Abstract

A temperature-dependent magneto-optical study of three different temperature-sensitive fluids was carried out. The changes in transmitted intensity as a function of applied magnetic field and temperature are recorded. The study provides an alternative technique for determining the Curie temperature of such fluids, and it is also feasible for the monitoring of temperature changes of such fluids optically within a limited time span.

© 2004 Optical Society of America

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References

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  1. R. E. Rosensweig, Ferrohydrodynamics (Cambridge U. Press, Cambridge, 1985), and references therein.
  2. C. W. Heaps, “Optical and magnetic properties of a magnetic suspension,” Phys. Rev. 57, 528–531 (1940).
    [CrossRef]
  3. W. C. Elmore, “Theory of optical and magnetic properties of ferromagnetic suspension,” Phys. Rev. 60, 593–596 (1941).
    [CrossRef]
  4. W. Heller, G. Quimfe, Y. Ta, “The magnetic and optical isotropy of selected microscopic crystals and the linear magneto-optical anisotropy of their suspensions,” Phys. Rev. 62, 479–486 (1942).
    [CrossRef]
  5. P. C. Scholten, “Magnetic measurements on particles in suspensions,” IEEE Trans. Magn. MAG-11, 1400–1402 (1975).
    [CrossRef]
  6. R. V. Mehta, “Optical properties of certain magnetic fluids,” in Thermomechanics of Magnetic Fluids, B. M. Berkovsky, ed., (Hemisphere, New York, 1977), pp. 139–148.
  7. S. Taketomi, “Magnetic fluid’s anomalous pseudo Cotton Mouton effects about 107 times larger than that of nitrobenzene,” Jpn. J. Appl. Phys. 22, 1137–1143 (1983).
    [CrossRef]
  8. R. V. Mehta, “Magneto optics of colloids,” in Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems, B. Sediacek, ed. (Walter de Gruyter and Co., Berlin, 1985), pp. 377–395.
  9. J. J. Muray, “Scattering of polarized light on magnetically aligned particles in multipole magnetic fields,” Appl. Opt. 4, 1011–1016 (1965).
    [CrossRef]
  10. R. V. Mehta, S. P. Vaidya, J. M. Patel, P. M. Vora, “Magnetically induced spatial filtering effect with magnetic fluids,” Appl. Opt. 26, 2297–2298 (1987).
    [CrossRef] [PubMed]
  11. J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
    [CrossRef]
  12. L. Sakhnini, J. Popplewell, “Interaction effects on magneto-optical birefringence in magnetic fluids,” J. Magn. Magn. Mater. 122, 142–145 (1993).
    [CrossRef]
  13. M. F. de Silva, F. A. Tourinho, L. Liebert, A. M. F. Neto, “Anisotropic magnetic fluid: a magnetic liquid crystal?” J. Magn. Magn. Mater. 122, 57–61 (1993).
    [CrossRef]
  14. A. Raval, “Development of ferrofluids in different carriers and to study their physical properties with special reference to application potentialities,” Ph.D. thesis (Bhavnagar University, Bhavnagar, India, 2000).
  15. T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
    [CrossRef]
  16. H. Matsuki, K. Mukrakami, “Performance of an automatic cooling device using temperative sensitive magnetic fluid,” J. Magn. Magn. Mater. 65, 363–365 (1987).
    [CrossRef]
  17. K. Nakatsuka, Y. Hama, J. Takahashi, “Heat transfer in temperature sensitive magnetic fluids,” J. Magn. Magn. Mater. 85, 207–209 (1990).
    [CrossRef]
  18. K. Parekh, R. V. Upadhyay, “Temperature dependence of magnetisation of magnetic fluid using Quincke’s Method,” Indian J. Pure Appl. Phys. 35, 523–528 (1997).
  19. T. Upadhyay, K. Parekh, “Thermomagnetic behaviour of Gd substituted ferrite magnetic fluids,” Indian J. Pure Appl. Phys. 40, 282–289 (2002).
  20. R. Patel, K. Parekh, R. V. Upadhyay, R. V. Mehta, “Rheology of transformer oil based ferrofluids,” Indian J. Eng. Mater. Sci., (to be published).
  21. K. Parekh, R. V. Upadhyay, R. V. Mehta, “Magnetocaloric effect in temperature-sensitive magnetic fluids,” Bull. Mater. Sci. 23, 91–95 (2000).
    [CrossRef]
  22. X. Xum, C. Peng, M. Mansuripur, “Estimation of thermal conductivity of magneto-optical media,” Appl. Opt. 39, 4355–4360 (2000), and references therein.
    [CrossRef]
  23. J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice Hall of India, New Delhi, India, 1988).
  24. D. Deirmendjian, Electromagnetic Scattering on Sperical Polydispersions (Elsevier, New York, 1969).
  25. H. S. Shah, R. V. Mehta, R. V. Desai, “Estimation of particle size distribution parameters with forward scattering techniques,” Appl. Opt. 18, 4173–4177 (1979).
    [CrossRef] [PubMed]

