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Frequency dependence and the electromagnetic susceptibility tensor in magnetic fabric studies — a discussion
PttYSICS OFTHE EARTH AN D PLANETARY INTERIORS
ELSEVIER
Physicsof the Earth and PlanetaryInteriors 89 (1995) 145-147
Frequency dependence and the electromagnetic susceptibility tensor in magnetic fabric studies a discussion R. Puranen *, A. Poikonen, M. Puranen Departmentof Geophysics, GeologicalSurveyof Finland, SF-02150Espoo, Finland Received 23 March 1993; revisionaccepted 23 August 1994
Abstract
It has been proposed that the magnetic susceptibilities measured with AC bridges at high frequencies ( f > 500 Hz) can be affected by high conductivities of samples. With model calculations ( f = 1000 Hz) we demonstrate for spherical samples (radius of 14 mm) that conductivity effects exceeding 0.1% are indeed expected in susceptibilities of high-conductivity samples (tr > 100 S m-1). However, the conductivity of most common rocks (~ < 0.01 S m-1) affects their susceptibility measurements by less than 0.00001%.
Ellwood et al. (1993) demonstrated with lowfield ( < 5 mT) torque meter measurements that palaeomagnetic sized samples containing highly conducting material show an increase in susceptibility at the high end of the frequency range 50-5000 Hz. They also proposed that such conductivity effects may be expected in AC bridge measurements of susceptibilities made at frequencies of 500 Hz or higher. We would like to elaborate on this proposal with the aid of results obtained while developing low-field ( < 0.05 mT) AC bridge apparatus ( [ = 1025 Hz) for measuring the susceptibility (Puranen and Puranen, 1977) and conductivity (Puranen et al., 1994) of geological samples. A sample inside the coil of an AC bridge can be modelled as a sphere placed in a homogeneous, alternating magnetic field H 0. The sphere
behaves as a dipole and its moment can be expressed in the form (Wait, 1951)
M = -27ra3Hoexp(itot)(X+ iY)
where a is the radius of the sphere, and to is the angular frequency of the magnetic field. The complex response function X + iY is determined by the formula
X + i v = [(/~o + Iz0g 2a2 + 21z) sinh (ga) - g a ( e l z + IZ0) cosh (ga)] × [(/~0 + ~ 0 g Z a 2 - / z ) sinh (ga)
+ga(t x - i z 0 ) cosh ( g a ) ] - '
(2)
where gt is the permeability of the sphere and iz 0 is the permeability of air. The quantity g2a2 is defined by relations
g2a2=itol~tra2=i2~rf(1 + K)lxo~ra2=iW * Corresponding author.
(1)
(3)
where K is susceptibility, tr is conductivity and W is the response parameter of the sphere. Near the
0031-9201/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0031-9201(94)02990-3
R. Puranen et al. / Physics of the Earth and Planetary Interiors 89 (1995) 145-147
146
resistive limit (at low frequencies) the higher order terms in the series expansions of the hyperbolic functions approach zero, together with the response parameter, and the in-phase and quadrature parts of function (2) then reduce into the approximate form
10
a~
/
"/
~Z~ ~/
/
/
iI
X= -2K/(3 +K) Y= (3/5)(1
(4)
+K)W/(3 + K ) 2
(5)
For purely conducting spheres (K = 0) we obtain the simple linear relation Y= WI15, which can be applied for estimation of conductivities (Puranen et al., 1994). Based on formula (2) the in-phase (X), quadrature (Y) and amplitude (Z) response of the spherical sample can be calculated as a function of the response parameter (14,'). The computed response curves are depicted in Fig. 1 for susceptibility K = 0.5 (SI), which can be regarded as a maximum value for rock samples. If the frequency, sample radius and susceptibility remain constant, the response parameter is directly proportional to sample conductivity. As W (and or) go to zero, the response curves attain the resistive limits X 0 = - 2 K / ( 3 + K ) , Y0=0 and Z0= 2K/(3 + K ) , which provide the basis for susceptibility measurements. At higher W values the responses become curved and the negative X 1.0
//
--~ 0.5 ~1~
...
Z It) C O
m rr
." /"
-0.5 0.001
I 0.01
I 0.1
"" " . .
