# A Search for Magnetized Quark Nuggets (MQNs), a Candidate for Dark Matter, Accumulating in Iron Ore

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{8}kg of taconite ore processed. The results are consistent with MQNs but there could be other, unknown explanations. We propose an experiment and calculations to definitively test the MQN hypothesis for dark matter.

## 1. Introduction

_{o}to 1.65 +/− 0.35 Tera Tesla (TT), which is approximately that at the surface of a neutron or proton. The corresponding computed mass range covers ~10

^{−23}kg to ~10

^{6}kg.

^{14}kg meet the theoretical and observational constraints of Cold Dark Matter (CDM)—unless the material converts normal matter to dark matter. They do not consider the broad mass distribution of MQNs nor their strong magnetic field. The work in [6] shows that >99.9999% of MQN mass falls within the allowed region for baryonic dark matter [4] and builds on [4] to show these two additional characteristics of MQNs are compatible with the conclusion of Jacobs, Starkman, and Lynn [4]. Of course, if MQNs convert normal matter to dark matter, the results reported in this paper cannot be attributed to MQNs. Therefore, the theory and discussions of [4,6] provide the theoretical foundation for stable MQNs consistent with dark matter and readers interested in that question are encouraged to examine that paper. However, unless MQNs are demonstrated to exist and to exist in sufficient abundance to account for dark matter, the discussion is premature. This paper provides additional evidence consistent with MQNs in numbers sufficient to their being dark matter, but the evidence is not dispositive. The proposed experiment described near the end of the paper should conclusively validate or invalidate the MQN hypothesis for dark matter and experimentally resolve the question.

_{3}O

_{4}) veins would be sufficient to overcome gravity for MQNs with mass ≥ approximately 30 mg. If MQNs exist, they may have accumulated in iron ore over geologic time.

^{9}y or 2 Ga. The weakly magnetic iron ore releases the MQNs when ore is crushed in a rod mill consisting of a horizontal, rotating, steel-lined barrel in which thick steel rods continually fall through circulating ore and crush the ore into powder. Any MQNs would initially be picked up by the ferromagnetic rods but would then be released by the violent impact with the barrel liner, so that they are transferred into the ferromagnetic liners. These liners are periodically sent to a furnace to be melted down to make fresh steel. On heating to melting the MQNs would be released at the Curie temperature, fall through insulating fire bricks, and be trapped in the ferromagnetic steel bottom of the furnace. Three furnace bottoms were scanned to look for evidence of anomalous magnetic activity resulting from the presence of MQNs. Simulations are described which support this sequence of events.

_{z}|, defined as the gradient in the absolute value of the vertical component of the magnetic field, on the bottom of these furnaces. We report the first measurements of the gradient in the magnetic field on the bottom of three such furnaces and compare the measurements with those of a control and with those of COMSOL computer simulations of MQNs in the furnace bottoms. The results are consistent with MQNs. However, the results are not dispositive since we cannot rule out the possibility of other, currently undiscovered explanations of the data. An experiment is proposed to definitively test the MQN hypothesis for dark matter.

_{z}| on furnace bottoms indicate there is about one ~0.3 kg MQN in about 4 liners, i.e., in about 3000 kg of liner steel. The corresponding fractional change in mass density is about 0.01%. We found the variation in the chemical composition of rod-mill rods (a steel with composition similar to that of rod-mill liners) accounts for the observed standard deviation in density of 0.18%, which is about 18 times the variation from an MQN in a liner. Consequently, it is not surprising that ~0.3 kg MQNS, if they exist in magnetite ore and recycled rod-mill liners, have not been previously observed in ore and liners.

## 2. Materials and Methods

_{mp}used for calculating the momentum transfer cross section $\pi {R}_{mp}^{2}$.

_{mqn}, velocity v

_{mqn}moving through material of density ρ

_{p}with a drag coefficient K ≈ 1, permeability μ

_{o}of free space, k = 10

^{−18}m

^{3}/kg = parameter inversely proportional to the dark matter density when the temperature of the Universe was ~100 MeV, and simulation parameter B

_{o}proportional to the surface magnetic field at the equator of the MQN, the momentum transfer cross section is

_{o}parameter to B

_{o}= 1.65 +/− 0.35 TT. Therefore, we can compute the stopping of MQNs, within the assumptions of the model explained in Appendix B, Calculation of acceptance fractions.

_{o}gives ~16% uncertainty in stopping power and range in rock for MQN masses of 0.1 kg to 1.0 kg.

^{−24}kg and ~10

^{6}kg. The mass distribution has been computed only to the precision of one order of magnitude in mass. Therefore, we will work with “decadal mass”, which is defined as MQN masses with the same value of the function Integer((log10 (mass)). Each decadal interval is represented by the approximate logarithmic mean of the interval, i.e., √10 kg ≈ 3.2 kg, times the minimum mass in the interval. For example, all MQNs with mass between 0.1 kg and ~1.0 kg are included as 0.32 kg decadal mass. In order to make the text easier to read, the term “decadal” may be dropped from the text. Unless otherwise stated, a mass of 3.2 times any power of 10 refers to the MQN mass in that decadal interval. We report that the 0.32 kg decadal mass interval is the most important one for detecting MQNs from iron ore because lower-mass MQNs do not produce sufficient magnetic field to be readily detectable above background and higher mass MQNs are fewer and are not as efficiently stopped by the rock above magnetite ore.

^{−22}kg/m

^{3}[16,17], the number of MQNs/m

^{3}in interstellar space is given in Table A1 in Appendix B, Calculation of acceptance fractions, for the lower, middle, and upper values of B

_{o}and for MQN decadal-mass from 0.003 kg to 300 kg.

