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over a broad range. Once in the final crossed beam trap, we ramp the magnetic field to select the scattering length a which gives the desired value of 1 / ( k$_{F}$ a ) . This allows us to tune k$_{F}$ and 1 / ( k$_{F}$ a ) independently. The cloud is then held at the final magnetic field for a time τ ≫ 10 ¯ ω − 1 to reach equilibrium before applying the Bragg pulse and imaging at that magnetic field. 

Bragg scattering is achieved by illuminating these $^{6}$Li clouds with two counter propagating laser beams with a very small frequency difference δ . This creates a standing wave which moves with velocity δ/k$_{Br}$ where k$_{Br}$ = 2 π/λ$_{Br}$ is the wave-vector and λ$_{Br}$ = 671 nm is the wavelength of the Bragg lasers. When the en- ergy difference between the two Bragg lasers equals the kinetic energy associated with a two-photon recoil, res- onant Bragg scattering occurs (i.e. δ = 2$_{ℏ}$ k 2 $_{Br}$/m ). The Bragg lasers are approximately 2 GHz red-detuned from the two ground hyperfine states which is large compared to the ∼ 80 MHz splitting between the | 1 / 2 , ± 1 / 2 〉 states. This means that both states are coupled approximately equally to the Bragg beams (to within 4 % ). 

To measure the static structure factor S ( k ) we begin by recording a Bragg spectrum. This consists of measuring the momentum transferred to the cloud by the Bragg lasers as a function of δ . The transferred momentum is directly proportional to the resultant center of mass displacement Δ X ( k, δ ) measured after 2 ms time of flight [29]. A Bragg spectrum for a unitary gas is shown in the inset of Fig. 2 for k/k$_{F}$ = 8 . 5 as a function of the Bragg frequency δ . At this momentum, the pair and free-atom excitations are clearly distinguished at frequencies of ∼ 150 kHz and ∼ 300 kHz, respectively. The Bragg pulse duration, 50 µ s, is short compared to the two-photon Rabi cycling period, ensuring that spectra are obtained in the linear response regime. 

The center of mass displacement, Δ X ( k, δ ) , is propor- tional to the convolution of the spectral content of the Bragg pulse and the dynamic structure factor of the gas S ( k, δ ) [33]. The proportionality constant depends upon the two-photon Rabi frequency Ω$_{Br}$ which can be dif- ficult to accurately measure. Integrating Δ X ( q, δ ) over δ yields a number proportional to the static structure factor, S ( k ) = $_{ℏ}$N − 1 ∫ S ( k, δ ) dδ , but Ω$_{Br}$ remains un- known. We overcome the need to find Ω$_{Br}$ by invok- ing the f -sum rule [24]: NE$_{r}$ = ∫ S ( k, δ ) δdδ , where E$_{r}$ = 2$_{ℏ}$ $^{2}$k2 $_{Br}$/m is the two-photon recoil energy. As both S ( k ) and the f -sum rule involve N , we normalise the area under each measured spectra using 

where the constant involving Ω$_{Br}$ appears before each integral and therefore cancels. Equation (3) is absolute, requiring only knowledge of the recoil energy, which can be determined with high accuracy, leading to an accurate measure of S ( k ) . In Fig. 2 we plot S ( k ) obtained in this 

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way for k/k$_{F}$ = 4 . 8 which shows excellent agreement with the zero temperature theoretical calculation. 

Next we come to the measurement of S ( k ) as a func- tion of k/k$_{F}$ . At unitarity we expect S ( k ) to vary linearly with k$_{F}$ /k according to Eq. (2). Rather than changing k directly by varying the angle between the two Bragg beams, we change k$_{F}$ = (48 N ) 1 / 6 $^{√}$m ¯ ω/$_{ℏ}$ by varying ¯ ω = ( ω$_{x}$ω$_{y}$ω$_{z}$ ) 1 / $^{3}$, the geometric mean frequency of the optical dipole trap. Universality allows us to change the relative length scale being probed simply by changing the density of the gas. Using the crossed beam opti- cal trap described above we can tune ¯ ω such that the Fermi wavevector can be tuned anywhere over the range k$_{F}$ = 2.1 µ m − 1 → 5.3 µ m − $^{1}$, or k/k$_{F}$ = 3.5 → 9.1. A sequence of Bragg spectra were obtained for gases at three values of 1 / ( k$_{F}$ a ) = +0.3, 0.0 and -0.2, while varying k$_{F}$ . From these, S ( k ) was extracted and the results are plotted in Fig. 3. The solid lines are the pre- diction from Eq. (2) using the zero temperature contact with no free parameters. The experimental points closely follow the theory, confirming the exact analytic depen- dence of S$_{↑↓}$ ( k ) ∼ = S ( k ) − 1 on q , Eq. (2). At unitarity 1 / ( k$_{F}$ a ) = 0, the dependence on a vanishes and a straight line fit (dashed line) shows the simple universal behav- ior of fermionic pairing. The fitted slope of 0 . 75 ± 0 . 03 is slightly below the zero temperature prediction of 0.81 