2002

T. Upadhyay, K. Parekh, “Thermomagnetic behaviour of Gd substituted ferrite magnetic fluids,” Indian J. Pure Appl. Phys. 40, 282–289 (2002).

2000

K. Parekh, R. V. Upadhyay, R. V. Mehta, “Magnetocaloric effect in temperature-sensitive magnetic fluids,” Bull. Mater. Sci. 23, 91–95 (2000).
[CrossRef]

X. Xum, C. Peng, M. Mansuripur, “Estimation of thermal conductivity of magneto-optical media,” Appl. Opt. 39, 4355–4360 (2000), and references therein.
[CrossRef]

1997

K. Parekh, R. V. Upadhyay, “Temperature dependence of magnetisation of magnetic fluid using Quincke’s Method,” Indian J. Pure Appl. Phys. 35, 523–528 (1997).

T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
[CrossRef]

1993

L. Sakhnini, J. Popplewell, “Interaction effects on magneto-optical birefringence in magnetic fluids,” J. Magn. Magn. Mater. 122, 142–145 (1993).
[CrossRef]

M. F. de Silva, F. A. Tourinho, L. Liebert, A. M. F. Neto, “Anisotropic magnetic fluid: a magnetic liquid crystal?” J. Magn. Magn. Mater. 122, 57–61 (1993).
[CrossRef]

1990

K. Nakatsuka, Y. Hama, J. Takahashi, “Heat transfer in temperature sensitive magnetic fluids,” J. Magn. Magn. Mater. 85, 207–209 (1990).
[CrossRef]

1987

J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
[CrossRef]

H. Matsuki, K. Mukrakami, “Performance of an automatic cooling device using temperative sensitive magnetic fluid,” J. Magn. Magn. Mater. 65, 363–365 (1987).
[CrossRef]

R. V. Mehta, S. P. Vaidya, J. M. Patel, P. M. Vora, “Magnetically induced spatial filtering effect with magnetic fluids,” Appl. Opt. 26, 2297–2298 (1987).
[CrossRef] [PubMed]

1983

S. Taketomi, “Magnetic fluid’s anomalous pseudo Cotton Mouton effects about 107 times larger than that of nitrobenzene,” Jpn. J. Appl. Phys. 22, 1137–1143 (1983).
[CrossRef]

1979

1975

P. C. Scholten, “Magnetic measurements on particles in suspensions,” IEEE Trans. Magn. MAG-11, 1400–1402 (1975).
[CrossRef]

1965

1942

W. Heller, G. Quimfe, Y. Ta, “The magnetic and optical isotropy of selected microscopic crystals and the linear magneto-optical anisotropy of their suspensions,” Phys. Rev. 62, 479–486 (1942).
[CrossRef]

1941

W. C. Elmore, “Theory of optical and magnetic properties of ferromagnetic suspension,” Phys. Rev. 60, 593–596 (1941).
[CrossRef]

1940

C. W. Heaps, “Optical and magnetic properties of a magnetic suspension,” Phys. Rev. 57, 528–531 (1940).
[CrossRef]

Aswal, V. K.

T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
[CrossRef]

Bacri, J. C.

J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
[CrossRef]

Cabuil, V.

J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
[CrossRef]

de Silva, M. F.

M. F. de Silva, F. A. Tourinho, L. Liebert, A. M. F. Neto, “Anisotropic magnetic fluid: a magnetic liquid crystal?” J. Magn. Magn. Mater. 122, 57–61 (1993).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Sperical Polydispersions (Elsevier, New York, 1969).

Desai, R. V.

Elmore, W. C.

W. C. Elmore, “Theory of optical and magnetic properties of ferromagnetic suspension,” Phys. Rev. 60, 593–596 (1941).
[CrossRef]

Goyal, P. S.

T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
[CrossRef]

Hama, Y.

K. Nakatsuka, Y. Hama, J. Takahashi, “Heat transfer in temperature sensitive magnetic fluids,” J. Magn. Magn. Mater. 85, 207–209 (1990).
[CrossRef]

Heaps, C. W.

