximately equal to the absorption cross section.

Another important theoretical expectation underlying the analysis of our experimental results is the alignment of the magnetic principal axes of the β-hematin crystals in an applied magnetic field. This anticipated result does depend on the magnetic field strength. Referring to the numbers in the Appendix for an average β-hematin volume and the published positive magnetic anisotropy difference Δχ = χ_xx − χ_zz, we expected the laboratory B-ON extinction cross sections to correspond to β-hematin hard axes restricted to and randomly oriented in the plane perpendicular to the magnetic field. All of the measurements are consistent with this picture. For example, the measured relative strengths 2: 5/3: 3/2 for the B-ON horizontal polarization: B-OFF (either polarization): B-ON vertical cases, respectively, within errors, is readily understood and easily calculated from a simple random-within-perpendicular-plane hard axis orientation for the B-ON cases and 3-D randomization for the B-OFF case.

From the above discussion, we saw that the result σ_x = 2σ_z translated into ${\sigma }_{z^{\prime }}=\frac{3}{2}{\sigma }_{x\prime }$ for a large enough magnetic field (for which the z axis is perpendicular to the field direction z′). If the anisotropy $\text{Δ}\chi ={\chi }_{xx}-{\chi }_{zz}$ were in fact negative, the average angle between the field and the hard axis would instead be zero (or 180°) for a sufficiently large magnetic field. If the electric polarization is not correlated with the magnetization, and we still have σ_x = 2σ_z, this would translate into ${\sigma }_{z^{\prime }}=\frac{1}{2}{\sigma }_{x^{\prime }}$ (this relation can be generally obtained from (12)-(13) through the dependence on 〈cos² θ〉). If this were the case, we would expect to see increased absorption for polarization perpendicular to the magnetic field compared to polarization parallel to the magnetic field, in contrast to our experimental results.

VI. Conclusion and future considerations

We have described a portable MOD instrument useful not only for the detection of the kinds of crystals present in malaria and other diseases, but also as a low-cost and rapid investigational tool for studying those crystals. The general detection of a given level of suspended magnetic crystals in a background fluid is the commercialized application. A marketed version of the instrument introduced in this paper is currently manufactured and in use for malaria detection. In this paper, we especially emphasize the additional experimental information that can be obtained from the dependence of the photodiode voltage signal on the external magnetic field, the polarization, and the crystal concentration. Let us consider the magnetic and electric crystal properties and extinction cross section models that can be probed.

On the magnetic properties, while we in this introductory study assumed the magnetic susceptibilities for β-hematin already published, finding the dependence of the laboratory cross sections on magnetic field strength (i.e., through 〈cos² θ〉) could give us a measure of Δχ. There is motivation for the study of the underlying magnetic moment, and hence spin direction, of the Fe³⁺. Based on the results presented herein, we already see that the induced effective magnetic dipole moment is enhanced in the porphyrin plane and future elucidation of that enhancement is of interest. It is also of interest to look for future chemistry analysis telling us about the magnetic moments perpendicular and parallel to the porphyrin plane.

On the electric properties, the dominance of the transverse cross section over the longitudinal cross section in the principal reference frame found in our work also motivates future study. The induced electric dipole moment coupling to the electric field of the incident beam is also enhanced in the porphyrin plane. The electrons are apparently freer (i.e., the charge distribution more polarizable) in the plane compared to the perpendicular direction. Of particular note, we determined from our experiments that the ratio of components of the polarizability tensor satisfy Im(α_xx)/Im(α_zz) = 2 to within a few percent. Interestingly it would also be of value to find the electric dipole moment components (at high frequencies) and the magnetic dipole moment components (at low frequencies or static) in and out of the porphyrin plane. On the theoretical side, first-principles calculations of the transition electric dipole moments for polarization perpendicular or parallel to the porphyrin plane could be compared to the simple answers found here. On the face of it, these calculations involving dozens of molecules are formidable and supercomputer intensive. Our simple results promise shortcuts may be found.

