R. BurrielInstituto de Ciencia de Materiales de AragonCSIC – University de Zaragoza50009 Zaragoza, Spain

Determination of the magnetocaloric parametersthrough magnetic and thermodynamic methods

in first-order transitions

Delft Days on Magnetocalorics, Delft, Netherlands, 30-31 October, 2008

Collaborators:- E. Palacios- L. Tocado- G. Wang

● Determination of magnetocaloric parameters� Calorimetric methods

� Heat capacity. Practical limitations

� Direct measurements of ∆∆∆∆TS and ∆∆∆∆ST

� Magnetic methods� Magnetization. Maxwell relation

� Isothermal. Isofield. Other thermal and field paths� First-order transitions. Overestimation of ∆∆∆∆ST

� Adiabatic pulsed field� Adiabatic and isothermal magnetization

● Applicability of the Maxwell relation. Hysteretic compounds� Errors with isothermal magnetization curves

� Solutions� Magnetization M(T) at constant field

� Isothermal magnetization of appropriate thermal and field history

Outline

� From heat capacity

∫=T

H dTT

TCTS Hp

0

,)(

)(

( )12

)()( HHS STSTT −=∆

∫−

=−=∆T HpHp

HHT dTT

TCTCTSTSS

0

,, 12

12

)()()()(

● Determination of the magnetocaloric parameters

�Calorimetric methods

• Good at low temperatures• Low precision at room T• Scarce C(T)H data

TC

S (T)B

Ent

rop

y, S

Temperature, T

S (T)B = 0

TC

S(T)B=0

S(T)B

∆TS

∆ST

∆ST

� From direct measurements

∆∆∆∆TS = [T(S)H2- T(S)H1

]

∆∆∆∆ST = ∆Q/T = [S(T)H2- S(T)H1

]

L. Tocado, E. Palacios, R. BurrielJ. Magn. Magn. Mater.290-291, 719 (2005).

2 0 0 4 0 0 6 0 0 8 0 00 ,0

1 ,5

3 ,0

4 ,5

t ( s )

Pow

er (

mW

)

Magne

tic F

ield

(T)

- 2 0

- 1 0

0

1 0

2 0

T (m

K)

T = 1 5 4 .2 7 3 K

0

2

4

6

Sample holder

Adiabatic shield

HpTp T

HTM

H

HTS

,,

),(),(

∂∂=

∂∂

� Magnetic methods

� From magnetizationdata

Using the Maxwell relation

dHT

HTM

HTC

TT

Hp

H

S

,0

),(

),(∫

∂∂×=∆

dHT

HTMS

Hp

H

T,

0

),(∫

∂∂=∆

● Limited to ∆∆∆∆ST

� Calculation of ∆∆∆∆TS requires C(T, H)

● Discontinuities in first derivatives� Computational errors

● Maxwell relations valid for state functions� M(T,H) is not a state function in first-order transitions � It depends on history (hysteresis)

● Easy measurements� Isothermal magnetization� At constant field� Other thermal and field history

● Problems

� Adiabatic pulsed fields

� Comparison of magnetization curves in adiabatic and isothermal processes

R.Z. Levitin, V.V. Snegirev, A.V. Kopylov, A.S. Lagutin, A. GerberJ. Magn. Magn. Mater.170, 223 (1997).

relation Maxwell,,

,

2

,

,

2

,

pBpT

pBpT

pTpB

T

M

B

S

T

M

TB

GM

B

G

B

S

BT

GS

T

G

∂∂=

∂∂

∂∂−=

∂∂∂

⇒−=

∂∂

∂∂−=

∂∂∂

⇒−=

∂∂

VdpMdBSdTdG −−−=

● Applicability of the Maxwell relation. Hysteretic compounds

300 320 340 3600

50

100

150

315 320 325 330

50

100

150

0 - 1 T 0 - 2 T 0 - 3 T 0 - 4 T 0 - 5 T 0 - 6 T 0 - 7 T 0 - 8 T 0 - 9 T

-∆S T

(J/

kg·K

)

T (K)

-∆S T

(J/

kg·K

)

T (K)

MnAs

300 320 340 3600

50

100

150

315 320 325 330

50

100

150

0 - 1 T 0 - 2 T 0 - 3 T 0 - 4 T 0 - 5 T 0 - 6 T 0 - 7 T 0 - 8 T 0 - 9 T

-∆S T

(J/

kg·K

)

T (K)

-∆S T

(J/

kg·K

)

T (K)

MnAs

310 320 330 340

7.5

8.0

8.5

9.0

6 T5 T4 T

3 T2 T

1 T0 T

S /

R

T(K)

Application to MnAs

∂S/∂T > 0

0 2 4 6 80

40

80

120

isotherm 1 isotherm 2 isotherm 3 isotherm 4 isotherm 5 isotherm 6

M (

em

u/g

)B (T)

(dM)B

300 310 320 330 3400

2

4

6 65432

dx/dT

ParamagneticOrthorhombicB

(T

)

