Hi, I read the answer to a similar question as yours:

With respect @Palash Gangopadhyay says : "Fitting magnetization data with Langevin function is the way to go; there are variations based on particles systems and target properties, but the basic Langevin function of L(x) = 1/tanh(x) – 1/x for M(H) = N µ L(µH / kT) works well where L is the Langevin function. Before fitting check on the dimensions of all parameter to make sure you are using right values of the constants. You can also normalize the magnetization curve before fitting. Check whether Ms(+B) = Ms(-B), if not there could be several of many things going on -- there are many known artifacts in vsm measurements that can result in the mismatch, I will not go into the details of it, you can check standard literature for that. Another common reason that could lead to this difference is "too much material" used for measurements. In that case you can use a constant C as in L(x) = 1/tanh(x) – 1/x +C, C is a parameter that you fit. If you have done your measurements in solution, then use different concentrations to check on C, C increases with increasing concentration in the interaction regime. If Ms(+B) = Ms(-B), then you can fit the data with Langevin equation alone. Any of Excel, Origin, MATLAB, Mathematica, any software that can handle user input in fitting equation will do. Some of them though does not handle Tanh and Coth functions well, if the software crashes out, you can try removing B=0 data point and refit, and use may be only +B or -B side of the data. Remember only the saturation magnetization Ms = N can be estimated in the lab, but not the number of particles N and the mean moment separately. Ideally, M(infinite) = N µ, which we assume as the saturation field at high magnetic field. Use any of the least-squares criterion for (non-linear) best-fit, if you are using MATLAB or Mathematica, get a crude fit done in excel first and then use that crude fit parameters in there to get fine fitting. Find µ0, calculate sigma for Ms fit value, get sigma~0, with as best fit you can get. Once you find µ0, assume spherical magnetic core with diameter D. Using the spherical volume formula, V = pi D^3 /6, and the known magnetic density of the particle system you are using, relate core diameter D to magnetic moment µ. You can calculate size distribution : sigma (ln D) = sigma (ln µ) / 3.

Coming back to your original question / comment, you will always (most likely) find the magnetization diameter smaller than TEM size. This is because there is always a amorphous magnetic dead layer on top of the crystalline core that does not show up (or reduces magnetization) in magnetization measurements. This dead layer forms most likely due to surface interaction, nature of synthesis, formation of the particles and the material itself (almost all particles, not just magnetic, will have this dead layer). There are also occurrences when there is different orientation of magnetic moments on the top layers than its core along with outer dead layer -- these are complications that can be dealt with case to case using various langevin function variations. But as far as size determination is concerned, magnetic core size determined using vsm (or for that matter squid) is always less than that measured using TEM."

But I suggest using DLS to measure the hydrodynamic diameter of your particles, if they are bar and need length and diameter, TEM can be used. If you use DLS, I am just studying on how to measure the diameter of a rod-shaped particle with DLS I will be happy to share my findings with you.

Hi, I read the answer to a similar question as yours:

With respect @Palash Gangopadhyay says : "Fitting magnetization data with Langevin function is the way to go; there are variations based on particles systems and target properties, but the basic Langevin function of L(x) = 1/tanh(x) – 1/x for M(H) = N µ L(µH / kT) works well where L is the Langevin function. Before fitting check on the dimensions of all parameter to make sure you are using right values of the constants. You can also normalize the magnetization curve before fitting. Check whether Ms(+B) = Ms(-B), if not there could be several of many things going on -- there are many known artifacts in vsm measurements that can result in the mismatch, I will not go into the details of it, you can check standard literature for that. Another common reason that could lead to this difference is "too much material" used for measurements. In that case you can use a constant C as in L(x) = 1/tanh(x) – 1/x +C, C is a parameter that you fit. If you have done your measurements in solution, then use different concentrations to check on C, C increases with increasing concentration in the interaction regime. If Ms(+B) = Ms(-B), then you can fit the data with Langevin equation alone. Any of Excel, Origin, MATLAB, Mathematica, any software that can handle user input in fitting equation will do. Some of them though does not handle Tanh and Coth functions well, if the software crashes out, you can try removing B=0 data point and refit, and use may be only +B or -B side of the data. Remember only the saturation magnetization Ms = N can be estimated in the lab, but not the number of particles N and the mean moment separately. Ideally, M(infinite) = N µ, which we assume as the saturation field at high magnetic field. Use any of the least-squares criterion for (non-linear) best-fit, if you are using MATLAB or Mathematica, get a crude fit done in excel first and then use that crude fit parameters in there to get fine fitting. Find µ0, calculate sigma for Ms fit value, get sigma~0, with as best fit you can get. Once you find µ0, assume spherical magnetic core with diameter D. Using the spherical volume formula, V = pi D^3 /6, and the known magnetic density of the particle system you are using, relate core diameter D to magnetic moment µ. You can calculate size distribution : sigma (ln D) = sigma (ln µ) / 3.

Coming back to your original question / comment, you will always (most likely) find the magnetization diameter smaller than TEM size. This is because there is always a amorphous magnetic dead layer on top of the crystalline core that does not show up (or reduces magnetization) in magnetization measurements. This dead layer forms most likely due to surface interaction, nature of synthesis, formation of the particles and the material itself (almost all particles, not just magnetic, will have this dead layer). There are also occurrences when there is different orientation of magnetic moments on the top layers than its core along with outer dead layer -- these are complications that can be dealt with case to case using various langevin function variations. But as far as size determination is concerned, magnetic core size determined using vsm (or for that matter squid) is always less than that measured using TEM."

But I suggest using DLS to measure the hydrodynamic diameter of your particles, if they are bar and need length and diameter, TEM can be used. If you use DLS, I am just studying on how to measure the diameter of a rod-shaped particle with DLS I will be happy to share my findings with you.

Dear Zaid H. Mahmoud ,

If you have nanoparticles that are superparamagnetic at RT, you can perform VSM measurements of the magnetisation as a function of temperature, the so-called field-cooled (FC) and zero-field-cooled (ZFC) curves, to estimate the particle size. Assuming a simple model with uniaxial magnetic anisotropy, the magnetic moment of each nanoparticle has only two stable orientations antiparallel to each other, which are separated by an energy barrier given by K x V, with K being the magnetic anisotropy and V being the volume of the nanoparticle. The thermal energy is given by k_B x T, with k_B being the Boltzmann constant and T being the temperature, and the Blocking temperature is the transition between the superparamagnetic state (above) and the blocked one (below), i.e. the temperature at which those energies are almost equal (there is a factor that depends on the measurement time). However, this is a simplified model. I suggest reading this article and the references therein:

Article Beyond the blocking model to fit nanoparticle ZFC/FC magneti...

In addition, you can try estimate particle size using saturation magnetization. For example see Article Spherical magnetic nanoparticles fabricated by laser target ...

Eq.3 and fig.13