Many other relevant questions arise. Here we see that the experimental results support the dominance of absorption over scattering, indicating that the interaction between light and β-hematin is in agreement with the Rayleigh model. In general. the efficiency (Q = σ/πa²) of absorption versus scattering depends on the wavelength λ of the light and the size of the attenuating particles. In the Rayleigh limit, the efficiency for scattering is of order (ka)⁴ while that for absorption is of order ka, where k = 2π/λ is the wave number and a is the size scale of the interacting particle. This result combined with the experimentally supported dominance of absorption over scattering implies that the light is interacting with the individual porphyrin units, which are on the order of a ≈ 1nm, rather than with the entire β-hematin crystal. Discrete dipole moment models [33] may be used to clarify why we find a long-wavelength (Rayleigh model) approximation useful even though the wavelengths of light used in this work are small compared to the overall crystal body size. By changing the wavelength of light used in the MOD, the Rayleigh-Gans-Mie scattering model could be tested further.

We can also look for new and interesting experiments to be studied with our probe. For other crystals, such as gout – see the pioneering work in [34]–[38] – we can study the principal cross section relations at a given wavelength, as we have done for β-hematin.

Appendix

We relate the extinction cross section in the principal basis to thermal averages in the laboratory frame in this Appendix. References for the amplitude analysis for scattering and absorption can be found in [31] and for the error function in https://mathworld.wolfram.com/Erf.html.

The transmitted intensity measured in voltage at the photodiode is the magnitude of the total Poynting vector for the light wave. The extinction intensity arises from the interference of the incident fields (electric and magnetic) with the scattered fields. The key ingredient is thus the scattered electric field amplitude, which is found in terms of the polarization vector (induced electric dipole density). This inducement is thus in terms of the incident electric field $\overrightarrow{P}=\overrightarrow{\alpha }\cdot \overrightarrow{E}$ through the polarizability tensor $\overleftrightarrow{\alpha }$. The induced dipoles defined along the magnetic principal axes can be expressed in terms of the diagonal tensor for β-hematin in that frame. We have the dyadic matrix

$$(\overleftrightarrow{\alpha })=\left(\begin{array}{ccc}{\alpha }_{xx}& 0& 0\\ 0& {\alpha }_{yy}& 0\\ 0& 0& {\alpha }_{zz}\end{array}\right)$$ (A1)

Now the goal is to find, through the optical theorem, the extinction cross section for an incident electric field (whose amplitude cancels out in our definition). In the Rayleigh limit, the efficiency ($Q=\sigma /\pi a^2$) for scattering is of order (ka)⁴ while that for absorption is of order ka, where k = 2π/λ and a is the size of the interacting unit. Then, because ka < 1 for visible light (λ ≈ 100nm − 1000nm) interacting with a porphyrin unit in a β-Hematin crystal (a ≈ 1nm), the scattering efficiency is much less than the absorption efficiency. Therefore, the extinction cross section is assumed to be dominated by absorption and the contribution to extinction from scattering is negligible (as supported by arguments in the main text). The extinction cross section can then be found from the optical theorem, which with linear light polarization $\widehat{\epsilon }$ is given by

$$\sigma (\widehat{\epsilon })=k\text{Im}\{\widehat{\epsilon }\cdot \overleftrightarrow{\alpha }\cdot \widehat{\epsilon }\}$$ (A2)

in terms of the polarizability tensor. (A2) represents the diagonal element for a coordinate system with $\widehat{\epsilon }$ as one of the axes.

