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Low-frequency magnetic dynamics of thulium cubic intermetallic compounds in vicinity of the quadrupolar phase transition point
Physica 142B (1986) 41-49 North-Holland, Amsterdam
LOW-FREQUENCY MAGNETIC DYNAMICS OF THULIUM CUBIC INTERMETALLIC COMPOUNDS IN VICINITY OF THE QUADRUPOLAR PHASE TRANSITION POINT”’
L. KOWALEWSKI
and R. WOJCIECHOWSKI
Solid State Division, Institute of Physics, A. Mickiewicz University, Pornan’, Polandb’ Received
14 February
1986
Behaviour of longitudinal magnetic dipolar fluctuations in vicinity of the quadrupolar phase transition point in the paramagnetic range of some thulium cubic intermetallic compounds has been discussed. It was shown that halfwidth of the magnetic central peak at the temperature of quadrupolar phase transition diminishes evidently in a static external magnetic field. Temperature dependence of the width of the magnetic central peak in a strong magnetic field shows up the minimum value at the quadrupolar phase transition point. The analysis of the width of the central magnetic mode in scattering experiments provides a new method of observation of the quadrupolar phase transition.
1. Introduction
The magnetic and quadrupolar phase transitions in rare-earth intermetallics have been studied for the last few years [l, 21. In the vicinity of these transition points the appropriate fluctuations of dipolar or quadrupolar order parameters exhibit a critical behaviour. The single ion dipolar and quadrupolar momenta consist of intra- as well as of intermanifold contributions. Both single ion momenta and bilinear or quadrupolar interactions strongly depend on the sequence of the crystal field levels. Therefore a great variety of magnetic properties of different rare-earth intermetallics has been observed [2-61. Description of dynamics of the fluctuations of both order parameters requires to assume a Hamiltonian which includes the crystalline electric field, the bilinear exchange and quadrupolar interactions. Analysis of the dynamical correlation functions of the fluctuations of longitudinal dipolar as well as quadrupolar order parameters may allow one to draw conclusions about the existence and the mechanism of magnetic and quadrupolar phase transitions. In some trulium compounds (TmZn, TmCd) the quadrupolar interactions are relatively strong and produce pure quadrupolar phase transition in the paramagnetic range (e.g. T, = 8.12 K and T, = 8.55 K for TmZn; T, = 0 K and T, = 3.16 K for TmCd [l]). That transition is manifested by a distortion of the lattice which reduces the cubic symmetry to the tetragonal one [6]. In a strong applied magnetic field a coupling between elastic and magnetic degrees of freedom may be induced [7] and an influence of the quadrupolar phase transition on the longitudinal magnetic response may be observed. It is known to be very difficult to induce electronic quadrupole excitations and to observe the quadrupolar phase transitions in inelastic neutron scattering experiments. However in an external magnetic field the quadrupolar phase transition can change the behaviour of magnetic peaks in the scattering experiments. The aim of our paper is to discuss the low frequency longitudinal magnetic response of some thulium cubic intermetallic compounds on an external magnetic field in vicinity of the quadrupolar phase a) Sponsored in part by the Institute for Low Temperature ‘) Matejki 48149, 60-769 Poznati, Poland.
and Structure
0378-4363 /86/$03.50 @ Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
Research,
Polish
Academy
of Sciences.
42
L. Kowalewski and R. Wojciechowski
I Low-frequency
magnetic dynamics of thulium
transition point. In particular let us analyze the influence of the quadrupolar phase transition on behaviour of the magnetic central peak of the spectral intensity function S( q, w) or of the relaxation function of the longitudinal magnetic moment.
