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Electron spin resonance in the nearly one-dimensional magnetic system CuCl2 · 2H2O
Volume 73A, number 1
PHYSICS LETTERS
20 August 1979
ELECTRON SPIN RESONANCE IN THE NEARLY ONE-DIMENSIONAL MAGNETIC SYSTEM
CuCl, - 2H,O J.S. KARRA and G.E. KEMMERER Temple University,Philadelphia,PA 19122, USA Received 10 April 1979 Revised manuscript received 31 May 1979
The line shape of electron spin resonance in CuCl2 - 2H20 exh&its one-dimensional character for 0 = 0” where B is the angle between the axis of the crystal and the magnetic field. The line shapes for 0 = 54” and 90” are lorentzian.
Although it has been reported [l-5] that CuCl, 2H20 is a nearly one-dimensional (1 d) Heisenberg magnetic system, DeIong et al. [6] have recently interpreted the specific heat data [1] to suggest that the compound is three dimensional (3d). Therefore, we have studied the line widths and the line shapes of electron spin resonance (ESR) in the CuCl, * 2H20 crystal to determine its Id character. There are several theoretical treatments [7-91 of ESR line shapes and line widths in nearly Id systems and we have followed in this paper the one presented by Hennessey et al. [7] to interpret our experimental results. The measurements were made at room temperature, at x-band microwave frequency and at different angles of orientation (0) of the external magnetic field ZZowith respect to the linear magnetic chain axis (c-axis) of the crystal. The line shapes at 0 = 54” and 90” are lorentzian. However, the line shape at 19= 0’ is highly non-lorentzian and has the characteristics of a one-dimensional magnetic system whose relaxation function @(t) = exp[-(Tt)3/2] (eq. (6) in ref. [7]), where r2j3 is proportional to the half width (AZZ1i2)at half intensity. The reader is referred to ref. [7] (eqs. (7) and (SO)) for the definitions of 7 and n, the respective contributions to the half width from the secular and non-secular parts of the dipolar interaction. Previous studies of this compound by ESR [lo] report the spin “S” of the Cu* ion to be a half and the g values along the three axes to be ga = 2.187, gb = 2.037, and gc = 2.252. There are two CUE ions in each l
l
unit cell [ 1l] and they occupy two different lattice sites (0, 0,O) and (i ,i, a). The lattice constants of theunitcellarea=7.38A,b=8.04Aandc=3.72A. The pure absorption modes of the ESR lines were recorded without using the external field modulation to eliminate the distortion of the line shape. The ESR line shapes for 19= 0”, 54” and 90” are presented in figs. 1 and 2, and their line widths (AH,,,) are presented in table 1. The ratio of the intrachain exchange interaction constant Jo along the axis to the interchain exchange interaction constant J along the a and b axes has been estimated by several authors and these estimates range from 0.13 to 0.55. In interpreting the present data we made use of the value 6.78 K for Jo/ZcB given in ref. [5], which has been calculated using the Green’s function method. For J’ the following relation given in ref. [7] was used, assuming the interchain exchange along the a and b axes to be equal:
with TN = 4.33 K and Z = 0.64(.Zo/.Z’)1Z2. The factor fhas different values depending on which one of the three decoupling procedures, RPA, Callen’s or Tahir-Kheli’s (TK) was used. In the present calculation of J’, all three decoupling schemes were used resulting in three different values for J which are presented in table 1. It has been suggested by Hennessey et al. [7] that 63
Volume 73A, number 1
PHYSICS LETFERS 50
-
45
-
40
-
20 August 1979
I
35-
B A
30-
~ 25
-
20
-
5
-
10
-
•°6°54°
5I
IC
5
I ....I....
I....I
,,..I..,,
5 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 2
U
Fig. 1. Abscissa ~,, the angular frequency, and the ordinate (I/by’, where ‘~is the maximum height of ESR at w = 0 and I is the height of ESR at c~.Solid line A is the lorentzian line (theory) with a half width at halfintensity i~H 112= 28 G. Solid line B is the gaussian line (theory) 2 and oneforgauss the is same 1.~6 half x 10” width. Hz/sSolid along circles theeare axisthe and experimental 1.8 X iO~Hz/s dataalong pointsthe at b0 axis. 54°.One div. on the horizontal axis is (20 G) 50’
~
4
EXP-(3t) D
~
A
40
B
35
-, —
30 ,
-... 25
——
-
,
ID
e~90~
$ w°5
Ae,oo
-
5. .,.,I..~
5
10
15
20
I.,..I,,,,I,,,.I,...1
25 30 35 40 45 50 55 60
.,,.I.,,.