C. W. Heaps, “Optical and magnetic properties of a magnetic suspension,” Phys. Rev. 57, 528–531 (1940).
[CrossRef]

Heller, W.

W. Heller, G. Quimfe, Y. Ta, “The magnetic and optical isotropy of selected microscopic crystals and the linear magneto-optical anisotropy of their suspensions,” Phys. Rev. 62, 479–486 (1942).
[CrossRef]

Liebert, L.

M. F. de Silva, F. A. Tourinho, L. Liebert, A. M. F. Neto, “Anisotropic magnetic fluid: a magnetic liquid crystal?” J. Magn. Magn. Mater. 122, 57–61 (1993).
[CrossRef]

Mansuripur, M.

Massart, R.

J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
[CrossRef]

Matsuki, H.

H. Matsuki, K. Mukrakami, “Performance of an automatic cooling device using temperative sensitive magnetic fluid,” J. Magn. Magn. Mater. 65, 363–365 (1987).
[CrossRef]

Mehta, R. V.

K. Parekh, R. V. Upadhyay, R. V. Mehta, “Magnetocaloric effect in temperature-sensitive magnetic fluids,” Bull. Mater. Sci. 23, 91–95 (2000).
[CrossRef]

T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
[CrossRef]

R. V. Mehta, S. P. Vaidya, J. M. Patel, P. M. Vora, “Magnetically induced spatial filtering effect with magnetic fluids,” Appl. Opt. 26, 2297–2298 (1987).
[CrossRef] [PubMed]

H. S. Shah, R. V. Mehta, R. V. Desai, “Estimation of particle size distribution parameters with forward scattering techniques,” Appl. Opt. 18, 4173–4177 (1979).
[CrossRef] [PubMed]

R. V. Mehta, “Magneto optics of colloids,” in Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems, B. Sediacek, ed. (Walter de Gruyter and Co., Berlin, 1985), pp. 377–395.

R. Patel, K. Parekh, R. V. Upadhyay, R. V. Mehta, “Rheology of transformer oil based ferrofluids,” Indian J. Eng. Mater. Sci., (to be published).

R. V. Mehta, “Optical properties of certain magnetic fluids,” in Thermomechanics of Magnetic Fluids, B. M. Berkovsky, ed., (Hemisphere, New York, 1977), pp. 139–148.

Meyer-Arendt, J. R.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice Hall of India, New Delhi, India, 1988).

Mukrakami, K.

H. Matsuki, K. Mukrakami, “Performance of an automatic cooling device using temperative sensitive magnetic fluid,” J. Magn. Magn. Mater. 65, 363–365 (1987).
[CrossRef]

Muray, J. J.

Nakatsuka, K.

K. Nakatsuka, Y. Hama, J. Takahashi, “Heat transfer in temperature sensitive magnetic fluids,” J. Magn. Magn. Mater. 85, 207–209 (1990).
[CrossRef]

Neto, A. M. F.

M. F. de Silva, F. A. Tourinho, L. Liebert, A. M. F. Neto, “Anisotropic magnetic fluid: a magnetic liquid crystal?” J. Magn. Magn. Mater. 122, 57–61 (1993).
[CrossRef]

Parekh, K.

T. Upadhyay, K. Parekh, “Thermomagnetic behaviour of Gd substituted ferrite magnetic fluids,” Indian J. Pure Appl. Phys. 40, 282–289 (2002).

K. Parekh, R. V. Upadhyay, R. V. Mehta, “Magnetocaloric effect in temperature-sensitive magnetic fluids,” Bull. Mater. Sci. 23, 91–95 (2000).
[CrossRef]

K. Parekh, R. V. Upadhyay, “Temperature dependence of magnetisation of magnetic fluid using Quincke’s Method,” Indian J. Pure Appl. Phys. 35, 523–528 (1997).

R. Patel, K. Parekh, R. V. Upadhyay, R. V. Mehta, “Rheology of transformer oil based ferrofluids,” Indian J. Eng. Mater. Sci., (to be published).

Patel, J. M.

Patel, R.

R. Patel, K. Parekh, R. V. Upadhyay, R. V. Mehta, “Rheology of transformer oil based ferrofluids,” Indian J. Eng. Mater. Sci., (to be published).

Peng, C.

Perzynski, R.

J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
[CrossRef]

Popplewell, J.

