M. El- Hilo a , J. Al Saei b and R. W Chantrell c

1. Dipolar Interactions in SuperparamagneticNano-Granular Magnetic Systems M. El-Hiloa, J. Al Saeib and R. W Chantrellc aDept. of Physics, College of Science, University of Bahrain, P.O. Box 32038, Sakhir, Bahrain bDept. of Physics, Imperial College London, London, SW7 2AZ c Department of Physics, The University of York,York, YO10 5DD, UK

2. Introduction • Dipolar interaction effects are complex many-body problems, especially when the particles are poly-dispersed, located randomly, and have their easy axes randomly oriented. • Many phenomenological models have been suggested to account for the interaction effects in fine particle systems. • One of these models represents interactions via a mean statistical field which can be added or subtracted from the applied field [1]. • Other models, the so-called T*model, in which a fictitious temperature is introduced in the Langevin response of the superparamagnetic system so that L(H/kT) is replaced by the modified function L(H/k(T+T*)) [2]. [1] S. Shtrikmann and E.P. Wohlfarth, Phys. Lett. 85 A , 457 (1981). [2] P. Allia, M. Coisson, M. Knobel, P. Tiberto and F. Vinai, Phys. Rev. B 60, 12 207 (1999).

3. Techniques such as Monte-Carlo simulations (MC) was found to be very useful in examining the effects of dipolar interaction in random systems. • Using MC simulations, Mao et al[3] have questioned the T* model and concluded that this model does not adequately describe the magnetization of interacting superparamagnetic assemblies. • The MC simulations of Chantrell et al[4] have shown that magnetization curves are depressed with increasing particle concentration which was attributed to the formation of flux closure. • In [4], the simulations of the initial susceptibility also predicted an apparent negative ordering temperature. It has been noted that this kind of order is not indicative of antiferomagnetic ordering. [3] Z. Mao, D. Chen and Z. He, J. Magn. Magn. Mater. 320 (19), 2335 (2008). [4] R.W. Chantrell, N. Walmsely, J. Gore and M. Maylin, Phys. Rev. B 63, 24410 (2000).

4. In a recent MC simulations study [5], the magnetization curves of a nano-granular superparamgentic system are predicted to be always depressed with increasing particle packing densities. • In [5], the distributions of dipolar interaction fields along the x, y and z directions are predicted to be symmetric and Gaussian in form with an average very close to zero. • In [5], the reduction in magnetization is attributed to the non-linear response of the magnetization to the applied field, which weights the negative interaction fields more strongly than the positive fields. • In this study, the effect of a Gaussian distribution of longitudinal dipolar fields on the magnetization response of a nano-granular superparmagnetic system in the limit of weak anisotropy is examined, analytically and numerically. [5] J. Al Saei, M. El-Hilo and R.W. Chantrell, Submitted to J. Appl. Phys. (2011)

5. The Model • For a distribution of longitudinal dipolar fields and in the limit of weak anisotropy, the magnetization of identical superparamagnetic particles is simply described by a Langevin function weighted by the distribution of interaction fields, and is simply expressed as; L(b) is the Langevin function • The distribution of longitudinal dipolar fields is assumed to remain invariant with applied field. In real systems, f(Hi) is expected to becomes narrower at high fields but it wont affect the results since;

6. Analytical Approach • At low fields, where b1, the Langevin function can be expanded and then the magnetic response of the system can be expressed as: • Since f(Hi) is symmetric, then the above equation can be simplified to: • Thus every term in the Langevin expansion is depressed as a result of a symmetric Gaussian distribution of longitudinal dipolar fields. • The reduction in magnetization increases when the standard deviation of dipolar longitudinal fields (Hi) is increased.

7. Thus, this reduction in the magnetization of the system cannot be used to infer that the local interaction field is negative, since both negative and positive interaction fields are equally probable. • When the magnetic response of the system is linear with applied field i.e. M(Ha)(Ha+Hi), then a symmetric distribution of interaction fields can have no effect on the magnetization because the weighted average of the interaction field <Hi>=0. • Thus the reduction in magnetization for the system examined is attributed to the non-linear response of the magnetization to the applied field, which weights the negative interaction fields more strongly than the positive fields.

8. The Initial Susceptibility • In the case where the standard deviation of dipolar fields is small i.e. Hi<<kBT/, the third term in the above equation becomes negligible, hence • Over a certain small range of temperatures, T0(T) varies slowly with temperature, then • The predicted negative average ordering temperature is an apparent measure of negative ordering in the system, since positive and negative fields are equally probable.

9. Numerical Results • The reduction in magnetization is due to the non-linear response of the magnetization to the applied field, which weights the negative interaction fields more strongly than the positive fields. Dm=6 nm The calculated magnetization curves for a symmetric Gaussian distribution function of longitudinal dipolar fields f(Hi) at different values of the standard deviation Hi=300, 600 and 900 Oe. The solid line represents the Langevin function, the case where Hi0.

10. Numerical Results • when Hi=30 Oe, i.e very small compared to the range of applied fieldsfor which the magnetization is still linear, the variation of i-1 vs. T extrapolates to almost zero ordering temperature The inverse of the calculated reduced initial susceptibility using a Gaussian distribution function of longitudinal dipolar fields f(Hi) at different values of the standard deviation Hi=30, 100 and 200 Oe.

11. Conclusions • The results show that interaction effects cannot be deduced from changes in the average magnetization of the system. • For a symmetric distribution of longitudinal dipolar fields, the magnetization of a superparamagnetic system is predicted to be always depressed as the spread of dipolar fields is increased. • The analytical analysis shows that this effect arises from the non-linear response of magnetization, which weights negative interaction fields more strongly than the positive fields. • Thus the idea of describing dipolar interactions in terms of a mean field that could be added or subtracted from the applied field in random systems is not justified.

12. Conclusions • Due to the non-linear effect of magnetization, the temperature variation of the initial susceptibility predicts an ordering temperature that varies with temperature. • The predicted negative ordering temperature cannot be considered as indicative of anti-ferromagnetic order. • Thus the idea of using an effective temperature (T*) to represent interaction effects does not adequately describe the magnetization of interacting superparamagnetic random systems.