2. Model In order to describe the dynamics of magnetic dipolar and quadrupolar fluctuations in a rare-earth intermetallic system perturbed by an external magnetic field, let us take the following Hamiltonian [2]
where XcEF is the cubic crystal field Hamiltonian, F,, is the dipolar exchange integral, G, is the quadrupolar interaction parameter, J, is the angular momentum of the ith ion, J: is its z-component, 0: is the second-order Stevens operator, g, is the Landi g-factor, pr, is the Bohr magneton and H is the magnetic field. Let us restrict our analysis to the thulium systems the ground state of which in the cubic paramagnetic range is the triplet Ts(‘) . This ground state is composed of one nonmagnetic but quadrupolar state and two other states which exhibit the magnetic as well as the quadrupolar momenta. Usually the quadrupolar interactions are relatively weak and the elastic and magnetic properties only weakly influence each other. In our paper we will analyze systems with strong quadrupolar interactions in which the coupling between magnetic and elastic degrees of freedom is induced by an external magnetic field [3, 6, 71. The analyzed system is characterized by two order parameters, namely by the magnetic moment
M=g,dJ’h and the quadrupolar Q = (0%~
one
= ([3(J’)*
- J(J +
l>lh,
where ( * * . ),, denotes the equilibrium average and J is the quantum number of angular momentum. The total angular and quadrupolar momenta J 5 = J”( q = 0) and Oi,, = 0 i( q = 0) can be divided with the aid of the projection operator P into the low and high frequency parts: J;
= PJ;
+ (1-
P)J;
and
o;,,=PO;,,+ (1 - m;.,, where q denote a wavevector and s = z, +, -. The low frequency dipolar variable PJ’( q = 0) can be written according to the Mori’s projection formalism [8] in the following form [9]: PJ’(q
=0) = c (J’, A;)MFA(A, p.y
A+),,’
MFAA,(q
= o),
L. Kowalewski
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
of thulium
43
where (. , . )MFA denotes the Kubo product in Molecular Field Approximation (MFA). Similarly, we express the variable PO i( q = 0). For example, when we consider only the ground triplet r$‘) and the next excited doublet r, in a thulium system we get for H = 0 1
PJ”(q=O)=
1-
Fo [xk
- 7
Jyq
= O),
&)
where F, = F( q = 0) is the uniform space Fourier transform of the exchange integral; the symbols xJ”,, xFUr, denote the linear, total and r, intramanifold single-ion magnetic susceptibilities, respectively, and
where
is the single-ion standard basis operator. The low frequency part of the dipolar variable in an external magnetic field is a linear combination of the variables Jpcl), 0: ,_(,) and O!&, with the coefficients being 5 15 the functions of static susceptibilities. The essential contributions to the low frequency longitudinal magnetic fluctuations near the quadrupolar phase transition point come from their intramanifold transitions. Let us choose therefore a set of initial dynamical variables which generate longitudinal dipolar as well as quadrupolar intramanifold transitions in an external magnetic field and which are arranged in a column matrix
A = {Av(q =‘4)
-col{J;$q
= 01,O&u(q = O), O&(q
= O)},
where
and
Oi,& =O)= F* uh4O~lr:~)L,,,r”*(~= 0). Let us also assume
that
all the variables
describe
appropriate
fluctuations
and are defined
as
A, =: A, - (A,),.
3. An analysis of the relaxation functions Let us discuss the behaviour of the relaxation function (PJi=,(t), PJz=,) in vicinity of the quadrupolar phase transition in a disordered phase. In the zero external magnetic field the dipolar and quadrupolar transitions can be treated separately. However in a strong static magnetic field a coupling
L. Kowalewski
44
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
of thulium
between elastic and magnetic degrees of freedom may be induced. Therefore in the magnetic field we can expect that the quadrupolar phase transition may be revealed in the longitudinal magnetic response of a system and in the behaviour of appropriate peak of spectral intensity function. Let us analyze therefore the zone-center relaxation function of the Curie-Langevin part of the order parameter Ji in an external magnetic field (H;(t),
PJ’,)
(J;, A;)(&
= c
F’.U
k +),;(A,,,
J’,)(A,
A +)J:JA&),
A:.).
The time evolution of the relaxation function (PJi(t), PJ’,) is the same as that of the relaxation function R,_,,(t) = (A.(t), A:.) and is described by the matrix Langevin equation f
di(t) dt
v - in . Z?(t) +
f(t - S) . Z?(s) ds = 0
I
(3)
where
(Jf.p(& Jf.$
et>=
(qr$)>
Jy
(O&(4 J’r$ the frequency
matrix fi is defined
ifi=(iLA, L denotes
A+),
the Liouville operator,
f(t) G
(AQ
eeifQL,
QA
+)(A,
the matrix memory function p takes the form A +)-’
and all Kubo products are appropriate for a system in a strong external magnetic field. The Laplace transform of the relaxation function R(t) fulfills the exact matrix equation
where the elements independently of H:
L!rv = -i c (A,, h
The frequency namely
of the frequency
A,)‘H’(A,
matrix in an external
.d ‘)L,’ W) = -c
h
([A,,
A,])iH’(A,
matrix is the zero matrix because all the commutators
It is easy to see that the above relations
field H are equal to zero
magnetic
A ‘);i [A,,
hold true because the standard
w = 0. Ah] are equal to zero;
basis operators
fulfill the
L. Kowalewski and R. Wojciechowski
commutator
I Low-frequency
45
magnetic dynamics of thulium
relation
Thus, the behaviour of a thulium system in the low frequency The damping matrix
region is purely relaxational.
