Abstract

Confocal microscopy is a potentially powerful technique for obtaining equation-of-state (EOS) data for fluids in a diamond anvil cell. Unlike conventional microscopy, a confocal microscope scans the cell in three dimensions. From the intensity profile of the reflected laser light, we calculated the index of refraction and optical thickness of the sample contained in the cell. These measurements, combined with the cross-sectional area of the sample, enabled us to calculate the volume. As a test of the experimental technique and analysis, we produced a pressure–volume curve for liquid water at 300K. The results agree with published EOS data within experimental error.

© 2009 Optical Society of America

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References

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  1. A. Jayaraman, “Diamond anvil cell and high-pressure physical investigations,” Rev. Mod. Phys. 55, 65-108 (1983).
    [CrossRef]
  2. A. Zaitzev, Optical Properties of Diamond: A Data Handbook, (Springer, 2001), pp. i-9.
  3. Y. Fei, H. Mao, and R. Hemley, “Thermal expansivity, bulk modulus, and melting curve of H2O-ice VII to 20 GPa,” J. Chem. Phys. 99, 5369-5373 (1993).
    [CrossRef]
  4. A. Dewaele, J. Eggert, P. Loubeyre, and R. LeToullec, “Measurement of refractive index and equation of state in dense He, H2, H2O, and Ne under high pressure in a diamond anvil cell,” Phys. Rev. B 67, 094112 (2003).
    [CrossRef]
  5. W. Evans and I. Silvera, “Index of refraction, polarizability, and equation of state of solid molecular hydrogen,” Physical Review B 57, 14105-14109 (1998).
    [CrossRef]
  6. J. Pawley, Handbook of Biological Confocal Microscopy (Springer, 1995), p. 4.
  7. A. E. Siegman, Lasers (University Science Books, 1986), pp. 663-697.
  8. S. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 658-661 (1983).
    [CrossRef] [PubMed]
  9. M. McCluskey, K. Zhuravlev, B. Davidson, and R. Newman, “Acceptor-hydrogen in semiconductors under pressure,” Physica B 308-310, 780-783 (2001).
    [CrossRef]
  10. W. S. Rasband, ImageJ (U.S. National Institutes of Health, 1997-2008), http://rsb.info.nih.gov/ij/.
  11. M. J.Weber, Handbook of Optical Materials (CRC Press, 2003), pp. 373-405.
  12. R. Courchinoux and P. Lalle, “Dynamic properties of water: sound velocity and refractive index,” in AIP Conference Proceedings (American Institute of Physics, 1996), Vol. 370, pp. 61-64.
    [CrossRef]
  13. H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
    [CrossRef]
  14. M. Choukron and O. Grasset, “Thermodynamic model for water and high-pressure ices up to 2.2 GPa and down to the metastable domain,” J. Chem. Phys. 127, 124506 (2007).
    [CrossRef]
  15. E. Noya, C. Menduiña, J. Aragones, and C. Vega, “Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI as obtained from computer simulation,” J. Phys. Chem. C 111, 15877-15888 (2007).
    [CrossRef]
  16. E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
    [CrossRef]
  17. M. D. McCluskey, B. D. Riley, A. M. Perenchio, and M. Knoblauch, “Confocal microscopy of water under static pressure,” Shock Compression of Condensed Matter--2007, AIP Conference Proceedings, M. Elert, M. D. Furnish, R. Chau, N. Holmes, and J. Nguyen, eds. (American Institute of Physics, 2007), Vol. 955, pp. 1109-1112.

2007 (2)

M. Choukron and O. Grasset, “Thermodynamic model for water and high-pressure ices up to 2.2 GPa and down to the metastable domain,” J. Chem. Phys. 127, 124506 (2007).
[CrossRef]

E. Noya, C. Menduiña, J. Aragones, and C. Vega, “Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI as obtained from computer simulation,” J. Phys. Chem. C 111, 15877-15888 (2007).
[CrossRef]

2003 (1)

A. Dewaele, J. Eggert, P. Loubeyre, and R. LeToullec, “Measurement of refractive index and equation of state in dense He, H2, H2O, and Ne under high pressure in a diamond anvil cell,” Phys. Rev. B 67, 094112 (2003).
[CrossRef]

2001 (1)