To connect the laboratory cross sections to the principal cross sections, we need the appropriate rotation transformation between primed and unprimed coordinates. Although the electric field has only two independent directions in the laboratory, which are x′ and z′ for forward transmission along y′, it will have components along all three unprimed Cartesian coordinates for an arbitrary crystal orientation. (It remains perpendicular to the wave direction ${\widehat{y}}^{\prime }$ which itself will also have three unprimed components.) The general rotation of the principal basis amplitude into alignment with the laboratory frame is ${\overrightarrow{E}}^{\prime }=\overleftrightarrow{R}\cdot \overrightarrow{E}$ for the matrix $\overleftrightarrow{R}$ in terms of, say, three Euler angles. The rotation can be reduced to two spherical angles for a 2-D axisymmetric unprimed system, as we did for the magnetization vector in Sec. II. While we make this simplification, it should be noted that we have carried out the following analysis for all three Euler angles in Fig. 5. The matrices in this general case can be found, for example, in Goldstein et al. [39]. Upon the averages over ϕ and ψ, the results are the same.

The unprimed $\overleftrightarrow{\alpha }$ elements give the principal cross sections via (A1). To find the connection to the laboratory cross sections, we need to find ${\overleftrightarrow{a}}^{\prime }$ the polarizability tensor in the laboratory reference frame (the primed frame); we refer back to Fig. 5 and the text discussion for the primed and unprimed axes. Because of the noted symmetry between x and y, we use the two spherical angles θ, ϕ to define the principal orientation. The lab frame is then related to the principal frame through a) a rotation about x by the angle θ defined in the text (between the z and z′ axes) yielding an intermediate rotated coordinate system (x, y″, z′) followed by b) a rotation about z′ by an angle ϕ finally to reach (x′, y′, z′). (This is opposite to the order described in Fig. 5 because the primed axes are obtained by rotations relative to the unprimed frame here.) The transformation matrix is $(\overleftrightarrow{R}(\theta ,\varphi ))=R_z(\varphi )R_x(\theta )$ or

$$\begin{gathered} (\overleftrightarrow{R})=\left(\begin{array}{ccc}\cos \varphi & \sin \varphi & 0\\ -\sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & \sin \theta \\ 0& -\sin \theta & \cos \theta \end{array}\right) \\ =\left(\begin{array}{ccc}\cos \varphi & \sin \varphi \cos \theta & \sin \varphi \sin \theta \\ -\sin \varphi & \cos \varphi \cos \theta & \cos \varphi \sin \theta \\ 0& -\sin \theta & \cos \theta \end{array}\right) \end{gathered}$$ (A3)

Therefore, the transformed polarizability dyad is given by: ${\overleftrightarrow{\alpha }}^{\prime }=\overleftrightarrow{R}\cdot \overleftrightarrow{\alpha }\cdot {\overleftrightarrow{R}}^T$ from which the diagonal elements in the laboratory frame can be found for the two possible polarization directions, ${\alpha }_{x^{\prime }{x}^{\prime }}$ and ${\alpha }_{z^{\prime }{z}^{\prime }}$. From (A1)-(A3) these elements give relations between the extinction cross sections (e.g., ${\sigma }_{x^{\prime }}=k\text{Im}\left\{{\alpha }_{x^{\prime }{x}^{\prime }}\right\}$ etc.) in the lab and principal frames

$$\begin{gathered} {\sigma }_{x^{\prime }}={\sigma }_x\left({\cos }^2\varphi +{\sin }^2\varphi {\cos }^2\theta \right)+{\sigma }_z{\sin }^2\varphi {\sin }^2\theta \\ {\sigma }_{z^{\prime }}={\sigma }_x{\sin }^2\theta +{\sigma }_z{\cos }^2\theta \end{gathered}$$ (A4)

pertaining to the polarizations in the x′ and z′ directions, respectively. (Note that by our recipe we find ${\sigma }_{y^{\prime }}={\sigma }_{x^{\prime }}(\varphi \to \varphi +\pi /2)\ne 0$ because the electric field component E_y′, which is zero, is factored out in (A2).) To relate the thermal average of these cross sections in the primed frame to the cross sections in the unprimed principal frame, we use the Boltzmann probability distribution (2)–(4), in which the magnetic energy is independent of ϕ. Thus the average over ϕ is random and we obtain (12) and (13)

$$\langle {\sigma }_{x^{\prime }}\rangle =\frac{{\sigma }_x+{\sigma }_z}{2}+\frac{{\sigma }_x-{\sigma }_z}{2}\langle {\cos }^2\theta \rangle$$ (12)

$$\langle {\sigma }_{z^{\prime }}\rangle ={\sigma }_x-\left({\sigma }_x-{\sigma }_z\right)\langle {\cos }^2\theta \rangle$$ (13)

