Materials Properties II Thermal Electrical Characteristics Sergio Calatroni

Materials & Properties II: Thermal & Electrical Characteristics Sergio Calatroni - CERN

Outline (we will discuss mostly metals) • Electrical properties Electrical conductivity o Temperature dependence o Limiting factors Surface resistance - • o Relevance for accelerators o Heat exchange by radiation (emissivity) Thermal properties Thermal conductivity o Temperature dependence, electron & phonons o Limiting factors Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 3

The electrical resistivity of metals changes with temperature Copper T Constant T-5 Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 4

All pure metals… 100 Electrical resistivity of Be Electrical resistivity of Al 10 10 1 1 Electrical resistivity [µ. cm] 10 10 -1 1 10 -2 Electrical resistivity of Ag 10 -2 10 -3 1 10 100 Temperature [K] 1000 1 10 Temperature [K] Properties II: Thermal & Electrical 1000 Temperature [K] CAS Vacuum 2017 - S. C. 5

Alloys? Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 6

Some resistivity values (in µ. cm) (pure metals) Variation of a factor ~70 for pure metals at room temperature Even alloys have seldom more than a few 100 s of µ cm We will not discuss semiconductors (or in general effects not due to electron transport) Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 7

Definition of electrical resistivity The electrical resistance of a real object (for example, a cable) The electrical resistivity is measured in Ohm. m Its inverse is the conductivity measured in S/m Changes with: temperature, impurities, crystal defects Electron relaxation time Electron mean free path Constant for a given material Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 8

Basics (simplified free electron Drude model) Electrical current = movement of conduction electrons - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S. C. 9

Defects - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S. C. 10

Possible defects: phonons Crystal lattice vibrations: phonons Temperature dependent - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S. C. 11

Possible defects: phonons Crystal lattice vibrations: phonons Temperature dependent - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S. C. 12

Possible defects: impurities Can be inclusions of foreign atoms, lattice defects, dislocations Not dependent on temperature - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S. C. 13

Possible defects: grain boundaries Grain boundaries, internal or external surfaces Not dependent on temperature - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S. C. 14

The two components of electrical resistivity Temperature dependent part It is characteristic of each metal, and can be calculated Proportional to: - Impurity content - Crystal defects - Grain boundaries Does not depend on temperature Total resistivity Properties II: Thermal & Electrical Varies of several orders of magnitude between room temperature and “low” temperature CAS Vacuum 2017 - S. C. 15

Temperature dependence: Bloch-Grüneisen function Debye temperature: ~ maximum frequency of crystal lattice vibrations (phonons) Given by total number of high-energy phonons proportional ~T Given by total number of phonons at low energy ~T 3 and their scattering efficiency T 2 Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 16

Low-temperature limits: Matthiessen’s rule Or in other terms Every contribution is additive. Physically, it means that the different sources of scattering for the electrons are independent Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 17

Effect of added impurities (copper) (Cu)(300 K)=1. 65 µ. cm Note: alloys behave as having a very large amount of impurities embedded in the material Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 18

An useful quantity: RRR Fixed number Depends only on “impurities” Dominant in alloys Practical formula Experimentally, we have a very neat feature remembering that Independent of the geometry of the sample. Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 19

Final example: copper RRR 100 Copper This is Cu-OFE Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 20

Estimates of mean free path Typical values? Example of Cu at room temperature Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 21

Interlude: LHC 8. 33 T dipoles (nominal field) @ 1. 9 K Beam screen operating from 4 K to 20 K SS + Cu colaminated, RRR ≈ 60 Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 22

Magnetoresistance The LHC Electron trajectories are bent due to the magnetic field Cyclotron radius: B x RRR [T] B-field Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 23

Fermi sphere • Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 24

The speed of conduction electrons • Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 25

Square resistance and surface resistance • a current a d Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 27

Square resistance and surface resistance And now imagine that instead of DC we have RF, and the RF current is confined in a skin depth: a current a d This is a (simplified) definition of surface resistance Rs (We will discuss this in more details at the tutorials) Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 28

Surface impedance in normal metals • Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 29

Why the surface resistance (impedance)? • It is used for all interactions between E. M. fields and materials • In RF cavities: quality factor • In beam dynamics (more at the tutorials): - Longitudinal impedance and power dissipation from wakes is where is a summation of over the bunch frequency spectrum - Transverse impedance: Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 30