. -...
/
//
/
/ I 1
"...
/
/
Y Y 0 ..................................... '
X
/
/ '7..
I I Z 2 = × 2 + y 2l
~ 10
~ 100
1000
Response parameter W
Fig. 1, In-phase (X), quadrature (Y) and amplitude (Z) response of a conducting,permeable (K = 0.5) sphere in a homogeneous alternating magneticfield. K is susceptibility and 1¥ is response parameter definedby Eq. (3).
>
"~
o
== o 0
/I
/~ ol
/
I
/
1
/,'1 l
'
'R/
/
,
/,',f
,'1
~ / ? /,' //~ // //Z' // / ,%o-/ / / /, /,/~!,,
'/, "
o.ol 10
/,
100
Conductivity
1000
// " 104
1
105
(S/m)
Fig. 2. Dependence between sample conductivity and conductivity effects (see text) in measurements of susceptibility ( K ) based on in-phase response (dashed lines) and amplitude response (solid lines). Measuring frequency is 1000 Hz and the volume of spherical samples is 11.5 cm 3.
response changes its sign (Fig. 1). As a result, with increasing conductivities the susceptibility estimates based on Z response become increasingly too high, whereas the estimates based on X response become increasingly too low. After fixing the measuring frequency ( f = 1000 Hz) and sample size (a = 14 mm), the X and Z responses can be calculated as a function of conductivity for samples with different susceptibilities. These responses are then compared with the responses X 0 and Z 0 of purely resistive samples to determine the conductivity effects ( X Xo)/I X0 I and (Z - Zo)/Z o of Fig. 2. The figure also illustrates the effect of conductivity on susceptibility measurements. The measurements of weakly magnetic samples ( K = 0.0001) are affected at the lowest conductivities, and the Z response measurements are affected first. To observe effects of 1% and 0.01% on such low susceptibilities, the sensitivity of susceptibility measurements must be 10 - 6 and 10 -8 (SI), respectively. The former sensitivity is that of our AC bridges, and the latter is achieved by the most sensitive instruments. A weak conductivity effect just exceeding 0.01% is observed in the X response of weakly magnetic samples when or > 1000 S m -1 and in
R. Puranen et al. / Physics o f the Earth and Planetary Interiors 89 (1995) 145-147
the Z response when ~r> 10 S m -1 (Fig. 2). Weak effects are thus possible for samples rich in graphite, pyrrhotite or compact ore material. Significant conductivity effects can also be expected if a small copper sphere (a = 2 mm, K = 0, ~r = 10 8 S m - 1 ) is embedded into a palaeomagnetic sized sample (cf. Ellwood et al., 1993). The response parameter of the copper sphere is so high (W--3.2) that it clearly affects the X and Z responses (Fig. 1) of even strongly magnetic (K = 0.5) samples. Nevertheless, the conductivity of most common rocks (or < 0.01 S m - l ) affects the measurement of their susceptibility ( K < 0.1) by less than 0.00001%. Both the in-phase and quadrature responses are measured with our AC bridge apparatus so that susceptibilities, conductivities and conductivity effects can be estimated. In practice, the evaluation of conductivity effects can be complicated by magnetic viscosity effects (Vincenz, 1965). However, if only the in-phase or amplitude response is determined with a measuring system,
147
the conductivity effects cannot be evaluated unless the measurements are made at several frequencies, as demonstrated by Ellwood et al. (1993).
References Ellwood, B.B., Terrell, G.E. and Cook, W.J., 1993. Frequency dependence and the electromagnetic susceptibility tensor in magnetic fabric studies. Phys. Earth Planet. Inter., 80: 65-74. Puranen, M. and Puranen, R., 1977. Apparatus for the measurement of magnetic susceptibility and its anisotropy. Report of Investigation, Vol. 28, Geological Survey of Finland, 46 pp. Puranen, R., Puranen, M. and Sulkanen, K., 1994. Inductive resistivity measurement with AC-bridge apparatus. J. Appl. Geophys., 32: 99-104. Vincenz, S.A., 1965. Frequency dependence of magnetic susceptibility of rocks in weak alternating fields. J. Geophys. Res., 70: 1371-1377. Wait, J.R., 1951. A conducting sphere in a time varying magnetic field. Geophysics, 16: 666-672.
