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Outline (we will discuss mostly metals)

Electrical properties

Electrical conductivity

Temperature dependence
Limiting factors

Surface resistance

Relevance for accelerators
Heat exchange by radiation (emissivity)

Thermal properties

Thermal conductivity

Temperature dependence, electron & phonons
Limiting factors

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Properties II: Thermal & Electrical

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The electrical resistivity of metals changes with temperature

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Properties II: Thermal & Electrical

Copper

Constant

T^-5

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All pure metals…

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Properties II: Thermal & Electrical

10^-1

10^-2

10^-3

10^-1

100

10^-2

Electrical resistivity ρ [µΩ.cm]

10^-1

10^-2

Electrical resistivity ρ [µΩ.cm]

10^-3

Electrical resistivity

of Be

Electrical resistivity

of Al

Temperature [K]

100

1000

Temperature [K]

100

1000

Temperature [K]

100

1000

Electrical resistivity

of Ag

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Alloys?

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Properties II: Thermal & Electrical

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Some resistivity values (in µΩ.cm) (pure metals)

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Properties II: Thermal & Electrical

Variation of a factor ~70

for pure metals at room temperature

Even alloys have seldom more than a few 100s of µΩcm

We will not discuss semiconductors (or in general effects not due to electron transport)

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Definition of electrical resistivity ρ

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Properties II: Thermal & Electrical

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

Constant for a given material

Changes with: temperature, impurities, crystal defects

Electron relaxation time

Electron mean free path

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Basics (simplified free electron Drude model)

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Properties II: Thermal & Electrical

Electrical current = movement of conduction electrons

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Defects

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Possible defects: phonons

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Properties II: Thermal & Electrical

Crystal lattice vibrations: phonons

Temperature dependent

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Possible defects: phonons

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Properties II: Thermal & Electrical

Crystal lattice vibrations: phonons

Temperature dependent

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Possible defects: impurities

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Properties II: Thermal & Electrical

Can be inclusions of foreign atoms, lattice defects, dislocations

Not dependent on temperature

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Possible defects: grain boundaries

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Properties II: Thermal & Electrical

Grain boundaries, internal or external surfaces

Not dependent on temperature

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The two components of electrical resistivity

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Properties II: Thermal & Electrical

Temperature dependent part

It is characteristic of each metal, and can be calculated

Varies of several orders of magnitude between room temperature and “low” temperature

Proportional to:

- Impurity content

- Crystal defects

- Grain boundaries

Does not depend on temperature

Total resistivity

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Temperature dependence: Bloch-Grüneisen function

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Properties II: Thermal & Electrical

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 ~T³ and their scattering efficiency T²

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Low-temperature limits: Matthiessen’s rule

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Properties II: Thermal & Electrical

Or in other terms

Every contribution is additive.

Physically, it means that the different sources of scattering for the electrons are independent

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Effect of added impurities (copper)

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Properties II: Thermal & Electrical

ρ_(Cu)(300K)=1.65 µΩ.cm

Note: alloys behave as having a very large amount of impurities embedded in the material

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An useful quantity: RRR

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Properties II: Thermal & Electrical

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.

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Final example: copper RRR 100

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Properties II: Thermal & Electrical

Copper

This is Cu-OFE

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Estimates of mean free path

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Properties II: Thermal & Electrical

Typical values? Example of Cu at room temperature

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Interlude: LHC

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Properties II: Thermal & Electrical

8.33 T dipoles (nominal field) @ 1.9 K

Beam screen operating from 4 K to 20 K

SS + Cu colaminated, RRR ≈ 60

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Magnetoresistance

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Properties II: Thermal & Electrical

B-field

B x RRR [T]

Cyclotron radius:

The LHC

Electron trajectories are bent

due to the magnetic field

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Fermi sphere

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Properties II: Thermal & Electrical

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The speed of conduction electrons

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Properties II: Thermal & Electrical

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Size effect

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Properties II: Thermal & Electrical

(Fuchs’ equation)

Case of a very thin metallic film

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Square resistance and surface resistance

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Properties II: Thermal & Electrical

current

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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:

This is a (simplified) definition of surface resistance R_s

(We will discuss this in more details at the tutorials)

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current

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Surface impedance in normal metals

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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:

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From RF to infrared: the blackbody

Thermal exchanges by radiation are mediated by EM waves in the infrared regime.

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Properties II: Thermal & Electrical

Schematization of a blackbody

Peak ≈ 3000 µm x K

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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:

At thermal equilibrium:
ε is the emissivity (blackbody=1)

A “grey” body will obey:
Thus for a grey body:

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Properties II: Thermal & Electrical

σ ≈ 5.67 × 10⁻⁸W/(m²K⁴)

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From RF to infrared in metals

Thermal exchanges by radiation are mediated by EM waves in the infrared regime.

At 300 K, λ_peak ≈ 10 µm of wavelength -> ≈ 10¹³ Hz or τ_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

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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

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Practical case: 316 LN

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Thermal conductivity of metals

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Copper

peak

constant

T^-1

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Thermal conductivity: insulators

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Determined by phonons (lattice vibrations). Phonons behave like a “gas”

peak

constant

T^-3

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Thermal conductivity: insulators

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Thermal conductivity from heat capacity (as in thermodynamics of gases)

∅ = max dimension of specimen

for ultra-pure crystals

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Thermal conductivity: metals

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Thermal conductivity from heat capacity

Determined by both electrons and phonons.

impurities

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Thermal conductivity of metals: total

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Properties II: Thermal & Electrical

Copper

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Wiedemann-Franz

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Properties II: Thermal & Electrical

L = 2.45x10^-8 WΩK^-2

(Lorentz number)

Proportionality between thermal conductivity and electrical conductivity

Useful for simple estimations, if one or the other quantity are known

Useful also (very very approximately) to estimate contact resistances

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The LHC collimator

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Contact resistance (both electrical and thermal)

Complicated… and no time left ☹

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)

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Properties II: Thermal & Electrical

Contact area:

n depends on:

Plastic deformation

Elastic deformation

Roughness “height” and “shape”

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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 & DeWitt, “Thermophysical Properties of Matter”

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The end. Questions?

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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

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Plane waves in normal metals

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With is the damping coefficient of the wave inside a metal, and δ is also called the field penetration depth.

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:

More generally, in metals:

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Surface impedance

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Properties II: Thermal & Electrical

;

S=d²

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Normal metals in the local limit

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Limits for conductivity and skin effect

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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

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Mean free path and skin depth

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Skin depth

Mean free path

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Anomalous skin effect

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Properties II: Thermal & Electrical

Understood by Pippard, Proc. Roy. Soc. A191 (1947) 370

Exact calculations Reuter, Sondheimer, Proc. Roy. Soc. A195 (1948) 336

Normal skin effect

Anomalous skin effect

Asymptotic value

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Debye temperatures

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Heat capacity of solids: Dulong-Petit law

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Low-temperature heat capacity of phonon gas

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(simplified plot in 2D)

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Phonon spectrum and Debye temperature

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Properties II: Thermal & Electrical

Density of states :

How many elemental oscillators of frequency

Assuming constant speed of sound