Abstract

We introduce a method for optical trap calibration that is suitable for viscoelastic material. The method is designed for use on experimental setups with two optical tweezers and is based on pulling a trapped particle with one trap while holding it with the other. No piezo stage is needed, and only one optical trap must be movable with galvo mirrors, piezo mirrors, or acousto-optical deflectors. The method combines advantages of commonly known PSD-fitting and fast-sweeping methods, allowing calibration of a completely fixed trap in a fluid of unknown viscosity/viscoelasticity. A detailed method description, a theoretical derivation, and an experimental comparison to other methods are reported.

© 2013 Optical Society of America

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  1. T. Mason and D. Weitz, Phys. Rev. Lett. 74, 1250 (1995).
    [CrossRef]
  2. T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
    [CrossRef]
  3. J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
    [CrossRef]
  4. B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Macromolecules 30, 7781 (1997).
    [CrossRef]
  5. F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
    [CrossRef]
  6. A. Cordoba, T. Indei, and J. D. Schieber, Soft Matt. 9, 3521 (2013).
    [CrossRef]
  7. B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
    [CrossRef]
  8. T. Savin and P. Doyle, Biophys. J. 88, 623 (2005).
    [CrossRef]
  9. Y. Kimura, J. Phys. Soc. Jpn. 78, 041005 (2009).
    [CrossRef]
  10. A. Ashkin, J. Dziedzic, J. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986).
    [CrossRef]
  11. D. Mizuno, C. Tardin, C. Schmidt, and F. MacKintosh, Science 315, 370 (2007).
    [CrossRef]
  12. G. Koenderink, M. Atakhorrami, F. MacKintosh, and C. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
    [CrossRef]
  13. JPK Instruments AG, Characterizing quantitative measurements of force and displacement with optical tweezers on the NanoTracker http://www.jpk.com .
  14. M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
    [CrossRef]
  15. K. Berg-Sorensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
    [CrossRef]
  16. M. Fischer and K. Berg-Sorensen, J. Opt. A 9, S239 (2007).
    [CrossRef]
  17. R. Simmons, J. Finer, S. Chu, and J. Spudich, Biophys. J. 70, 1813 (1996).
    [CrossRef]
  18. A. Kachynski, A. Kuzmin, H. Pudavar, D. Kaputa, A. Cartwright, and P. Prasad, Opt. Lett. 28, 2288 (2003).
    [CrossRef]
  19. K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
    [CrossRef]
  20. K. Visscher and S. Block, Methods Enzymol. 298, 460 (1998).
    [CrossRef]
  21. L. Starrs and P. Bartlett, Faraday Discuss. 123, 323 (2003).
    [CrossRef]
  22. M. M. Shindel, J. W. Swan, and E. M. Furst, Rheol. Acta 52, 455 (2013).
    [CrossRef]
  23. A. Wozniak, J. van Mameren, and S. Ragona, Pharm. Biotechnol. 10, 467 (2009).
    [CrossRef]
  24. W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge University, 1986).

2013

A. Cordoba, T. Indei, and J. D. Schieber, Soft Matt. 9, 3521 (2013).
[CrossRef]

M. M. Shindel, J. W. Swan, and E. M. Furst, Rheol. Acta 52, 455 (2013).
[CrossRef]

2009

A. Wozniak, J. van Mameren, and S. Ragona, Pharm. Biotechnol. 10, 467 (2009).
[CrossRef]

Y. Kimura, J. Phys. Soc. Jpn. 78, 041005 (2009).
[CrossRef]

2008

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

2007

D. Mizuno, C. Tardin, C. Schmidt, and F. MacKintosh, Science 315, 370 (2007).
[CrossRef]

M. Fischer and K. Berg-Sorensen, J. Opt. A 9, S239 (2007).
[CrossRef]

2006

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

G. Koenderink, M. Atakhorrami, F. MacKintosh, and C. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
[CrossRef]

2005

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

T. Savin and P. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

2004

K. Berg-Sorensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

2003

1998

K. Visscher and S. Block, Methods Enzymol. 298, 460 (1998).
[CrossRef]

M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
[CrossRef]

1997

B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Macromolecules 30, 7781 (1997).
[CrossRef]

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

1996

R. Simmons, J. Finer, S. Chu, and J. Spudich, Biophys. J. 70, 1813 (1996).
[CrossRef]

1995

T. Mason and D. Weitz, Phys. Rev. Lett. 74, 1250 (1995).
[CrossRef]

1986

Allersma, M.

M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
[CrossRef]

Ashkin, A.

Atakhorrami, M.

G. Koenderink, M. Atakhorrami, F. MacKintosh, and C. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
[CrossRef]

Bartlett, P.

L. Starrs and P. Bartlett, Faraday Discuss. 123, 323 (2003).
[CrossRef]

Berg-Sorensen, K.

M. Fischer and K. Berg-Sorensen, J. Opt. A 9, S239 (2007).
[CrossRef]

K. Berg-Sorensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

Bjorkholm, J.

Block, S.

