Apr26_2_Duthil - CERN Accelerator School

Report
Materials properties at low temperature
CERN Accelerator School
Erice (Sicilia) - 2013
Contact :
Patxi DUTHIL
[email protected]
Contents
 Thermal properties
•
•
•
Heat capacity
Thermal conductivity
Thermal expansion
 Electrical properties
•
•
•
Electrical resistivity
RRR
Insulation properties
 Mechanical properties
•
•
Tensile behaviour
Material
 Magnetic properties
•
•
Introduction
Dia, para, ferro, antiferromagnets
CERN Accelerator School – 2013
Material properties at low temperature
2
THERMAL PROPERTIES
 Introduction
Thermal properties are related to:
•
atoms vibrations around their equilibrium position (in lattice crystal):
o vibrations amplitude diminishes with temperature
o vibrations may propagate at the sound speed and are studied as plane waves
to witch phonons are associated
•
movements of negative charges (electrons) and positive charges
(vacancies) for conductor materials
•
other effects: magnetic properties, superconducting state... (see specific
lectures)
CERN Accelerator School – 2013
Material properties at low temperature
3
THERMAL PROPERTIES
 Heat capacity C
•
 H 
 U 
C

CV  



(JK-1)
p
 T  p
 T V
quantity of energy (heat) extracted/introduced from/into 1kg of material
to decrease/increase by 1K its temperature.
Definition:
NB1 - Specific heat c: heat capacity or thermal capacity per unit of mass (Jkg-1K-1).
Molar heat capacity (Jmol-1K-1).
NB2 - The difference cp – cv is generally negligible for solids at low temperature.
•
Physical behaviour: capacity of a material to stock or release heat energy
•
as T  0, c  0
•
Heat capacity is important in cool-down or warm-up processes:
o to estimate the energy involved (and cost);
o to asses the transient states of thermal heat transfers as it relates to thermal
diffusivity.
CERN Accelerator School – 2013
Material properties at low temperature
4
THERMAL PROPERTIES
 Heat capacity c
•
Crystal lattice contribution: cph
U
 9 R T 
NkB
 D 
Debye model:
 3 T 

 D 
3R
c ph
3 D

0
T
3 D

0
T
3
x3



dx  3R D3  D 
x
 T
e 1
 D 
dx

D
 T
2
x


e  1
x 4e x
can be represented by a unique function:
o For T>2D: cph~3R
o For T<D/10: cph T3
CERN Accelerator School – 2013
Material properties at low temperature
D3 is the third Debye function
R is the gas constant
The Debye temperature is given by:
1/ 3
hv  3N 
D  s 

k B  4V 
h: Planck constant
kB: Boltzmann constant
vs: sound speed in the material
N/V: number of atoms per unit volume
Material
D (K)
Copper
Aluminium
Titanium
Niobium
SS 304
SS 316
340
430
420
265
470
500
12π  T 
 
c ph 
5  θ D 
4
3
5
THERMAL PROPERTIES
 Heat capacity c
•
Electron contribution: ce
For solid conductor : ce=T
•
 (10-3 Jkg-1K-2)
Copper
Aluminium
Titanium
Niobium
11.0
50.4
74.2
94.9
Heat capacity of metallic conductors:
o
o
o
o
•
Material
c = cph + ce
For T>2D:
For T<D/10:
Bellow 10K:
(cph~3R )  c  T and diminishes slowly as T decreases ( <<1)
c=cph + ce=T3 + T
cph<<1  c  T
Heat capacity of thermal insulator:
o cph is predominant
o For T>2D: cph~3R
o For T<D/10: cph  T3
•
Heat capacity of superconductors:
c=   Tc a e(-b Tc/T) for T < Tc, Tc the critical temperature
 : coefficient of the electronic term and determined at T> Tc
a, b: coefficients
CERN Accelerator School – 2013
Material properties at low temperature
6
THERMAL PROPERTIES
 Specific heat capacity curves for some materials
104
103
102
101
100
10-1
10-2
10-3
CERN Accelerator School – 2013
Material properties at low temperature
7
THERMAL PROPERTIES
 Specific heat capacities of some materials
Temp.
(K)