T (K)

dx/dB

FerromagneticHexagonal

1

Isothermal magnetization measurements:M(B)T

Ferromagnetic phase (x = 1) � Isotherms 1 and 2

Ferromagnetic phase (x) + Paramagnetic phase (1-x) � Isotherms 3 and 4

Paramagnetic phase (x = 0) � Isotherms 5 and 6

Phase diagram from M and Cp measurements

MnAs

310 320 330

2 3 4 5 61

(dM)B(dS)T

310 320 330 340 3500

50

100

150 0 - 1 T 0 - 2 T 0 - 3 T 0 - 4 T 0 - 6 T 0 - 9 T

-∆S T

(J/

kg·K

)

T (K)

M = x·MF + (1-x)·MPB

PF

B

F

B

P

B T

xMM

T

Mx

T

Mx

T

M

∂∂−+

∂∂+

∂∂−=

∂∂

)()1(

dBB

xMMdB

T

MxdB

T

MxdS

TPF

B

F

B

P

∂∂−+

∂∂+

∂∂−= )()1(

∫

∂∂−

Bt

BPF dB

T

xMM

0

)(

( )( ) 0

0,

0 =

=

= ∆=

∂∂

B

anomBp

B H

C

T

x

Calculation in a real process (isothermal magnetization)

dBT

MdS

B

∂∂=

dS = dS’ + dSex

Solutions for hysteretic compounds

● Magnetization at constant fields M(T)B

● Isothermal magnetization (for increasing fields) after forcing the P state by heating and cooling

● Isothermal magnetization (for decreasing fields) after forcing the F state by cooling and heating

0

2

4

6

8

220 230 240 250 260 270 280

T (K)

B (

T)

Cooling

HeatingField

Temperature

0

2

4

6

8

220 230 240 250 260 270 280

T (K)

B (

T)

Cooling

HeatingField

Temperature

Mn1.1Fe0.9P0.82Ge0.18

200 220 240 260 280 3000

30

60

90

120

150

M (

Am

2 /kg)

T (K)

Cooling B = 1 T B = 2 T B = 3 T B = 4 T B = 5 T B = 6 T B = 7 T B = 8 T B = 9 T

200 220 240 260 280 3000

30

60

90

120

150

M (

Am

2 /kg)

T (K)

Heating B = 1 T B = 2 T B = 3 T B = 4 T B = 5 T B = 6 T B = 7 T B = 8 T B = 9 T

210 220 230 240 250 260 270 280 2900

10

20

30

40

-∆S

(J/

kgK

)

T (K)

∆B = 1 T ∆B = 2 T ∆B = 3 T ∆B = 4 T ∆B = 5 T ∆B = 6 T ∆B = 7 T ∆B = 8 T ∆B = 9 T

Heating

210 240 2700

10

20

30

-∆S

(J/

kgK

)

T (K)

∆B = 1 T ∆B = 2 T ∆B = 3 T ∆B = 4 T ∆B = 5 T ∆B = 6 T ∆B = 7 T ∆B = 8 T ∆B = 9 T

Cooling

Mn1.1Fe0.9P0.82Ge0.18

200 220 240 260 280 3000

5

10

15

20

25

30

35

-∆S

(J/

kgK

)

T (K)

∆B=1T ∆B=2T ∆B=3T ∆B=4T ∆B=5T ∆B=6T ∆B=7T ∆B=8T ∆B=9T

Increasing Field

220 230 240 250 260 270 280 290 3000

10

20

30

40

-∆S

(J/

kgK

)

T (K)

∆B=1T ∆B=2T ∆B=3T ∆B=4T ∆B=5T ∆B=6T ∆B=7T ∆B=8T ∆B=9T

Decreasing Field

220 240 260 2800

20

40

60

80

-∆S

(J/

kgK

)

T (K)

∆B=1T ∆B=2T ∆B=3T ∆B=4T ∆B=5T ∆B=6T ∆B=7T ∆B=8T ∆B=9T

Decreasing Field

210 220 230 240 250 260 270 280 2900

20

40

60

80

-∆S

(J/

kgK

)

T (K)

∆B=1T ∆B=2T ∆B=3T ∆B=4T ∆B=5T ∆B=6T ∆B=7T ∆B=8T ∆B=9T

Increasing Field

Isothermal magnetizationIsothermal magnetization

from P or F state

MnAs

Adiabatic temperature changefor MnAs

• Direct ∆TS (symbols)• From Cp (lines)

MnAs

Isothermal entropy changefor MnAs• Direct results (symbols)• From Cp (lines)

L. Tocado, E. Palacios, R. Burriel

J. Therm. Anal. Calorim.84, 213 (2006).

� Importance of reliable estimates of ∆∆∆∆TS and ∆∆∆∆ST

� Practical results from M(T,H) in paramagnets and in second order transitions

� Materials with giant MCE: first-order transitions� Incorrect results from M(T,H)

� Overvaluedcalculations

� Limitations from Cp results for high temperature ∆∆∆∆ST calculations

� Direct measurement of ∆∆∆∆TS and ∆∆∆∆ST

� Importance of thermal history in M(H) measurements� Use ofisofield magnetization curves orcontrolled isothermal

curves� Practical results in first-order transition compounds

� Absence of spurious spikes, no evidence of colossal MCE

� Calculation of the spurious spikes

Conclusions