With $E(\theta )/k_{B}T=A{\cos }^2\theta$, the cos²θ average is

$$\langle {\cos }^2\theta \rangle =-\frac{dz}{dA}=\frac{1}{2A}-\frac{e^{-A}}{\sqrt{\pi A}\text{erf}(\sqrt{A})}$$ (A5)

in which the partition function is

$$Z=2\pi {\int }_0^{\pi }{e}^{-A{\cos }^2\theta }\sin \theta d\theta =2\pi \frac{\sqrt{\pi }\text{erf}(\sqrt{A})}{\sqrt{A}}$$ (A6)

For small A (the B-OFF case), we have from the Maclaurin-Taylor series

$$\langle {\cos }^2\theta \rangle =\frac{1}{3}-\frac{4}{15}A+O\left(A^2\right)$$ (A7)

and for large A, we have the asymptotic series,

$$\langle {\cos }^2\theta \rangle =\frac{1}{2A}-\frac{e^{-A}}{\sqrt{\pi A}}+O\left(\frac{e^{-2A}}{A}\right)$$ (A8)

In the limit of small A, the use of (A7) in (12) and (13) yields

$$\langle {\sigma }_{x^{\prime }}\rangle =\frac{2}{3}{\sigma }_x+\frac{1}{3}{\sigma }_z-\frac{2}{45}\left({\sigma }_x-{\sigma }_z\right)A+O\left(A^2\right)$$ (A9)

$${\sigma }_{z^{\prime }}=\frac{2}{3}{\sigma }_x+\frac{1}{3}{\sigma }_z+\frac{4}{45}\left({\sigma }_x-{\sigma }_z\right)A+O\left(A^2\right)$$ (A10)

Therefore, we reach (8) as the leading term in

$$\langle {\sigma }_{x^{\prime }}\rangle =\langle {\sigma }_{z^{\prime }}\rangle =\frac{2}{3}{\sigma }_x+\frac{1}{3}{\sigma }_z+O(A)$$ (A11)

In the limit of large A, the numerics of (A8) tells us

$$\langle {\cos }^2\theta \rangle <\frac{1}{10},\text{if}>A5$$ (A12)

The large A limit (small $\langle {\cos }^2\theta \rangle$) corresponds to the perpendicularity of the crystal hard axes with respect to the magnetic field, as emphasized in the text.

How large is A in our experiment for the B-ON case? Recall from (2) and $E(\theta )/k_{B}T=A{\cos }^2\theta$

$$A=\frac{\left({\chi }_{xx}-{\chi }_{zz}\right)B^{2}V}{2{\mu }_0{k}_{B}T}$$ (A13)

First, note that A is a positive constant for ${\chi }_{zz}<{\chi }_{xx}$ which seems to be the case experimentally. (If it were negative, the perpendicularity of z and z′ would be changed to parallel alignment. The experiments thus have sensitivity to the relative values of magnetic susceptibility.) At room temperature (295K)

$$A\cong 0.1\left({\chi }_{xx}-{\chi }_{zz}\right)B^{2}V$$ (A14)

where B is in Tesla, V is in nm³, and χ is in SI units. If we consider a general scale for the crystal dimension of 1000 nm × 100 nm × 100nm as described in the text, the volume is $1\times {10}^7{\text{nm}}^3$. For $B=0.58\text{T}$ and $\left({\chi }_{xx}-{\chi }_{zz}\right)=0.66\times {10}^{-4}$ and $V={10}^7{\text{nm}}^3$, we obtain $A\simeq 22\text{or}\langle {\cos }^2\theta \rangle \simeq 0.02$.