^{−6}, 5 × 10

^{−6}, and 1.2 × 10

^{−6}for MQN mass distributions calculated [6] for B

_{o}= 1.3 TT, 1.65 TT, and 2.0 TT, respectively.

_{o}is represented in the last column of Table A1 as the variation in the Log

_{10}of the MQN number density. The upper and lower uncertainty factors are, respectively, ~4.8 larger and ~1/4th as large as the value at B

_{o}= 1.65 TT. This uncertainty is obviously much larger than the +/−16% uncertainty in the stopping power.

^{3}. The depth at which each MQN in the sample stops is stored. The program for the Monte Carlo calculation is available in the data upload accompanying this paper.

_{IS}of interstellar MQNs of decadal mass m

_{dec}in Table A1 and the acceptance fractions f

_{acc}in Table A2 to calculate the rate $\frac{d{F}_{mqn}}{dt}$ of accumulating interstellar MQNs of decadal mass m

_{dec}per gigaton (Gt = 10

^{12}kg) of taconite ore per gigayear (Ga = 10

^{9}y) of accumulation time as a function of decadal MQN mass for B

_{o}= 1.65 TT. The units are chosen to represent convenient industrial-scale quantities of mined ore and typical geologic accumulation times.

_{acc}is the is the fraction of the MQNs which stop in the rock per 100 m depth interval and the factor $\frac{{f}_{acc}}{100}$ gives the fraction of the MQNs which stop in the rock per meter depth, the factor $\frac{{10}^{12}}{2700}$ gives the number of cubic meters of rock with mass density 2700 kg/m

^{3}in a gigaton of rock, and the factor 3.154 × 10

^{16}is the number of seconds in a Ga. Interstellar MQNs are assumed to have mean speed v

_{mqn}= 2.3 × 10

^{5}m/s. The results are shown in Table A3.

_{dec}= 0.3 kg and MQNs from 0 to 200 m collecting in a 50 m thick taconite deposit at ~150 to 200 m depth, the accumulation rate is ~0.8 kg/Gt/Ga. Since taconite in the Iron Range of Minnesota USA may have been accumulating MQNs for 1.8 Ga, a gigaton of ore would be expected to yield only ~1.4 kg of ~0.3 kg MQNs with upper and lower uncertainty factors (in the right most column of Table A1), respectively, ~5 and 1/4th from the uncertainty in B

_{o}. Even with the processing of taconite consolidating these MQNs, as discussed in subsequent sections, this accumulation rate would make detecting interstellar MQNs very challenging.

^{4+/−1}times the background mass density of the dark matter halo if the dark matter were accumulated without losses for 4.5 Ga, the age of our solar system.

_{dec}= 0.3 kg and MQNs from 0 to 200 m collecting in the 50 m thick taconite deposit at ~150 to 200 m depth, the accumulation rate is ~2.5 kg/Gt/Ga, only ~3 times the rate for interstellar MQNs if the solar MQN number density equals the interstellar number density. Potential enhancement of the solar MQN number density will be considered in Section 4.

^{4}S/m [22] is too low for eddy-current braking to be significant and was not included. The results are presented as the height h

_{eject}of a free-fall drop in meters that will eject MQNs to well beyond the magnetic attraction of a sphere of magnetite, as a function of MQN mass m in kg and magnetite radius r

_{magnetite}in meters:

^{6}S/m +/− 0.14 × 10

^{6}S/m, which is ~ a factor of ~400 larger than the 2.2 × 10

^{4}S/m electrical conductivity of magnetite [22]. The difference is sufficient to make eddy-current forces important for slowing MQN motion through rods. In addition, an MQN only has to be ejected out of the surface of the rod and into the steel of a liner to be transferred—not ejected to a large distance from the rod, as required for ejection from magnetite. Neglecting the additional complication of rod rotation, we find that a 0.32 kg MQN would be ejected from a rod by a drop height h

_{eject_rods}~0.62 m without eddy-current forces and 0.68 m with eddy current forces. Our limited simulations provide an approximate mass dependence of

## 3. Results

#### 3.1. Scanning Furnace Bottoms for MQN Signatures

_{z},

_{1}and B

_{2}spaced 1 cm apart (ruggedized Micro Altitude Heading Reference System (uAHRS) by Inertial Sense Inc., Provo, UT, USA with their EvalTool software 1.8.4 b101_2021_03_30_172532) was used to scan the magnetic field profiles, as illustrated in Figure 2. The magnetic field readings of the B1 and B2 magnetometers were automatically recorded at slightly different times. The data of B2 was time synchronized to the times of B1 by interpolating the B2 data to give the B2 reading at the same time as the nearest B1 reading, thereby time synchronizing the two measurements of magnetic field separated by a constant 1 cm distance. Then, |gradB

_{z}| was calculated from Equation (5).

_{z}data to be accurately recorded at approximately 35,000 imprecisely-defined locations (x,y) for each furnace. We interpreted the data as accurate measurements of gradB

_{z}with approximately random sampling within the scanned area.

^{−1}m

^{2}) × gradB

_{z}(T m

^{−1}) versus x (m), in which the 28.5 (T

^{−1}m

^{2}) multiplier was varied to adjust the apparent amplitude of gradB

_{z}. Results for furnaces 1 and 2 are shown in Figure 3a. The plots indicated the approximate spatial distribution of gradB

_{z}even though the exact position of the probe for each measurement was not accurately determined. The raw data files and the program used to synchronize the B

_{z}(t) measurements, compute gradB

_{z}and plot the data are available for examination in the data archive.