can be simplified when thulium ions are treated as independent and by introducing the parameters yAA which describe the linewidth associated with the single-ion appropriate intramanifold fluctuations. So, we get
where xfV denote the Curie-Langevin static susceptibilities in an external field H. Now, let us calculate their linear and nonlinear parts in molecular field approximation. The value of the Curie-Langevin longitudinal moment in MFA associated with the ground triplet ri” and induced by a strong external magnetic field is equal to
(4) where MFA xr,r,
=
1 _‘;;a
[I0
Fo(xk
-
x;dI
JJ
is the linear and (3) xr,r,
MF*
(3)
1
_ c1
-
(3)
FOx;J)4
xl
x2
+
1-
G,(&
0;)
I
is the nonlinear ground intratriplet static susceptibility (see eq. (A.3)), both are calculated single-ion static linear symbols xyJ and x& denote the total and Curie-Langevin respectively while ~(1~)and xy’ the third-order sing le-ion susceptibilities (see Appendix), in the absence of exchange field. If we restrict the analyzed system to the ground triplet get the following expression for a halfwidth of the magnetic central peak: Y r5x22
r=
7 XllX22
where
in MFA. The susceptibilities all calculated ril) only, we
-
x12x21
(7)
L. Kowalewski and R. Wojciechowski
46
I Low-frequency
magnetic dynamics of thulium
It is easy to see that if a system evolves towards the quadrupolar phase transition (T-+ third-order susceptibility XEROX”” tends to infinity due to the condition of existence of T,: 1 - G&2:‘,
O;),=,,
= 0.
Therefore, if we substitute eq. (4) into (6) and eq. (8) into magnetic central peak takes the minimum value at Toin an behaviour of the magnetic central peak in the vicinity of T, quadrupolar phase transition. Both linear and nonlinear static susceptibilities exhibit namely they both tend to infinity due to the relation 1 - F&T
+ To) the
eq. (7) we will see that the linewidth of the external magnetic field (fig. 1). This unusual can be used for experimental verification of proper
extrapolation
behaviour
for T+
= T,) = 0
T,;
(9)
which is valid for T-+ T,.
Fig. 1. Calculated temperature (T) and external field (23) dependence of the halfwidth (r) of the magnetic T 2 T,. The halfwidth r has a minimum value at T, and again reaches zero at the Curie point T,( < T,).
4. Concluding
central
peak
for
remarks
The low frequency longitudinal magnetic excitations in a thulium system in a paramagnetic range and in the vicinity of quadrupolar phase transition are purely relaxational independently of the external magnetic field. The temperature behaviour of halfwidth of the central magnetic mode exhibits a noteworthy dependence on the external magnetic field. It shows up an evident minimum value at the temperature of the quadrupolar phase transition which is proportional to the intensity of the external field. This minimum does not exist without any external magnetic field. We can say, therefore, that the analysis of width of the central magnetic mode, in neutron scattering experiments provides a new method of observation of the quadrupolar phase transition. The magnetic and quadrupolar Curie-Langevin linear and nonlinear susceptibilities within MFA show noteworthy temperature behaviour near T, and T,. The linear results are the same as those
L. Kowalewski and R. Wojciechowski
I Low-frequency
magnetic dynamics of thulium
47
obtained by Huber [9]. The Curie-Langevin nonlinear susceptibilities are fully contained in the total nonlinear susceptibilities given by Morin and Schmitt [l]. The Curie-Langevin dipolar fluctuations are critically slowing down at the Curie point which follows from the eqs. (5) and (6) and the relation 1 - F(O)&’
= 0.
Let us calculate the elastic (Curie-Langevin) parts of the magnetic and quadrupolar nonlinear susceptibilities of a thulium system. All these susceptibilities may be obtained by applying a perturbation theory to the zeroth-order Hamiltonian XcEF. We start with the definiton of the susceptibility xAB given by the relation
where H is an external magnetic field acting on the dynamical variable A is given by the formula (A) =
$ c (iklAljf)(
jf(ik) e-BEn,
variable B. The mean value of the
(A.11
r,k
iJ
where 2 is a partition function, Ei are the eigenvalues and ]ik), )jl) are the eigenvectors of the perturbed Hamiltonian determined by the perturbation procedure. For n = 1 we obtain the same linear susceptibilities as in Huber’s paper [9], and for n = 3, we have the third-order susceptibilities. To obtain the elastic third-order magnetic and quadrupolar or mixed dipolar-quadrupolar susceptibilities we use a perturbation theory up to the third and second order in the expressions of the perturbed energies and perturbed states, respectively. First, we calculate the third-order elastic magnetic susceptibilities for the triplet-doublet system. In that case we put into eq. (A.l) the following variables A = JFgl, and B = .I;:,, where g,p,JFjl, denotes the elastic part of the magnetic moment. For simplicity, let us introduce the following notation for the crystal field states: IP,
n) = In)
n = 1,2,3, m=4,5
Ir3,m)=lm)
and also assume that E r11j = 0 and Er3 = A. The Hamiltonian gf = XCEF -
of the system in MFA takes the form
g,p&+,
- F,( J”)J’ - G,( 0;) 0;.