ELSEVIER
Physicsof the Earth and PlanetaryInteriors 89 (1995) 145-147
Frequency dependence and the electromagnetic susceptibility tensor in magnetic fabric studies a discussion R. Puranen *, A. Poikonen, M. Puranen Departmentof Geophysics, GeologicalSurveyof Finland, SF-02150Espoo, Finland Received 23 March 1993; revisionaccepted 23 August 1994
Abstract
It has been proposed that the magnetic susceptibilities measured with AC bridges at high frequencies ( f > 500 Hz) can be affected by high conductivities of samples. With model calculations ( f = 1000 Hz) we demonstrate for spherical samples (radius of 14 mm) that conductivity effects exceeding 0.1% are indeed expected in susceptibilities of high-conductivity samples (tr > 100 S m-1). However, the conductivity of most common rocks (~ < 0.01 S m-1) affects their susceptibility measurements by less than 0.00001%.
Ellwood et al. (1993) demonstrated with lowfield ( < 5 mT) torque meter measurements that palaeomagnetic sized samples containing highly conducting material show an increase in susceptibility at the high end of the frequency range 50-5000 Hz. They also proposed that such conductivity effects may be expected in AC bridge measurements of susceptibilities made at frequencies of 500 Hz or higher. We would like to elaborate on this proposal with the aid of results obtained while developing low-field ( < 0.05 mT) AC bridge apparatus ( [ = 1025 Hz) for measuring the susceptibility (Puranen and Puranen, 1977) and conductivity (Puranen et al., 1994) of geological samples. A sample inside the coil of an AC bridge can be modelled as a sphere placed in a homogeneous, alternating magnetic field H 0. The sphere
behaves as a dipole and its moment can be expressed in the form (Wait, 1951)
M = -27ra3Hoexp(itot)(X+ iY)
where a is the radius of the sphere, and to is the angular frequency of the magnetic field. The complex response function X + iY is determined by the formula
X + i v = [(/~o + Iz0g 2a2 + 21z) sinh (ga) - g a ( e l z + IZ0) cosh (ga)] × [(/~0 + ~ 0 g Z a 2 - / z ) sinh (ga)
+ga(t x - i z 0 ) cosh ( g a ) ] - '
(2)
where gt is the permeability of the sphere and iz 0 is the permeability of air. The quantity g2a2 is defined by relations
g2a2=itol~tra2=i2~rf(1 + K)lxo~ra2=iW * Corresponding author.
(1)
(3)
where K is susceptibility, tr is conductivity and W is the response parameter of the sphere. Near the
0031-9201/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0031-9201(94)02990-3
R. Puranen et al. / Physics of the Earth and Planetary Interiors 89 (1995) 145-147
146
resistive limit (at low frequencies) the higher order terms in the series expansions of the hyperbolic functions approach zero, together with the response parameter, and the in-phase and quadrature parts of function (2) then reduce into the approximate form
10
a~
/
"/
~Z~ ~/
/
/
iI
X= -2K/(3 +K) Y= (3/5)(1
(4)
+K)W/(3 + K ) 2
(5)
For purely conducting spheres (K = 0) we obtain the simple linear relation Y= WI15, which can be applied for estimation of conductivities (Puranen et al., 1994). Based on formula (2) the in-phase (X), quadrature (Y) and amplitude (Z) response of the spherical sample can be calculated as a function of the response parameter (14,'). The computed response curves are depicted in Fig. 1 for susceptibility K = 0.5 (SI), which can be regarded as a maximum value for rock samples. If the frequency, sample radius and susceptibility remain constant, the response parameter is directly proportional to sample conductivity. As W (and or) go to zero, the response curves attain the resistive limits X 0 = - 2 K / ( 3 + K ) , Y0=0 and Z0= 2K/(3 + K ) , which provide the basis for susceptibility measurements. At higher W values the responses become curved and the negative X 1.0
//
--~ 0.5 ~1~
...
Z It) C O
m rr
." /"
-0.5 0.001
I 0.01
I 0.1
"" " . .
. -...
/
//
/
/ I 1
"...