_{i},y

_{i}) in its integral form, i.e., in a set that is ordered from the lowest (i = 1) to highest (i = n) value of |gradB

_{z}|

_{i}, so that y

_{i}= i/n, expressed as %, for x

_{i}= |gradB

_{z}|

_{i}. The corresponding plots are shown as “% of measurements < |gradB

_{z}|” versus |gradB

_{z}|. Therefore, for each data set, y ranges from 0% to 100% as x ranges from the minimum to the maximum value of |gradB

_{z}| for that data set. The resulting plots are shown in Figure 4a.

^{2}for MQN separation d, 1/d

^{2}scales as t, and 1/d scales as t

^{0.5}. Since |gradB

_{z}| = the change in B

_{z}per unit length, |gradB

_{z}| scales as t

^{0.5}for a constant accumulation rate, as it does in Figure 4b, consistent with the hypothesis that MQNs have accumulated in the mill liners from the processed taconite at a nearly constant rate.

_{z}| versus |gradB

_{z}| provided a convenient way to visualize the data. The results of the scans of the control are shown in Figure 5.

_{z}| decreases with increasing temperature from 13 °C to ~38 °C and becomes insensitive to temperature above 110 °C.

_{z}| distribution is insensitive to temperature, as shown in Figure 5, we conclude that the scan at 166 °C adequately represents an MQN-free control to compare with the scans of the furnace bottoms.

_{z}| distributions in 2.54 cm thick A36 steel as a function of MQN mass and areal number density.

#### 3.2. Computer Simulations of MQNs in Steel Plates

_{z}| as the key diagnostic, as defined by Equation (5) and illustrated in Figure 2.

_{z}| distributions. Since the accuracy of the simulations was insufficient to determine the pattern of magnet moments in an array, we explored four patterns, partially derived from laboratory emulations with magnetized spheres, and compared the results with the data shown in Figure 4a. The simulations and analysis are described in detail in Appendix D, Effect of magnetic-moment orientation on |gradB

_{z}| distributions. As shown in Figure A3, |gradB

_{z}| increases rapidly with decreasing d whenever d < 1.9 cm, as the coupling between the MQNs increases. Figure A2a shows that only the pattern from MQNs (1) with mass ~0.32 kg, (2) with separation d between 1.43 cm and 1.5 cm, and (3) with all magnetic moments aligned (Figure A1d) is consistent with the data in Figure 4a for furnaces 1 and 2, which have been in service for 41 years.

_{z}| distributions are only weakly dependent on d for 1.95 cm ≤ d ≤ 3.0 cm, and Figure A2b shows that all four patterns of magnetic moment are similar for d = 2.92 cm. Very similar results were obtained with 2.5 cm ≤ d ≤3.5 cm. Insensitivity to d and to pattern of magnetic moments is expected in this range of d values, in which the |gradB

_{z}| diagnostic measures isolated MQNs since d is much larger than the ~1 cm shielding distance.

_{z}| below the furnace bottoms is caused by uniformly spaced MQNs of 0.32 kg mass.

## 4. Discussion

_{z}| from the furnace scans are consistent with simulations of MQNs from magnetite processing and are inconsistent with the corresponding scans of the steel control at the same temperature. However, the physics of centimeter-scale variations in the magnetic field perpendicular to a thick steel plate has not been well researched; therefore, there could be some unknown phenomenon consistent with the data and not requiring MQNs.

_{z}| with a steel-plate temperature of ~13 °C resembles those of furnace 3 after 9 years of operation. However, the temperature of furnace 3 was inconsistent with 13 °C. In addition, the distribution of |gradB

_{z}| in the control plate is not consistent with those of furnaces 1 and 2 after 41 years of operation. We conclude that the variation in the distribution of |gradB

_{z}| with temperature is not a viable alternative explanation of the results.

_{z}| for furnace 3. There is no way of knowing for certain.

_{z}| diagnostic for MQN mass < 0.1 kg means that our observations should be most sensitive to MQNs with mass between 0.1 kg and 1 kg, i.e., with 0.32 kg decadal mass. The data for furnaces 1 and 2 in Figure 8 are consistent with the simulations for single-mass 0.32 kg MQNs, with all MQN magnetic moments aligned and with a single inter-MQN spacing between 1.43 and 1.50 cm.

_{z}| below the furnace bottoms is caused by MQNs, then these results imply 1490 +/− 70 kg/m

^{2}of MQNs were accumulated during its 41 years of operation and about 200 + 100/−50 kg/y of MQNs accumulating per furnace with an effective area of ~6 + 3/−1.5 m

^{2}for the bottom. The large uncertainty in area and corresponding accumulation rate reflects the uncertainty in the MQN distribution outside the 4.5-m

^{2}scanned area.

_{z}| should be proportional to the square root of furnace age, as shown in Figure 4b.

_{z}| for the four simulated orientations shown in Figure A1, for MQN separation of 2.92 cm, is 0.0021 T/m with a standard deviation of 0.0004 T/m. This is compatible with the measured value of 0.0025 T/m for furnace 3. Hence the data from furnaces 1, 2 and 3 are compatible with a uniform accumulation rate of MQNs.

_{z}| below the furnace bottoms is caused by MQNs. At 4 tons per batch and 3 batches a day for 250 days per year, ME Elecmetal processes approximately 2700 metric tons of steel per furnace per year, including the 30% to 40% new steel to make up for liner wear. Therefore, each furnace would process approximately 1760 tons of liner material to accumulate about 200 kg of MQNs. The corresponding mean MQN concentration in used liner material is approximately 0.12 kg/ton of liner.