M. McCluskey, K. Zhuravlev, B. Davidson, and R. Newman, “Acceptor-hydrogen in semiconductors under pressure,” Physica B 308-310, 780-783 (2001).
[CrossRef]

1998 (1)

W. Evans and I. Silvera, “Index of refraction, polarizability, and equation of state of solid molecular hydrogen,” Physical Review B 57, 14105-14109 (1998).
[CrossRef]

1997 (1)

E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
[CrossRef]

1993 (1)

Y. Fei, H. Mao, and R. Hemley, “Thermal expansivity, bulk modulus, and melting curve of H2O-ice VII to 20 GPa,” J. Chem. Phys. 99, 5369-5373 (1993).
[CrossRef]

1983 (2)

A. Jayaraman, “Diamond anvil cell and high-pressure physical investigations,” Rev. Mod. Phys. 55, 65-108 (1983).
[CrossRef]

S. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 658-661 (1983).
[CrossRef] [PubMed]

1973 (1)

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

Aragones, J.

E. Noya, C. Menduiña, J. Aragones, and C. Vega, “Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI as obtained from computer simulation,” J. Phys. Chem. C 111, 15877-15888 (2007).
[CrossRef]

Canny, B.

E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
[CrossRef]

Chand, D.

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

Chervin, J.

E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
[CrossRef]

Choukron, M.

M. Choukron and O. Grasset, “Thermodynamic model for water and high-pressure ices up to 2.2 GPa and down to the metastable domain,” J. Chem. Phys. 127, 124506 (2007).
[CrossRef]

Courchinoux, R.

R. Courchinoux and P. Lalle, “Dynamic properties of water: sound velocity and refractive index,” in AIP Conference Proceedings (American Institute of Physics, 1996), Vol. 370, pp. 61-64.
[CrossRef]

Davidson, B.

M. McCluskey, K. Zhuravlev, B. Davidson, and R. Newman, “Acceptor-hydrogen in semiconductors under pressure,” Physica B 308-310, 780-783 (2001).
[CrossRef]

Dewaele, A.

A. Dewaele, J. Eggert, P. Loubeyre, and R. LeToullec, “Measurement of refractive index and equation of state in dense He, H2, H2O, and Ne under high pressure in a diamond anvil cell,” Phys. Rev. B 67, 094112 (2003).
[CrossRef]

Eggert, J.

A. Dewaele, J. Eggert, P. Loubeyre, and R. LeToullec, “Measurement of refractive index and equation of state in dense He, H2, H2O, and Ne under high pressure in a diamond anvil cell,” Phys. Rev. B 67, 094112 (2003).
[CrossRef]

Evans, W.

W. Evans and I. Silvera, “Index of refraction, polarizability, and equation of state of solid molecular hydrogen,” Physical Review B 57, 14105-14109 (1998).
[CrossRef]

Fei, Y.

Y. Fei, H. Mao, and R. Hemley, “Thermal expansivity, bulk modulus, and melting curve of H2O-ice VII to 20 GPa,” J. Chem. Phys. 99, 5369-5373 (1993).
[CrossRef]

Gauthier, M.

E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
[CrossRef]

Grasset, O.

M. Choukron and O. Grasset, “Thermodynamic model for water and high-pressure ices up to 2.2 GPa and down to the metastable domain,” J. Chem. Phys. 127, 124506 (2007).
[CrossRef]

Gupta, B.

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

Hemley, R.

Y. Fei, H. Mao, and R. Hemley, “Thermal expansivity, bulk modulus, and melting curve of H2O-ice VII to 20 GPa,” J. Chem. Phys. 99, 5369-5373 (1993).
[CrossRef]

Jayaraman, A.

A. Jayaraman, “Diamond anvil cell and high-pressure physical investigations,” Rev. Mod. Phys. 55, 65-108 (1983).
[CrossRef]

Knoblauch, M.

M. D. McCluskey, B. D. Riley, A. M. Perenchio, and M. Knoblauch, “Confocal microscopy of water under static pressure,” Shock Compression of Condensed Matter--2007, AIP Conference Proceedings, M. Elert, M. D. Furnish, R. Chau, N. Holmes, and J. Nguyen, eds. (American Institute of Physics, 2007), Vol. 955, pp. 1109-1112.

Lalle, P.

