AJAX request on link click in Wordpress

I am trying to start a PHP function in Wordpress on the click of a link. I've managed to do it using a form-submit, but I want to do it on a link click.

I've pulled together an AJAX request but it doesn't seem to work - any thoughts much appreciated.

HTML Link Code

<a href="#link_id"> Ajax Click </a>

Javascript Function

<script>

// Create a function to pick up the link click

$('#link_id').click(function(event){
    event.preventDefault(); // prevent default behaviour of link click

    var data = {
        action: 'test_response_php', // This is the PHP function to call - note it must be hooked to AJAX
    };

    // Post to The Wordpress URL

    jQuery.post("<?php echo admin_url('admin-ajax.php'); ?>", data, function(response) {
        alert(response);
    });

});

PHP Function

function php_function() {
    echo 'Hello World';
    // Some interesting server side stuff

}

add_action('wp_ajax_nopriv_test_response_php', 'php_function');
add_action('wp_ajax_test_response_php', 'php_function');