(A.21
After simple but tedious calculation we obtain -&MFA =
1 (1 - F&)4
(3) + ~G,x~,~x~)I, Ix ’
(A-3)
L. Kowalewski
48
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
xz,+ XtL
0
where x& = xJzJz = magnetic susceptibility,
is a single ion magnetic susceptibility, Xi, a single-ion magnetic susceptibility, and X& a single-ion Curie-Langevin
F, = c Flj, Go = c G, i
i
For the triplet-doublet
system we have
XI(3) = (1 - FoX;J3(1 - F,X:)J [ X:2 - ; P(xP,,‘] + F;x:“,‘x::x&
+
Fo(1
+
;
PF;U
-
Fox:,)U
-
;
PF,(l
-
Fox:Jxtd3,
-
-
4,xl’,>x%!x:,
Fox;,>(x:,>‘xl’,
x?’ = Cl- F,x:JU - F,x;,>x% + F,x&(l - F,,x:,)x;% + F;(x:r.)*(x?’- x%), =
xr,,Q
(1 -
F,,x:,)x~~~L + Fi(xkJ2(x? -x&J 1-
Gox2
where
Jim = (nlJ’lm>, ~knn 0 XCL
=
z(, =
P(1J521’
;
x:,=,1 2 1 = ;
x%_
= ;
(3) _ X vv -
- kT
,
1
P’
P2
;
1 - ePdikT),
(1J;214
$
+ 1J5314>,
(1 J5,120;,2,
+ t J:,120;,,3)>
2iJf,14(P- A-‘> cf _f )
x2 = P c
xl”’
A
i
+ 1J5312)>
Jf4/‘(
A
x$?
3 + 2exp
(ml@ln>,
=
A2
1
of thulium
wL,12~
= A-‘1Jf41’[P(O:,44fG
r,
1; )
f. = exp(-
O%-<)
$1
zi’, + Ap’(O’:,44
-
O;,,,>(&
-&,I.
Van Vleck
L. Kowalewski
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
of thulium
49
In a similar way, using the Hamiltonian (A.2), we can also obtain other third-order magnetic susceptibilities. Especially, if we denote the intratriplet and intradoublet parts of 0: by Q5 and Q3 respectively, we get the following expressions for the nonlinear mixed dipolar-quadrupolar susceptibilities. 1 NXXOCL)2[X ?),l - G~(x~x?)‘~ - x2.d~~’ x2:& = (1 - F,,x;,)~(~ - G,,x2)
+(I - ~ox,oJ2x~f~J~ - %(x2 x&
x~‘~d>l
-
x%M
- x2,,>1>~
1 = (1 _ FoxJDJ)2(1_ GOX2) mXDCd2[X2 (2)x2- G,(x~x~)‘~ - x2,2(x?
+(I-
-
&X~“)2X3%X2,2~~
where
References [l] [2] [3] [4] [5] [6] [7] [8] [9]
P. Morin and D. Schmitt, Phys. Rev. B27 (1983) 4412. P. Morin and D. Schmitt, Proc. Int. Conf. on Magnetism of Rare-Earth and Actinides, Bucharest (1983) p. 60. P. Morin and D. Schmitt, Phys. Rev. B23 (1981) 5936. K.A. Gschneider, Handbook on the Physics and Chemistry of Rare Earths, L.R. Eyring, ed. (North-Holland, New York, 1978). Vols. l-4. J. Rossat-Mignod, Systematics and the Properties of the Lanthanides, S.P. Sinha, ed. (D. Reidel, Dordrecht, 1983) pp. 255-310. P. Morin, D. Schmitt and E. du Tremolet de Lacheisserie, Phys. Rev. B21 (1980) 1742. P. Thalmeier and P. Fulde, 2. Phys. B22 (1975) 359. H. Mori, Prog. Theor. Phys. 33 (1965) 423. D. Huber, Phys. Rev. B18 (1978) 429.