65 70
I
75
I
80 85
90 95
00
U2
Fig. 2. Abscissa and ordinate axe the same as in fig. 1. Solid line A (0 0°)is the theoretical plot of the Fourier transform of exp[—(i’t)3”2 I for ~ 29 G. Solid line B (0 = 90°)and the solid and broken line C (0 = 0°)are the theoretical lorentzian line shapes with half widths 26 G and 29 G, respectively. Solid line D (0 0°)is the theoretical gaussian line for half width 29 G. Circles and triangles are the experimental data points at 0 = 90° and 0°,respectively. The arrow at w = 5 denotes the cut-offpoint for the id diffusive behavior at 0 = 0°.
64
Volume 73A, number 1
PHYSICS LET1~ERS
20 August 1979
Table 1 0
j/J
observed width (G)
t
0
7t0
(s)
calculated (G)
0°
32 32 32
0.06 (Callen) 0.16 (RPA) 0.20 (TK)
5.7 X 10-11 1.7 X lO~~ 1.05 X 10-11
0.24 0.07 0.04
82b) 45 b) 35 b) 85 C)
90°
13 a)
0.06 (Callen) 0.16 (RPA) 0.20 (TK)
5.7 X 10—11 1.7 x 10—11 1.05 x i0~’
3.1 x 10—2 0.9 x 10—2 0.57 X 10—2
16 9
54°
7
28
a) Contribution to the line width from the secular part of the dipolar interaction only. The observed half width is actually 26 G. b) Assuming the line shape is lorentzian. 3 /21 for ‘y = 29 G. 85 G is the value calcuC) Theassuming experimental curve clearly lated the line to be id. approximates to the Fourier transform of exp [—(vt)
in an ideal ld system the spin correlation function tli(r) decays by spin diffusion as r 1/2 corresponding to the one-dimensional diffusive process. However, in systems which are nearly one dimensional, the interchain exchange J’ gives rise to a three-dimensional behavior after a certain time “t0” defined as the cut-off time for the ld decay to persist. It is given by 3/j’\—4/3 =~ / 32S(S 9 + 1))\2/3 /~\l/ ~j) (3)
~
where D/C2
=
dipolar interaction. However, the nonsecular part contributes to the second moment, Mr~,and consequently to the line width, where the index (M) = ±1, ±2. Therefore, the relaxation function ‘t’(t) is a simple exponential, exp(—~ 54ot), where ~i54°is the half width given by eq. (50) in ref. [7], 0/J 054° = kPiM~ 0, (4) where k is a constant of order unity. The line shape is then lorentzian. The observed line shape (fig. 1) is also lorentzian up to seven half widths and departs from it only slightly thereafter.
ir1/2J
0/h for a spin half system and D is the spin diffusion constant. Corresponding to the three different values of J’, there are three different values for “t0”, which are listed in table 1. If “t0” is sufficiently long sofrom thatthe yt0id ~ line 1 one observes no significant departure shape and width. However, if 7t 0 ~ 1 the line shape tends to be lorentzian as explained in ref. [7]. In calculating the value of ‘y for CuCl2 21120 with the aid of eqs. (7) and (39) in ref. [7], the contributions to the second moment from the these next nearest neighbors are taken to account. Since equations overestimate the ‘yin.
