Solution to Exercise 2.2-1"Properties for Johnson Complexes"

	Discuss the equation for the concentration of vacancy - impurity atom complexes (Johnson complexes).
		Consider an impurity atom concentration of 1 % and 1 ppm, a vacancy formation enthalpy of 1 eV (neglect the formation entropy) and several binding energies (including extremes).
		Discuss the concentration of Johnson complexes as a function of temperature and in relation to the concentration of the impurity atoms and the equilibrium concentration of vacancies.
	Use approximations, order-of- magnitude considerations and reasonable numbers whenever possible.

The basic equation for the concentration of Johnson complexes is

c_C	=	z · c_F · c_V 1 – z · c_F	· exp	∆S_C k	· exp	H_C kT

We first need to chose a coordination number, we take z = 12 for fcc and hcp crystals. All other coordination numbers are smaller; we thus have the maximal effect of z.

The given concentration of impurity atoms of 1 % and 1 ppm correspond to c_F = 10^–2 and c_F = 10^–6, respectively.

First we note that the factor 1 – z · c_F equals 0,88 or 0,999..; i.e. we can forget it - at least for the low concentration.

Next we calculate the ratios c_C / c_F and c_C / c_V in order to get a feeling how the Johnson complex concentration relates to the (fixed) concentration of impurity atoms and the (temperature dependent) equilibrium concentration of vacancies. We have

c_C c_F	=	(12 ... 13,6) · c_V	· exp	∆S_C k	· exp	H_C kT	=	(12 ... 13,6) · exp –	(H_F^V – G_B) kT

c_C c_V	=	(12 ... 13,6) · c_F	· exp	∆S_C k	· exp	H_C kT	=	(12 ... 13,6) · c_F	· exp	G_B kT

The numbers in the bracket span the range of the given c_F concentrations.

Our first result thus is simple: The ratios asked for are directly proportional to the concentration of vacancies or foreign atoms, respectively. The proportionality factor is about 2 times the Boltzmann factor of the free enthalpy of complex formation. So let's look at the role of the binding energy.

Let's look at binding energies (more precisely: binding free enthalpies G_B) of – ∞ eV (i.e. extreme repulsion between a vacancy and the foreign atom), 0 eV (no interaction), ½ H_F^V (strong interaction), and H_F^V (extreme interaction). This gives us

G_B	– ∞	0	½ H_F^V	H_F^V

c_C c_F	0	≈ 12c_V	≈ 12 · (c_V)^½	≈ 12

c_C c_V	0	≈ 12c_F	≈ 12 · c_F · (c_V)^{– ½}	≈	12 · c_F c_V

What does it mean?

First, for extreme repulsion, we simply do not form Johnson complexes as we would expect.

Second, for zero interaction, we form Johnson complexes just at random - a vacancy just does not care if it sits next to an impurity atom or not. The concentration thus is directly given by the product of the concentrations of the partners (the factor 12 just accounts for the 12 different ways to form a Johnson complex with one vacancy).

Third, for appreciable but not extreme binding energies the quotient c_C / c_F is always < 1, because (c_v)^½ << 1; it decreases rapidly with temperature. This means that in equilibrium only a small part of the foreign atoms will form Johnson complexes.

Fourth, for appreciable but not extreme binding energies the quotient c_C / c_V can be >1 or <1, depending on 12c_F being larger or smaller than (c_V)^½. Below some temperature the vacancy concentration will always be so low that the ratio is >1, we then have more Johnson complexes than free vacancies. But that does not mean we have many - just more then the extremely few vacancies.

Fifth, for extreme binding energies we have a problem. The relations given just must be wrong - we cannot for example, have 12 times as many Johnson complexes as we have foreign atoms. What went wrong?

Well, our starting formula is only valid under the assumption that c_C << c_F. This assumption is obviously violated for binding energies too large; we then must not use the simple formula.

If we take the correct formula, we simply find that c_V times the exponential vanishes (i.e. c_C /c_V does not make sense anymore), and c_C / c_F ≈ z /(1 + z) ≈ 1 under all conditions, as we would expect.

With frame

Equilibrium Concentration of theVacancy-Impurity Atom Complex

Exercise 2.2-1: Properties of Johnson Complexes

Exercise 2.2-2 Quick Questions