Sample of Dual Biological Substances
and Sensor Coil under Very High Fields

Simulation of NMR Spectrometers

©2004 Magneticasoft

The analysis of chemical and biological substances is now widely practiced with Nuclear Magnetic Resonance spectrometers (in short NMR spectro). This implies the action of a very high magnetic field on a tiny substance sample. The NMR spectro signal is picked up by a miniature coil around the sample. The field range is from 5 teslas for general-use instruments to more than 20 teslas for the most advanced machines. Biological tests and medical research rely heavily on NMR spectros (particularly in cancer research).

Increasing field intensity improves resolution, but also increases the deleterious signals due to the material composition of the coil winding. The difficulty of determining a precise analysis for the various magnetic signals results from the fact that theses signals are very small compared to the polarizing field  (relatively less than (10^(-6)), down to (10^(-10))). Magnetica, with its basic high-numerical precision, is able to deal easily with this situation.

This demonstration is based on a 10 teslas machine with a tiny sample made of two materials representating a biological subtance (50 micrometers in diameter and length) inside a coil of 4 turns and a diameter of 100 micrometers). The demonstration shows results for coils made of either gold (diamagnetic) or a copper aluminum composite (quasi amagnetic). In order  to facilitate the reading of the results, units are set in micrometers for length and nanoteslas for magnetic field and magnetization.

The proposed symbolic names for these magnetic systems are spectro1 for the calculation of the biological sample without the coil, spectro2g for the gold coil, and spectro2c for the composite coil.

Magnetic System Definition

MagneticaUserUnits[LengthUnit -> Micro Meter, 
        FieldUnit -> Nano Tesla];

This shows the units for the magnetic currents.

MagneticaCurrentUnits[]

spectro1 = 
        AXSMagneticSystem[{AXSLargeMagneticSource[10^10], 
            AXSSoftMagneticComponent[WaterSea, 0, 25, 30, 0, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[SiliconResin, 25.1, 50, 50, 0, 
              LinearPermeability -> True]}, MeshingLevel -> 1, 
          SystemLabel -> "simulation biological sample (Sea \n\
water/SiliconResin)"];

spectro2g = 
        AXSMagneticSystem[{AXSLargeMagneticSource[10^10], 
            AXSSoftMagneticComponent[Gold, 100, 102, 10, -30, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[Gold, 100, 102, 10, -10, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[Gold, 100, 102, 10, 10, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[Gold, 100, 102, 10, 30, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[WaterSea, 0, 25, 30, 0, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[SiliconResin, 25.1, 50, 50, 0, 
              LinearPermeability -> True]}, MeshingLevel -> 1, 
          SystemLabel -> "simulation biological sample and spectro \n\
coil (Gold)"];

spectro2c = 
        AXSMagneticSystem[{AXSLargeMagneticSource[10^10], 
            AXSSoftMagneticComponent[Composit1, 100, 102, 10, -30, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[Composit1, 100, 102, 10, -10, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[Composit1, 100, 102, 10, 10, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[Composit1, 100, 102, 10, 30, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[WaterSea, 0, 25, 30, 0, 
              LinearPermeability -> True], 
            AXSSoftMagneticComponent[SiliconResin, 25.1, 50, 50, 0, 
              LinearPermeability -> True]}, MeshingLevel -> 1, 
          SystemLabel -> "simulation biological sample and spectro \n\
coil (Cu/Al)"];

Drawing the Magnetic Systems

DrawSystem[spectro2g];

DrawSystem2D[spectro2g];

Material Properties

The function Properties shows the physical and magnetic properties of the material, and the function MaterialMagnetizationCurves displays the magnetic properties as a function of the exciting magnetic field (called the magnetizing field to express its action on the matter).

Magnetization is a local property of the matter that arises from the tiny dipoles of the atomic substances. Volume magnetization, which represents the density of cooperative dipoles, has the same dimensional unit as the magnetic field. Therefore, it can be expressed in any relevant unit: gauss, telsa, ampere per meter, and so on.