value in TMMC, an almost perfect id system, by a factor of 1.7, our values of ‘y were corrected by the same factor. ESR at 8 = 54°and 90°.In a 1d system there is no contribution to the line width of ESR and to its second moment, ~ at 0 = 54°from the secular part of the
To separate the secular and nonsecular contributions to the line width at 0 = 90°,the following reladon was used to calculate the nonsecular contribution: 0(0= 540)/M~M)(0 = 90°) 054°/fl90° = M~’ where for 054°the experimentally observed half width (29 G) was substituted. 090°thus determined is 13 G and therefore the line width due to the secular part has turned out to be 13 G also. The2is calculated width 16 G, when due to the secular Callen’s value for tpart, 1.5 73/2t,~ 0 is used. There is therefore reasonable agreement between the theory and experiment with regard to the line width. This line is also lorentzian for up to seven line widths and in good agreement with theory. At 0 = 0°,in an ideal 1 d system only the secular part contributes to n4o). Nearest neighbor contribudon to is less than 5%. Even at 0 = 0°,7t0 is still 65
Volume 73A, number 1
PHYSICS LETTERS
less than 1 and the line shape is expected to be lorentzian. However, the observed shape is nonlorentzian and is in good3/2] agreement the calculated Fourier transform for ~ 29with G. The value of offor exp0 [—(‘yt) ~ = 0°is 85 G which is far greater than the deduced value (29 G). This discrepancy may be explained in the following way. The line may be divided into two segments. The first segment is from w = 0 to w = 5 (as indicated by an arrow in fig. 2) and the second segment is for values of w greater than ~ We have decided upon this particular choice because the first segment is lorentzian in shape for up to 3!2 line widths (M1 1/2). Corresponding0s.For to the point times at less cut-offthe time S X l0~ to be id in characthan5,the this value, linet0isisexpected ter. The second segment closely approximates the id line shape. Moreover, this value of t .‘=
0 ascertained from
the line gives for ~t0 a value slightly larger than 1. Therefore, we conclude that the region of observation includes both 3d as well as ld line shapes.
66
20 August 1979
The spin dynamics indicate that CuC12 21120 is a nearly one-dimensional magnetic system. We thank Prof. R. Tahir-Kheli for his help in computer analysis of the line shapes. .
References [1] S.A. Freidberg, Physica 18 (1952) 714. [2] T. Oguchi, Prog. Theor. Phys. 13 (1958) 148. [3] W. Marshall, Phys. Chem. Solids 7 (1958) 159. [4] Nagai, J. Phys. Soc. (Japan) 18 (1963) [5] 0. A.C. Hewson et al., Phys. Rev. 137 (1965)570. 1465. [6] M.J. L.J. De Jong et al., Adv. Phys. 23 B7 (1974) 145. [7] Hennessey et al., Phys. Rev. (1973) 930.
[8] G.F. Reiter and J.P. Boucher, Phys. Rev. Bil (1975) [9] 1823. A. Lagendijk and H. DeRaedt, Phys. Rev. B16 (1977) 293. [10] H.J. Gerritsen et al., Physica 21(1955)629. [11] M. Date et al., J. Appl. Phys. 34 (1961) 1038.
PHYSICS LETTERS
20 August 1979
ELECTRON SPIN RESONANCE IN THE NEARLY ONE-DIMENSIONAL MAGNETIC SYSTEM
CuCl, - 2H,O J.S. KARRA and G.E. KEMMERER Temple University,Philadelphia,PA 19122, USA Received 10 April 1979 Revised manuscript received 31 May 1979
The line shape of electron spin resonance in CuCl2 - 2H20 exh&its one-dimensional character for 0 = 0” where B is the angle between the axis of the crystal and the magnetic field. The line shapes for 0 = 54” and 90” are lorentzian.
Although it has been reported [l-5] that CuCl, 2H20 is a nearly one-dimensional (1 d) Heisenberg magnetic system, DeIong et al. [6] have recently interpreted the specific heat data [1] to suggest that the compound is three dimensional (3d). Therefore, we have studied the line widths and the line shapes of electron spin resonance (ESR) in the CuCl, * 2H20 crystal to determine its Id character. There are several theoretical treatments [7-91 of ESR line shapes and line widths in nearly Id systems and we have followed in this paper the one presented by Hennessey et al. [7] to interpret our experimental results. The measurements were made at room temperature, at x-band microwave frequency and at different angles of orientation (0) of the external magnetic field ZZowith respect to the linear magnetic chain axis (c-axis) of the crystal. The line shapes at 0 = 54” and 90” are lorentzian. However, the line shape at 19= 0’ is highly non-lorentzian and has the characteristics of a one-dimensional magnetic system whose relaxation function @(t) = exp[-(Tt)3/2] (eq. (6) in ref. [7]), where r2j3 is proportional to the half width (AZZ1i2)at half intensity. The reader is referred to ref. [7] (eqs. (7) and (SO)) for the definitions of 7 and n, the respective contributions to the half width from the secular and non-secular parts of the dipolar interaction. Previous studies of this compound by ESR [lo] report the spin “S” of the Cu* ion to be a half and the g values along the three axes to be ga = 2.187, gb = 2.037, and gc = 2.252. There are two CUE ions in each l