LOW-FREQUENCY MAGNETIC DYNAMICS OF THULIUM CUBIC INTERMETALLIC COMPOUNDS IN VICINITY OF THE QUADRUPOLAR PHASE TRANSITION POINT”’
L. KOWALEWSKI
and R. WOJCIECHOWSKI
Solid State Division, Institute of Physics, A. Mickiewicz University, Pornan’, Polandb’ Received
14 February
1986
Behaviour of longitudinal magnetic dipolar fluctuations in vicinity of the quadrupolar phase transition point in the paramagnetic range of some thulium cubic intermetallic compounds has been discussed. It was shown that halfwidth of the magnetic central peak at the temperature of quadrupolar phase transition diminishes evidently in a static external magnetic field. Temperature dependence of the width of the magnetic central peak in a strong magnetic field shows up the minimum value at the quadrupolar phase transition point. The analysis of the width of the central magnetic mode in scattering experiments provides a new method of observation of the quadrupolar phase transition.
1. Introduction
The magnetic and quadrupolar phase transitions in rare-earth intermetallics have been studied for the last few years [l, 21. In the vicinity of these transition points the appropriate fluctuations of dipolar or quadrupolar order parameters exhibit a critical behaviour. The single ion dipolar and quadrupolar momenta consist of intra- as well as of intermanifold contributions. Both single ion momenta and bilinear or quadrupolar interactions strongly depend on the sequence of the crystal field levels. Therefore a great variety of magnetic properties of different rare-earth intermetallics has been observed [2-61. Description of dynamics of the fluctuations of both order parameters requires to assume a Hamiltonian which includes the crystalline electric field, the bilinear exchange and quadrupolar interactions. Analysis of the dynamical correlation functions of the fluctuations of longitudinal dipolar as well as quadrupolar order parameters may allow one to draw conclusions about the existence and the mechanism of magnetic and quadrupolar phase transitions. In some trulium compounds (TmZn, TmCd) the quadrupolar interactions are relatively strong and produce pure quadrupolar phase transition in the paramagnetic range (e.g. T, = 8.12 K and T, = 8.55 K for TmZn; T, = 0 K and T, = 3.16 K for TmCd [l]). That transition is manifested by a distortion of the lattice which reduces the cubic symmetry to the tetragonal one [6]. In a strong applied magnetic field a coupling between elastic and magnetic degrees of freedom may be induced [7] and an influence of the quadrupolar phase transition on the longitudinal magnetic response may be observed. It is known to be very difficult to induce electronic quadrupole excitations and to observe the quadrupolar phase transitions in inelastic neutron scattering experiments. However in an external magnetic field the quadrupolar phase transition can change the behaviour of magnetic peaks in the scattering experiments. The aim of our paper is to discuss the low frequency longitudinal magnetic response of some thulium cubic intermetallic compounds on an external magnetic field in vicinity of the quadrupolar phase a) Sponsored in part by the Institute for Low Temperature ‘) Matejki 48149, 60-769 Poznati, Poland.
and Structure
0378-4363 /86/$03.50 @ Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
Research,
Polish
Academy
of Sciences.
42
L. Kowalewski and R. Wojciechowski
I Low-frequency
magnetic dynamics of thulium
transition point. In particular let us analyze the influence of the quadrupolar phase transition on behaviour of the magnetic central peak of the spectral intensity function S( q, w) or of the relaxation function of the longitudinal magnetic moment.
