Introduction
EHPRG-40 in Edinburgh
7. September 2002
Pressure and Temperature Dependence of the
Phonon Density-of-States of Fe
in the Invar Alloy Fe65Ni35
Hubertus Giefers
Physics Department, University of Paderborn, D-33095 Paderborn, Germany
H. Giefers, University of Paderborn
Survey
Introduction
• Nuclear Resonant Inelasic Scattering (NIS)
• HP cell and experimental setup
• some notes to Invar
Spectra of Fe65Ni35
• NIS spectra T and p dependence
• phonon DOS spectra of Fe65Ni35
• comparison: Fe65Ni35 / a-Fe
Derived parameters of Fe65Ni35
• thermodyn./elastic properties of Fe65Ni35
• sound velocity
• few notes to the Debye temperature
• Debye temperatures and Grüneisen parameters
Summary and Outlook
Acknowledgment
H. Giefers, University of Paderborn
Nuclear Resonant Inelastic Scattering (NIS)
Counts
prompt count rate
from SR puls
delayed count rate
57
from Fe
Time
• energy scans around the Mössbauer
resonance
• counting of delayed photons
• NIS: mostly secondary radiation (Fe Ka,b x-rays)
Phonon density of states
H. Giefers, University of Paderborn
Experimental Details for NIS
Sample:
• 15 µg Invar
• 200 µm Ø
Pressure cell:
• large opening
for detectors
• Be gasket for
Ka,b fluorescence
APD
APD
ID22N
H. Giefers, University of Paderborn
Invar
• detected 1896 by Ch.-Ed. Guillaume
• thermal expansion of aV  10-6 K-1 for
-5 -1
Fe65Ni35 compared to 10 K for Fe
• Nobel prize 1920
• Magnetism plays important role
H. Giefers, University of Paderborn
Normalized measured NIS spectra
Absorption probability Inorm (1/eV)
Temperature
12
Pressure
Fe65Ni35
Fe65Ni35
660 K
450 K
295 K
150 K
5K
10
8
0.0 GPa
5.3 GPa
10.4 GPa
17.3 GPa
6
4
2
0
-60
-40
-20
0
20
40
rel. Energy (meV)
60
-60
-40
-20
0
20
40
60
rel. Energy (meV)
H. Giefers, University of Paderborn
Derived Phonon Density-of-States
Temperature
60
Pressure
phonon DOS g(E) (1/eV)
0.0 GPa
1.5 GPa
3.1 GPa
5.3 GPa
7.1 GPa
10.4 GPa
17.3 GPa
5K
150 K
309 K
350 K
400 K
450 K
480 K
500 K
600 K
660 K
Fe65Ni35
50
40
30
Fe65Ni35
20
10
0
0
10
20
30
Energy (meV)
40
0
10
20
30
40
Energy (meV)
H. Giefers, University of Paderborn
Fe65Ni35
0.05 a -Fe
g(E) (1/meV)
0.05
0.04
0.03
0.02
0.01
0.00
0.05
0.04
0.03
0.02
0.01
0.00
0
5K
150 K
295 K
450 K
600 K
0.04
0.0 GPa
7.1 GPa
10.4 GPa
17.3 GPa
10
0.02
0.01
0.05
20
30
40
50
294 K
24 K
0.03
0
g(E) (1/meV)
g(E) (1/meV)
g(E) (1/meV)
Phonon DOS: Comparison of Fe65Ni35 with a-Fe
0.04
10 20 30 40
Energy (meV)
a -Fe
0.0 GPa
6.9 GPa
0.03
0.02
0.01
0
10 20 30 40
Energy (meV)
Energy (meV)
H. Giefers, University of Paderborn
Elastic / thermodyn. properties derived with g(E)
fLM: Lamb-Mössbauer factor
f LM



g E  1  e  E k BT

 exp  ER
dE 
 E k BT


E 1 e
0



F vib: Helmholtz free energy




Fvib  3 k BT g E  ln e E 2 kBT  e  E 2 kBT dE
0
D(s): mean force constant
S vib: entropy
along direction s=k/k