l
unit cell [ 1l] and they occupy two different lattice sites (0, 0,O) and (i ,i, a). The lattice constants of theunitcellarea=7.38A,b=8.04Aandc=3.72A. The pure absorption modes of the ESR lines were recorded without using the external field modulation to eliminate the distortion of the line shape. The ESR line shapes for 19= 0”, 54” and 90” are presented in figs. 1 and 2, and their line widths (AH,,,) are presented in table 1. The ratio of the intrachain exchange interaction constant Jo along the axis to the interchain exchange interaction constant J along the a and b axes has been estimated by several authors and these estimates range from 0.13 to 0.55. In interpreting the present data we made use of the value 6.78 K for Jo/ZcB given in ref. [5], which has been calculated using the Green’s function method. For J’ the following relation given in ref. [7] was used, assuming the interchain exchange along the a and b axes to be equal:
with TN = 4.33 K and Z = 0.64(.Zo/.Z’)1Z2. The factor fhas different values depending on which one of the three decoupling procedures, RPA, Callen’s or Tahir-Kheli’s (TK) was used. In the present calculation of J’, all three decoupling schemes were used resulting in three different values for J which are presented in table 1. It has been suggested by Hennessey et al. [7] that 63
Volume 73A, number 1
PHYSICS LETFERS 50
-
45
-
40
-
20 August 1979
I
35-
B A
30-
~ 25
-
20
-
5
-
10
-
•°6°54°
5I
IC
5
I ....I....
I....I
,,..I..,,
5 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 2
U
Fig. 1. Abscissa ~,, the angular frequency, and the ordinate (I/by’, where ‘~is the maximum height of ESR at w = 0 and I is the height of ESR at c~.Solid line A is the lorentzian line (theory) with a half width at halfintensity i~H 112= 28 G. Solid line B is the gaussian line (theory) 2 and oneforgauss the is same 1.~6 half x 10” width. Hz/sSolid along circles theeare axisthe and experimental 1.8 X iO~Hz/s dataalong pointsthe at b0 axis. 54°.One div. on the horizontal axis is (20 G) 50’
~
4
EXP-(3t) D
~
A
40
B
35
-, —
30 ,
-... 25
——
-
,
ID
e~90~
$ w°5
Ae,oo
-
5. .,.,I..~
5
10
15
20
I.,..I,,,,I,,,.I,...1
25 30 35 40 45 50 55 60
.,,.I.,,.
65 70
I
75
I
80 85
90 95
00
U2
Fig. 2. Abscissa and ordinate axe the same as in fig. 1. Solid line A (0 0°)is the theoretical plot of the Fourier transform of exp[—(i’t)3”2 I for ~ 29 G. Solid line B (0 = 90°)and the solid and broken line C (0 = 0°)are the theoretical lorentzian line shapes with half widths 26 G and 29 G, respectively. Solid line D (0 0°)is the theoretical gaussian line for half width 29 G. Circles and triangles are the experimental data points at 0 = 90° and 0°,respectively. The arrow at w = 5 denotes the cut-offpoint for the id diffusive behavior at 0 = 0°.
64
Volume 73A, number 1
PHYSICS LET1~ERS
20 August 1979
Table 1 0
j/J
observed width (G)
t
0
7t0
(s)
calculated (G)
0°
32 32 32
0.06 (Callen) 0.16 (RPA) 0.20 (TK)
5.7 X 10-11 1.7 X lO~~ 1.05 X 10-11
0.24 0.07 0.04
82b) 45 b) 35 b) 85 C)
90°
13 a)
0.06 (Callen) 0.16 (RPA) 0.20 (TK)
5.7 X 10—11 1.7 x 10—11 1.05 x i0~’
3.1 x 10—2 0.9 x 10—2 0.57 X 10—2
16 9
54°
7
28
a) Contribution to the line width from the secular part of the dipolar interaction only. The observed half width is actually 26 G. b) Assuming the line shape is lorentzian. 3 /21 for ‘y = 29 G. 85 G is the value calcuC) Theassuming experimental curve clearly lated the line to be id. approximates to the Fourier transform of exp [—(vt)
in an ideal ld system the spin correlation function tli(r) decays by spin diffusion as r 1/2 corresponding to the one-dimensional diffusive process. However, in systems which are nearly one dimensional, the interchain exchange J’ gives rise to a three-dimensional behavior after a certain time “t0” defined as the cut-off time for the ld decay to persist. It is given by 3/j’\—4/3 =~ / 32S(S 9 + 1))\2/3 /~\l/ ~j) (3)
~
where D/C2
=
dipolar interaction. However, the nonsecular part contributes to the second moment, Mr~,and consequently to the line width, where the index (M) = ±1, ±2. Therefore, the relaxation function ‘t’(t) is a simple exponential, exp(—~ 54ot), where ~i54°is the half width given by eq. (50) in ref. [7], 0/J 054° = kPiM~ 0, (4) where k is a constant of order unity. The line shape is then lorentzian. The observed line shape (fig. 1) is also lorentzian up to seven half widths and departs from it only slightly thereafter.