1
2
4
6
8
10
15
20
25
30
35
40
50
60
70
77
80
90
100
120
140
160
180
200
220
240
260
280
300
Cp
SS 304
0.464
0.931
1.880
2.860
3.900
5.020
8.120
12.60
19.60
29.30
42.00
57.80
100.0
128.0
167.0
188.0
197.0
230.0
250.0
290.0
329.0
364.0
395.0
419.0
431.0
439.0
447.0
459.0
470.0
Cu
0.012
0.028
0.090
0.218
0.460
0.870
2.930
7.270
15.30
26.60
41.80
59.00
95.0
135.0
170.0
195.0
205.0
230.0
251.0
286.0
312.0
332.0
346.0
356.0
364.0
371.0
377.0
382.0
386.0
Brass
0.012
0.035
0.142
0.364
0.780
1.470
4.970
11.900
24.40
40.70
60.40
81.40
121.0
160.0
191.0
212.0
221.0
247.0
267.0
298.0
321.0
337.0
349.0
357.0
365.0
370.0
373.0
376.0
380.0
(J/kg-K)
Constantan Manganin Inconel 718 K Monel
0.113
0.090
0.399
0.110
0.211
0.150
0.781
0.220
0.446
0.246
1.530
0.473
0.761
0.404
2.230
0.784
1.180
0.679
2.940
1.180
1.740
1.120
3.700
1.700
3.550
3.190
5.890
3.650
7.440
7.420
10.00
7.100
15.40
15.00
18.30
12.90
26.70
25.80
30.00
21.00
41.80
40.30
45.40
31.80
59.00
57.10
63.00
45.00
95.0
93.0
100.0
78.0
135.0
133.0
130.0
110.0
173.0
171.0
165.0
150.0
196.0
195.0
183.0
171.0
205.0
205.0
190.0
180.0
232.0
234.0
215.0
210.0
245.0
247.0
240.0
240.0
276.0
279.0
282.0
285.0
308.0
312.0
315.0
314.0
332.0
338.0
347.0
336.0
346.0
354.0
379.0
355.0
356.0
365.0
405.0
370.0
364.0
374.0
416.0
385.0
371.0
382.0
422.0
399.0
377.0
389.0
429.0
410.0
382.0
394.0
438.0
420.0
386.0
400.0
444.0
430.0
Metals and alloys
Invar-36
0.251
0.481
0.932
1.510
2.270
3.220
6.480
12.10
21.30
35.00
55.80
80.00
130.0
176.0
211.0
239.0
250.0
272.0
297.0
337.0
366.0
388.0
405.0
419.0
431.0
441.0
448.0
454.0
463.0
Ti-6Al-4V
0.001
0.008
0.066
0.221
0.525
1.030
3.490
8.20
16.10
27.00
41.30
58.30
99.5
144.0
188.0
217.0
229.0
267.0
301.0
356.0
398.0
430.0
457.0
478.0
494.0
508.0
519.0
530.0
539.0
Al
0.100
0.108
0.276
0.515
0.867
1.400
3.840
8.90
17.80
31.50
51.90
77.50
142.0
214.0
287.0
336.0
357.0
422.0
481.0
579.0
653.0
713.0
760.0
797.0
826.0
849.0
869.0
886.0
902.0
6061-T6
0.051
0.108
0.280
0.515
0.867
1.400
3.840
8.90
17.80
31.50
51.90
77.50
142.0
214.0
287.0
336.0
357.0
422.0
481.0
579.0
653.0
713.0
760.0
797.0
826.0
849.0
869.0
886.0
902.0
5083-T0
0.051
0.108
0.280
0.515
0.867
1.400
3.840
8.90
17.80
31.50
51.90
77.50
142.0
214.0
287.0
336.0
357.0
422.0
481.0
579.0
653.0
713.0
760.0
797.0
826.0
849.0
869.0
886.0
902.0
Niobium
0.060
0.175
0.422
0.768
1.290
2.050
5.840
12.00
20.40
32.00
49.30
68.00
99.0
127.0
152.0
167.0
173.0
189.0
202.0
222.0
234.0
243.0
250.0
254.0
258.0
261.0
264.0
266.0
268.0
NbTi
0.002
0.027
0.523
1.270
2.320
3.800
10.30
21.00
37.30
58.00
83.80
110.0
150.0
200.0
240.0
261.0
270.0
300.0
307.0
329.0
357.0
378.0
390.0
400.0
410.0
417.0
422.0
425.0
426.0
Constantan: Cu-Ni
Manganin: Cu-Mn-Ni
Monel: Ni-Cu-Fe
CERN Accelerator School – 2013
Material properties at low temperature
8
THERMAL PROPERTIES
 Specific heat capacities of some materials
Cp
Temp.

1
2
4
6
8
10
15
20
25
30
35
40
50
60
70
77
80
90
100
120
140
160
180
200
220
240
260
280
300
(J/kg-K)
Thermal Insulators
Pyrex Glass
Teflon
(PTFE)
Polycarbonate
Amorphous
Nylon
0.003
0.025
0.197
0.883
2.19
4.19
13.7
27.4
44.3
62.8
80.5
98.4
136
0.04
0.32
2.62
6.33
11.4
18
44.5
76
102
125
146
165
202
0.024
0.192
1.54
6.1
13.7
24
60.4
102
132
159
185
210
255
0.018
0.144
1.15
3.97
8.85
16
48.3
93
143
197
248
300
410
G-10
(normal to
cloth lay)
0.00986
0.0613
0.538
2.09
4.76
8.47
22.7
41.5
62.5
84.6
105
126
170
174
209
229
237
264
272
321
396
459
502
539
570
606
659
714
737
238
274
301
312
350
386
453
517
589
668
741
785
832
901
973
1010
286
325
352
364
403
439
515
595
675
754
832
910
989
1070
1150
1230
500
587
653
680
750
820
953
1080
1200
1330
1450
1570
1690
1800
1900
2000
213
252
277
288
321
355
420
481
539
595
648
698
745
791
836
880
(K)
CERN Accelerator School – 2013
Material properties at low temperature
0.03
0.17
1.56
5.72
12.4
21.3
49.8
84
117
150
180
210
270
Carbon
ReinforcedPlastic,
CRFP normal
0.0184
0.24
1.61
3.79
6.63
10
19.4
30
40.1
50.6
61.4
73.5
105
328
380
422
440
490
540
638
733
826
915
1000
1080
1150
1230
1290
1360
140
173
200
211
239
274
341
405
467
528
592
666
741
808
881
980
Epoxy
Mylar, PET
0.013
0.104
0.829
3.37
7.55
13
27
48
84.3
125
162
195
240
282
323
349
360
400
442
531
621
704
777
845
906
967
1040
1110
1160
9
THERMAL PROPERTIES
 Heat capacity
T2
• During a thermodynamic process at constant pressure: h   c p dT
T
• The involved energy is then E= mh
• h can be seen as a heat stock per mass unit (Jkg-1)
1
106
105
104
103
102
101
100
10-1
10-2
10-3
At low temperature, it can be noticed:
CERN Accelerator School – 2013
Material properties at low temperature
- the high value of G10 (epoxy+glass fibers)
- the high value of stainless steel 304 L
- the high values of He and N2 gases
10
THERMAL PROPERTIES
T
Temp.
(K)