m
D s   2


g E  E dE
2
0

Svib  3 k B

0
E
 E  e E k BT  1
 2 kE T
2
B
 E k BT
g E  
 ln  e
 e k BT

1
 2k BT  e

D,HT: Debye temperature

 dE


cV: specific heat
high-temperature limit

 D ,HT
4

3 kB
 gE  E dE
0

cV  3 k B
 gE E k T  e
2
B
0
e E k BT
E k BT
 1
2
dE
H. Giefers, University of Paderborn
0.9
0.8
0.7
0.6
0.5
0.4
200
180
160
140
120
100
400
380
360
340
320
300
0
0.85
Pressure
0.80
0.75
0.70
200
180
160
140
120
100
460
440
420
400
380
360
200 400 600
0 3 6 9 12 15 18
Temperature (K)
Pressure (GPa)
50
0
-50
-100
-150
-200
cV (kB/atom)
D (K)
D (N/m)
fLM
Temperature
Svib(kB/atom) Fvib (meV/atom)
Elastic / thermodyn. properties: T / p dependence
Pressure
Temperature
12
8
4
0
-4
-8
3.6
6
3.4
4
3.2
2
3.0
0
2.8
2.8
3
2
2.7
1
0
0
2.6
200 400 600
Temperature (K)
0 3 6 9 12 15 18
Pressure (GPa)
H. Giefers, University of Paderborn
Average sound velocity vD
25
g(E) (1/eV)
20
15
10
0.0 GPa
3.1 GPa
5.3 GPa
7.1 GPa
10.4 GPa
17.3 GPa
in the Debye-model:
3
1 2
 3 3
3
v D v p vs
linear fit
2
g(E) = a·E
5
4.00
0
0
50
100
2
150
200
3.75
2
- strong variation of vD under
pressure
- “soft“ TA(110) mode hardens
strongly by pressure (known from
ultrasonic and neutron studies)
vD (km/s)
Energy (meV )
vD  3
3.50
V
2 2 3 a
3.25
3.00
2.75
2.50
0
3
6
9
12
15
18
p (GPa)
H. Giefers, University of Paderborn
Remark on the Debye-temperatures
Dispersion relation
(single crystal)
low energy region:
g(E)  a  E
vD  3
2
4
3 kB
 gE  E dE
0
in the Debye-model:
3
1 2

 3
3
3
v D v p vs
 vD 3 6 
kB
V

D , HT 
V
2 2 3 a
D, LT 
high-energy region:
D,LT
D,HT
2
H. Giefers, University of Paderborn
Debye-temperatures/Grüneisen parameters: HT vs. LT

4
3 kB

g E  E dE
0
derived from g(E) in the region
0  E  45 meV
D,HT
D,LT
500
D (K)
D , HT 
525
475
450
425
400
375
D, LT 
 vD 3 6 
kB
V
2
350
5
derived from g(E) in the region
0  E  10 meV
D = - d ln D/d ln V
D
4
3
“Pressure and Temperature Dependence
of the Fe Phonon Density of States
in the Invar alloy Fe65Ni35“
2
H. Giefers, K. Rupprecht, O. Leupold
and G. Wortmann (submitted)
0
1.00
LT
1
HT
0.98
0.96
0.94
0.92
0.90
0.88
V/V0
H. Giefers, University of Paderborn
Summary and Outlook
• The phonon DOS spectra of Fe65Ni35 as function of pressure and temperature
reflect, when compared with normal metals, the “anomalous” Invar properties.
• These “anomalous” properties are reflected in particular by the Debye
temperatures D,HT and D,LT and Grüneisen parameters HT and LT,
derived from the high-energy and low-energy part of the DOS; they disappear
with the suppression of the Fe moment around 12 GPa (in agreement with an
IXS study by Rueff et al., PRB 63, 132409 (2001))
Outlook:
• Comparison of local phonon DOS at the Fe sites measured here with
integral phonon DOS derived from IXS and neutron studies.
• Similar 57Fe-NIS studies of (ordered and disordered) Fe72Pt28 Invar.
H. Giefers, University of Paderborn
Acknowledgments
Co-workers:
Paderborn group
• Gerhard Wortmann
• Kirsten Rupprecht
• Ulrich Ponkratz
Grenoble/ESRF
• Olaf Leupold
• Alexandr Chumakov
• Bryan Doyle
• Rudolf Rüffer
Invar sample
• Uwe van Brück
• G.V. Smirnov
This work was supported by a joint BMBF proposal of the
Universities of Paderborn and Rostock (project 05 KS1PPB/3)
H. Giefers, University of Paderborn