ir1/2J
0/h for a spin half system and D is the spin diffusion constant. Corresponding to the three different values of J’, there are three different values for “t0”, which are listed in table 1. If “t0” is sufficiently long sofrom thatthe yt0id ~ line 1 one observes no significant departure shape and width. However, if 7t 0 ~ 1 the line shape tends to be lorentzian as explained in ref. [7]. In calculating the value of ‘y for CuCl2 21120 with the aid of eqs. (7) and (39) in ref. [7], the contributions to the second moment from the these next nearest neighbors are taken to account. Since equations overestimate the ‘yin.
value in TMMC, an almost perfect id system, by a factor of 1.7, our values of ‘y were corrected by the same factor. ESR at 8 = 54°and 90°.In a 1d system there is no contribution to the line width of ESR and to its second moment, ~ at 0 = 54°from the secular part of the
To separate the secular and nonsecular contributions to the line width at 0 = 90°,the following reladon was used to calculate the nonsecular contribution: 0(0= 540)/M~M)(0 = 90°) 054°/fl90° = M~’ where for 054°the experimentally observed half width (29 G) was substituted. 090°thus determined is 13 G and therefore the line width due to the secular part has turned out to be 13 G also. The2is calculated width 16 G, when due to the secular Callen’s value for tpart, 1.5 73/2t,~ 0 is used. There is therefore reasonable agreement between the theory and experiment with regard to the line width. This line is also lorentzian for up to seven line widths and in good agreement with theory. At 0 = 0°,in an ideal 1 d system only the secular part contributes to n4o). Nearest neighbor contribudon to is less than 5%. Even at 0 = 0°,7t0 is still 65
Volume 73A, number 1
PHYSICS LETTERS
less than 1 and the line shape is expected to be lorentzian. However, the observed shape is nonlorentzian and is in good3/2] agreement the calculated Fourier transform for ~ 29with G. The value of offor exp0 [—(‘yt) ~ = 0°is 85 G which is far greater than the deduced value (29 G). This discrepancy may be explained in the following way. The line may be divided into two segments. The first segment is from w = 0 to w = 5 (as indicated by an arrow in fig. 2) and the second segment is for values of w greater than ~ We have decided upon this particular choice because the first segment is lorentzian in shape for up to 3!2 line widths (M1 1/2). Corresponding0s.For to the point times at less cut-offthe time S X l0~ to be id in characthan5,the this value, linet0isisexpected ter. The second segment closely approximates the id line shape. Moreover, this value of t .‘=
0 ascertained from
the line gives for ~t0 a value slightly larger than 1. Therefore, we conclude that the region of observation includes both 3d as well as ld line shapes.
66
20 August 1979
The spin dynamics indicate that CuC12 21120 is a nearly one-dimensional magnetic system. We thank Prof. R. Tahir-Kheli for his help in computer analysis of the line shapes. .
References [1] S.A. Freidberg, Physica 18 (1952) 714. [2] T. Oguchi, Prog. Theor. Phys. 13 (1958) 148. [3] W. Marshall, Phys. Chem. Solids 7 (1958) 159. [4] Nagai, J. Phys. Soc. (Japan) 18 (1963) [5] 0. A.C. Hewson et al., Phys. Rev. 137 (1965)570. 1465. [6] M.J. L.J. De Jong et al., Adv. Phys. 23 B7 (1974) 145. [7] Hennessey et al., Phys. Rev. (1973) 930.
[8] G.F. Reiter and J.P. Boucher, Phys. Rev. Bil (1975) [9] 1823. A. Lagendijk and H. DeRaedt, Phys. Rev. B16 (1977) 293. [10] H.J. Gerritsen et al., Physica 21(1955)629. [11] M. Date et al., J. Appl. Phys. 34 (1961) 1038.