1
2
4
6
8
10
15
20
25
30
35
40
50
60
70
77
80
90
100
120
140
160
180
200
220
240
260
280
300
h   c p dT
Metals and alloys
(J/kg)
1K
SS 304
0
0.697
3.5
8.2
14.9
23.9
55
107
186
308
484
733
1520
2660
4140
5380
5960
8100
10500
15920
22100
29100
36700
44800
53200
62000
70700
79600
88800
Cu
0
0.0195
0.128
0.315
0.931
2.37
9.66
35.5
90.5
194
366
615
1380
2530
4060
5340
5940
8120
10500
15870
21900
28300
35100
42100
49300
56700
64200
71800
79500
Brass
0
0.0219
0.18
0.488
1.53
3.97
17
59.4
150
310
565
916
1930
3330
5090
6500
7150
9490
12100
17750
24000
30500
37400
44500
51700
59100
66500
74000
81600
Constantan Manganin
0
0
0.161
0.121
0.807
0.514
1.97
1.06
3.88
2.08
6.82
3.98
17.3
12.7
45.8
39.5
102
94
205
195
378
362
627
602
1390
1350
2540
2480
4090
4000
5380
5290
5980
5890
8170
8090
10600
10500
15860
15810
21700
21700
28000
28200
34800
35100
41900
42300
49100
49700
56500
57300
63900
65000
71500
72800
79200
80800
CERN Accelerator School – 2013
Material properties at low temperature
Inconel 718
0
0.591
2.9
6.58
11.7
18.5
39.6
80.5
150
269
459
727
1540
2690
4170
5390
5950
7980
10300
15500
21500
28200
35500
43300
51400
59800
68400
77000
85900
K Monel
0
0.164
0.848
2.03
3.96
6.9
18.9
45.9
94.6
179
310
501
1110
2050
3350
4470
5000
6950
9210
14440
20410
27010
33910
41210
48710
56610
64710
73010
81510
Invar-36
0
0.367
1.77
4.15
7.88
13.4
35.3
82.2
162
303
533
866
1920
3450
5390
6970
7700
10300
13200
19530
26600
34100
42000
50300
58800
67500
76400
85500
94600
Ti-6Al-4V
0
0.00384
0.0635
0.226
0.905
2.56
12.1
41.2
101
207
378
625
1410
2630
4280
5700
6370
8850
11700
18260
25800
34100
43000
52400
62100
72100
82400
92900
103600
Al
0
0.0992
0.457
1.1
2.42
4.83
15.5
47.6
110
234
441
761
1850
3620
6130
8310
9350
13300
17800
28400
40800
54400
69200
84800
101000
117800
134800
152800
170800
6061-T6
0
0.078
0.452
1.11
2.42
4.82
15.5
47.6
110
234
441
761
1850
3620
6130
8310
9350
13300
17800
28400
40800
54400
69200
84800
101000
117800
134800
152800
170800
5083-T0
0
0.078
0.452
1.11
2.42
4.82
15.5
47.6
110
234
441
761
1850
3620
6130
8310
9350
13300
17800
28400
40800
54400
69200
84800
101000
117800
134800
152800
170800
Niobium
0
0.118
0.703
1.7
3.65
7.18
25.9
69.5
147
279
487
774
1610
2740
4140
5260
5770
7580
9540
13770
18340
23140
28040
33095
38220
43440
48640
53940
59275
NbTi
0
0.0002
0.473
1.97
5.42
11.8
44.3
122
266
502
864
1340
2640
4390
6600
8350
9150
12000
15112
21530
28400
35700
43300
51306
59410
67700
76100
84500
93080
11
THERMAL PROPERTIES
T
Temp.
(K)

1
2
4
6
8
10
15
20
25
30
35
40
50
60
70
77
80
90
100
120
140
160
180
200
220
240
260
280
300
h   c p dT
(J/kg)
Thermal Insulators
1K
Polycarbonate
Pyrex Glass Teflon (PTFE) Amorphous
0
0
0
0.0128
0.126
0.0899
0.179
2.75
1.47
0.953
11
9.35
3.82
28.4
28.9
10.4
58.1
65.3
54.8
223
289
154
512
678
336
955
1260
600
1520
1990
957
2200
2860
1400
2980
3840
2580
4820
6180
4130
7020
8890
6050
9580
11900
7580
11600
14300
8280
12500
15400
10800
15800
19200
13500
19500
23400
19590
12360
32960
26700
31100
44100
35100
52600
56700
44700
77300
71000
55200
104500
86900
66100
134100
104300
77900
165900
123300
90600
200700
143400
104300
237000
166400
118500
276000
189400
CERN Accelerator School – 2013
Material properties at low temperature
Nylon
0
0.0655
1.13
5.32
17.5
42.9
206
547
1140
1990
3090
4470
8020
12600
18000
22400
24400
31500
39400
57100
77400
100300
125600
153400
183400
216400
251400
288400
327400
G-10 (normal
to cloth lay)
0
2.86E-02
0.504
3.02
9.69
22.7
101
257
520
884
1360
1940
3420
5330
7650
9510
10400
13400
16800
7740
16800
27000
38300
50700
64200
78600
94000
110000
127000
Epoxy
0
0.0757
1.48
9.22
27.3
59.5
241
567
1070
1740
2560
3540
5940
8930
12500
15300
16600
21200
26400
38200
51900
67500
84900
104100
124900
147400
171400
196400
222400
Carbon
ReinforcedPlastic,
CRFP normal
Mylar, PET
0
0
0.101
0.0489
1.78
0.794
7.34
5.65
17.8
16.7
33.9
35.7
108
126
230
317
405
659
632
1170
910
1890
1250
2780
2140
4970
3360
7580
4920
10600
6230
13000
6850
14000
9100
17800
11700
22000
17840
31770
25300
43300
34000
56500
43900
71300
55200
87600
67800
105000
81900
124000
97300
144000
114700
165000
132700
188000
12
THERMAL PROPERTIES
 Thermal conductivity
•
The Fourier’s law gives the quantity of heat through a unit surface and
diffusing during a unit of time within a material subjected to a temperature
gradient