L. Sakhnini, J. Popplewell, “Interaction effects on magneto-optical birefringence in magnetic fluids,” J. Magn. Magn. Mater. 122, 142–145 (1993).
[CrossRef]

Quimfe, G.

W. Heller, G. Quimfe, Y. Ta, “The magnetic and optical isotropy of selected microscopic crystals and the linear magneto-optical anisotropy of their suspensions,” Phys. Rev. 62, 479–486 (1942).
[CrossRef]

Raval, A.

A. Raval, “Development of ferrofluids in different carriers and to study their physical properties with special reference to application potentialities,” Ph.D. thesis (Bhavnagar University, Bhavnagar, India, 2000).

Rosensweig, R. E.

R. E. Rosensweig, Ferrohydrodynamics (Cambridge U. Press, Cambridge, 1985), and references therein.

Sakhnini, L.

L. Sakhnini, J. Popplewell, “Interaction effects on magneto-optical birefringence in magnetic fluids,” J. Magn. Magn. Mater. 122, 142–145 (1993).
[CrossRef]

Salid, D.

J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
[CrossRef]

Scholten, P. C.

P. C. Scholten, “Magnetic measurements on particles in suspensions,” IEEE Trans. Magn. MAG-11, 1400–1402 (1975).
[CrossRef]

Shah, H. S.

Ta, Y.

W. Heller, G. Quimfe, Y. Ta, “The magnetic and optical isotropy of selected microscopic crystals and the linear magneto-optical anisotropy of their suspensions,” Phys. Rev. 62, 479–486 (1942).
[CrossRef]

Takahashi, J.

K. Nakatsuka, Y. Hama, J. Takahashi, “Heat transfer in temperature sensitive magnetic fluids,” J. Magn. Magn. Mater. 85, 207–209 (1990).
[CrossRef]

Taketomi, S.

S. Taketomi, “Magnetic fluid’s anomalous pseudo Cotton Mouton effects about 107 times larger than that of nitrobenzene,” Jpn. J. Appl. Phys. 22, 1137–1143 (1983).
[CrossRef]

Tourinho, F. A.

M. F. de Silva, F. A. Tourinho, L. Liebert, A. M. F. Neto, “Anisotropic magnetic fluid: a magnetic liquid crystal?” J. Magn. Magn. Mater. 122, 57–61 (1993).
[CrossRef]

Upadhyay, R. V.

K. Parekh, R. V. Upadhyay, R. V. Mehta, “Magnetocaloric effect in temperature-sensitive magnetic fluids,” Bull. Mater. Sci. 23, 91–95 (2000).
[CrossRef]

K. Parekh, R. V. Upadhyay, “Temperature dependence of magnetisation of magnetic fluid using Quincke’s Method,” Indian J. Pure Appl. Phys. 35, 523–528 (1997).

T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
[CrossRef]

R. Patel, K. Parekh, R. V. Upadhyay, R. V. Mehta, “Rheology of transformer oil based ferrofluids,” Indian J. Eng. Mater. Sci., (to be published).

Upadhyay, T.

T. Upadhyay, K. Parekh, “Thermomagnetic behaviour of Gd substituted ferrite magnetic fluids,” Indian J. Pure Appl. Phys. 40, 282–289 (2002).

T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
[CrossRef]

Vaidya, S. P.

Vora, P. M.

Xum, X.

Appl. Opt.

Bull. Mater. Sci.

K. Parekh, R. V. Upadhyay, R. V. Mehta, “Magnetocaloric effect in temperature-sensitive magnetic fluids,” Bull. Mater. Sci. 23, 91–95 (2000).
[CrossRef]

IEEE Trans. Magn.

P. C. Scholten, “Magnetic measurements on particles in suspensions,” IEEE Trans. Magn. MAG-11, 1400–1402 (1975).
[CrossRef]

Indian J. Pure Appl. Phys.

K. Parekh, R. V. Upadhyay, “Temperature dependence of magnetisation of magnetic fluid using Quincke’s Method,” Indian J. Pure Appl. Phys. 35, 523–528 (1997).

T. Upadhyay, K. Parekh, “Thermomagnetic behaviour of Gd substituted ferrite magnetic fluids,” Indian J. Pure Appl. Phys. 40, 282–289 (2002).