From RF to infrared: the blackbody Thermal exchanges by radiation are mediated by EM waves in the infrared regime. Peak ≈ 3000 µm x K Schematization of a blackbody Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 31

Blackbody radiation • A blackbody is an idealized perfectly emitting and absorbing body (a cavity with a tiny hole) • Stefan-Boltzmann law of radiated power density: σ ≈ 5. 67 × 10− 8 W/(m 2 K 4) • At thermal equilibrium: • is the emissivity (blackbody=1) • A “grey” body will obey: • Thus for a grey body: Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 32

From RF to infrared in metals • Thermal exchanges by radiation are mediated by EM waves in the infrared regime. • At 300 K, peak ≈ 10 µm of wavelength -> ≈ 1013 Hz or RF ≈ 10 -13 s • The theory of normal skin effect is usually applied for: • But it can be applied also for: • In the latter case it means: • For metals at moderate T we can then use the standard skin effect theory to calculate emissivity Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 33

Emissivity of metals • From: • Thus we can calculate emissivity from reflectivity: • The emissivity of metals is small • The emissivity of metals depends on resistivity • Thus, the emissivity of metals depends on temperature and on frequency Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 34

Practical case: 316 LN Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 35

Thermal conductivity of metals Copper peak T-1 constant Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 36

Thermal conductivity: insulators Determined by phonons (lattice vibrations). Phonons behave like a “gas” peak T-3 constant Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 37

Thermal conductivity: insulators Thermal conductivity from heat capacity (as in thermodynamics of gases) for ultra-pure crystals = max dimension of specimen Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 38

Thermal conductivity: metals Determined by both electrons and phonons. Thermal conductivity from heat capacity impurities Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 39

Thermal conductivity of metals: total Copper Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 40

Wiedemann-Franz Proportionality between thermal conductivity and electrical conductivity L = 2. 45 x 10 -8 W K-2 (Lorentz number) Useful for simple estimations, if one or the other quantity are known Useful also (very approximately) to estimate contact resistances Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 41

The LHC collimator Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 42

Contact resistance (both electrical and thermal) • Complicated… and no time left n depends on: Plastic deformation Elastic deformation Contact area: • Roughness “height” and “shape” Contacts depend also on oxidation, material(s) properties, temperature… Example for electric contacts: • • Theoretically: - R P-1/3 in elastic regime - R P-1/2 in plastic regime Experimentally: - R P-1 -1/2 (same as for thermal contacts) Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 43

References • Charles Kittel, “Introduction to solid state physics” • Ashcroft & Mermin, “Solid State Physics” • S. W. Van Sciver, “Helium Cryogenics” • M. Hein, “HTS thin films at µ-wave frequencies” • J. A. Stratton, “Electromagnetic Theory” • Touloukian & De. Witt, “Thermophysical Properties of Matter” Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 44

The end. Questions? 45

Plane waves in vacuum Plane wave solution of Maxwell’s equations in vacuum: Where (in vacuum): So that: The ratio is often called impedance of the free space and the above equations are valid in a continuous medium Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 46

Plane waves in normal metals More generally, in metals: This results from taking the full Maxwell’s equations, plus a supplementary equation which relates locally current density and field: In metals and the wave equations become: With is the damping coefficient of the wave inside a metal, and is also called the field penetration depth. Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 47

V~ Surface impedance z x S=d 2 ; y Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 48

Normal metals in the local limit Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 49

Limits for conductivity and skin effect 1. Normal skin effect if: e. g. : high temperature, low frequency 2. Anomalous skin effect if: e. g. : low temperature, high frequency Note: 1 & 2 valid under the implicit assumption 1 & 2 can also be rewritten (in advanced theory) as: It derives that 1 can be true for and also for Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 50

Mean free path and skin depth Mean free path Skin depth Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 51

Anomalous skin effect Asymptotic value Normal skin effect Understood by Pippard, Proc. Roy. Soc. A 191 (1947) 370 Exact calculations Reuter, Sondheimer, Proc. Roy. Soc. A 195 (1948) 336 Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 52

Debye temperatures Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 53

Heat capacity of solids: Dulong-Petit law Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 54

Low-temperature heat capacity of phonon gas (simplified plot in 2 D) Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 55

Phonon spectrum and Debye temperature Density of states : How many elemental oscillators of frequency Assuming constant speed of sound Properties II: Thermal & Electrical CAS Vacuum 2017 - S. C. 56