q  k T
•
(J/s/m²W/m²)
Example: heat conduction (diffusion) into a lineic support
L: length (m); A: cross section area (m²)
TH
TC
QHC
0
TC
Q HC
L
dx  Q HC    k (T )dT
Thus we can write 
A
A
0
TH
L
L x
and (if k=cst) :
•
•
QHC  k
TH  TC
A0
L
k is the thermal conductivity (W/m/K). It relates to the facility with which
heat can diffuse into a material.
However, k is non constant especially on the cryogenic temperature range.
CERN Accelerator School – 2013
Material properties at low temperature
13
THERMAL PROPERTIES
 Thermal conductivity
•
•
Similarly simplified, heat is transported in solids by electrons and phonons
(lattice vibration)  k = ke + kph
Lattice contribution:
o kph=1/3  cph vs lph Vm,
Vm is the material density (Kg/m3)
lph is the mean free path of the phonons
o At very low T (T<<D) kp~ T3
•
Electronic contribution:
o ke=1/3  ce vF le Vm,
o At very low T (T<<D) ke~ T
•
•
Vm is the material density
le is the mean free path of the electrons
vF is the Fermi velocity
In semi-conductors, heat conduction is a mixture of phonons and electrons
contribution
Other interactions may occur (electron-vacancy...)
CERN Accelerator School – 2013
Material properties at low temperature
14
THERMAL PROPERTIES
 Thermal conductivity
•
5<RRR<150
100<RRR<200
200<RRR<5000
For pure metals:
o kph is negligible
o k has a maximum at low temperature
o At low T°, k is affected by impurities
o The more is the purity of the material,
o
o
the higher is this maximum
the lower is the T° of this maximum
o k  T at low temperature
•
104
Ordinary copper:
OFHC copper:
Very pure copper
103
102
For metallic alloys:
o k decreases as T decreases
101
o k  T at low temperature
o Wiedemann-Franz law:
relates ke and the electric resistivity  :  ·ke /T = 2.44510-8 (W/K²)
•
For superconductors:
o T > Tc (normal state)  cf. behaviour of metals
o T < Tc (Meissner state): ks  T3 and ks(T) << kn(T)  thermal interrupter
CERN Accelerator School – 2013
Material properties at low temperature
15
THERMAL PROPERTIES
 Thermal conductivity
•
For thermal insulators
o k is smaller than for metals (by several orders of magnitude)
o k  T3 (for crystallized materials)
•
Thermal conductivities
103
102
(RRR=30)
101
100
10-1
10-2
10-3
NB: LHe at 4K or He at 300 K (gas), has smaller thermal conductivity than an insulator like G10.
CERN Accelerator School – 2013
Material properties at low temperature
16
THERMAL PROPERTIES
 Thermal conductivity
Temp.
(K)