_{z}| distributions are caused by MQNs. According to the official history of the Erie Mining Company (LTV site, now Polymet Mining Inc., Hoyt Lakes, MN, USA) [24], approximately 909 million tons of taconite were processed from 1957 to 2001 in about 30 parallel mill lines at the Erie plant. Therefore, each mill line processed about 0.69 million tons of taconite per year.

_{z}| below the furnace bottoms is caused by MQNs, then

^{6}-taconite-tons) =

^{−6}MQN-kg/taconite-ton = 8.5 × 10

^{+3}MQN-kg/ore-gigaton.

^{3}MQN-kg/Gt/Ga.

^{3}MQN-kg/Gt/Ga (for the first 200 m of ore deposit in a typical 200 m deep mine) is ~6000 times the value expected from interstellar dark matter which is 0.81 MQN-kg/Gt/Ga (from columns 1 and 2 of Table A3 for 0.32 kg MQN). It is also ~2000 times the solar mass accumulation rate of ~2.5 MQN-kg/Gt/Ga (from columns 1 and 2 of Table A5 for 0.32 kg MQN) unless the solar accumulation processes described above produce a mass enhancement of a factor of ~2000 for Bo = 1.65 TT. Including the factor of ~5 uncertainty in the number density of 0.32 kg MQNs, still requires a factor of ~400 enhancement from the solar accumulation process. Both of these calculations assume that all dark matter would be composed of MQNs. However, the enhancement factor of the proposed solar-MQN accumulation process is currently unknown. If the enhancement factor is <<2000, then the data from the furnaces cannot be caused by MQNs or the simulation results [16] for the local density of dark matter must be in error. If the enhancement factor is >>2000 and if the data from the furnaces are determined to be caused by MQNs, then the fraction of dark matter attributable to MQNs must be <<1.

^{15}kg ≈ 10

^{−9}Earth masses. The radius of that MQN would be about 2 m and the magnetic field from the MQN would be about 0.0004 of Earth’s current magnetic field. Therefore, the MQN hypothesis is consistent with available information about Earth’s mass profile and surface magnetic field.

^{7}-y lifetime [18,19] of asteroids near Earth and well away from the Jupiter resonances near Mars orbit. Therefore, the theoretical processes for slowing down and transitioning interstellar MQNs from their hyperbolic orbits to elliptical orbits near Earth must have a

^{3.0+/−0.7}mass enhancement for solar MQN ×

^{−22}kg/m

^{3}mass density of interstellar MQNs divided by

^{7}y lifetime of near-Earth bodies in solar orbit ≈

^{−26+/−0.7}kg m

^{−3}y

^{−1}

^{4+/−1}mass enhancement for solar MQN ×

^{−22}kg/m

^{3}mass density of interstellar MQNs divided by

^{9}y age of the solar system ≈

^{−27+/−1}kg m

^{−3}y

^{−1}.

_{z}| data and the COMSOL simulations summarized in Figure 8 are consistent with the MQN hypothesis, but these results are not incontrovertible since we cannot eliminate some unidentified cause of the |gradB

_{z}| distribution that does not involve MQNs. Extracting and isolating ~0.3 kg MQNs with nuclear-density and a ~1 TT magnetic field from rod-mill liners would be definitive. We attempted to isolate and extract MQNs from 52 rod-mill liners, each of which weighed approximately 600 kg, before they reach the furnaces at ME Elecmetal. The liner mass per year processed by the foundry and the inferred MQN-mass accumulation rate in the furnaces let us estimate that the 52 liners should contain approximately 6 MQNs. We suspended each liner at 45 degrees for at least 100 s so an MQN could flow through the steel to the bottom corner of the liner. That corner was then cut off with a plasma cutter while checking that the temperature of the bottom most edge of the corner remained ferromagnetic so any MQN would remain in the sample. The sample masses varied from 95 to 400 g, so the addition of a ~300 g MQN would be evident. None were found. The weights of each sample in air and in water were measured. The variation in the density was only ~1% and is consistent with the variation in the chemical composition from similar material. Our process depended on MQNs flowing through the steel under gravitational and eddy-current forces. If so, MQNs would fall through each vertically suspended liner to its bottom, which was cut off and processed for excess mass and for mass change upon heating to above the Curie temperature. No MQNs were detected. However, we subsequently found that hysteresis losses associated with MQN motion through the high-carbon steel of the liners produced a resistance to motion that was >8 times the gravitational force. MQNs could not flow through the liners to be concentrated and collected, so the null result did not invalidate the MQN hypothesis.

## 5. Conclusions

_{z}| from the three furnaces and a control sample are found to be consistent with uniform accumulation with time of MQNs from magnetite processing. Simulations are able to reproduce the observed effect. The best agreement of data and simulations are consistent with the theoretical mass distribution of MQNs and with the mass dependencies in acceptance factors in rock and transfer efficiencies in ore processing. Consistency does not imply proof, of course, since there may be other, yet unidentified, causes of the same |gradB

_{z}| distributions. If the measured |gradB

_{z}| distributions are caused by MQNs, then the data and simulations indicate ~200 kg of approximately 0.3 kg MQNs accumulate per year per furnace. The corresponding accumulation rate is consistent with ~1 kg of ~0.32 kg MQNs per 1.2 × 10

^{8}kg of taconite ore processed.

