Van der Waals interaction

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Today - the Van der Waals interaction between two atoms.

In this course, we mainly discuss single atoms, or many atoms. This chapter is an exception --- we discuss interactions between two atoms. There are several reasons for it: The van der Waals force belongs to the class of Casimir forces, which can be regarded as either being due to the exchange of virtual photons, or due to the modified zero-point energy of the electromagnetic field. We will present both viewpoints. The latter has attracted recent interest in the search for the origin of the dark energy of the universe. A second reason to discuss van der Waals forces is that it provides a non-trivial example for the power of the diagrammatic approach, and finally, it shows how the exchange of virtual photons can lead to attraction. So we can encounter in AMO physics an example for forces mediated by virtual exchange of field quanta.

Two physical pictures of the van der Waals interaction: classical and quantum

Two physical pictures illuminate the richness of the problem. Consider two atoms $a$ and $b$ separated by distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} . The Coulomb interaction between these is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_I = \int \frac{\rho_a \rho_b}{r_{ab}} d^3 r_a d^3 r_b \,. }

Integrating leaves us with two interacting dipoles

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H' = \frac{1}{R^3}( \vec{d}_a \cdot \vec{d}_b-3(\vec{d}_a \cdot \hat{R} ) ( \vec{d_b}\cdot \hat{R}) ) \,. }

In the quantum picture, these dipoles become operators. First order perturbation theory tells us that if one atom is excited and the other is in the ground state, this gives an interaction term

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{C_3}{R^3} \,. }

But if both atoms are in the ground state, then the leading term is given by second order perturbation theory, as

$-{\frac {C_{6}}{R^{6}}}\,.$

Physical interpretation

Let us interpret this result now (see Kleppner's class notes) - the physics of the van der Waals interaction are all due to quantum fluctuations. Consider two resonant circuits that are capactively coupled:

The coupling goes as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/R^3} . Each resonator has frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} , and classically coupled oscillators split their frequencies, giving

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_\pm = \omega_0 \pm (term)\frac{1}{R^3} - (term)\frac{1}{R^6} \,. }

The ground state energy is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar(\omega_+ + \omega_-)/2 } . The two $1/R^{3}$ add up to zero leaving just the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/R^6} term.

An equivalent description focusing on the atomic dipole moment is as follows. A neutral atom has no expectation for the dipole operator, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle d_a{\rangle}=0} , but the variance for this is nonzero, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle d_a^2{\rangle}} . This is just as the simple harmonic oscillator in its ground state has zero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle p{\rangle}} , but nonzero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle p^2{\rangle}} in the ground state. The fluctuating dipole moment of atom Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} creates a fluctuating field at position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_b \propto \frac{d_a}{R^3} \,. }

This induces a dipole in atom Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . The interaction energy between two dipoles is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d_a d_b}{R^3} = \frac{ \langle d_a^2{\rangle}}{R^6} \,. }

Origin in fluctuations of the electromagnetic field

So far, we've neglected fluctuations of the electromagnetic field. What could the vacuum fluctuations of the field do to neutral atoms? Fluctuations at position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} can induce dipole moments, since the atoms are polarizable:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} d_a &=& \alpha_a E(r_a) \\ d_b &=& \alpha_b E(r_b) \,. \end{array}}

The corresponding interaction is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_I \propto \frac{\alpha_a \alpha_b \langle E(r_a) E(r_b){\rangle}}{R^3} \,. }

See Larry Spuch, Physics Today article, posted on web site for more about this. Conceptually, one sums over all modes of the electromagnetic field - this time in free space, without the boundary condition as we do in the Casimir force case. The long wavelengths do not average out to zero, when they are of wavelength comparable to the distance between the two atoms. Their contribution, when density of states are included, and cutoff appropriately, gives

$H_{I}\propto {\frac {1}{R^{7}}}\,.$

So how does the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/R^6} term disappear? If you focus on the fluctuations of the instantaneous dipole moments, then you get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/R^6} ; this is the correct result for short range interactions. It can be derived, as we showed above, by conceptually using the uncertainty principle for the atomic dipoles. In contrast, when the uncertainty principle for the electromagnetic field is used, we get a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/R^7} force; this is applicable at long ranges, and is known as the Casimir-Polder effect, or the retarded van der Waals interaction. Note that a full quantum theory of both atoms and field are needed to get both of these interactions properly. The crossover distance between the two interactions is set by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R\sim \lambda_{res}} , the resonant wavelength of that atoms.

The entirety of the van der Waals interaction is a quantum phenomenon (thus, the reason your table is solid is because of quantum mechanics!). Classically, there is no average value of the dipole moment, and also no fluctuations - the classical van der Waals force is zero. But quantum-mechanically, vacuum fluctuations are always present. It doesn't cost extra energy to create them. For the van der Waals force, the dipole fluctuations simply become correlated and lower the total energy of the system (by leading to an attractive interaction energy).

