Spin–lattice relaxation in a dense paramagnet CuF2·2H2O

C.H. Lee, C.E. Lee*

Department of Physics, Korea University, Anam-dong, Sungbuk-ku, Seoul 136-701, South Korea

Received 4 June 2003; received in revised form 10 January 2004; accepted 13 January 2004 By A. Pinczuk

Abstract

NMR spin–lattice relaxation was investigated in a dense paramagnet CuF2·2H2O. We find that the 1H nuclear spin–lattice

relaxation is dominated by the electron spin-flip and the Raman process of the electron spin–lattice relaxation.

q 2004 Elsevier Ltd. All rights reserved.

PACS: 75.30.Hx; 76.60. 2 k

Keywords: A. Magnetically ordered materials; E. Nuclear resonances

1. Introduction

Nuclear magnetic resonance (NMR) has been employed

as a powerful tool for the study of magnetic materials. The

resonant nucleus in the NMR measurements and unpaired

electrons in the paramagnetic material may be on the same

atom, or on different atoms [1]. When the resonant nucleus

is bonded directly to a paramagnetic ion, the transferred

hyperfine interaction [2] between the resonant nucleus and

the electron that is partly transferred to the resonant atom

will be dominant, as in the case of 19F NMR of MnF2 [3].

The magnetic dipole–dipole interaction will dominate,

however, when the resonant nuclei are not bonded directly

to a paramagnetic ion, as in the cases of 1H NMR of

CuSO4·5H2O [4], CuCl2·2H2O [5], and CuF2·2H2O [6].

However, the issue of nuclear magnetic relaxation was not

fully discussed in these previous works, and it is the purpose

of this work to carry out a comprehensive study of the

mechanism and temperature dependence of the spin–lattice

relaxation in CuCl2·2H2O.

Valuable information may be obtained on the dynamics

of unpaired electrons by investigating nuclear spin–lattice

relaxation in paramagnetic insulators. When the dipole–

dipole interactions are dominant, the magnetic hyperfine

interaction and the electron correlation time may be

estimated with relative ease, since the delocalized unpaired

electrons play only a minor role. In the present investigation,

we have measured the temperature dependence of the 1H

NMR spin–lattice relaxation in the Heisenberg paramagnet

CuF2·2H2O in order to identify the electron spin–lattice

relaxation mechanism and to obtain the spin-flip exchange

energy in this system of dense paramagnetic ions [7–10].

The NMR relaxation due to the magnetic dipole–dipole

interaction has long been established [11]. For a resonant

nucleus with I ¼ 1=2 in a powder sample, the spin–lattice

relaxation time T1 by electron spins S is given by [12]

T211 ¼ 2

a

15

Xr26

i

3te

1 þ v2I t

2e

þ7te

1 þ v2St

2e

" #; ð1Þ

where a ¼ g2I g2m2

BSðS þ 1Þ: g; v; and ri are the gyromag-

netic ratio, the Larmor frequency, and the distance between

the resonant nucleus and the ith electron spin, respectively.

te denotes the electron-spin correlation time, and g and mB

indicate the g-factor and the Bohr magneton, respectively.

Since our T1 measurements were carried out at 45 MHz and

the magnitude of te is on the other of 10211 s as will be

verified afterwards, the conditions v2s t

2e q 1 and v2

I t2e p 1

are satisfied. Thus, the spin–lattice relaxation rate

T211 ¼

2

5aX

r26i te ð2Þ

is finally obtained.

0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ssc.2004.01.014

Solid State Communications 130 (2004) 41–43

www.elsevier.com/locate/ssc

* Corresponding author. Tel.: þ82-2-3290-3098; fax: þ82-2-

927-3292.

E-mail address: rscel@korea.ac.kr (C.E. Lee).

2. Experiment

For the spin–lattice relaxation time ðT1Þ measurements,

the inversion recovery method was used at the 1H NMR

frequency of 45 MHz in the temperature range of 77 K to

room temperature. According to the elemental analysis the

commercially available sample of CuF2·2H2O includes

some nonmagnetic calcium (Ca2þ) (0.45%) and magnesium

(Mg2þ) (0.18%) ions.

3. Results and discussion

The temperature dependence of the 1H spin–lattice

relaxation rate is shown in Fig. 1, in which a decrease with

increasing temperature is noticed. The temperature depen-

dence of T1 may arise from the electron-spin correlation

time te; with contributions from the electron spin–lattice

relaxation time T1e and the electron spin-flip correlation

time tf :

t21e ¼ T21

1e þ t21f : ð3Þ

Since the electron-spin flips are caused by the exchange

interaction among the neighboring electron spins, tf is

nearly independent of temperature. On the other hand, T1e

can be ascribed to the modulation of the crystal electric field

or the ligand field through the motions of the electrically

charged ions under the action of the lattice vibration. As a

result, several mechanisms may give rise to the temperature

dependence, which has been given by [13]

T211e ¼ b coth

hne

2kT

� �þ gTn þ

z

expD

kT

� �2 1

: ð4Þ

The first term represents the direct process, where ne is the

Larmor frequency of the magnetic ion. The second term

corresponds to the Raman process, where the exponent n can

vary depending on the electronic states of the magnetic ion.