1
2
4
6
8
10
15
20
25
30
35
40
50
60
70
77
80
90
100
120
140
160
180
200
220
240
260
280
300
k (Wm-1K-1)
SS 304 Cu-RRR=30 Brass Constantan Manganin Inconel 718 K Monel Invar-36
0.042
46
0.626 9.50E-02
7.30E-02
0.107
0.138
1.98E-02
0.103
92
1.5
0.285
0.18
0.233
0.35
6.05E-02
0.227
184
3.59
0.878
0.484
0.504
0.889
0.185
0.381
276
6.08
1.64
0.848
0.809
1.53
0.345
0.565
367
8.86
2.52
1.26
1.14
2.26
0.533
0.77
457
11.8
3.5
1.7
1.48
3.05
0.74
1.33
670
18.5
6.07
2.86
2.25
5.09
1.26
1.95
848
24.6
8.7
4.1
2.95
7.1
1.8
2.61
957
29.2
11
5.38
3.53
8.68
2.36
3.3
1000
32.8
13
6.6
4
10
2.9
4.02
974
35.8
14.4
7.66
4.35
11.2
3.38
4.7
903
38.3
15.5
8.6
4.65
12.2
3.85
5.8
732
42
18.1
10.1
5.3
13.5
4.9
6.8
598
46.3
18.3
11.2
5.7
14.2
5.8
7.6
514
51
18.8
12
6.1
14.9
6.6
8.07
477
53.7
19
12.6
6.35
15.3
7.17
8.26
465
55
19.1
12.9
6.45
15.5
7.4
8.86
438
60.6
19.5
13.5
6.8
16
8
9.4
422
65
20
14
7.1
16.5
8.5
10.4
403
72.9
20.8
14.8
7.63
17.4
9.33
11.2
396
79.7
21.4
15.4
8.09
18.2
10
11.9
394
85
21.9
16
8.5
18.8
10.6
12.5
392
89.3
22.4
16.6
8.86
19.5
11.2
13
391
93.3
22.8
17.2
9.2
20
11.7
13.5
390
98.2
23.2
18
9.52
20.4
12.3
13.9
389
103
23.6
19
9.84
20.8
12.8
14.3
388
108
23.9
20.1
10.2
21.3
13.1
14.6
387
113
24.2
21.1
10.5
21.8
13.4
14.9
386
116
24.9
22
11
22.2
13.7
CERN Accelerator School – 2013
Material properties at low temperature
Metals and alloys
Ti-6Al-4V Al-RRR=30
0.124
28.6
0.223
57.1
0.403
114
0.569
171
0.725
228
0.87
284
1.21
420
1.5
541
1.71
636
1.9
698
2.11
711
2.3
690
2.6
579
2.9
467
3.2
386
3.39
344
3.46
329
3.7
291
3.98
265
4.45
235
4.83
227
5.19
226
5.55
225
5.9
226
6.22
226
6.54
226
6.87
227
7.24
228
7.7
229
6061-T6
2.2
4.7
9.53
14.3
19
23.8
37
50.1
61.1
71
80.3
88.5
100
111
113
117
118
120
121
124
128
131
133
135
141
147
152
156
160
5083-T0 Niobium
NbTi
0.677
0.4
2.60E-02
1.45
1.8
6.00E-02
3.12
8.99
0.176
4.89
21.6
0.304
6.73
37.7
0.436
8.6
55
0.57
13.2
75.9
0.858
17.8
85
1.13
22.3
87.9
1.42
26.6
86
1.7
30.6
82.3
1.98
34.5
77
2.26
42
66
2.81
48.2
61
3.37
54
58
3.93
57.6
56.5
4.32
59
56
4.49
63
55
5.04
67
54.5
5.5
74
54.5
6.28
80.1
54.5
6.89
86
54.5
7.38
92.1
54.5
7.76
98
54.5
8.1
104
54.5
8.5
110
54.5
8.85
116
54.5
9.12
122
54.5
9.32
128
54.5
9.5
17
THERMAL PROPERTIES
 Thermal conductivity integrals
T
•
one must integrates the
thermal conductivity over
the considered temperature
range in order to evaluate
the diffused heat quantity.
Q HC
•
T
A C
  k (T )dT
L TH
Thermal conduction integrals
are evaluated from a
reference temperature TREF
(1K for example). Thus
conduction integrals of
interest over a given
temperature range is given
by the difference:
TH
TH
TC
TC
TREF
TREF
 k (T )dT   k (T )dT   k (T )dT
CERN Accelerator School – 2013
Material properties at low temperature
 k (T )dT
Temp.
(K)

1
2
4
6
8
10
15
20
25
30
35
40
50
60
70
77
80
90
100
120
140
160
180
200
220
240
260
280
300
(W/m)
Thermal Insulators
T1 K
Pyrex Glass Teflon (PTFE)
0
0
0.0302
0.00831
0.165
0.0646
0.358
0.171
0.592
0.32
0.857
0.504
1.49
1.05
2.2
1.72
2.99
2.47
3.87
3.3
4.88
4.2
6.01
5.15
8.59
7.16
11.7
9.29
15.3
11.5
18.1
13.1
19.4
13.8
24.1
16.2
29.3
18.7
41.1
23.66
54.7
28.7
69.8
33.8
86.2
39
103.8
44.2
122.2
49.4
141.3
54.6
161.3
59.8
181.3
65
201.3
70.2
Polycarbonate
Amorphous
0
0.0226
0.079
0.143
0.214
0.294
0.54
0.849
1.23
1.66
2.11
2.6
3.66
4.84
6.1
7.03
7.43
8.84
10.3
13.41
16.75
20.29
24
27.9
32
36.3
40.7
45.3
50
Nylon
0
0.00271
0.0154
0.041
0.0803
0.134
0.337
0.637
1.04
1.54
2.14
2.84
4.47
6.29
8.3
9.79
10.4
12.7
15.1
20.04
25.2
30.6
36.1
41.8
47.5
53.4
59.4
65.5
71.7
G-10
(normal to
cloth lay)
0
0.0148
0.0901
0.214
0.381
0.584
1.19
1.93
2.78
3.74
4.8
5.95
8.48
11.2
14.3
16.7
17.7
21.3
25.2
33.61
42.8
52.6
63
73.9
85.1
96.8
108.7
120.9
133.2
Epoxy
0
0.0262
0.112
0.212
0.322
0.438
0.74
1.07
1.43
1.82
2.24
2.67
3.62
4.67
5.79
6.63
7
8.3
9.71
12.86
16.37
20.11
23.91
27.91
32.01
36.11
40.31
44.61
48.91
Carbon
ReinforcedPlastic,
CRFP normal
0
0.00709
0.031
0.065
0.109
0.165
0.356
0.622
0.968
1.39
1.87
2.41
3.68
5.26
7.13
8.62
9.32
12
15.1
22.2
30.6
40.3
51.2
62.8
74.5
86.4
98.6
111.1
124.1
Mylar, PET
0
0.00174
0.0115
0.0342
0.0704
0.12
0.309
0.57
0.885
1.24
1.63
2.04
2.89
3.8
4.74
5.42
5.72
6.74
7.79
9.96
12.22
14.54
16.91
19.29
21.69
24.19
26.59
28.99
31.49
18
THERMAL PROPERTIES
 Thermal conductivity integrals
T