_{z}| below the furnace bottoms is caused by MQNs, consistency with dark matter requires MQNs to have accumulated in the solar system to a mass density of ~2 × 10

^{3.0+/−0.7}(i.e., ~400 to ~10,000) times the mass density of interstellar dark matter by a yet-to-be quantified combination of (1) purely gravitational scattering with the planets and (2) first slowing down during passage through a portion of the Sun’s corona and/or chromosphere at grazing incidence and subsequently scattering by planets into near-Earth orbits. Calculating the net density of MQNs accumulating near Earth through these processes is necessary but not sufficient to connect the furnace data with the MQN hypothesis for dark matter.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Method and Validation of COMSOL Simulations

_{o}on the equator. The value of B

_{o}= 1.65 TT +/− 0.35 TT, as inferred in References [9,10,12]. To avoid overstressing the meshing capability of the code, the sub-micron diameter of MQNs was approximated by a 500-micron radius magnet with the same magnetic moment of the MQN. Runs maintaining a constant magnetic moment of the MQN and varying the radius of the MQN between 0.5 microns and 500 microns gave the same force (within ~26%) on the MQN at a fixed position inside the steel plate.

^{18}A/m. The extension ensured that the space immediately around the MQN and extending to where B ~2.4 T at H ~3.5 × 10

^{5}A/m is magnetically saturated. MQNs with the same magnetic moment and sufficiently small to saturate the same volume of space around it will generate comparable magnetic fields in the steel and air.

_{o}, the permeability of free space, for 0.35 T ≤ B ≤ 2.0 TT.

_{net}= F

_{BH}− F

_{µr}

_{1}, in which F

_{BH}is the force computed with the magnetic properties of the steel given by the B-H curve and F

_{µr}

_{1}is the force computed with those magnetic properties given by the relative permeability set to 1 (through the Magnetic Flux Conservation setting).

_{net}was validated by experiments. The calculated force agrees with the observed force attracting a spherical neodymium magnet to a slab of steel within the 10% uncertainty of the sphere’s magnetization. The same procedure gives the force and torque on the magnet as a function of depth of the magnet in a hole in a 2.54 cm thick iron slab. Friction with the walls of the hole prevented quantitative measurements. However, the results agree qualitatively with the observed equilibrium position (center of slab) and orientation (magnetic moment perpendicular to the hole axis).

_{z}divided by the magnetic field that would be there in vacuum |B

_{z}/B

_{vac}| at the sensor located at 7.5 +/− 1 mm from the steel-air surface is

_{mqn}between 0.1 kg and 10 kg, respectively.

_{z}at the sensor is

_{mqn}between 0.1 kg and 10 kg, respectively. For m

_{mqn}< 0.0001 kg, Earth’s magnetic field and variations in the magnetic domains in the steel preclude observing B

_{z}from the MQN. The spatial variation in both of these background fields occur on a larger scale length than the B

_{z}from the MQN, so |gradB

_{z}| provides a more reliable diagnostic for MQNs. The COMSOL simulations give

_{mqn}between 0.1 kg and 10 kg, respectively. We find that a two-magnetometer sensor (with magnetometers separated by ~1.0 cm) and positioned on a rolling platform (to maintain a steel-to-magnetometer distance of 7.5 +/− 1 mm) to measure the gradient |gradB

_{z}| can reliably detect MQNs of mass > 0.1 kg.

## Appendix B. Calculation of Acceptance Fractions

_{mqn}and mass m

_{mqn}, velocity v

_{mqn}moving through material of density ρ

_{p}with a drag coefficient K ≈ 1, and surface magnetic field at the equator B

_{S}, the geometric radius of the MQN is

_{o}is the permeability of free space. The cross section for momentum transfer is given by the area of the MQN magnetopause and the stopping power is given by

_{S}and implicitly on the unknown MQN mass density ρ

_{mqn}. However, the Monte Carlo simulation [6] that produced the MQN mass distribution as a function of the magnetic field parameter B

_{o}showed that B

_{S}, ρ

_{mqn}, and B

_{o}are related by

_{DM_T=100MeV}= 1.6 × 10

^{8}kg/m

^{3}= the density of dark matter at time ~65 μs, when the temperature of the Universe was ~100 MeV, k = 10

^{−18}.

_{mp}that is independent of ρ

_{mqn}:

_{o}parameter to B

_{o}= 1.65 +/− 0.35 TT. The B

_{o}parameter characterizes the MQN mass distribution which was computed [6] only by decadal mass interval, defined as all MQN masses with the same value of the function Integer((log

_{10}(mass)). Each decadal interval is represented by the logarithmic mean of the interval, i.e., $\sqrt{10}$ times the minimum mass in the interval. Assuming the interstellar MQN mass density for all MQN masses is the mass density of the local dark matter halo = 4 × 10

^{−22}kg/m

^{3}[16,17], the number of MQNs m

^{−3}in interstellar space is given in Table A1 for the lower, middle, and upper values of B

_{o}and for the 0.3 kg decadal mass interval of most interest for this paper. Table A1 provides a wider decadal-mass range (0.003 kg to 300 kg) to help assess other iron ore deposits.

**Table A1.**Calculated MQN number density in interstellar space as a function of B

_{o}parameter and MQN decadal mass interval assuming the total MQN mass density equals the local mass density of the dark matter halo.