The last term describes the Orbach process, in which

transitions between the two low-lying states of the magnetic

ion occur via an excited state whose energy is less than the

maximum phonon energy but greater than the energies of

the ground states, by D: In the Raman process, the exponent

n in Eq. (4) has been shown to be 7 in the case T p uD

(Debye temperature) and 2 in the case T q uD [13].

Therefore, if T1e is determined by the Raman process, T1

is supposed to vary as Tn with the exponent n lying between

2 and 7. The uD for CuF2·2H2O is about 135 K [14]. In our

case, the least-squares fit of the experimental data to the

equation, with the Raman process only from Eq. (4),

T1 ¼5

2aP

r26i

½gTn þ t21f �; ð5Þ

gave the values for fitting parameters, g ¼ 4:00 £ 105;

tf ¼ 1:84 £ 10211 s and n ¼ 2:0 ^ 0:1: Our T1 data happens

to be best fitted to a T2 dependence, which is depicted in Fig.

1 by a solid curve, and the T1e thus appears to be dominated

by the Raman process.

Isotropic exchange interactions give rise to the electron-

spin flips of the form

Hex ¼X

JijSi·Sj; ð6Þ

where Jij is the exchange coupling constant. Using the

general expression [15] for the exchange frequency vex; the

relation

v2ex ¼

2

3

J2

h2zSðS þ 1Þ

has been derived for the nearest neighbor interaction [16],

where z is the number of the nearest neighbors of the

paramagnetic ion [17]. Then, the exchange parameter J can

be obtained from the relation v2ex ¼ p=t2

f [15], which gives

J=kB ¼ 1:8 K in our case.

In conclusion, the temperature dependence of the 1H

NMR spin–lattice relaxation time in a sample of dense

paramagnet CuF2·2H2O was investigated. As a result, it was

found that the 1H NMR spin–lattice relaxation is dictated by

the modulation of the electron spin–lattice relaxation

(Raman process) and electron spin-flip.

Acknowledgements

This work was supported by the KISTEP (National

Research Laboratory and M102KS010001-02K1901-

01814) and by the Brain Korea 21 Project in 2003.

Measurements at the Korea Basic Science Institute (Seoul)

are gratefully acknowledged.Fig. 1. Temperature dependence of 1H spin–lattice relaxation time

in CuF2·2H2O. The solid line is the fit to Eq. (5) with n ¼ 2:

C.H. Lee, C.E. Lee / Solid State Communications 130 (2004) 41–4342

References

[1] A. Abragam, Principles of Nuclear Magnetism, Clarendon

Press, Oxford, 1961.

[2] R.G. Shulman, Phys. Rev. Lett. 2 (1959) 459.

[3] R.G. Shulman, V. Jaccarino, Phys. Rev. 103 (1956) 1126.

[4] N. Bloembergen, Physica 16 (1950) 95.

[5] N.J. Poulis, G.E.C. Hardeman, Physica 18 (1952) 201.

[6] R.G. Shulman, B.J. Wyluda, J. Chem. Phys. 35 (1961) 1498.

[7] K.W. Lee, C.E. Lee, Phys. Rev. B67 (2003) 134424.

[8] S. Torre, J. Ziolo, Phys. Rev. B39 (1989) 8915.

[9] L.J. de Jongh, Magnetic Properties of Layered Transition

Metal Compounds, Kluwer, Dordrecht, 1986.

[10] K.W. Lee, C.H. Lee, C.E. Lee, J.K. Kang, Phys. Rev. B62

(2000) 95.

[11] N. Bloembergen, E.M. Purcell, R.V. Pound, Phys. Rev. 73

(1948) 679.

[12] M.O. Norris, J.H. Strange, J.G. Powels, M. Rhodes, K.

Marsden, K. Krynicki, J. Phys. C1 (1968) 422.

[13] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of

Transition Ions, Clarendon Press, Oxford, 1970.

[14] L.J. de Jongh, A.R. Miedema, Adv. Phys. 23 (1974) 1.

[15] R. Kubo, K. Tomita, J. Phys. Soc. Jpn 9 (1954) 888.

[16] P.W. Anderson, P.R. Weiss, Rev. Mod. Phys. 25 (1953) 269.

[17] T. Moriya, Prog. Theor. Phys. (Kyoto) 16 (1956) 23 see also

page 641.

C.H. Lee, C.E. Lee / Solid State Communications 130 (2004) 41–43 43