2. Model In order to describe the dynamics of magnetic dipolar and quadrupolar fluctuations in a rare-earth intermetallic system perturbed by an external magnetic field, let us take the following Hamiltonian [2]
where XcEF is the cubic crystal field Hamiltonian, F,, is the dipolar exchange integral, G, is the quadrupolar interaction parameter, J, is the angular momentum of the ith ion, J: is its z-component, 0: is the second-order Stevens operator, g, is the Landi g-factor, pr, is the Bohr magneton and H is the magnetic field. Let us restrict our analysis to the thulium systems the ground state of which in the cubic paramagnetic range is the triplet Ts(‘) . This ground state is composed of one nonmagnetic but quadrupolar state and two other states which exhibit the magnetic as well as the quadrupolar momenta. Usually the quadrupolar interactions are relatively weak and the elastic and magnetic properties only weakly influence each other. In our paper we will analyze systems with strong quadrupolar interactions in which the coupling between magnetic and elastic degrees of freedom is induced by an external magnetic field [3, 6, 71. The analyzed system is characterized by two order parameters, namely by the magnetic moment
M=g,dJ’h and the quadrupolar Q = (0%~
one
= ([3(J’)*
- J(J +
l>lh,
where ( * * . ),, denotes the equilibrium average and J is the quantum number of angular momentum. The total angular and quadrupolar momenta J 5 = J”( q = 0) and Oi,, = 0 i( q = 0) can be divided with the aid of the projection operator P into the low and high frequency parts: J;
= PJ;
+ (1-
P)J;
and
o;,,=PO;,,+ (1 - m;.,, where q denote a wavevector and s = z, +, -. The low frequency dipolar variable PJ’( q = 0) can be written according to the Mori’s projection formalism [8] in the following form [9]: PJ’(q
=0) = c (J’, A;)MFA(A, p.y
A+),,’
MFAA,(q
= o),
L. Kowalewski
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
of thulium
43
where (. , . )MFA denotes the Kubo product in Molecular Field Approximation (MFA). Similarly, we express the variable PO i( q = 0). For example, when we consider only the ground triplet r$‘) and the next excited doublet r, in a thulium system we get for H = 0 1
PJ”(q=O)=
1-
Fo [xk
- 7
Jyq
= O),
&)
where F, = F( q = 0) is the uniform space Fourier transform of the exchange integral; the symbols xJ”,, xFUr, denote the linear, total and r, intramanifold single-ion magnetic susceptibilities, respectively, and
where
is the single-ion standard basis operator. The low frequency part of the dipolar variable in an external magnetic field is a linear combination of the variables Jpcl), 0: ,_(,) and O!&, with the coefficients being 5 15 the functions of static susceptibilities. The essential contributions to the low frequency longitudinal magnetic fluctuations near the quadrupolar phase transition point come from their intramanifold transitions. Let us choose therefore a set of initial dynamical variables which generate longitudinal dipolar as well as quadrupolar intramanifold transitions in an external magnetic field and which are arranged in a column matrix
A = {Av(q =‘4)
-col{J;$q
= 01,O&u(q = O), O&(q
= O)},
where
and
Oi,& =O)= F* uh4O~lr:~)L,,,r”*(~= 0). Let us also assume
that
all the variables
describe
appropriate
fluctuations
and are defined
as
A, =: A, - (A,),.
3. An analysis of the relaxation functions Let us discuss the behaviour of the relaxation function (PJi=,(t), PJz=,) in vicinity of the quadrupolar phase transition in a disordered phase. In the zero external magnetic field the dipolar and quadrupolar transitions can be treated separately. However in a strong static magnetic field a coupling
L. Kowalewski
44
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
of thulium
between elastic and magnetic degrees of freedom may be induced. Therefore in the magnetic field we can expect that the quadrupolar phase transition may be revealed in the longitudinal magnetic response of a system and in the behaviour of appropriate peak of spectral intensity function. Let us analyze therefore the zone-center relaxation function of the Curie-Langevin part of the order parameter Ji in an external magnetic field (H;(t),
PJ’,)
(J;, A;)(&
= c
F’.U
k +),;(A,,,
J’,)(A,
A +)J:JA&),
A:.).
The time evolution of the relaxation function (PJi(t), PJ’,) is the same as that of the relaxation function R,_,,(t) = (A.(t), A:.) and is described by the matrix Langevin equation f
di(t) dt
v - in . Z?(t) +
f(t - S) . Z?(s) ds = 0
I
(3)
where
(Jf.p(& Jf.$
et>=
(qr$)>
Jy
(O&(4 J’r$ the frequency
matrix fi is defined
ifi=(iLA, L denotes
A+),
the Liouville operator,
f(t) G
(AQ
eeifQL,
QA
+)(A,
the matrix memory function p takes the form A +)-’
and all Kubo products are appropriate for a system in a strong external magnetic field. The Laplace transform of the relaxation function R(t) fulfills the exact matrix equation
where the elements independently of H:
L!rv = -i c (A,, h
The frequency namely
of the frequency
A,)‘H’(A,
matrix in an external
.d ‘)L,’ W) = -c
h
([A,,
A,])iH’(A,
matrix is the zero matrix because all the commutators
It is easy to see that the above relations
field H are equal to zero
magnetic
A ‘);i [A,,
hold true because the standard
w = 0. Ah] are equal to zero;
basis operators
fulfill the
L. Kowalewski and R. Wojciechowski
commutator
I Low-frequency
45
magnetic dynamics of thulium
relation
Thus, the behaviour of a thulium system in the low frequency The damping matrix
region is purely relaxational.