k (T )dT
(W/m)
Temp.
T
(K)
1K

SS304 Cu-RRR=300 Cu-RRR=30 Brass Constantan Manganin Inconel 718 K Monel
1
0
0
0
0
0
0
0
0
2
0.0726
69
1.05
0.183
0.124
0.169
0.241
4
0.4
3560
345
6.07
1.31
0.773
0.901
1.46
6
1.02
8360
807
16
3.87
2.12
2.24
3.91
8
1.96
14900
1450
31
8.03
4.22
4.19
7.72
10
3.28
22800
2270
51.2
13.9
7.15
6.77
13
15
8.51
46600
5130
128
38.2
18.6
16.2
33.6
20
16.7
72900
8910
235
74.8
35.9
29.2
63.8
25
28.1
95800
13500
370
124
59.7
45.4
103
30
42.8
115000
18400
525
184
89.6
64.3
150
35
61.2
130000
23300
697
252
125
85
204
40
82.9
140000
28000
883
328
166
108
262
50
136
155000
36200
1280
497
260
158
391
60
199
164000
42900
1730
679
367
213
530
70
271
171000
48400
2210
865
483
272
675
77
326
176000
51800
2580
997
569
315
781
80
350
177000
53300
2740
1050
607
334
828
90
436
182000
57800
3320
1250
739
401
985
100
527
187000
62000
3950
1440
877
470
1150
120
725
196200
70270
5330
1847
1165
617
1489
140
940
204900
78200
6860
2269
1467
775
1845
160
1170
213300
86100
8500
2700
1781
941
2210
180
1414
221700
94000
10240
3140
2107
1114
2600
200
1667
229900
101800 12080
3600
2447
1295
2990
220
1937
238200
109600 13950
4060
2797
1480
3400
240
2207
246300
117400 16050
4530
3167
1680
3810
260
2487
254400
125100 18150
5000
3557
1880
4230
280
2777
262500
132900 20350
5480
3967
2080
4660
300
3077
270500
140600 22650
5970
4397
2300
5100
CERN Accelerator School – 2013
Material properties at low temperature
Metals and alloys
Invar-36 Ti-6Al-4V Al-RRR=30 6061-T6 5083-T0
0
0
0
0
0
0.0388
0.174
42.8
3.46
1.06
0.276
0.804
214
17.7
5.61
0.819
1.78
501
41.4
13.7
1.7
3.07
900
74.6
25.3
2.95
4.67
1410
118
40.5
7.93
9.91
3190
272
95.2
15.6
16.7
5590
487
173
26
24.7
8560
765
273
39.1
33.8
11900
1100
395
54.7
43.8
15400
1480
538
72.9
54.8
18900
1900
701
117
79.4
25300
2840
1080
170
107
30500
3900
1540
232
137
34800
5020
2050
281
160
37300
5830
2440
302
171
38300
6180
2610
379
207
41400
7370
3220
462
245
44200
8580
3870
640
329
49240
11040
5280
834
422
53830
13560
6820
1040
522
58300
16130
8490
1258
630
62800
18780
10270
1482
744
67300
21480
12170
1732
865
71800
24180
14170
1982
993
76400
27080
16370
2242
1127
80900
30080
18570
2502
1265
85400
33180
20970
2772
1415
90000
36380
23470
Niobium NbTi
0
0
0.968
0.04
10.9
0.27
46.2
0.756
107
1.5
192
2.5
515
6.04
932
11.1
1360
17.4
1800
25.2
2220
34.4
2620
45
3340
70.4
3970
101
4560
138
4960
167
5130
180
5690
228
6230
280.49
7320
398
8400
530
9490
673
10580
824
11665
983.3
12750
1150
13840
1323
14930
1503
16020
1687
17119 1875.4
19
THERMAL PROPERTIES
 Thermal diffusivity
•
Heat conduction equation (non stationary):

T 
ρc p
   (k T )
t

Isotropic
Cst coefficients
T
k

 2T  κ  2T
t ρc p
Thermal diffusivity:
•
•
•
κ
k
ρc p
[m²/s]
The thermal diffusivity allows to asses the time constant of heat to diffuse
over a characteristic length L (time to warm-up or cool-down by a system
by heat conduction)
For metals, at low T°: k  T and cp  T3  k rises as T decreases
(especially for highly pure metals for which k is strongly affected by purity
at low T° ; not cp)
Generally speaking Cp rises as T decreases
CERN Accelerator School – 2013
Material properties at low temperature
20
THERMAL PROPERTIES
 Thermal diffusivity
101
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
NB: 304L thermal diffusivity is two order of magnitude lower than G10
CERN Accelerator School – 2013
Material properties at low temperature
21
THERMAL PROPERTIES
 Thermal expansion/contraction
•
•
1  dV 
Coefficient of thermal expansion (cf. Basics thermodynamics): V   
V  dT  P
Generally speaking, V>0 and so at constant pressure, a temperature
decrease induces a reduction of the physical dimensions (size) of a body.
 Thermal expansion/contraction of solids
•
•
For solid, we can ignore the effect of pressure
In cryogenic systems, components can be submitted to large temperature
difference:
o because they are links to both cold and warm surfaces (cold mass supports) ;
o during cool-downs or warm-ups transient states.
•
Being a function of the temperature, thermal expansion can affect:
o the resistance of an assembly, generating large stresses;
o the dimensional stability of an assembly (buckling).
CERN Accelerator School – 2013
Material properties at low temperature
22
THERMAL PROPERTIES
 Thermal expansion/contraction of solids
1 d
• Linear expansion coefficient:  (T ) 
•
For a crystallized solid, it varies as cph
(K-1)
 dT
o At very low temperature:   T3
o Tends to a constant value as T increases towards ambient temperature
•
In practice, the expansion coefficient is computed from a reference
temperature (300K):
T
    REF