Approximate Decadal Mass m_{dec} (kg) | Interstellar MQNs/m^{3} in Decadal Mass Interval for B _{o} = 1.3 TT | Interstellar MQNs/m^{3} in Decadal Mass Interval for B _{o} = 1.65 TT | Interstellar MQNs/m^{3} in Decadal Mass Interval for B _{o} = 2.0 TT | Mean and Variation in Log_{10} of Number Density in Units of MQNs/m^{3} |
---|---|---|---|---|

0.003 | 5.0 × 10^{−26} | 1.1 × 10^{−26} | 2.8 × 10^{−27} | −25.97 +0.66/−0.58 |

0.03 | 1.9 × 10^{−26} | 5.2 × 10^{−27} | 1.5 × 10^{−27} | −26.28 +0.57/−0.55 |

0.3 | 3.0 × 10^{−26} | 6.3 × 10^{−27} | 1.5 × 10^{−27} | −26.20 +0.68/−0.61 |

3 | 2.0 × 10^{−26} | 5.0 × 10^{−27} | 1.4 × 10^{−27} | −26.30 +0.59/−0.57 |

30 | 3.6 × 10^{−27} | 1.3 × 10^{−27} | 3.6 × 10^{−28} | −26.87 +0.43/−0.57 |

300 | 1.6 × 10^{−27} | 7.6 × 10^{−28} | 1.7 × 10^{−28} | −27.12 +0.33/−0.65 |

_{o}is represented in the last column of Table A1 as the variation in the Log

_{10}of the MQN number density. The uncertainties are approximately a factor of ~4.8 larger or ~1/4th as large as the value at B

_{o}= 1.65 TT.

_{o}= 1.65 TT stopped in 100 m increments are shown as a function of MQN decadal mass in Table A2.

**Table A2.**Computed acceptance fractions of interstellar MQNs stopping in rock of density 2700 kg/m

^{3}for B

_{o}= 1.65 TT as a function of MQN decadal mass and for depth intervals of 100 m.

Approximate Decadal Mass m_{dec} (kg) | Depth Interval 0 to 100 m | Depth Interval 100 to 200 m | Depth Interval 200 to 300 m | Depth Interval 300 to 400 m | Depth Interval 400 to 500 m |
---|---|---|---|---|---|

0.003 | 0.27392 | 0.4234 | 0.22934 | 0.0294158 | 0.00041393 |

0.03 | 0.051 | 0.0927 | 0.152 | 0.189 | 0.208 |

0.3 | 0.00999 | 0.0221 | 0.031 | 0.0421 | 0.0471 |

3 | 0.00227 | 0.00341 | 0.00656 | 0.00874 | 0.01 |

30 | 0.000433 | 0.000664 | 0.0016 | 0.00137 | 0.00196 |

_{o}= 1.65 TT are shown in Table A3.

**Table A3.**Estimated $\frac{d{F}_{mqn}}{dt}=$ rate of mass accumulation of interstellar MQNs in a taconite or magnetite mine for five decadal mass intervals in units of kg mass of MQNs per gigaton of rock per Ga accumulation time, assuming the MQN speed v

_{mqn}= 230 km/s for all MQNs. The fraction that is stopped in the air is included in the 0 to 100 m interval since these would drift to the surface and fall into first magnetite layer.

Approximate Decadal Mass m_{dec} (kg) | Depth Interval 0 to 100 m | Depth Interval 100 to 200 m | Depth Interval 200 to 300 m | Depth Interval 300 to 400 m | Depth Interval 400 to 500 m |
---|---|---|---|---|---|

0.003 | 1.0 | 0 | 0 | 0 | 0 |

0.03 | 0.979 | 0.021 | 0 | 0 | 0 |

0.3 | 0.33 | 0.454 | 0.157 | 0.0337 | 0.0008 |

3 | 0.093 | 0.100 | 0.200 | 0.183 | 0.106 |

30 | 0.016 | 0.025 | 0.040 | 0.054 | 0.051 |

**Table A4.**Computed acceptance fractions of solar MQNs stopping in rock of density 2700 kg/m

^{3}for B

_{o}= 1.65 TT as a function of MQN decadal mass and for depth intervals of 100 m.

Approximate Decadal Mass m_{dec} (kg) | Depth Interval 0 to 100 m | Depth Interval 100 to 200 m | Depth Interval 200 to 300 m | Depth Interval 300 to 400 m | Depth Interval 400 to 500 m |
---|---|---|---|---|---|

0.003 | 0.999 | 0.00 | 0.00 | 0.00 | 0.00 |

0.03 | 0.979 | 0.02 | 0.00 | 0.00 | 0.00 |

0.3 | 0.331 | 0.45 | 0.16 | 0.03 | 0.00 |

3 | 0.096 | 0.10 | 0.20 | 0.18 | 0.00 |

30 | 0.016 | 0.025 | 0.04 | 0.054 | 0.052 |

300 | 0.003 | 0.006 | 0.009 | 0.011 | 0.011 |

**Table A5.**Estimated $\frac{d{F}_{mqn}}{dt}=$ rate of mass accumulation of solar MQNs in a taconite or magnetite mine for five decadal mass intervals in units of kg of decadal mass of MQNs per gigaton of rock per Ga accumulation time, assuming the MQN number density in the solar system is the same as it is in interstellar space and the speed v

_{mqn}= 30 km/s for all MQNs.