can be simplified when thulium ions are treated as independent and by introducing the parameters yAA which describe the linewidth associated with the single-ion appropriate intramanifold fluctuations. So, we get
where xfV denote the Curie-Langevin static susceptibilities in an external field H. Now, let us calculate their linear and nonlinear parts in molecular field approximation. The value of the Curie-Langevin longitudinal moment in MFA associated with the ground triplet ri” and induced by a strong external magnetic field is equal to
(4) where MFA xr,r,
=
1 _‘;;a
[I0
Fo(xk
-
x;dI
JJ
is the linear and (3) xr,r,
MF*
(3)
1
_ c1
-
(3)
FOx;J)4
xl
x2
+
1-
G,(&
0;)
I
is the nonlinear ground intratriplet static susceptibility (see eq. (A.3)), both are calculated single-ion static linear symbols xyJ and x& denote the total and Curie-Langevin respectively while ~(1~)and xy’ the third-order sing le-ion susceptibilities (see Appendix), in the absence of exchange field. If we restrict the analyzed system to the ground triplet get the following expression for a halfwidth of the magnetic central peak: Y r5x22
r=
7 XllX22
where
in MFA. The susceptibilities all calculated ril) only, we
-
x12x21
(7)
L. Kowalewski and R. Wojciechowski
46
I Low-frequency
magnetic dynamics of thulium
It is easy to see that if a system evolves towards the quadrupolar phase transition (T-+ third-order susceptibility XEROX”” tends to infinity due to the condition of existence of T,: 1 - G&2:‘,
O;),=,,
= 0.
Therefore, if we substitute eq. (4) into (6) and eq. (8) into magnetic central peak takes the minimum value at Toin an behaviour of the magnetic central peak in the vicinity of T, quadrupolar phase transition. Both linear and nonlinear static susceptibilities exhibit namely they both tend to infinity due to the relation 1 - F&T
+ To) the
eq. (7) we will see that the linewidth of the external magnetic field (fig. 1). This unusual can be used for experimental verification of proper
extrapolation
behaviour
for T+
= T,) = 0
T,;
(9)
which is valid for T-+ T,.
Fig. 1. Calculated temperature (T) and external field (23) dependence of the halfwidth (r) of the magnetic T 2 T,. The halfwidth r has a minimum value at T, and again reaches zero at the Curie point T,( < T,).
4. Concluding
central
peak
for
remarks
The low frequency longitudinal magnetic excitations in a thulium system in a paramagnetic range and in the vicinity of quadrupolar phase transition are purely relaxational independently of the external magnetic field. The temperature behaviour of halfwidth of the central magnetic mode exhibits a noteworthy dependence on the external magnetic field. It shows up an evident minimum value at the temperature of the quadrupolar phase transition which is proportional to the intensity of the external field. This minimum does not exist without any external magnetic field. We can say, therefore, that the analysis of width of the central magnetic mode, in neutron scattering experiments provides a new method of observation of the quadrupolar phase transition. The magnetic and quadrupolar Curie-Langevin linear and nonlinear susceptibilities within MFA show noteworthy temperature behaviour near T, and T,. The linear results are the same as those
L. Kowalewski and R. Wojciechowski
I Low-frequency
magnetic dynamics of thulium
47
obtained by Huber [9]. The Curie-Langevin nonlinear susceptibilities are fully contained in the total nonlinear susceptibilities given by Morin and Schmitt [l]. The Curie-Langevin dipolar fluctuations are critically slowing down at the Curie point which follows from the eqs. (5) and (6) and the relation 1 - F(O)&’
= 0.
Let us calculate the elastic (Curie-Langevin) parts of the magnetic and quadrupolar nonlinear susceptibilities of a thulium system. All these susceptibilities may be obtained by applying a perturbation theory to the zeroth-order Hamiltonian XcEF. We start with the definiton of the susceptibility xAB given by the relation
where H is an external magnetic field acting on the dynamical variable A is given by the formula (A) =
$ c (iklAljf)(
jf(ik) e-BEn,
variable B. The mean value of the
(A.11
r,k
iJ
where 2 is a partition function, Ei are the eigenvalues and ]ik), )jl) are the eigenvectors of the perturbed Hamiltonian determined by the perturbation procedure. For n = 1 we obtain the same linear susceptibilities as in Huber’s paper [9], and for n = 3, we have the third-order susceptibilities. To obtain the elastic third-order magnetic and quadrupolar or mixed dipolar-quadrupolar susceptibilities we use a perturbation theory up to the third and second order in the expressions of the perturbed energies and perturbed states, respectively. First, we calculate the third-order elastic magnetic susceptibilities for the triplet-doublet system. In that case we put into eq. (A.l) the following variables A = JFgl, and B = .I;:,, where g,p,JFjl, denotes the elastic part of the magnetic moment. For simplicity, let us introduce the following notation for the crystal field states: IP,
n) = In)
n = 1,2,3, m=4,5
Ir3,m)=lm)
and also assume that E r11j = 0 and Er3 = A. The Hamiltonian gf = XCEF -
of the system in MFA takes the form
g,p&+,
- F,( J”)J’ - G,( 0;) 0;.