K. Visscher and S. Block, Methods Enzymol. 298, 460 (1998).
[CrossRef]

Cartwright, A.

Chu, S.

R. Simmons, J. Finer, S. Chu, and J. Spudich, Biophys. J. 70, 1813 (1996).
[CrossRef]

A. Ashkin, J. Dziedzic, J. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986).
[CrossRef]

Cordoba, A.

A. Cordoba, T. Indei, and J. D. Schieber, Soft Matt. 9, 3521 (2013).
[CrossRef]

deCastro, M.

M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
[CrossRef]

Dekker, C.

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

Doyle, P.

T. Savin and P. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

Dziedzic, J.

Finer, J.

R. Simmons, J. Finer, S. Chu, and J. Spudich, Biophys. J. 70, 1813 (1996).
[CrossRef]

Fischer, M.

M. Fischer and K. Berg-Sorensen, J. Opt. A 9, S239 (2007).
[CrossRef]

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge University, 1986).

Florin, E.-L.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

Flyvbjerg, H.

K. Berg-Sorensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

Forró, L.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

Furst, E. M.

M. M. Shindel, J. W. Swan, and E. M. Furst, Rheol. Acta 52, 455 (2013).
[CrossRef]

Gittes, F.

M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
[CrossRef]

B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Macromolecules 30, 7781 (1997).
[CrossRef]

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Hage, S.

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

Indei, T.

A. Cordoba, T. Indei, and J. D. Schieber, Soft Matt. 9, 3521 (2013).
[CrossRef]

Jeney, S.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

Kachynski, A.

Kanaar, R.

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

Kaputa, D.

Kimura, Y.

Y. Kimura, J. Phys. Soc. Jpn. 78, 041005 (2009).
[CrossRef]

Koenderink, G.

G. Koenderink, M. Atakhorrami, F. MacKintosh, and C. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
[CrossRef]

Kulik, A. J.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

Kuzmin, A.

Lukic, B.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

MacKintosh, F.

D. Mizuno, C. Tardin, C. Schmidt, and F. MacKintosh, Science 315, 370 (2007).
[CrossRef]

G. Koenderink, M. Atakhorrami, F. MacKintosh, and C. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
[CrossRef]

MacKintosh, F. C.

B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Macromolecules 30, 7781 (1997).
[CrossRef]

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Mason, T.

T. Mason and D. Weitz, Phys. Rev. Lett. 74, 1250 (1995).
[CrossRef]

Mizuno, D.

D. Mizuno, C. Tardin, C. Schmidt, and F. MacKintosh, Science 315, 370 (2007).
[CrossRef]

Modesti, M.

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

Olmsted, P. D.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

Peterman, E.

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

Prasad, P.

Press, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge University, 1986).

Pudavar, H.

Ragona, S.

A. Wozniak, J. van Mameren, and S. Ragona, Pharm. Biotechnol. 10, 467 (2009).
[CrossRef]

Savin, T.

T. Savin and P. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

Schieber, J. D.

A. Cordoba, T. Indei, and J. D. Schieber, Soft Matt. 9, 3521 (2013).
[CrossRef]

Schmidt, C.

D. Mizuno, C. Tardin, C. Schmidt, and F. MacKintosh, Science 315, 370 (2007).
[CrossRef]

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

G. Koenderink, M. Atakhorrami, F. MacKintosh, and C. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
[CrossRef]

M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
[CrossRef]

Schmidt, C. F.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Macromolecules 30, 7781 (1997).
[CrossRef]

Schnurr, B.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Macromolecules 30, 7781 (1997).
[CrossRef]

Shindel, M. M.

M. M. Shindel, J. W. Swan, and E. M. Furst, Rheol. Acta 52, 455 (2013).
[CrossRef]

Simmons, R.

R. Simmons, J. Finer, S. Chu, and J. Spudich, Biophys. J. 70, 1813 (1996).
[CrossRef]

Spudich, J.

R. Simmons, J. Finer, S. Chu, and J. Spudich, Biophys. J. 70, 1813 (1996).
[CrossRef]

Starrs, L.

L. Starrs and P. Bartlett, Faraday Discuss. 123, 323 (2003).
[CrossRef]

Stewart, R.

M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
[CrossRef]

Stienen, G.

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

Swan, J. W.

M. M. Shindel, J. W. Swan, and E. M. Furst, Rheol. Acta 52, 455 (2013).
[CrossRef]

Tardin, C.

D. Mizuno, C. Tardin, C. Schmidt, and F. MacKintosh, Science 315, 370 (2007).
[CrossRef]

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge University, 1986).

Tischer, C.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

van der Heijden, T.

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

van Mameren, J.

A. Wozniak, J. van Mameren, and S. Ragona, Pharm. Biotechnol. 10, 467 (2009).
[CrossRef]

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

Vermeulen, K.

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge University, 1986).

Visscher, K.

K. Visscher and S. Block, Methods Enzymol. 298, 460 (1998).
[CrossRef]

Weitz, D.

T. Mason and D. Weitz, Phys. Rev. Lett. 74, 1250 (1995).
[CrossRef]

Wozniak, A.