/
/
Y Y 0 ..................................... '
X
/
/ '7..
I I Z 2 = × 2 + y 2l
~ 10
~ 100
1000
Response parameter W
Fig. 1, In-phase (X), quadrature (Y) and amplitude (Z) response of a conducting,permeable (K = 0.5) sphere in a homogeneous alternating magneticfield. K is susceptibility and 1¥ is response parameter definedby Eq. (3).
>
"~
o
== o 0
/I
/~ ol
/
I
/
1
/,'1 l
'
'R/
/
,
/,',f
,'1
~ / ? /,' //~ // //Z' // / ,%o-/ / / /, /,/~!,,
'/, "
o.ol 10
/,
100
Conductivity
1000
// " 104
1
105
(S/m)
Fig. 2. Dependence between sample conductivity and conductivity effects (see text) in measurements of susceptibility ( K ) based on in-phase response (dashed lines) and amplitude response (solid lines). Measuring frequency is 1000 Hz and the volume of spherical samples is 11.5 cm 3.
response changes its sign (Fig. 1). As a result, with increasing conductivities the susceptibility estimates based on Z response become increasingly too high, whereas the estimates based on X response become increasingly too low. After fixing the measuring frequency ( f = 1000 Hz) and sample size (a = 14 mm), the X and Z responses can be calculated as a function of conductivity for samples with different susceptibilities. These responses are then compared with the responses X 0 and Z 0 of purely resistive samples to determine the conductivity effects ( X Xo)/I X0 I and (Z - Zo)/Z o of Fig. 2. The figure also illustrates the effect of conductivity on susceptibility measurements. The measurements of weakly magnetic samples ( K = 0.0001) are affected at the lowest conductivities, and the Z response measurements are affected first. To observe effects of 1% and 0.01% on such low susceptibilities, the sensitivity of susceptibility measurements must be 10 - 6 and 10 -8 (SI), respectively. The former sensitivity is that of our AC bridges, and the latter is achieved by the most sensitive instruments. A weak conductivity effect just exceeding 0.01% is observed in the X response of weakly magnetic samples when or > 1000 S m -1 and in
R. Puranen et al. / Physics o f the Earth and Planetary Interiors 89 (1995) 145-147
the Z response when ~r> 10 S m -1 (Fig. 2). Weak effects are thus possible for samples rich in graphite, pyrrhotite or compact ore material. Significant conductivity effects can also be expected if a small copper sphere (a = 2 mm, K = 0, ~r = 10 8 S m - 1 ) is embedded into a palaeomagnetic sized sample (cf. Ellwood et al., 1993). The response parameter of the copper sphere is so high (W--3.2) that it clearly affects the X and Z responses (Fig. 1) of even strongly magnetic (K = 0.5) samples. Nevertheless, the conductivity of most common rocks (or < 0.01 S m - l ) affects the measurement of their susceptibility ( K < 0.1) by less than 0.00001%. Both the in-phase and quadrature responses are measured with our AC bridge apparatus so that susceptibilities, conductivities and conductivity effects can be estimated. In practice, the evaluation of conductivity effects can be complicated by magnetic viscosity effects (Vincenz, 1965). However, if only the in-phase or amplitude response is determined with a measuring system,
147
the conductivity effects cannot be evaluated unless the measurements are made at several frequencies, as demonstrated by Ellwood et al. (1993).
References Ellwood, B.B., Terrell, G.E. and Cook, W.J., 1993. Frequency dependence and the electromagnetic susceptibility tensor in magnetic fabric studies. Phys. Earth Planet. Inter., 80: 65-74. Puranen, M. and Puranen, R., 1977. Apparatus for the measurement of magnetic susceptibility and its anisotropy. Report of Investigation, Vol. 28, Geological Survey of Finland, 46 pp. Puranen, R., Puranen, M. and Sulkanen, K., 1994. Inductive resistivity measurement with AC-bridge apparatus. J. Appl. Geophys., 32: 99-104. Vincenz, S.A., 1965. Frequency dependence of magnetic susceptibility of rocks in weak alternating fields. J. Geophys. Res., 70: 1371-1377. Wait, J.R., 1951. A conducting sphere in a time varying magnetic field. Geophysics, 16: 666-672.