R. Courchinoux and P. Lalle, “Dynamic properties of water: sound velocity and refractive index,” in AIP Conference Proceedings (American Institute of Physics, 1996), Vol. 370, pp. 61-64.
[CrossRef]

LeToullec, R.

A. Dewaele, J. Eggert, P. Loubeyre, and R. LeToullec, “Measurement of refractive index and equation of state in dense He, H2, H2O, and Ne under high pressure in a diamond anvil cell,” Phys. Rev. B 67, 094112 (2003).
[CrossRef]

Loubeyre, P.

A. Dewaele, J. Eggert, P. Loubeyre, and R. LeToullec, “Measurement of refractive index and equation of state in dense He, H2, H2O, and Ne under high pressure in a diamond anvil cell,” Phys. Rev. B 67, 094112 (2003).
[CrossRef]

Mao, H.

Y. Fei, H. Mao, and R. Hemley, “Thermal expansivity, bulk modulus, and melting curve of H2O-ice VII to 20 GPa,” J. Chem. Phys. 99, 5369-5373 (1993).
[CrossRef]

McCluskey, M.

M. McCluskey, K. Zhuravlev, B. Davidson, and R. Newman, “Acceptor-hydrogen in semiconductors under pressure,” Physica B 308-310, 780-783 (2001).
[CrossRef]

McCluskey, M. D.

M. D. McCluskey, B. D. Riley, A. M. Perenchio, and M. Knoblauch, “Confocal microscopy of water under static pressure,” Shock Compression of Condensed Matter--2007, AIP Conference Proceedings, M. Elert, M. D. Furnish, R. Chau, N. Holmes, and J. Nguyen, eds. (American Institute of Physics, 2007), Vol. 955, pp. 1109-1112.

Menduiña, C.

E. Noya, C. Menduiña, J. Aragones, and C. Vega, “Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI as obtained from computer simulation,” J. Phys. Chem. C 111, 15877-15888 (2007).
[CrossRef]

Murty, D.

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

Newman, R.

M. McCluskey, K. Zhuravlev, B. Davidson, and R. Newman, “Acceptor-hydrogen in semiconductors under pressure,” Physica B 308-310, 780-783 (2001).
[CrossRef]

Noya, E.

E. Noya, C. Menduiña, J. Aragones, and C. Vega, “Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI as obtained from computer simulation,” J. Phys. Chem. C 111, 15877-15888 (2007).
[CrossRef]

Pawley, J.

J. Pawley, Handbook of Biological Confocal Microscopy (Springer, 1995), p. 4.

Perenchio, A. M.

M. D. McCluskey, B. D. Riley, A. M. Perenchio, and M. Knoblauch, “Confocal microscopy of water under static pressure,” Shock Compression of Condensed Matter--2007, AIP Conference Proceedings, M. Elert, M. D. Furnish, R. Chau, N. Holmes, and J. Nguyen, eds. (American Institute of Physics, 2007), Vol. 955, pp. 1109-1112.

Pruzan, Ph.

E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
[CrossRef]

Rasband, W. S.

W. S. Rasband, ImageJ (U.S. National Institutes of Health, 1997-2008), http://rsb.info.nih.gov/ij/.

Riley, B. D.

M. D. McCluskey, B. D. Riley, A. M. Perenchio, and M. Knoblauch, “Confocal microscopy of water under static pressure,” Shock Compression of Condensed Matter--2007, AIP Conference Proceedings, M. Elert, M. D. Furnish, R. Chau, N. Holmes, and J. Nguyen, eds. (American Institute of Physics, 2007), Vol. 955, pp. 1109-1112.

Self, S.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), pp. 663-697.

Silvera, I.

W. Evans and I. Silvera, “Index of refraction, polarizability, and equation of state of solid molecular hydrogen,” Physical Review B 57, 14105-14109 (1998).
[CrossRef]

Sinha, K.

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

Vega, C.

E. Noya, C. Menduiña, J. Aragones, and C. Vega, “Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI as obtained from computer simulation,” J. Phys. Chem. C 111, 15877-15888 (2007).
[CrossRef]

Verma, S.

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

Weber, M. J.

M. J.Weber, Handbook of Optical Materials (CRC Press, 2003), pp. 373-405.

Wolanin, E.

E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
[CrossRef]

Yadav, H.

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

Zaitzev, A.