   (T )
TREF

 REF
l
where denotes for the length of the
body at the reference temperature
o around ambient temperature:
l / l  T
o at low temperature (4-77K ):
l / l  T4 (in practice the coefficient of
proportionality is negligible)
CERN Accelerator School – 2013
Material properties at low temperature
23
THERMAL PROPERTIES
 Thermal expansion/contraction of solids
•
We note that most of the thermal expansion/contraction is effective between
300K and 77K (temperature of boiling LN2 at P=1atm).
CERN Accelerator School – 2013
Material properties at low temperature
24
THERMAL PROPERTIES
 Thermal expansion/contraction of solids
•
Example:
B
Tamb
A ( for example Cu)
Cu
T << Tamb
Induces:
- Large stress
- Mechanical instability (buckling)
CERN Accelerator School – 2013
Material properties at low temperature
T << Tamb
Induces large stress
25
ELECTRICAL PROPERTIES
 Electric conductivity
•
•
Within metals, electrical charge is transported by the "free electrons".
The parameters determining the electrical conductivity of metals are:
o
o
o
o
o
N: the number of electrons per unit volume
e: the charge carried by an electron
m: the mass of an electron
v: the average velocity of "conduction electrons"
le : the average distance the electrons travel before being scattered by atomic
lattice perturbation (the mean free path)
•
Only the mean free path le is temperature dependant.
•
At high (ambient) temperature, the electron free path le is dominated by
electron scattering from thermal vibrations (phonons) of the crystal lattice.
The electrical conductivity is linearly temperature-dependant.
At low temperature, the free path le is limited mainly by scattering off
chemical and physical crystal lattice imperfections (impurities, vacancies,
dislocations). The electrical conductivity tends to a constant value.
•
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Material properties at low temperature
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ELECTRICAL PROPERTIES
 Electric resistivity of metals
• (T)=0+i(T), 0 =cst and i relates to the electron-phonon interaction
•
It can be shown that:
o For T>2D:
o For T<D/10:
i(T)  T
i(T)  T5 and in practice i(T)  Tn with 1<n<5
103
102
101
100
10-1
NB: electrical resistance: R(T)=L/S ()
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Material properties at low temperature
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ELECTRICAL PROPERTIES
 Electric resistivity of metals
•
An indication of metal purity is provided by the determination of a
 (273K )  (273K )
Residual (electrical) Resistivity Ratio: RRR 

 (4,2 K )
0
101
Ordinary copper:
OFHC copper:
Very pure copper
5<RRR<150
100<RRR<200
200<RRR<5000
100
10-1
10-2
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Material properties at low temperature
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ELECTRICAL PROPERTIES
 Electric resistivity
•
•
•
Resistivity of semiconductors is very non linear
It typically increases with decreasing the temperature due to fewer electron
in the conduction band (used to make temperature sensors: thermistor)
Around high (ambient) temperature, electrical properties are not modified
by impurities and:
ρ(T )  A  e
δ
2 k BT
where
A is an experimental constant
δ energy band depending on the
material
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Material properties at low temperature
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MECHANICAL PROPERTIES
 Introduction
•
Tensile test:
F/2
Stress
s=F/s0 (N/m²Pa)
cross section s0
L
Ultimate tensile strength
UTS
Fracture
F/2
YS0.2
0.2%
offset line
Yield tensile strength
YS
Slop:
Young modulus
E = Re  L/L
Plastic deformation
(irreversible)
NB: stiffness k=EA/L
Necking
Strain
Elastic deformation
(reversible)
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Material properties at low temperature
L/L (%)
30
MECHANICAL PROPERTIES
 Introduction
•
• Brittle behaviour
Ductile behaviour
(think about lead, gold...)
(think about glass)
Stress
Stress
Strain
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Material properties at low temperature
Strain
31
MECHANICAL PROPERTIES
 Introduction
T1
>
F/S0
T2
>
F/S0
T3
Fragile fracture
F/S0
F/S0
UTS
YS
A%
•
A%
A%
T3
T2
T1
T
When temperature goes down, a material tends to become brittle
(fragile) even if it is ductile at ambient temperature.
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Material properties at low temperature
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MECHANICAL PROPERTIES
 Mechanical behaviour
•
The mechanical behaviour at cold temperature of metals and metallic
alloys depends on their crystal structure.
• For face-centered cubic crystal structure (FCC):
(Cu-Ni alloys, aluminium and its alloys, stainless steel (300
serie), Ag, Pb, brass, Au, Pt),
they belongs ductile until low temperatures and do not
present any ductile-brittle transition.
• For body-centered cubic cristal structure (BCC):
(ferritic steels, carbon steel, steel with Ni (<10%), Mo, Nb,
Cr, NbTi)
a ductile-brittle transition appears at low T°.
• For compact hexagonal structure (HCP):
(Zn, Be, Zr ,Mg, Co, Ti alloys (TA5E)...)
no general trend comes out.
mechanical properties depends on interstitial components
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Material properties at low temperature
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MECHANICAL PROPERTIES
 Mechanical behaviour
FCC
BCC
HCP
Copper
Aluminium
Nickel
Silver
Gold
Austenitic stainless steel
(304, 304L, 316, 316L, 316LN)
Lead
Platinium
Iron
Carbon steel
Nickel Stell
Niobium
Chromium
Niobium-Titanium
Zinc
Titanium
Magnesium
Cobalt
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Material properties at low temperature
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MECHANICAL PROPERTIES
 Yield, ultimate strength
•
•
Young Modulus slightly change with temperature
Yield and ultimate strengths increases at low temperature
From: Ekin, J.W. Experimental Techniques for Low Temperature Measurements
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Material properties at low temperature
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MECHANICAL PROPERTIES
 General behaviours
From: Ekin, J. Experimental Techniques for
Low Temperature Measurements
Young Modulus
1:
2:
3:
4:
2024 T4 aluminium
copper-beryllium
K monel
Titanium
5 : SS 304
6 : Carbon Steal C 1020
7 : Steal 9% Ni
From: Technique de l’Ingénieur
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Material properties at low temperature
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MAGNETIC PROPERTIES
 Introduction
•
•
In vacuum:


B  0 H
B (TVs m-²N A-1 m-1); 0=4 10-7 (N A-2); H (Vs/Am A m-1)
In a material: B=μ0 H + μ0 M
M (Vs/Am A m-1)
M = χ H is the magnetization and represents how strongly a region of material is
magnetized. It is defined as the net magnetic dipole moment per unit volume.
•
Thus: B= μ0(1 + χ) H = μ0 μr H
The magnetic moment of a free atom depends on:
o electrons spin
o orbital kinetic moment of the electrons around the nucleus
o kinetic moment change induced by the application of a magnetic field
•
5 types of magnetic behaviour can be distinguished:
o Diamagnetism and paramagnetism due to isolated atoms (ions) and free
electrons
o Ferromagnetism, anti-ferromagnetism and ferrimagnetism due to collective
behaviour of atoms
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Material properties at low temperature
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MAGNETIC PROPERTIES
 Diamagnetic materials
•
•
•
•
•
•
•
If magnetic susceptibility  = R-1 <0 where R is the relative
magnetic permeability
It causes a diamagnet to create a magnetic field in opposition to an
externally applied magnetic field
When the field is removed the effect disappears
Examples: Silver, Mercury, Diamond, Lead, Copper
If the (small) field H is applied then:
M=H
 does not depend on temperature
NB: type I superconductors are perfect diamagnets for T<TC
Ex.: Cu, Nb
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Material properties at low temperature
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MAGNETIC PROPERTIES
 Paramagnetic materials
•  = R-1 >0
• Paramagnets are attracted by an externally applied magnetic field
•  is small  slight effect
• Different models of paramagnetic systems exist
• Relation to electron spins
o Permanent magnetic moment (dipoles) due to the spin of unpaired
electrons in the atoms’ orbitals. But randomization  no effect
o If a magnetic field is applied, the dipoles tend to align with the applied
field  net magnetic moment
o When the field is removed the effect disappears
o For low levels of magnetization, M =   H = C / T H
( = C / T )
where C = N 0 mu²/(3kBT) is the Curie constant (mu is the permanent magnetic
moment)
Thus  increases as T decreases
(Application: magnetic thermometers)
o Ex.: Al
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Material properties at low temperature
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MAGNETIC PROPERTIES
 Ferromagnetic materials
•
•
•
•
•
•
Unpaired electron spins (cf. paramagnets)
+ electrons’ intrinsic magnetic moment ; tendency to be parallel to an
applied field and parallel to each other
 Magnetization remains
 = Cst / (T-C ) ; C =Curie temperature
Ferromagnets loose their ferromagnetic properties above C .
For classical ferromagnets, C > Tamb
Examples: Fe, Ni or Co alloys (not austenitic steels)
When an increase in the applied external magnetic field H cannot
increase the magnetization M the material reaches saturation state :
Bellow C :
M (0)  M (T )
M (0)
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Material properties at low temperature
 T 3/ 2
T/C
40
MAGNETIC PROPERTIES
 Antiferromagnetic materials
•
•
•
•
for antiferromagnets, the tendency of intrinsic magnetic moments of
neighboring valence electrons is to point in opposite directions.
A substance is antiferromagnetic when all atoms are arranged so that
each neighbor is 'anti-aligned'.
Antiferromagnets have a zero net magnetic moment below a critical
temperature called Néel temperature N  no field is produced by
them.
Above Néel temperature, antiferromagnets can exhibit diamagnetic
and ferrimagnetic properties:
cst

T  N
 Ferrimagnetic materials
•
•
Ferrimagnets keep their magnetization in the absence of an applied
field (like ferromagnets)
Neighboring pairs of electron spins like to point in opposite directions
(like antiferromagnets)
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Material properties at low temperature
41
REFERENCES
•
CRYOCOMP, CRYODATA software (based on standard reference data from NIST),
Cryodata Inc. (1999).
•
BUI A., HÉBRAL B., KIRCHER F., LAUMOND Y., LOCATELLI M., VERDIER J., Cryogénie : propriétés
physiques aux basses températures, B 2 380 − 1 (1993).
•
EKIN J.W., Experimental Techniques for Low Temperature Measurements, Oxford
University Press, ISBN 978-0-19-857054-7 (2006).
•
Amand J.-F., Casas-Cubillos J., Junquera T., Thermeau J.-P., Neutron Irradiation Tests in
Superfluid Helium of LHC Cryogenic Thermometers, ICEC'17 Bournemouth (UK), July
(1998)
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Thank you for your attention
THERMAL CONDUCTIVITY OF CRYOFLUIDS
 Liquids
•
•
As liquids Tamb, cryogenic liquids are bad thermal conductors (small k)
LHe:
o LHe thermal conductivity is lower than thermal insulator like G10
o LHe II (superfluid helium, T<2,17 K) is a heatsuperconductor (kLHe II 
2kW/(mK)
o Maximum of thermal conductivity arround 1.95K (k is 100 larger that the
thermal conductivity of a high pure copper)
 Gases
•
Small thermal conduction
1
k  ρcv  p v
3
μ
 p cm  8,6  10
p
3
T
M
12
 8RT 
v 

 M 
1/2
•
v
∝
T
(ℓp is limited by molecules collisions)
• cV ∝ 
• ℓp ∝ 1/ ∝ 1/P
o At P=Patm, k 
o Low pressure:
ℓp comparable with distance between hot and
cold surfaces (free-molecule regime)  k  T
T1/2
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