Approximate Decadal Mass m_{dec} (kg) | Depth Interval 0 to 100 m | Depth Interval 100 to 200 m | Depth Interval 200 to 300 m | Depth Interval 300 to 400 m | Depth Interval 400 to 500 m |
---|---|---|---|---|---|

0.003 | 0.06 | 0.00 | 0.00 | 0.00 | 0.00 |

0.03 | 0.27 | 0.01 | 0.00 | 0.00 | 0.00 |

0.3 | 1.1 | 1.5 | 0.52 | 0.11 | 0.00 |

3 | 2.5 | 2.6 | 5.3 | 4.81 | 0.00 |

30 | 1.1 | 30. | 24. | 4.7 | 3.6 |

300 | 1.1 | 2.3 | 3.4 | 4.5 | 4.5 |

## Appendix C. Holding Force of Steel and Magnetite on Embedded MQNs

_{mqn}= 0, increases with distance from the slab center, and increases with increasing MQN mass. The distance between the steel–air interface (at z

_{mqn}= −12.7 mm) and the equilibrium position (under gravitational and magnetic forces) increases with increasing mass. The ratio of magnetic force to gravitational force near the air-steel interface also increases with increasing MQN mass. Consequently, steel components of magnetite processing equipment may accumulate MQNs through static forces for sufficiently large MQN mass. Simulations show the magnetic attraction of MQNs to magnetite is greater than the gravitational attraction of the MQN to the Earth if the MQN mass is ≥approximately 30 mg.

^{6}S/m +/− 15%) and the hysteresis force (defined by the hysteresis losses in the ferromagnetic material per unit distance the MQN moves through the material) transfer from steel rods to steel liners upon impact. Since each mill has its own and often proprietary process, sufficient information to calculate the probability of accumulation of MQNs in a mill’s liners is not usually available. In addition, the COMSOL computer code understandably has some difficulty accurately calculating forces on MQNs.

^{−3}+/− 0.3%. The results implied that either the furnace data are caused by some phenomena not related to MQNs or the great majority of MQNs are transferred to the liners.

## Appendix D. Effect of Magnetic-Moment Orientation on |gradB_{z}| Distributions

_{z}| distributions that would be measured if the MQNs had each of four patterns of magnetic moments within the x-y plane at the equilibrium distance z from the bottom surface of a 2.54 cm thick steel plate. The four patterns are shown in Figure A1. The resulting |gradB

_{z}| distributions were compared to the data shown in Figure 4a.

**Figure A1.**Patterns of the x-y orientation (shown as arrows) of magnetic moments examined with simulations: (

**a**) adjacent rows have opposite magnetic moments, (

**b**) clusters of three MQNs in a cell, (

**c**) emulation in non-magnetic lattice of 14 neodymium magnetic spheres, and (

**d**) all magnetic moments are in the same (+x) direction. (

**e**) shows the pattern of the 14 MQNs (dots) with the area in the rectangle over which values for B

_{1}are computed in COMSOL simulations discussed below. B

_{2}is recorded at 1 cm to the right of the position of B

_{1}. MQNs outside the scanned area contribute to the field as next nearest neighbors. The results approximate the scan of a much larger area with the same pattern.

_{z}| distributions from simulations of all four configurations with separations d = 2.92 cm and 1.46 cm represent separations somewhat more than the 1.0 cm shielding distance described above. The values of B1 were computed over the area shown in Figure A1e; the value of B2 was extracted from the position 1 cm to the right of the position of each B1 data point. The |gradB

_{z}| distributions were calculated from Equation (5). The results are shown in Figure A2.

**Figure A2.**Representative |gradB

_{z}| distributions from the 9-year old furnace 3 (solid blue), the first scan of a 41-year-old furnace 2 (solid red), control at 165 °C (black), and the |gradB

_{z}| distributions from COMSOL simulations of the four patterns of magnetic moments shown in Figure A1: adjacent rows have opposite magnetic moments (light-blue dashes), clusters of three MQNs in a cell (green dashes), emulation in non-magnetic lattice (orange dashes), and all magnetic moments are in the same (+x) direction (red dashes). (

**a**,

**b**), respectively, show the results with d = 1.46 cm and d = 2.92 cm. The significance the error bars is described in the caption to Figure 4.

_{z}| greater than some minimum value and consistent with the data from furnace 2. The emulation curve (orange dashes) intercepts the x-axis in the correct place but the slope is inconsistent with the furnace 2 data. Changing the MQN mass from 0.32 kg to 0.1 kg gives the correct slope but moves the intercept to |gradB

_{z}| = 0.0012, which is inconsistent with the furnace-1 and furnace-2 data. Only the distribution from “all in +x direction” has a shape consistent with the data from the furnace-scans shown in Figure 4a and illustrated by the data from furnace 2 (solid red line) in Figure A2a. The results of these simulations and of the emulation with neodymium magnets in a steel plate, as discussed above, support the “all in the +x direction” as the best approximation to the pattern of magnetic moments inside the steel furnace bottom. Simulations of additional values of d with the “all in +x direction” pattern are shown in Figure A3.

**Figure A3.**Comparison of data from furnace scans and COMSOL simulations of those scans as a function of MQN spacing d for MQN mass of 0.32 kg and with all MQNs’ magnetic moments aligned in the +x direction. Solid lines show data: control at 165 °C (black), furnace 3 after 9 years of operation (blue), furnace 1 after 41 years of operation (orange), furnace 2 after 41 years of operation (dark red), repeat scan of furnace 2 showing variation in measurement technique (bright red). Dashed lines show simulation results as a function of MQN spacing d: with d = 3.0 cm (light blue), d = 2.5 cm (gray), d = 1.95 cm (dark blue), d = 1.9 cm (purple), d = 1.5 cm (orange), d = 1.46 cm (dark red), and d = 1.43 cm (bright red with small dashes). The significance the error bars is described in the caption to Figure 4.