A. Wozniak, J. van Mameren, and S. Ragona, Pharm. Biotechnol. 10, 467 (2009).
[CrossRef]

Wuite, G.

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

Wyman, C.

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

Biophys. J.

T. Savin and P. Doyle, Biophys. J. 88, 623 (2005).
[CrossRef]

M. Allersma, F. Gittes, M. deCastro, R. Stewart, and C. Schmidt, Biophys. J. 74, 1074 (1998).
[CrossRef]

R. Simmons, J. Finer, S. Chu, and J. Spudich, Biophys. J. 70, 1813 (1996).
[CrossRef]

Faraday Discuss.

L. Starrs and P. Bartlett, Faraday Discuss. 123, 323 (2003).
[CrossRef]

J. Opt. A

M. Fischer and K. Berg-Sorensen, J. Opt. A 9, S239 (2007).
[CrossRef]

J. Phys. Soc. Jpn.

Y. Kimura, J. Phys. Soc. Jpn. 78, 041005 (2009).
[CrossRef]

Macromolecules

B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Macromolecules 30, 7781 (1997).
[CrossRef]

Methods Enzymol.

K. Visscher and S. Block, Methods Enzymol. 298, 460 (1998).
[CrossRef]

Mol. Cell

T. van der Heijden, M. Modesti, S. Hage, R. Kanaar, C. Wyman, and C. Dekker, Mol. Cell 30, 530 (2008).
[CrossRef]

Nature

J. van Mameren, M. Modesti, R. Kanaar, C. Wyman, E. Peterman, and G. Wuite, Nature 457, 745 (2008).
[CrossRef]

Opt. Lett.

Pharm. Biotechnol.

A. Wozniak, J. van Mameren, and S. Ragona, Pharm. Biotechnol. 10, 467 (2009).
[CrossRef]

Phys. Rev. Lett.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Phys. Rev. Lett. 79, 3286 (1997).
[CrossRef]

T. Mason and D. Weitz, Phys. Rev. Lett. 74, 1250 (1995).
[CrossRef]

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
[CrossRef]

G. Koenderink, M. Atakhorrami, F. MacKintosh, and C. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
[CrossRef]

Rev. Sci. Instrum.

K. Berg-Sorensen and H. Flyvbjerg, Rev. Sci. Instrum. 75, 594 (2004).
[CrossRef]

K. Vermeulen, J. van Mameren, G. Stienen, E. Peterman, G. Wuite, and C. Schmidt, Rev. Sci. Instrum. 77, 013704 (2006).
[CrossRef]

Rheol. Acta

M. M. Shindel, J. W. Swan, and E. M. Furst, Rheol. Acta 52, 455 (2013).
[CrossRef]

Science

D. Mizuno, C. Tardin, C. Schmidt, and F. MacKintosh, Science 315, 370 (2007).
[CrossRef]

Soft Matt.

A. Cordoba, T. Indei, and J. D. Schieber, Soft Matt. 9, 3521 (2013).
[CrossRef]

Other

JPK Instruments AG, Characterizing quantitative measurements of force and displacement with optical tweezers on the NanoTracker http://www.jpk.com .

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge University, 1986).

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Figures (3)

Fig. 1.
Fig. 1.

Trap calibration method: (a) displacing the bead with movable trap while holding with fixed trap; holding bead with (b) fixed trap and (c) movable trap. (d) QPD signals during trap movement; thermal fluctuations of the bead in (e) fixed trap and (f) movable trap. 1, QPD of movable trap; 2, QPD of fixed trap; 3, detection objective; 4, movable trap axis; 5, fixed trap axis (both trap axes are drawn nonvertical for better layout only); 6, bead; 7, trapping objective; 8, laser beam of movable trap; 9, laser beam of fixed trap.

Fig. 2.
Fig. 2.

Comparison of (a) absolute values of trap sensitivity and (b) trap stiffness for DI water (X channel only).

Fig. 3.
Fig. 3.

Comparison of (a) absolute values of trap sensitivity and (b) trap stiffness for 2% w/w 5–6 MDa polyacrylamide solution. Note that the PSD-fitting method is not applicable to viscoelastic fluid. Calibration of fixed trap in viscoelastic fluid is only possible with the dual trapping method (X channel only).

Equations (5)

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

Vxi(t)=cxi[xb(t)xi(t)]+dxi+δVxi(t),
αxi=1υxmdVxi¯(t)dt,
αxf=cxfkxmkxf+kxm;αxm=cxmkxfkxf+kxm,
(σxf)2=(cxf)2kBTkxf;(σxm)2=(cxm)2kBTkxm.
cxf=(σxm)2(αxf)2+(σxf)2(αxm)2(σxm)2αxfcxm=(σxm)2(αxf)2+(σxf)2(αxm)2(σxf)2αxmkxf=kBT((σxm)2(αxf)2+(σxf)2(αxm)2)2(σxf)2(σxm)4(αxf)2kxm=kBT((σxm)2(αxf)2+(σxf)2(αxm)2)2(σxf)4(σxm)2(αxm)2.

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