J. Magn. Magn. Mater

J. C. Bacri, V. Cabuil, R. Massart, R. Perzynski, D. Salid, “Ionic ferrofluid: optical Properties,” J. Magn. Magn. Mater, 65, 285–288 (1987).
[CrossRef]

J. Magn. Magn. Mater.

L. Sakhnini, J. Popplewell, “Interaction effects on magneto-optical birefringence in magnetic fluids,” J. Magn. Magn. Mater. 122, 142–145 (1993).
[CrossRef]

M. F. de Silva, F. A. Tourinho, L. Liebert, A. M. F. Neto, “Anisotropic magnetic fluid: a magnetic liquid crystal?” J. Magn. Magn. Mater. 122, 57–61 (1993).
[CrossRef]

H. Matsuki, K. Mukrakami, “Performance of an automatic cooling device using temperative sensitive magnetic fluid,” J. Magn. Magn. Mater. 65, 363–365 (1987).
[CrossRef]

K. Nakatsuka, Y. Hama, J. Takahashi, “Heat transfer in temperature sensitive magnetic fluids,” J. Magn. Magn. Mater. 85, 207–209 (1990).
[CrossRef]

Jpn. J. Appl. Phys.

S. Taketomi, “Magnetic fluid’s anomalous pseudo Cotton Mouton effects about 107 times larger than that of nitrobenzene,” Jpn. J. Appl. Phys. 22, 1137–1143 (1983).
[CrossRef]

Phys. Rev.

C. W. Heaps, “Optical and magnetic properties of a magnetic suspension,” Phys. Rev. 57, 528–531 (1940).
[CrossRef]

W. C. Elmore, “Theory of optical and magnetic properties of ferromagnetic suspension,” Phys. Rev. 60, 593–596 (1941).
[CrossRef]

W. Heller, G. Quimfe, Y. Ta, “The magnetic and optical isotropy of selected microscopic crystals and the linear magneto-optical anisotropy of their suspensions,” Phys. Rev. 62, 479–486 (1942).
[CrossRef]

Phys. Rev. B.

T. Upadhyay, R. V. Upadhyay, R. V. Mehta, V. K. Aswal, P. S. Goyal, “Characterization of a temperature-sensitive magnetic fluid,” Phys. Rev. B. 55, 5585–5588 (1997).
[CrossRef]

Other

R. E. Rosensweig, Ferrohydrodynamics (Cambridge U. Press, Cambridge, 1985), and references therein.

R. V. Mehta, “Magneto optics of colloids,” in Physical Optics of Dynamic Phenomena and Processes in Macromolecular Systems, B. Sediacek, ed. (Walter de Gruyter and Co., Berlin, 1985), pp. 377–395.

R. Patel, K. Parekh, R. V. Upadhyay, R. V. Mehta, “Rheology of transformer oil based ferrofluids,” Indian J. Eng. Mater. Sci., (to be published).

R. V. Mehta, “Optical properties of certain magnetic fluids,” in Thermomechanics of Magnetic Fluids, B. M. Berkovsky, ed., (Hemisphere, New York, 1977), pp. 139–148.

A. Raval, “Development of ferrofluids in different carriers and to study their physical properties with special reference to application potentialities,” Ph.D. thesis (Bhavnagar University, Bhavnagar, India, 2000).

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice Hall of India, New Delhi, India, 1988).

D. Deirmendjian, Electromagnetic Scattering on Sperical Polydispersions (Elsevier, New York, 1969).

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Figures (4)

Fig. 1
Fig. 1

Schematic of experimental setup.

Fig. 2
Fig. 2

Reduced intensity (ΔI/I 0) as a function of the applied magnetic field. Solid curves are fitted to Eq. (6); the best-fit parameters are listed in Table 1. (●, MZ4 fluid; ■, MZ5 fluid; σ, MZG fluid).

Fig. 3
Fig. 3

Variation of reduced intensity (ΔI/I 0) as a function of temperature. The solid symbols are a variation of the normalized change in intensity with temperature, and the open symbols are a variation of the reduced magnetization with temperature. (● and ○, MZ4 fluid; ■ and □, MZ5 fluid).

Fig. 4
Fig. 4

Variation of normalized change in intensity (ΔI/I 0) as a function of temperature for MZG fluid. The solid symbols (σ) are a variation of the reduced intensity with temperature, and the open symbols (△) are a variation of the reduced magnetization with temperature.

Tables (1)

Tables Icon

Table 1 Parameters Obtained from the Langevin Fit to the Experimental Results

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

I1=I0coth α-1α,
α=μH/kT.
I1=I0α/3.
μ=Mdμ0πD36,
fD=1Dσ2π1/2exp-lnD/D022σ2,
0 fDdD=1,
I=0I1DfDdD.

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