A. Zaitzev, Optical Properties of Diamond: A Data Handbook, (Springer, 2001), pp. i-9.

Zhuravlev, K.

M. McCluskey, K. Zhuravlev, B. Davidson, and R. Newman, “Acceptor-hydrogen in semiconductors under pressure,” Physica B 308-310, 780-783 (2001).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

H. Yadav, D. Murty, S. Verma, K. Sinha, B. Gupta, and D. Chand, “Measurement of refractive index of water under high dynamic pressures,” J. Appl. Phys. 44, 2197-2200 (1973).
[CrossRef]

J. Chem. Phys. (2)

M. Choukron and O. Grasset, “Thermodynamic model for water and high-pressure ices up to 2.2 GPa and down to the metastable domain,” J. Chem. Phys. 127, 124506 (2007).
[CrossRef]

Y. Fei, H. Mao, and R. Hemley, “Thermal expansivity, bulk modulus, and melting curve of H2O-ice VII to 20 GPa,” J. Chem. Phys. 99, 5369-5373 (1993).
[CrossRef]

J. Phys. Chem. C (1)

E. Noya, C. Menduiña, J. Aragones, and C. Vega, “Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI as obtained from computer simulation,” J. Phys. Chem. C 111, 15877-15888 (2007).
[CrossRef]

Phys. Rev. B (2)

E. Wolanin, Ph. Pruzan, J. Chervin, B. Canny, and M. Gauthier, “Equation of state of ice VII up to 106 GPa,” Phys. Rev. B 56, 5781-5785 (1997).
[CrossRef]

A. Dewaele, J. Eggert, P. Loubeyre, and R. LeToullec, “Measurement of refractive index and equation of state in dense He, H2, H2O, and Ne under high pressure in a diamond anvil cell,” Phys. Rev. B 67, 094112 (2003).
[CrossRef]

Physica B (1)

M. McCluskey, K. Zhuravlev, B. Davidson, and R. Newman, “Acceptor-hydrogen in semiconductors under pressure,” Physica B 308-310, 780-783 (2001).
[CrossRef]

Physical Review B (1)

W. Evans and I. Silvera, “Index of refraction, polarizability, and equation of state of solid molecular hydrogen,” Physical Review B 57, 14105-14109 (1998).
[CrossRef]

Rev. Mod. Phys. (1)

A. Jayaraman, “Diamond anvil cell and high-pressure physical investigations,” Rev. Mod. Phys. 55, 65-108 (1983).
[CrossRef]

Other (7)

A. Zaitzev, Optical Properties of Diamond: A Data Handbook, (Springer, 2001), pp. i-9.

M. D. McCluskey, B. D. Riley, A. M. Perenchio, and M. Knoblauch, “Confocal microscopy of water under static pressure,” Shock Compression of Condensed Matter--2007, AIP Conference Proceedings, M. Elert, M. D. Furnish, R. Chau, N. Holmes, and J. Nguyen, eds. (American Institute of Physics, 2007), Vol. 955, pp. 1109-1112.

J. Pawley, Handbook of Biological Confocal Microscopy (Springer, 1995), p. 4.

A. E. Siegman, Lasers (University Science Books, 1986), pp. 663-697.

W. S. Rasband, ImageJ (U.S. National Institutes of Health, 1997-2008), http://rsb.info.nih.gov/ij/.

M. J.Weber, Handbook of Optical Materials (CRC Press, 2003), pp. 373-405.

R. Courchinoux and P. Lalle, “Dynamic properties of water: sound velocity and refractive index,” in AIP Conference Proceedings (American Institute of Physics, 1996), Vol. 370, pp. 61-64.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Piston-cylinder DAC, not to scale: (a) piston, (b) cylinder, (c) and (d) backing plates that allow alignment of diamonds, and (e) gasket. Dashed lines, allen screws that are tightened to increase pressure. In this work, the sample volume (between the diamond faces) is typically 100 μm thick and 300 μm diameter.

Fig. 2
Fig. 2

Schematic of confocal microscope. Collimated laser light (a) is focused by the objective lens (b) onto the sample (c). Reflected light (arrows) is gathered by the objective and passes through the tube lens (d). The pinhole (e) rejects light emitted away from the laser focus and admits light into the detector (f). The coordinates are listed on the left side.