The substances involved in this demonstration are common materials (not ferromagnetic). They all shows a small linear response to an exciting magnetic field; their relative susceptibility lies in the range of (10^(-6)).

Properties[Gold]

Gold is diamagnetic (its susceptibility is negative), but under a magnetizing field of 10 teslas, the magnetization M reaches -27.4 microteslas.

MaterialMagnetizationCurves[Gold, {M, Susceptibility}, 
      MagnetizingFieldRange -> {0, 10^10}]

The following composite is made of copper (diamagnetic) and aluminum (paramagnetic). For a given proportion of these two metals, the resulting composite is practically amagnetic.

Properties[Composit1]

The composite is slightly paramagnetic (its susceptibility is positive), and now under a magnetizing field of 10 teslas, the magnetization M is only 0.292 microtesla, which is a decrease of 100 times in comparison with gold.

MaterialMagnetizationCurves[Composit1, M, 
      MagnetizingFieldRange -> {0, 10^10}]

The next two materials simulate the substances of a biological cell. Both are diamagnetic and the difference in their susceptibility is small. Nevertheless, this difference is what must be displayed.

Properties[WaterSea]

MaterialMagnetizationCurves[WaterSea, M, 
      MagnetizingFieldRange -> {0, 10^10}]

Properties[SiliconResin]

MaterialMagnetizationCurves[SiliconResin, M, 
      MagnetizingFieldRange -> {0, 10^10}]

Punctual Magnetic Field Calculation

The function FieldPointOrigin shows the complete magnetic field contributions of the various substances, including the coil matter, at the center of the sample.

FieldPointOrigin[spectro2g, ComponentsList -> All]

The contributions of the gold from the four turns are, respectively, 7, 44, 44, and 7 nanoteslas.

FieldPointOrigin[spectro2c, ComponentsList -> All]

The contributions of the Cu/Al composite from the four turns are, respectively, 0.08, 0.47, 0.47, and 0.08 nanotesla. The values are much smaller than for the gold windings.

Functional Magnetization

The functional magnetization is the local value of the magnetization inside the biological substances.

The results can be expressed in either tabular form with B (flux density), M (magnetization), and H (magnetizing field) for each cell of the component or by curves as a function of cell position.

MagnetizationTable[spectro2g, CellsList  ->  {{1, 5}}]

These are the magnetization states of the first gold turn.

MagnetizationTable[spectro2c, CellsList  ->  {{16, 20}}]

These are the magnetization states of the inner substance (simulated by sea water and silicon resin) of the biological cell.

MagnetizationCurves[spectro2c, {Mz}, PlotRows -> {1}, 
      ComponentsList -> {6, 7}]

This curve shows the variation of the magnetization at the interface of the two substances of the biological cell.

Field Map Graphics

Field of the Biological Materials Taking into Account the Sensor Coil 
Material: The Case of Gold Winding

This defines a new function (using Mathematica) that gives the field generated by the gold material of the winding. The currently defined units are nanoteslas and micrometers.

bz2g[r_, 
          z_] := N[
            FieldPoint[spectro2g, {r, z}] - 10^10][[2]];

Here is a field plot along an axial line at r = 45 (micrometers).

Plot[bz2g[45, z], {z, -50, 65}];

This graph shows the sensor signal in the case of the gold winding. Notice the perturbation due to the winding material.

Field of the Biological Materials and Taking into Account the Sensor Coil 
Material: The Case of Cu/Al Composite Winding

This defines a new function (using Mathematica) that gives the field generated by the composite material of the winding. The currently defined units are nanoteslas and micrometers.

bz2c[r_, 
          z_] := N[
            FieldPoint[spectro2c, {r, z}] - 10^10][[2]];

Here is a field plot along an axial line at r = 45  (micrometers).

Plot[bz2c[45, z], {z, -50, 65}];

This graph shows the sensor signal in the case of the Cu/Al composite. No perturbation due to the winding is detected.