(A.21
After simple but tedious calculation we obtain -&MFA =
1 (1 - F&)4
(3) + ~G,x~,~x~)I, Ix ’
(A-3)
L. Kowalewski
48
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
xz,+ XtL
0
where x& = xJzJz = magnetic susceptibility,
is a single ion magnetic susceptibility, Xi, a single-ion magnetic susceptibility, and X& a single-ion Curie-Langevin
F, = c Flj, Go = c G, i
i
For the triplet-doublet
system we have
XI(3) = (1 - FoX;J3(1 - F,X:)J [ X:2 - ; P(xP,,‘] + F;x:“,‘x::x&
+
Fo(1
+
;
PF;U
-
Fox:,)U
-
;
PF,(l
-
Fox:Jxtd3,
-
-
4,xl’,>x%!x:,
Fox;,>(x:,>‘xl’,
x?’ = Cl- F,x:JU - F,x;,>x% + F,x&(l - F,,x:,)x;% + F;(x:r.)*(x?’- x%), =
xr,,Q
(1 -
F,,x:,)x~~~L + Fi(xkJ2(x? -x&J 1-
Gox2
where
Jim = (nlJ’lm>, ~knn 0 XCL
=
z(, =
P(1J521’
;
x:,=,1 2 1 = ;
x%_
= ;
(3) _ X vv -
- kT
,
1
P’
P2
;
1 - ePdikT),
(1J;214
$
+ 1J5314>,
(1 J5,120;,2,
+ t J:,120;,,3)>
2iJf,14(P- A-‘> cf _f )
x2 = P c
xl”’
A
i
+ 1J5312)>
Jf4/‘(
A
x$?
3 + 2exp
(ml@ln>,
=
A2
1
of thulium
wL,12~
= A-‘1Jf41’[P(O:,44fG
r,
1; )
f. = exp(-
O%-<)
$1
zi’, + Ap’(O’:,44
-
O;,,,>(&
-&,I.
Van Vleck
L. Kowalewski
and R. Wojciechowski
I Low-frequency
magnetic
dynamics
of thulium
49
In a similar way, using the Hamiltonian (A.2), we can also obtain other third-order magnetic susceptibilities. Especially, if we denote the intratriplet and intradoublet parts of 0: by Q5 and Q3 respectively, we get the following expressions for the nonlinear mixed dipolar-quadrupolar susceptibilities. 1 NXXOCL)2[X ?),l - G~(x~x?)‘~ - x2.d~~’ x2:& = (1 - F,,x;,)~(~ - G,,x2)
+(I - ~ox,oJ2x~f~J~ - %(x2 x&
x~‘~d>l
-
x%M
- x2,,>1>~
1 = (1 _ FoxJDJ)2(1_ GOX2) mXDCd2[X2 (2)x2- G,(x~x~)‘~ - x2,2(x?
+(I-
-
&X~“)2X3%X2,2~~
where
References [l] [2] [3] [4] [5] [6] [7] [8] [9]
P. Morin and D. Schmitt, Phys. Rev. B27 (1983) 4412. P. Morin and D. Schmitt, Proc. Int. Conf. on Magnetism of Rare-Earth and Actinides, Bucharest (1983) p. 60. P. Morin and D. Schmitt, Phys. Rev. B23 (1981) 5936. K.A. Gschneider, Handbook on the Physics and Chemistry of Rare Earths, L.R. Eyring, ed. (North-Holland, New York, 1978). Vols. l-4. J. Rossat-Mignod, Systematics and the Properties of the Lanthanides, S.P. Sinha, ed. (D. Reidel, Dordrecht, 1983) pp. 255-310. P. Morin, D. Schmitt and E. du Tremolet de Lacheisserie, Phys. Rev. B21 (1980) 1742. P. Thalmeier and P. Fulde, 2. Phys. B22 (1975) 359. H. Mori, Prog. Theor. Phys. 33 (1965) 423. D. Huber, Phys. Rev. B18 (1978) 429.