Fig. 3
Fig. 3

Reflections and optical thickness. R 01 is the reflection coefficient for interface 01; R 12 is defined analogously. The incident laser (a) has power P 0 . The light reflected (b) from 01 has power P 0 R 01 . The light transmitted (c) through 01 has power P 0 ( 1 R 01 ) . It is then reflected (d) from interface 12 with power P 0 ( 1 R 01 ) R 12 , then transmitted (e) through 01 with power P 0 ( 1 R 01 ) 2 R 12 . If d is the true distance between two interfaces, then z 2 = z 1 + d / n 1 ; the dashed lines show the apparent origin of the beam reflected from 12.

Fig. 4
Fig. 4

One slice of the image stack for the diamond anvil cell. The dark area is the shadow of the ruby chip used for pressure measurement. The light circle is the reflection from the second diamond–water interface. The surrounding gray ring is the reflection from the steel gasket.

Fig. 5
Fig. 5

Reflected intensity profile: 1, diamond–air interface; 2, diamond–sample interface; and 3, diamond–sample interface. Solid lines are the fitted reflected intensity profile.

Fig. 6
Fig. 6

Calibration curve for refractive index. The peak ratios Q 2 / Q 1 were the quantities actually fitted, but the resulting index measurements [Eqs. (20, 21)] are very nearly linear themselves.

Fig. 7
Fig. 7

The measured refractive index as a function of density. See Table 1 for comparison with published values.

Fig. 8
Fig. 8

Volume of the DAC, calculated from our measurements of optical thickness, refractive index, and area, is compared to equations of state for water [14], ice VI [15], and ice VII [16] (solid lines).

Tables (2)

Tables Icon

Table 1 Refractive Index as a Linear Function of Density

Tables Icon

Table 2 Equation-of-State Parameters for Liquid Water at 300 K [Eq. (22)] a

Equations (22)

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

d i f = d o f ( d o f ) 2 + R o 2 f 2 ,
R i = f 2 ( d o f ) 2 + R o 2 R o ,
w i = f [ ( d o f ) 2 + R o 2 ] 1 / 2 w o .
z 1 = 2 ( z z i ) 4 ( z z i ) 2 + R 2 f 1 2 + z + 2 f 1 ,
R 1 = f 1 2 4 ( z z i ) 2 + R 2 R ,
w 1 = f 1 [ 4 ( z z i ) 2 + R 2 ] 1 / 2 w ,
z 2 = L + q 2 f 2 2 p q 2 ( 1 2 q 2 + f 1 2 ) ( q 2 R 2 ) 1 / 2 [ p q 2 ( 1 2 q 2 + f 1 2 ) ( q 2 R 2 ) 1 / 2 ] 2 + f 1 4 R 2 ,
R 2 = f 1 2 f 2 2 q 2 [ p q 2 ( 1 2 q 2 + f 1 2 ) ( q 2 R 2 ) 1 / 2 ] 2 + f 1 4 R 2 R ,
w 2 = f 1 f 2 q { [ p q 2 ( 1 2 q 2 + f 1 2 ) ( q 2 R 2 ) 1 / 2 ] 2 + f 1 4 R 2 } 1 / 2 w ,
z 2 L 2 f 2 2 f 1 2 ( z z i ) ,
R 2 f 2 2 f 1 2 R ,
w 2 f 2 f 1 w .
P ( z 2 ) P = Q i [ 1 exp ( D 2 2 w 2 2 R 2 2 ( L z 2 ) 2 + R 2 2 ) ] .
f i ( z ) = Q i [ 1 exp ( D 2 2 w 2 f 1 2 f 2 2 R 2 4 ( z z i ) 2 + R 2 ) ] .
f ( z ) = i = 1 4 f i ( z ) .
Q 1 = R 10 ,
Q 2 = ( 1 R 10 ) 2 R 12 ,
Q 3 = ( 1 R 10 ) 2 ( 1 R 12 ) 2 R 12 ,
Q 4 = ( 1 R 10 ) 2 ( 1 R 12 ) 2 ( 1 R 12 ) 2 R 10 .
n 2 = ( 1 x 1 + x ) n 1 ,
x 2 Q 2 Q 1 R 10 ( 1 R 10 ) 2 ,
V V 0 = a P 1 a P 2 ( P 0 P ) a P 3 ( P 0 P ) a P 4 .

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