Van der Waals forces due to absorption and emission of virtual photons

Returning to our theme of perturbation theory for this chapter, let us consider the van der Waals interaction from a diagrammatic approach. (This follows API pp 121-126). The fundamental idea is an exchange of virtual photons. The Hamiltonian in the dipole approximation is

$H'_{I}=-dE_{\perp }(R)-d'E_{\perp }(R')\,.$

In deriving the QED Hamiltonian, we discarded the longidutional fields because they were determined by the charges and not a degree of freedom, and we kept the transverse fields, as the radiation field. In the qualitative discussion above, we obtained the dipole moments Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{d}} and their interactions from a Coulomb integral. However, in our QED approach, outside the neutral atoms there only exists the radiation field. We want to understand now how this can result in the van der Waals interaction.

The only way two neutral atoms can interact with each other is by exchange of virtual photons. See Cohen-Tanoudji for hundreds of pages of illuminative discussion about this.

The language of Feynman diagrams can be used to understand the van der Waals interaction. Consider two identical atoms spaced apart by distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} . The atoms initially start in state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , the ground state. One emits a virtual photon, going into a virtual excited state (with an energy below the ground state). This photon is absorbed by the second atom. After a short time, the first atom emits another photon, which is absorbed by the second one, returning both atoms to the ground state:

This picture indicates that to leading order, we need diagrams with two photons, and four vertices. The dominant contribution comes from photons with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\sim D} , that is $\omega \sim 2\pi c/D$. Another important piece of guidance is that the virtual process is admixing states of higher energy to the ground state, lowering the total energy; energy shifts in second order processes go as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E \sim \frac{V_{kj} V_{0i}}{E_i-E_j} \,, }

characterizing the smaller amplitude of mixing with higher energy states. With extra intermediate states this becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E \sim \frac{V_{lk}V_{kj} V_{0i}}{(E_l-E_k)(E_i-E_j)} \,. }

When we emit virtual photons, there are two energies involved; one is the atomic excitation energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} , the other one is the exchanged photon energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} . The energy defects in the above picture are thus as shown in the figure above: initially $\omega _{0}+\omega$, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\omega_0} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0+\omega} .

A second diagram involved is as if one atom emitted two photons simultaneously:

Which of these two diagrams is preferable depends on how large Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} is compared to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} . Ultimately, that distinguishes short and long range interactions, and $1/R^{6}$ versus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/R^7} . The energy defects in the three regions are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega+\omega_0} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\omega} (versus $2\omega _{0}$ for the first diagram), and then $\omega +\omega _{0}$.

The contribution of each diagram is proportional to the inverse product of the energy defects:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{first} \sim \frac{1}{\omega_0+\omega}\frac{1}{2\omega_0}\frac{1}{\omega_0+\omega} \sim \frac{1}{\omega_0 \omega^2} \,. }

In contrast, for the second diagram, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{second} \sim \sim \frac{1}{\omega_0^2 \omega} \,. }

Focusing on these two diagrams, let us consider what happens at short and long ranges. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_0} be the wavelength of the radiation. Short range is defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\ll \lambda_0} (meaning $\omega \gg \omega _{0}$), and long range is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\gg \lambda_0} (meaning Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega\ll\omega_0} ). Thus for long range, the first diagram dominates, whereas for short range, the second. Thus, going from long range to short range involves an additional factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} in the interaction. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \sim 1/D} , this gives the change from $1/D^{6}$ to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/D^7} in the potential.

Casimir forces revisted

Let us now look at Casimir forces for different systems.

Consider three scenarios:

• Atom-atom. At long distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\propto \frac{\alpha_1\alpha_2}{R^7}} .
• Atom-wall. Recall that the polarizibility of an atom has units of volume, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^3} . A large metal sphere also has polarizability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^3} . Thus, at long distances, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\propto\frac{\alpha R^3}{R^7} \sim\frac{\alpha}{R^4}} .

• Two walls. Two walls can be regarded as two spheres, with a radius similar to their distance. The potential is now 1/R, but we have to normalize it to the area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{V}{A}\propto \frac{1}{R R^2} \sim \frac{1}{R^3}} .

At short distances:

Both the atom-atom and the atom-wall interaction change in the power law by one factor of R, whereas the wall-wall interaction is always in the far field (A metallic wall corresponds to an atom with high excitation energy and therefore small wavelength).

• Atom-atom. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\sim\frac{1}{R^6}}
• Atom-wall. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\sim\frac{1}{R^3}}
• Two walls. Always in the far field.

References

• see API pp. 121-126
• four pages course notes from Dan Kleppner
• Physics Today paper by L. Spruch (Nov. 1986, p. 37)