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**Figure 1.**Photos illustrating the challenges and method of recording the magnetic field data on furnaces surfaces in the industrial environment. (

**a**) View of ~3.8 m diameter furnace 1 in operation, (

**b**) furnace 1 bottom with projected pattern to guide manual scanning, and (

**c**) first author (JPV) in the pit scanning the B-field of the furnace shell in preparation for scanning the furnace bottom.

**Figure 2.**The probe with two magnetometers B

_{1}and B

_{2}is shown passing across the region of increased magnetic field near a simulated MQN inside a 2.54 cm thick steel plate.

**Figure 3.**(

**a**) Raster plots of y(m) + 28.5 (T

^{−1}m

^{2}) × gradB

_{z}(T m

^{−1}) versus x(m) for furnace 1 (

**left**) and furnace 2 (

**right**) illustrate the approximate spatial distributions of gradB

_{z}and (

**b**) histograms of the distribution of |gradB

_{z}|, the absolute value of gradB

_{z}, color coded as furnace 1 (orange), furnace 2 first scan (dark red), furnace 2 s scan (bright red), furnace 3 (blue), and control at 166 °C (black).

**Figure 4.**(

**a**) Cumulative distributions of |gradB

_{z}| measurements on the three furnaces and the control: furnace 1 (orange), furnace 2 first scan (dark red), furnace 2 s scan (bright red), furnace 3 (blue), and control at 166 °C (black). (

**b**) Plot of median of the |gradB

_{z}| distributions versus the square root of the age of the furnace; the linear relationship is consistent with a constant accumulation rate of whatever is causing the |gradB

_{z}| distributions. Error bars represent +/− 2σ calculated uncertainty. The standard deviation σ

_{x}in the value of |gradB

_{z}| (i.e., ×coordinate) at the median (50%) for the three data sets from 41-year-old furnaces 1 and 2 in (

**a**) was calculated and assumed to be representative of the corresponding uncertainty for the 9-year-old furnace 3 in (

**a**). The standard deviation σ

_{y}≈ 0.3% in the median position (i.e., y coordinate) in (

**a**) was calculated from counting statistics for a binomial distribution with ~35,000 data points, which is typical for these curves, and for the distributions for furnaces 1 and 2 shown in Figure 3b. The error bar in the y coordinate is ~one third of the line width, so it does display. However, the paucity of runs and the unquantifiable uncertainties in the data make σ a lower limit to the uncertainty as a measure of confidence level. We invite the reader to interpret them as only qualitatively indicative of the real uncertainty and as consistent with the difference between the median of the data from the 41-year-old furnaces 1 and 2 and the median of the data from the 9-year-old furnace 3. The error bars in (

**b**) were derived from those in (

**a**) and from the +/− 0.5 y uncertainty in the operational age of the respective furnaces.

**Figure 5.**Cumulative distributions of |gradB

_{z}| measurements on the control plate for plate temperatures of 13 °C (light blue), 13 °C (blue), 38 °C (orange), 110 °C (red), and 166 °C (black).

**Figure 6.**Upward magnetic force on an MQN inside a 25.4 mm thick steel plate as a function of distance from the plate center (z

_{mqn}= 0) for MQN masses between 0.1 kg (light green), 0.32 kg (purple), 1.0 kg (yellow), 10 kg (red), and 100 kg (blue). The circles show the position at which the downward gravitational force is balanced by the upward magnetic force. The air–steel boundary is at z

_{mqn}= −12.7 mm.

**Figure 7.**COMSOL simulation of the shielding effect of steel on the force between two 1 kg MQNs separated by distance d with their magnetic moments aligned. The attractive force F

_{steel}in steel (blue) and the attractive force F

_{vacuum}in vacuum (red) are shown on the left axis. The shielding coefficient (black) = 1 − F

_{steel}/F

_{vacuum}and is shown on the right axis.

**Figure 8.**Cumulative distributions of |gradB

_{z}| measurements on the three furnaces and the control are shown: furnace 1 (orange), furnace 2 first scan (dark red), furnace 2 s scan (bright red), furnace 3 (blue), and control at 166 °C (black). Best-fit COMSOL simulation results are shown: (1) with MQN spacing d = 2.92 cm, MQN mass of 0.32 kg, and with MQNs’ adjacent rows with opposite magnetic moments as shown in Figure A1a (dashed blue), and (2) with MQN spacing d = 1.46 cm, MQN mass of 0.32 kg, and with all MQN magnetic moments aligned, as shown in Figure A1d (dashed red). The different configurations of magnetic moments are tentatively attributed to the weak coupling for d = 2.92 cm and strong coupling for d = 1.46 cm since the shielding distance is ~1 cm, as shown in Figure 7. The significance the error bars is described in the caption to Figure 4.

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**MDPI and ACS Style**

VanDevender, J.P.; Sloan, T.; Glissman, M.
A Search for Magnetized Quark Nuggets (MQNs), a Candidate for Dark Matter, Accumulating in Iron Ore. *Universe* **2024**, *10*, 27.
https://doi.org/10.3390/universe10010027

**AMA Style**

VanDevender JP, Sloan T, Glissman M.
A Search for Magnetized Quark Nuggets (MQNs), a Candidate for Dark Matter, Accumulating in Iron Ore. *Universe*. 2024; 10(1):27.
https://doi.org/10.3390/universe10010027

**Chicago/Turabian Style**

VanDevender, J. Pace, T. Sloan, and Michael Glissman.
2024. "A Search for Magnetized Quark Nuggets (MQNs), a Candidate for Dark Matter, Accumulating in Iron Ore" *Universe* 10, no. 1: 27.
https://doi.org/10.3390/universe10010027